Some topics in analysis related to topological groups and Lie algebras

Stephen Semmes Rice University

Preface

ii

Contents

I Basic notions and objects 1

1 Absolute values and norms 2 1.1 Metrics and ultrametrics . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Absolute value functions . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Equivalent absolute value functions . . . . . . . . . . . . . . . . . 5 1.4 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Discreteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 p-Adic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Residue fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.8 Norms and ultranorms . . . . . . . . . . . . . . . . . . . . . . . . 11 1.9 Bounded linear mappings . . . . . . . . . . . . . . . . . . . . . . 13 1.10 Some norms on k^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.11 Inner products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.12 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.13 Bounded bilinear mappings . . . . . . . . . . . . . . . . . . . . . 20 1.14 Minkowski functionals . . . . . . . . . . . . . . . . . . . . . . . . 22 1.15 Balanced subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Some basic notions related to Lie algebras 26 2.1 Modules and homomorphisms . . . . . . . . . . . . . . . . . . . . 26 2.2 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 The adjoint representation . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7 More on multiplication operators . . . . . . . . . . . . . . . . . . 36 2.8 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.9 Traces of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.10 Vector spaces and linear mappings . . . . . . . . . . . . . . . . . 40 2.11 Ideals and quotients . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.12 Bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.13 Dual spaces and mappings . . . . . . . . . . . . . . . . . . . . . . 44 2.14 Nondegenerate bilinear forms . . . . . . . . . . . . . . . . . . . . 45 2.15 Sesquilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3 Submultiplicativity and invertibility 50

3.1 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Submultiplicative seminorms . . . . . . . . . . . . . . . . . . . . 52 3.3 Some matrix seminorms . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 Continuity of inverses . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6 Invertible linear mappings . . . . . . . . . . . . . . . . . . . . . . 59 3.7 Isometric linear mappings . . . . . . . . . . . . . . . . . . . . . . 60 3.8 Hilbert space isometries . . . . . . . . . . . . . . . . . . . . . . . 62 3.9 Preserving bilinear forms . . . . . . . . . . . . . . . . . . . . . . 63 3.10 Preserving nondegenerate bilinear forms . . . . . . . . . . . . . . 65 3.11 Preserving sesquilinear forms . . . . . . . . . . . . . . . . . . . . 66 3.12 Bilinear forms and matrices . . . . . . . . . . . . . . . . . . . . . 68 3.13 Sesquilinear forms and matrices . . . . . . . . . . . . . . . . . . . 71 3.14 Invertibility and involutions . . . . . . . . . . . . . . . . . . . . . 73 3.15 Invertibility and seminorms . . . . . . . . . . . . . . . . . . . . . 75

4 Formal power series 77 4.1 Direct sums and products . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Bilinear mappings and Cauchy products . . . . . . . . . . . . . . 79 4.3 Formal power series and modules . . . . . . . . . . . . . . . . . . 81 4.4 Sums and extensions . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 Extending bilinear mappings . . . . . . . . . . . . . . . . . . . . 84 4.6 Formal power series and algebras . . . . . . . . . . . . . . . . . . 86 4.7 Invertibility in A[[T]] . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.8 Homomorphisms over k[T] . . . . . . . . . . . . . . . . . . . . . . 88 4.9 Formal power series and homomorphisms . . . . . . . . . . . . . 90 4.10 Extensions and compositions . . . . . . . . . . . . . . . . . . . . 92 4.11 Two-step extensions . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.12 Extending algebra homomorphisms . . . . . . . . . . . . . . . . . 95 4.13 Some remarks about commutators . . . . . . . . . . . . . . . . . 97 4.14 Formal power series and involutions . . . . . . . . . . . . . . . . 98

5 Some related notions 100 5.1 Adjoining nilpotent elements . . . . . . . . . . . . . . . . . . . . 100 5.2 Adjoining nilpotent elements to algebras . . . . . . . . . . . . . . 101 5.3 Invertibility in A[ϵ_{n}] . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4 Adjoining two nilpotent elements . . . . . . . . . . . . . . . . . . 104 5.5 Invertibility in A[ϵ_{1}, η_{1}] . . . . . . . . . . . . . . . . . . . . . . . 106 5.6 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.7 Polynomial functions . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.8 Several commuting indeterminates . . . . . . . . . . . . . . . . . 110 5.9 Polynomial functions in several variables . . . . . . . . . . . . . . 113 5.10 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.11 Formal differential operators . . . . . . . . . . . . . . . . . . . . . 117 5.12 First-order differential operators . . . . . . . . . . . . . . . . . . 118 5.13 Homogeneous formal polynomials . . . . . . . . . . . . . . . . . . 120 5.14 Homogeneous differential operators . . . . . . . . . . . . . . . . . 122 5.15 Some algebras of differential operators . . . . . . . . . . . . . . . 123

6 Bilinear actions and representations 125 6.1 Bilinear actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2 Subactions and homomorphisms . . . . . . . . . . . . . . . . . . 126 6.3 Quotient actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.4 Representations of associative algebras . . . . . . . . . . . . . . . 129 6.5 Representations of Lie algebras . . . . . . . . . . . . . . . . . . . 130 6.6 Subrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.7 Homomorphisms between representations . . . . . . . . . . . . . 133 6.8 Quotient representations . . . . . . . . . . . . . . . . . . . . . . . 134 6.9 Sums of representations . . . . . . . . . . . . . . . . . . . . . . . 135 6.10 Compatible bilinear mappings . . . . . . . . . . . . . . . . . . . . 136 6.11 Representations and formal power series . . . . . . . . . . . . . . 138 6.12 Opposite algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.13 Matrices and associative algebras . . . . . . . . . . . . . . . . . . 141 6.14 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.15 Representations and differential operators . . . . . . . . . . . . . 144

7 Representations and multilinear mappings 146 7.1 Some remarks about subalgebras . . . . . . . . . . . . . . . . . . 146 7.2 Representations on linear mappings . . . . . . . . . . . . . . . . . 147 7.3 Multilinear mappings . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.4 Boundedness and multilinearity . . . . . . . . . . . . . . . . . . . 150 7.5 Representations on multilinear mappings . . . . . . . . . . . . . . 152 7.6 Centralizers and invariant elements . . . . . . . . . . . . . . . . . 153 7.7 Invariant multilinear mappings . . . . . . . . . . . . . . . . . . . 154 7.8 Traces of linear mappings . . . . . . . . . . . . . . . . . . . . . . 156 7.9 The Killing form . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.10 Invariant subspaces and traces . . . . . . . . . . . . . . . . . . . 159 7.11 Radicals of bilinear mappings . . . . . . . . . . . . . . . . . . . . 160 7.12 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.13 Functions on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.14 Some remarks about sums, products . . . . . . . . . . . . . . . . 164 7.15 Some natural isomorphisms . . . . . . . . . . . . . . . . . . . . . 165

8 Formal series and ordered rings 167 8.1 Poles of finite order . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.2 Sequences and series . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.3 Formal series and module homomorphisms . . . . . . . . . . . . . 170 8.4 Extending module homomorphisms . . . . . . . . . . . . . . . . . 171 8.5 Homomorphisms from A((T)) into B((T)) . . . . . . . . . . . . . 172 8.6 Formal series and bilinear mappings . . . . . . . . . . . . . . . . 174 8.7 Bilinear mappings on A((T)) × B((T)) . . . . . . . . . . . . . . . 176 8.8 Algebras and modules over k((T)) . . . . . . . . . . . . . . . . . 177 8.9 Absolute values on k((T)) . . . . . . . . . . . . . . . . . . . . . . 179 8.10 Formal series and algebra homomorphisms . . . . . . . . . . . . . 181 8.11 Involutions and formal series . . . . . . . . . . . . . . . . . . . . 182 8.12 Ordered rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.13 Some additional features . . . . . . . . . . . . . . . . . . . . . . . 185 8.14 Ordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

II Solvability, nilpotence, and semisimplicity 188

9 Solvability and nilpotence 189 9.1 Some basic isomorphism theorems . . . . . . . . . . . . . . . . . 189 9.2 Products of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 9.3 Solvable Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.4 The solvable radical . . . . . . . . . . . . . . . . . . . . . . . . . 192 9.5 Nilpotent Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 194 9.6 Two-dimensional Lie algebras . . . . . . . . . . . . . . . . . . . . 195 9.7 Nilpotency conditions . . . . . . . . . . . . . . . . . . . . . . . . 196 9.8 Maximal Lie subalgebras . . . . . . . . . . . . . . . . . . . . . . . 198 9.9 Nilpotent linear mappings . . . . . . . . . . . . . . . . . . . . . . 199 9.10 Engel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 9.11 Flags and matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 202 9.12 A useful lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 9.13 Lie’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 9.14 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . 207 9.15 Another corollary . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

10 Matrices and traces 210 10.1 Some remarks about gl_{n}(k) . . . . . . . . . . . . . . . . . . . . . 210 10.2 Some basic properties of sl_{2}(k) . . . . . . . . . . . . . . . . . . . 212 10.3 Scalar and diagonal matrices . . . . . . . . . . . . . . . . . . . . 213 10.4 Centrality in gl_{n}(k), sl_{n}(k) . . . . . . . . . . . . . . . . . . . . . 214 10.5 Solvability and traces . . . . . . . . . . . . . . . . . . . . . . . . 215 10.6 Diagonalizable linear mappings . . . . . . . . . . . . . . . . . . . 216 10.7 Nilpotent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 10.8 Jordan–Chevalley decompositions . . . . . . . . . . . . . . . . . . 219 10.9 Some related situations . . . . . . . . . . . . . . . . . . . . . . . 221 10.10Replicas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 10.11Two lemmas about traces . . . . . . . . . . . . . . . . . . . . . . 224 10.12Cartan’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 10.13Comparing radicals . . . . . . . . . . . . . . . . . . . . . . . . . . 227 10.14Complementary ideals . . . . . . . . . . . . . . . . . . . . . . . . 228 10.15Simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 229 11 Some examples and related properties 231

11.1 Simplicity and solvability . . . . . . . . . . . . . . . . . . . . . . 231 11.2 Traces on M_{n}(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11.3 A nice criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 11.4 Ideals and structure constants . . . . . . . . . . . . . . . . . . . . 235 11.5 Nondegeneracy and bilinear forms . . . . . . . . . . . . . . . . . 236 11.6 Bilinear forms and adjoints . . . . . . . . . . . . . . . . . . . . . 237 11.7 Bilinear forms and symmetry conditions . . . . . . . . . . . . . . 239 11.8 Traces and involutions . . . . . . . . . . . . . . . . . . . . . . . . 240 11.9 Inner products over ordered fields . . . . . . . . . . . . . . . . . . 241 11.10Self-adjoint Lie subalgebras . . . . . . . . . . . . . . . . . . . . . 244 11.11Comparing involutions . . . . . . . . . . . . . . . . . . . . . . . . 245 11.12Some projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 11.13Antisymmetric bilinear forms . . . . . . . . . . . . . . . . . . . . 248 11.14Symmetric bilinear forms . . . . . . . . . . . . . . . . . . . . . . 250

12 Some complex versions 252 12.1 Complexifying ordered fields . . . . . . . . . . . . . . . . . . . . . 252 12.2 Sesquilinearity over k[\sqrt{−a}] . . . . . . . . . . . . . . . . . . . . . 253 12.3 Inner products over k[\sqrt{−a}] . . . . . . . . . . . . . . . . . . . . . 255 12.4 Self-adjointness in gl(V) . . . . . . . . . . . . . . . . . . . . . . . 257 12.5 Projections and sesquilinear forms . . . . . . . . . . . . . . . . . 259 12.6 Nondegenerate Hermitian forms . . . . . . . . . . . . . . . . . . . 260 12.7 Real parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 12.8 Bilinear forms over k[\sqrt{−a}] . . . . . . . . . . . . . . . . . . . . . 264 12.9 Conjugate-linearity and adjoints . . . . . . . . . . . . . . . . . . 265 12.10Antisymmetric forms over k[\sqrt{−a}] . . . . . . . . . . . . . . . . . 266 12.11Symmetric forms over k[\sqrt{−a}] . . . . . . . . . . . . . . . . . . . . 268 12.12Nonnegative self-adjoint operators . . . . . . . . . . . . . . . . . 269 12.13Polar decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 270 12.14Bilinear forms and inner products . . . . . . . . . . . . . . . . . . 272 12.15Complex bilinear forms . . . . . . . . . . . . . . . . . . . . . . . 273

13 Semisimplicity 276 13.1 Semisimple modules . . . . . . . . . . . . . . . . . . . . . . . . . 276 13.2 Complete reducibility . . . . . . . . . . . . . . . . . . . . . . . . 278 13.3 Casimir elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 13.4 Another perspective . . . . . . . . . . . . . . . . . . . . . . . . . 281 13.5 A more particular situation . . . . . . . . . . . . . . . . . . . . . 283 13.6 A splitting theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 283 13.7 Weyl’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 13.8 Symmetric forms and tensors . . . . . . . . . . . . . . . . . . . . 286 13.9 Reductive Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 288 13.10Semisimple ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 13.11Another approach . . . . . . . . . . . . . . . . . . . . . . . . . . 289 13.12Inner derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 14 Nilpotence and diagonalizability 292

14.1 A duality argument . . . . . . . . . . . . . . . . . . . . . . . . . . 292 14.2 A criterion for nilpotence . . . . . . . . . . . . . . . . . . . . . . 293 14.3 ad-Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . 294 14.4 Semisimple subalgebras of gl(V) . . . . . . . . . . . . . . . . . . 296 14.5 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 14.6 Induced linear mappings . . . . . . . . . . . . . . . . . . . . . . . 298 14.7 Invariant tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 14.8 Diagonalizability and quotients . . . . . . . . . . . . . . . . . . . 300 14.9 Diagonalizability and representations . . . . . . . . . . . . . . . . 302 14.10Exponentiating nilpotents . . . . . . . . . . . . . . . . . . . . . . 303 14.11Nilpotent derivations . . . . . . . . . . . . . . . . . . . . . . . . . 305 14.12Exponentials and homomorphisms . . . . . . . . . . . . . . . . . 307 14.13Representations and structure constants . . . . . . . . . . . . . . 308 14.14Lie’s theorem and nilpotence . . . . . . . . . . . . . . . . . . . . 309

15 Complexifications and sl_{2}(k) modules 311 15.1 Basic properties of sl_{2}(k) modules . . . . . . . . . . . . . . . . . 311 15.2 Maximal or primitive vectors . . . . . . . . . . . . . . . . . . . . 312 15.3 Related submodules . . . . . . . . . . . . . . . . . . . . . . . . . 313 15.4 Constructing modules W (m) . . . . . . . . . . . . . . . . . . . . 315 15.5 More on W (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 15.6 Another construction . . . . . . . . . . . . . . . . . . . . . . . . . 318 15.7 Some exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . 319 15.8 Exponentials and W (m) . . . . . . . . . . . . . . . . . . . . . . . 321 15.9 Additional properties of sl_{2}(k) modules . . . . . . . . . . . . . . 322 15.10Complexifying real vector spaces . . . . . . . . . . . . . . . . . . 323 15.11Spaces of linear mappings . . . . . . . . . . . . . . . . . . . . . . 324 15.12Complexifying algebras over R . . . . . . . . . . . . . . . . . . . 326 15.13Conjugate-linear involutions . . . . . . . . . . . . . . . . . . . . . 327

16 Module homomorphisms 330 16.1 Bilinear actions and homomorphisms . . . . . . . . . . . . . . . . 330 16.2 Homomorphisms and semisimplicity . . . . . . . . . . . . . . . . 331 16.3 Linear mappings and dimensions . . . . . . . . . . . . . . . . . . 333 16.4 Homomorphisms and dimensions . . . . . . . . . . . . . . . . . . 333 16.5 Semisimplicity and dimensions . . . . . . . . . . . . . . . . . . . 334 16.6 Absolutely simple Lie algebras . . . . . . . . . . . . . . . . . . . 335 16.7 Algebras over subrings . . . . . . . . . . . . . . . . . . . . . . . . 337 16.8 Lie algebras over subfields . . . . . . . . . . . . . . . . . . . . . . 338

III Toral subalgebras and roots 340

17 Subalgebras and diagonalizability 341 17.1 Toral subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 17.2 Simultaneous diagonalizability . . . . . . . . . . . . . . . . . . . 342 17.3 Related bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . 344 17.4 Maximal toral subalgebras . . . . . . . . . . . . . . . . . . . . . . 345 17.5 Self-centralizability and diagonalizability . . . . . . . . . . . . . . 347 17.6 Characteristic 0, Z(A) = {0} . . . . . . . . . . . . . . . . . . . . 349 17.7 The dimension of A_{α} . . . . . . . . . . . . . . . . . . . . . . . . . 350 17.8 Other elements of Φ_{B} . . . . . . . . . . . . . . . . . . . . . . . . 353 17.9 Adding elements of Φ_{B} . . . . . . . . . . . . . . . . . . . . . . . . 355 17.10The dual of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 17.11Using the Killing form . . . . . . . . . . . . . . . . . . . . . . . . 359 17.12Vector spaces over Q . . . . . . . . . . . . . . . . . . . . . . . . . 360 17.13Vector spaces over R . . . . . . . . . . . . . . . . . . . . . . . . . 361 17.14Normalizers and centralizers . . . . . . . . . . . . . . . . . . . . . 363 17.15Semisimplicity and diagonalizability . . . . . . . . . . . . . . . . 364

18 Cartan subalgebras 366 18.1 Subspaces related to ad_{x} . . . . . . . . . . . . . . . . . . . . . . . 366 18.2 Some remarks about characteristic polynomials . . . . . . . . . . 367 18.3 Characteristic polynomials and ad_{x} . . . . . . . . . . . . . . . . . 368 18.4 Engel subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 18.5 Self-normalizing subalgebras . . . . . . . . . . . . . . . . . . . . . 371 18.6 Orthogonality conditions . . . . . . . . . . . . . . . . . . . . . . . 372 18.7 Nilpotence and normalizing elements . . . . . . . . . . . . . . . . 373 18.8 Minimal Engel subalgebras . . . . . . . . . . . . . . . . . . . . . 374 18.9 Nilpotent vectors and sums . . . . . . . . . . . . . . . . . . . . . 375 18.10Cartan subalgebras and semisimplicity . . . . . . . . . . . . . . . 377 18.11Subalgebras and homomorphisms . . . . . . . . . . . . . . . . . . 379 18.12Unions of hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . 380 18.13Centralizers and diagonalizability . . . . . . . . . . . . . . . . . . 381

19 Root systems 383 19.1 Symmetries and reflections . . . . . . . . . . . . . . . . . . . . . 383 19.2 Invariant inner products . . . . . . . . . . . . . . . . . . . . . . . 384 19.3 Defining root systems . . . . . . . . . . . . . . . . . . . . . . . . 385 19.4 The Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 19.5 Isomorphic root systems . . . . . . . . . . . . . . . . . . . . . . . 388 19.6 Duality and symmetries . . . . . . . . . . . . . . . . . . . . . . . 389 19.7 Adjoints and reflections . . . . . . . . . . . . . . . . . . . . . . . 391 19.8 Inverse roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 19.9 Angles between roots . . . . . . . . . . . . . . . . . . . . . . . . . 395 19.10A criterion for linear independence . . . . . . . . . . . . . . . . . 397 19.11Bases for root systems . . . . . . . . . . . . . . . . . . . . . . . . 398 19.12Positive roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 19.13Bases and duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 19.14Bases and the Weyl group . . . . . . . . . . . . . . . . . . . . . . 404 19.15Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 20 Root systems, 2 410

20.1 Products in the Weyl group . . . . . . . . . . . . . . . . . . . . . 410 20.2 The Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 411 20.3 Coxeter graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 20.4 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 20.5 Root strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 20.6 Roots and linear subspaces . . . . . . . . . . . . . . . . . . . . . 418 20.7 More on irreducibility . . . . . . . . . . . . . . . . . . . . . . . . 418 20.8 Maximal roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 20.9 More on maximality . . . . . . . . . . . . . . . . . . . . . . . . . 422 20.10Root systems of type A_{n} . . . . . . . . . . . . . . . . . . . . . . . 424 20.11Types B_{n} and C_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 20.12Type D_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 20.13Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 431 20.14Type A_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 20.15Dynkin diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

21 Root systems, 3 435 21.1 More on type A_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 21.2 Type G_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 21.3 Another description . . . . . . . . . . . . . . . . . . . . . . . . . 437 21.4 Type F_{4} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 21.5 Inverse systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 21.6 Type E_{8} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 21.7 Type E_{7} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 21.8 Type E_{6} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 21.9 Irreducible root systems . . . . . . . . . . . . . . . . . . . . . . . 450 21.10Connected Coxeter graphs . . . . . . . . . . . . . . . . . . . . . . 451 21.11Lengths in the Weyl group . . . . . . . . . . . . . . . . . . . . . . 452 21.12Nonnegative linear functionals . . . . . . . . . . . . . . . . . . . . 454 21.13The mapping α ↦ −α . . . . . . . . . . . . . . . . . . . . . . . . 456 21.14Another mapping on V . . . . . . . . . . . . . . . . . . . . . . . 457 21.15Invertibility of Cartan matrices . . . . . . . . . . . . . . . . . . . 458

22 Roots and Lie algebras 460 22.1 A basic situation . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 22.2 Inverse roots and B . . . . . . . . . . . . . . . . . . . . . . . . . 462 22.3 Semigroups and submodules . . . . . . . . . . . . . . . . . . . . . 464 22.4 Semigroups and algebras . . . . . . . . . . . . . . . . . . . . . . . 465 22.5 Some nilpotency properties . . . . . . . . . . . . . . . . . . . . . 467 22.6 Roots and submodules . . . . . . . . . . . . . . . . . . . . . . . . 469 22.7 Borel subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 22.8 Some diagonalizability conditions . . . . . . . . . . . . . . . . . . 472 22.9 Diagonalizability and subspaces . . . . . . . . . . . . . . . . . . . 474 22.10Injectivity on Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 22.11Some additional conditions . . . . . . . . . . . . . . . . . . . . . 476 22.12Standard Borel subalgebras . . . . . . . . . . . . . . . . . . . . . 478 22.13Subalgebras and derivations . . . . . . . . . . . . . . . . . . . . . 479 22.14Reducibility and semigroups . . . . . . . . . . . . . . . . . . . . . 480 22.15Reducibility and roots . . . . . . . . . . . . . . . . . . . . . . . . 483

23 Roots and Lie algebras, 2 485 23.1 Inverse roots and homomorphisms . . . . . . . . . . . . . . . . . 485 23.2 Bases and submodules . . . . . . . . . . . . . . . . . . . . . . . . 486 23.3 Some generating submodules . . . . . . . . . . . . . . . . . . . . 487 23.4 Some generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 23.5 Some relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 23.6 Isomorphisms and diagonalizability . . . . . . . . . . . . . . . . . 492 23.7 Automorphisms and sl_{2}(k) . . . . . . . . . . . . . . . . . . . . . . 493 23.8 Automorphisms and roots . . . . . . . . . . . . . . . . . . . . . . 494 23.9 Automorphisms and bases for Φ_{B} . . . . . . . . . . . . . . . . . . 496 23.10Sums and diagonalizability . . . . . . . . . . . . . . . . . . . . . 498 23.11Reducibility and diagonalizability . . . . . . . . . . . . . . . . . . 500

24 Strong ad-nilpotence and automorphisms 504 24.1 Invariant subspaces and nilpotent vectors . . . . . . . . . . . . . 504 24.2 Finite-dimensional spaces . . . . . . . . . . . . . . . . . . . . . . 505 24.3 Strong ad-nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . 506 24.4 Strong ad-nilpotence and homomorphisms . . . . . . . . . . . . . 507 24.5 Another criterion for nilpotence . . . . . . . . . . . . . . . . . . . 509 24.6 Nilpotence and automorphisms . . . . . . . . . . . . . . . . . . . 509 24.7 Conjugacy and solvability . . . . . . . . . . . . . . . . . . . . . . 510 24.8 The case where A = B_{2} . . . . . . . . . . . . . . . . . . . . . . . 511 24.9 Conjugacy in arbitrary Lie algebras . . . . . . . . . . . . . . . . . 514

24.9.1 The beginning of the proof of the theorem . . . . . . . . . 514 24.9.2 Standard Borel subalgebras of A . . . . . . . . . . . . . . 515 24.9.3 The second induction hypothesis . . . . . . . . . . . . . . 516

24.10The first subcase . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 24.10.1 Nilpotent elements of B_{Δ} ∩ B . . . . . . . . . . . . . . . . 517 24.10.2 Some Borel subalgebras of M and A . . . . . . . . . . . . 519

24.11The second subcase . . . . . . . . . . . . . . . . . . . . . . . . . . 520 24.11.1 Reducing to T ⊆ A_{0} . . . . . . . . . . . . . . . . . . . . . 521 24.11.2 The case where T = A_{0} . . . . . . . . . . . . . . . . . . . 522 24.11.3 The case where T ≠ A_{0}, B ⊆ C_{A}(T) . . . . . . . . . . . . 522 24.11.4 The case where T ≠ A_{0}, B ⊈ C_{A}(T) . . . . . . . . . . . . 523

24.12The second case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 24.13Some consequences . . . . . . . . . . . . . . . . . . . . . . . . . . 526 24.14The case of sl_{2}(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 24.15Automorphisms and semisimplicity . . . . . . . . . . . . . . . . . 529

IV Some related constructions 531

25 Universal enveloping algebras 532 25.1 Polynomial algebras . . . . . . . . . . . . . . . . . . . . . . . . . 532 25.2 Tensor products of associative algebras . . . . . . . . . . . . . . . 534 25.3 Tensor algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 25.4 Defining universal enveloping algebras . . . . . . . . . . . . . . . 536 25.5 Constructing universal enveloping algebras . . . . . . . . . . . . . 537 25.6 Some properties of UA . . . . . . . . . . . . . . . . . . . . . . . . 538 25.7 Commuting Lie subalgebras . . . . . . . . . . . . . . . . . . . . . 539 25.8 Symmetric algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 541 25.9 A filtration on UA . . . . . . . . . . . . . . . . . . . . . . . . . . 543 25.10The Poincaré–Birkhoff–Witt theorem . . . . . . . . . . . . . . . . 545 25.11The rest of the proof . . . . . . . . . . . . . . . . . . . . . . . . . 547 25.12Some corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 25.13Some related filtrations . . . . . . . . . . . . . . . . . . . . . . . 552 25.14The diagonal mapping . . . . . . . . . . . . . . . . . . . . . . . . 554 25.15More on primitive elements . . . . . . . . . . . . . . . . . . . . . 556

26 Free Lie algebras 560 26.1 Magmas and their homomorphisms . . . . . . . . . . . . . . . . . 560 26.2 Magmas and algebras . . . . . . . . . . . . . . . . . . . . . . . . 561 26.3 Constructing free Lie algebras . . . . . . . . . . . . . . . . . . . . 561 26.4 Degree 2 in L_{X} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 26.5 Free associative algebras . . . . . . . . . . . . . . . . . . . . . . . 565 26.6 Injectivity and free modules . . . . . . . . . . . . . . . . . . . . . 566 26.7 Finite sets X, k = Z . . . . . . . . . . . . . . . . . . . . . . . . . 568 26.8 Arbitrary sets X, k = Z . . . . . . . . . . . . . . . . . . . . . . . 569 26.9 Arbitrary X, k . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 26.10Bases from P. Hall families . . . . . . . . . . . . . . . . . . . . . 571 26.11More on L_{X}^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

27 Some more Lie algebras 573 27.1 Generators and relations . . . . . . . . . . . . . . . . . . . . . . . 573 27.2 A module over L̂ . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 27.3 Some properties of L̃ . . . . . . . . . . . . . . . . . . . . . . . . . 578 27.4 Some brackets in L̃ . . . . . . . . . . . . . . . . . . . . . . . . . . 580 27.5 Eigenvectors in L̃ . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 27.6 Some more properties of L̃ . . . . . . . . . . . . . . . . . . . . . . 582 27.7 Some more relations in L̃ . . . . . . . . . . . . . . . . . . . . . . 584 27.8 Serre’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 27.9 Ideals I, J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 27.10Subalgebras X, H, and Y of L . . . . . . . . . . . . . . . . . . . 587 27.11Eigenvectors in L . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 27.12Locally nilpotent linear mappings . . . . . . . . . . . . . . . . . . 590 27.13Some properties of θ_{α} . . . . . . . . . . . . . . . . . . . . . . . . 591 27.14Roots and H′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 27.15Some dimensions of eigenspaces . . . . . . . . . . . . . . . . . . . 594

28 Some more Lie algebras, 2 596 28.1 Linear combinations of roots . . . . . . . . . . . . . . . . . . . . 596 28.2 The dimension of L . . . . . . . . . . . . . . . . . . . . . . . . . . 597 28.3 Automorphisms and the Weyl group . . . . . . . . . . . . . . . . 598 28.4 Semisimplicity of L . . . . . . . . . . . . . . . . . . . . . . . . . . 600 28.5 An isomorphism theorem . . . . . . . . . . . . . . . . . . . . . . 601 28.6 Some automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 602 28.7 Roots and diagonalizability . . . . . . . . . . . . . . . . . . . . . 604 28.8 Quotients of L̃ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 28.9 Additional identities in B . . . . . . . . . . . . . . . . . . . . . . 606 28.10Eigenspaces in L . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 28.11Subsets of Δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

29 Chevalley’s normalization 611 29.1 Linearly independent roots . . . . . . . . . . . . . . . . . . . . . 611 29.2 An identity for root strings . . . . . . . . . . . . . . . . . . . . . 613 29.3 Related properties of λ_{α}’s . . . . . . . . . . . . . . . . . . . . . . 615 29.4 Associated Lie algebras . . . . . . . . . . . . . . . . . . . . . . . 615 29.5 A lemma about brackets . . . . . . . . . . . . . . . . . . . . . . . 617 29.6 Some remarks about automorphisms . . . . . . . . . . . . . . . . 618 29.7 Diagonal automorphisms . . . . . . . . . . . . . . . . . . . . . . . 619 29.8 Basis vectors in A . . . . . . . . . . . . . . . . . . . . . . . . . . 620 29.9 Some properties of the c_{α,β}’s . . . . . . . . . . . . . . . . . . . . 622 29.10Uniqueness of Chevalley bases . . . . . . . . . . . . . . . . . . . . 623 29.11The complex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 29.12The corresponding Killing forms . . . . . . . . . . . . . . . . . . 627 29.13Using other commutative rings . . . . . . . . . . . . . . . . . . . 630 29.14Some mappings on A(Z) . . . . . . . . . . . . . . . . . . . . . . . 631 29.15Some related mappings on A(k_{1}) . . . . . . . . . . . . . . . . . . 633

30 Roots and abstract weights 636 30.1 Abstract weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 30.2 Some remarks about lattices . . . . . . . . . . . . . . . . . . . . . 637 30.3 Related positivity conditions . . . . . . . . . . . . . . . . . . . . 639 30.4 Automorphisms and Υ . . . . . . . . . . . . . . . . . . . . . . . . 641 30.5 Some partial orderings on V . . . . . . . . . . . . . . . . . . . . . 643 30.6 Some more properties of Υ . . . . . . . . . . . . . . . . . . . . . 644 30.7 Saturated subsets of Υ . . . . . . . . . . . . . . . . . . . . . . . . 646 30.8 More on highest weights . . . . . . . . . . . . . . . . . . . . . . . 647 30.9 More on saturated sets . . . . . . . . . . . . . . . . . . . . . . . . 649 30.10Another look at saturated sets . . . . . . . . . . . . . . . . . . . 650 30.11A property of saturated sets . . . . . . . . . . . . . . . . . . . . . 652 31 Regular elements and polynomials 653

31.1 Integral domains and polynomials . . . . . . . . . . . . . . . . . . 653 31.2 The Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . 654 31.3 Polynomial mappings . . . . . . . . . . . . . . . . . . . . . . . . . 656 31.4 More on the Zariski topology . . . . . . . . . . . . . . . . . . . . 657 31.5 Regular elements of Lie algebras . . . . . . . . . . . . . . . . . . 658 31.6 The set of regular elements . . . . . . . . . . . . . . . . . . . . . 660 31.7 Regularity and automorphisms . . . . . . . . . . . . . . . . . . . 661 31.8 Regular semisimple elements . . . . . . . . . . . . . . . . . . . . 662 31.9 Regularity and semisimplicity . . . . . . . . . . . . . . . . . . . . 663 31.10Symmetric algebras and homomorphisms . . . . . . . . . . . . . 664 31.11Some polynomial functions on V′ . . . . . . . . . . . . . . . . . . 665 31.12Polynomials and symmetric algebras . . . . . . . . . . . . . . . . 667 31.13Polynomials on Lie algebras . . . . . . . . . . . . . . . . . . . . . 668

32 Some remarks and helpful facts 670 32.1 Roots and vector spaces . . . . . . . . . . . . . . . . . . . . . . . 670 32.2 Roots and dual spaces . . . . . . . . . . . . . . . . . . . . . . . . 672 32.3 Dual spaces and abstract weights . . . . . . . . . . . . . . . . . . 673 32.4 Abstract weights and group rings . . . . . . . . . . . . . . . . . . 676 32.5 Homomorphisms and Lie algebras . . . . . . . . . . . . . . . . . . 676 32.6 Submodules of UA . . . . . . . . . . . . . . . . . . . . . . . . . . 678 32.7 Some automorphisms of UA . . . . . . . . . . . . . . . . . . . . . 679 32.8 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 32.9 The center of UA . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 32.10Constant-coefficient differential operators . . . . . . . . . . . . . 682

V Some representation theory 685

33 Representations and semisimplicity 686 33.1 Diagonalizability and roots . . . . . . . . . . . . . . . . . . . . . 686 33.2 Using a larger field . . . . . . . . . . . . . . . . . . . . . . . . . . 687 33.3 Related spaces of linear mappings . . . . . . . . . . . . . . . . . . 688 33.4 Larger fields and toral subalgebras . . . . . . . . . . . . . . . . . 690 33.5 Weights and modules . . . . . . . . . . . . . . . . . . . . . . . . . 693 33.6 Primitive or maximal vectors . . . . . . . . . . . . . . . . . . . . 694 33.7 Representations and UA . . . . . . . . . . . . . . . . . . . . . . . 696 33.8 Spanning standard cyclic modules . . . . . . . . . . . . . . . . . 697 33.9 Indecomposability . . . . . . . . . . . . . . . . . . . . . . . . . . 699 33.10Irreducible standard cyclic modules . . . . . . . . . . . . . . . . . 701 33.11Representations and left ideals . . . . . . . . . . . . . . . . . . . 703 33.12Tensor products over algebras . . . . . . . . . . . . . . . . . . . . 704 33.13Using modules over other algebras . . . . . . . . . . . . . . . . . 705 33.14Existence of standard cyclic modules . . . . . . . . . . . . . . . . 707 33.15Homomorphisms into other modules . . . . . . . . . . . . . . . . 709 34 Some related examples and properties 711

34.1 Some remarks about sl_{2}(k) . . . . . . . . . . . . . . . . . . . . . 711 34.2 Some modules over sl_{2}(k) . . . . . . . . . . . . . . . . . . . . . . 712 34.3 Some module homomorphisms . . . . . . . . . . . . . . . . . . . 714 34.4 Some properties of Z_{0}(μ_{0}) . . . . . . . . . . . . . . . . . . . . . . 715 34.5 Comparison with Z(μ) . . . . . . . . . . . . . . . . . . . . . . . . 716 34.6 Looking at x in UA . . . . . . . . . . . . . . . . . . . . . . . . . 718 34.7 Some related properties of x and h . . . . . . . . . . . . . . . . . 719 34.8 Irreducibility and weights . . . . . . . . . . . . . . . . . . . . . . 720 34.9 Weight spaces in Z(μ) . . . . . . . . . . . . . . . . . . . . . . . . 721

35 Formal Laurent series 723 35.1 Formal Laurent polynomials and series . . . . . . . . . . . . . . . 723 35.2 Multiplication and Laurent series . . . . . . . . . . . . . . . . . . 725 35.3 Laurent polynomial functions . . . . . . . . . . . . . . . . . . . . 727 35.4 Differentiating Laurent series . . . . . . . . . . . . . . . . . . . . 728 35.5 Derivatives and functions . . . . . . . . . . . . . . . . . . . . . . 729 35.6 Differential operators and Laurent series . . . . . . . . . . . . . . 731 35.7 First-order operators . . . . . . . . . . . . . . . . . . . . . . . . . 733 35.8 Homogeneous Laurent series . . . . . . . . . . . . . . . . . . . . . 735 35.9 Homogeneity and differential operators . . . . . . . . . . . . . . . 736 35.10Representations and Laurent series . . . . . . . . . . . . . . . . . 738 35.11A linear mapping onto A . . . . . . . . . . . . . . . . . . . . . . . 740 35.12Some related bilinear functionals . . . . . . . . . . . . . . . . . . 740 35.13Adjoints of differential operators . . . . . . . . . . . . . . . . . . 742

36 Formal Laurent series, 2 743 36.1 Laurent series and sl_{2}(k) . . . . . . . . . . . . . . . . . . . . . . . 743 36.2 Some submodules of LS_{A,d}(T_{1}, T_{2}) . . . . . . . . . . . . . . . . . 745 36.3 Submodules over s_{2}(k) . . . . . . . . . . . . . . . . . . . . . . . . 746 36.4 Characteristic 0, A = k . . . . . . . . . . . . . . . . . . . . . . . 747 36.5 Some helpful projections . . . . . . . . . . . . . . . . . . . . . . . 748 36.6 Projections and H, X, Y . . . . . . . . . . . . . . . . . . . . . . 750 36.7 Some related quotient modules . . . . . . . . . . . . . . . . . . . 751 36.8 Quotients and characteristic 0, A = k . . . . . . . . . . . . . . . 753 36.9 Some more submodules and quotients . . . . . . . . . . . . . . . 754 36.10Some additional submodules . . . . . . . . . . . . . . . . . . . . . 755

37 Formal Laurent series, 3 758 37.1 Some commutativity conditions . . . . . . . . . . . . . . . . . . . 758 37.2 Simpler coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 761 37.3 Some actions on LS_{A}(T_{1}, T_{2}, R) . . . . . . . . . . . . . . . . . . . 763 37.4 The action of s̃_{2}(k) . . . . . . . . . . . . . . . . . . . . . . . . . . 765 37.5 Some submodules of LS_{A,d}(T_{1}, T_{2}) R^{γ} . . . . . . . . . . . . . . . 767 37.6 Submodules over s̃_{2}(k) . . . . . . . . . . . . . . . . . . . . . . . . 768 37.7 Characteristic 0 and LP_{k,d}(T_{1}, T_{2}) R^{γ} . . . . . . . . . . . . . . . 770 37.8 Helpful projections on LS_{A,d}(T_{1}, T_{2}) R^{γ} . . . . . . . . . . . . . . 771 37.9 Projections and H̃, X̃, Ỹ . . . . . . . . . . . . . . . . . . . . . . 773 37.10Related quotients over s̃_{2}(k) . . . . . . . . . . . . . . . . . . . . . 775

38 Representations and semisimplicity, 2 777 38.1 Weights and finite dimension . . . . . . . . . . . . . . . . . . . . 777 38.2 Weights and a larger field . . . . . . . . . . . . . . . . . . . . . . 779 38.3 Conditions for finite dimension . . . . . . . . . . . . . . . . . . . 780 38.4 Some initial steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 38.5 Submodules over subalgebras . . . . . . . . . . . . . . . . . . . . 784 38.6 Some related exponentials on V . . . . . . . . . . . . . . . . . . . 785 38.7 Weights and the Weyl group . . . . . . . . . . . . . . . . . . . . . 787 38.8 Sums of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788 38.9 Weights and tensor products . . . . . . . . . . . . . . . . . . . . 790 38.10Fundamental weights and modules . . . . . . . . . . . . . . . . . 791 38.11Simple transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 792 38.12Weights and saturated sets . . . . . . . . . . . . . . . . . . . . . 793 38.13Formal characters in Z[Υ] . . . . . . . . . . . . . . . . . . . . . . 795 38.14Invariant elements of Z[Υ] . . . . . . . . . . . . . . . . . . . . . . 797

39 Dimensions of weight spaces 800 39.1 Casimir elements in UA . . . . . . . . . . . . . . . . . . . . . . . 800 39.2 Casimir elements and nice bases . . . . . . . . . . . . . . . . . . 801 39.3 Adjusted bases for sl_{2}(k) modules . . . . . . . . . . . . . . . . . 803 39.4 Some preliminary subspaces . . . . . . . . . . . . . . . . . . . . . 804 39.5 Some traces of ρ_{x_{α}} ∘ ρ_{z_{α}} . . . . . . . . . . . . . . . . . . . . . . . 806 39.6 The case where 2 l > m . . . . . . . . . . . . . . . . . . . . . . . 808 39.7 Some related expressions for traces . . . . . . . . . . . . . . . . . 810 39.8 Related identities for suitable ν . . . . . . . . . . . . . . . . . . . 812 39.9 Some remarks about dual bases . . . . . . . . . . . . . . . . . . . 813 39.10Traces and b′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815 39.11Using an algebraically closed field . . . . . . . . . . . . . . . . . . 817 39.12Freudenthal’s formula . . . . . . . . . . . . . . . . . . . . . . . . 820

Bibliography 822

Index 824

Part I

Chapter 1

Basic notions and objects Absolute values and norms

1.1 Metrics and ultrametrics

Let X be a set. A nonnegative real-valued function d(x, y) defined for x, y ∈ X is said to be a semimetric on X if it satisfies the following three conditions. First, d(x, x) = 0 (1.1.1)

for every x ∈ X. Second, d(x, y) = d(y, x) (1.1.2)

for every x, y ∈ X. Third,

d(x, z) ≤ d(x, y) + d(y, z) (1.1.3)

for every x, y, z ∈ X. If we also have that

d(x, y) > 0 (1.1.4)

for every x, y ∈ X with x ≠ y, then d(·, ·) is said to be a metric on X. The discrete metric is defined on X by putting d(x, y) equal to 0 when x = y, and equal to 1 when x ≠ y.

Similarly, a nonnegative real-valued function d(x, y) defined for x, y ∈ X is said to be a semi-ultrametric on X if if it satisfies (1.1.1), (1.1.2), and

d(x, z) ≤ max(d(x, y), d(y, z)) (1.1.5)

for every x, y, z ∈ X. Note that (1.1.5) implies (1.1.3), so that a semi-ultrametric on X is a semimetric on X in particular. If a semi-ultrametric d(x, y) on X satisfies (1.1.4), then d(x, y) is said to be an ultrametric on X. It is easy to see that the discrete metric on X is an ultrametric.

Let d(x, y) be a semimetric on X. The open ball in X centered at x ∈ X with radius r > 0 with respect to d(·, ·) is defined as usual by

B(x, r) = B_{d}(x, r) = {y ∈ X : d(x, y) < r}. (1.1.6)

Similarly, the closed ball in X centered at x ∈ X with radius r ≥ 0 is defined by \overline{B}(x, r) = \overline{B}_{d}(x, r) = \{y ∈ X : d(x, y) ≤ r\}. (1.1.7)

A subset U of X is said to be an open set with respect to d(·, ·) if for every x ∈ U there is an r > 0 such that

B(x, r) ⊆ U. (1.1.8)

This defines a topology on X, by standard arguments. One can check that open balls in X are open sets, and that closed balls are closed sets. If d(·, ·) is a metric on X, then X is Hausdorff with respect to the topology determined by d(·, ·).

Suppose that d(·, ·) is a semi-ultrametric on X. If x, y ∈ X satisfy d(x, y) < r for some r > 0, then it is easy to see that

B(x, r) ⊆ B(y, r). (1.1.9)

More precisely, B(x, r) = B(y, r), (1.1.10)

because we can interchange the roles of x and y in (1.1.9). Similarly, if x, y ∈ X satisfy d(x, y) ≤ r for some r ≥ 0, then

\overline{B}(x, r) ⊆ \overline{B}(y, r), (1.1.11)

and hence \overline{B}(x, r) = \overline{B}(y, r). (1.1.12)

This implies that closed balls in X with positive radius are open sets, and one can check that open balls in X are closed sets in this situation.

1.2 Absolute value functions

Let k be a field. A nonnegative real-valued function | · | on k is said to be an absolute value function on k if it satisfies the following conditions. First, |x| = 0 if and only if x = 0. Second, |x y| = |x| |y| (1.2.1)

for every x, y ∈ k. Third, |x + y| ≤ |x| + |y| (1.2.2)

for every x, y ∈ k. The standard absolute value functions on the fields R of real numbers and C of complex numbers are absolute value functions in this sense. The trivial absolute value function on any field k is defined by putting |x| equal to 0 when x = 0, and equal to 1 when x ≠ 0.

If |·| is any absolute value function on a field k, then |1| = 1, where the first 1 is the multiplicative identity element in k, and the second 1 is the multiplicative identity element in R. This uses the fact that 1^{2} = 1 in k, so that |1| = |1|^{2} by (1.2.1). If x ∈ k satisfies x^{n} = 1 for some positive integer n, then |x|^{n} = |1| = 1, and hence |x| = 1. In particular, | − 1| = 1, because (−1)^{2} = 1. It follows that

d(x, y) = |x − y| (1.2.3)

defines a metric on k, using | − 1| = 1 to get that (1.2.3) is symmetric in x and y.

A nonnegative real-valued function |·| on a field k is said to be an ultrametric absolute value function on k if it satisfies the first two conditions in the definition of an absolute value function, and

|x + y| ≤ max(|x|, |y|) (1.2.4)

for every x, y ∈ k. It is easy to see that (1.2.4) implies (1.2.2), so that an ultrametric absolute value function on k is an absolute value function on k. If | · | is an ultrametric absolute value function on k, then (1.2.3) is an ultrametric on k. The trivial absolute value function on any field k is an ultrametric absolute value function. The ultrametric associated to the trivial absolute value function as in (1.2.3) is the discrete metric.

Let p be a prime number. The p-adic absolute value |x|_{p} of a rational number x is defined as follows. If x = 0, then we put |x|_{p} = 0. Otherwise, if x ≠ 0, then x can be expressed as p^{j} (a/b) for some integers a, b, and j, where a, b ≠ 0, and neither a nor b is an integer multiple of p. In this case, we put

|x|_{p} = p^{−j}. (1.2.5)

One can check that this defines an ultrametric absolute value function on the field Q of rational numbers. The corresponding ultrametric

d_{p}(x, y) = |x − y|_{p} (1.2.6)

is known as the p-adic metric on Q.

Let k be any field again, and let Z_{+} be the set of positive integers, as usual. If x ∈ k and n ∈ Z_{+}, then let n · x be the sum of n x’s in k. An absolute value function | · | on k is said to be archimedean on k if there are n ∈ Z_{+} such that |n · 1| is arbitrarily large. Otherwise, | · | is said to be non-archimedean on k. If | · | is an ultrametric absolute value function on k, then it is easy to see that

|n · 1| ≤ 1 (1.2.7)

for every n ∈ Z_{+}, so that | · | is non-archimedean on k. Conversely, it is well known that a non-archimedean absolute value function on k is necessarily an ultrametric absolute value function on k. In particular, (1.2.7) holds for every n ∈ Z_{+} in this case, which can be verified more directly. More precisely, if | · | is any absolute value function on k, then one can check that

|n^{j} · 1| = |(n · 1)^{j}| = |n · 1|^{j} (1.2.8)

for all positive integers j, n. If |n · 1| > 1 for some n ∈ Z_{+}, then (1.2.8) tends to +∞ as j → ∞, so that | · | is archimedean on k.

1.3 Equivalent absolute value functions

If a is a positive real number with a ≤ 1, then it is well known that

(r + t)^{a} ≤ r^{a} + t^{a} (1.3.1)

for all nonnegative real numbers a, b. To see this, observe first that

max(r, t) ≤ (r^{a} + t^{a})^{1/a} (1.3.2)

for every a > 0. We also have that

r + t = r^{1−a} r^{a} + t^{1−a} t^{a} ≤ max(r^{1−a}, t^{1−a}) (r^{a} + t^{a}). (1.3.3)

If a ≤ 1, then it follows that

r + t ≤ max(r, t)^{1−a} (r^{a} + t^{a}) ≤ (r^{a} + t^{a})^{(1−a)/a+1} = (r^{a} + t^{a})^{1/a}, (1.3.4)

using (1.3.2) in the second step. This implies (1.3.1), as desired.

Let d(x, y) be a semimetric on a set X. If 0 < a ≤ 1, then one can check that d(x, y)^{a} (1.3.5)

also defines a semimetric on X. More precisely, one can verify that (1.3.5) satisfies the triangle inequality using (1.3.1) and the triangle inequality for d(x, y). If d(x, y) is a semi-ultrametric on X, then (1.3.5) is a semi-ultrametric on X for every a > 0.

Suppose that d(x, y) is a semimetric on X again, and that (1.3.5) is a semimetric on X too for some a > 0. Observe that

B_{d^{a}}(x, r^{a}) = B_{d}(x, r) (1.3.6)

for every x ∈ X and r > 0, and that

\overline{B}_{d^{a}}(x, r^{a}) = \overline{B}_{d}(x, r) (1.3.7)

for every x ∈ X and r ≥ 0. In particular, (1.3.6) implies that d(x, y) and (1.3.5) determine the same topology on X.

Let k be a field, and let | · | be an absolute value function on k. If 0 < a ≤ 1, then |x|^{a} also defines an absolute value function on k. As before, this uses (1.3.1) to get the triangle inequality for |x|^{a} from the triangle inequality for | · |. If | · | is an ultrametric absolute value function on k, then |x|^{a} is an ultrametric absolute value function on k for every a > 0.

Suppose that | · | is an absolute value function on k again, and that | · |^{a} is an absolute value function on k as well for some a > 0. Thus the metric associated to | · |^{a} on k is the same as the ath power of the metric associated to | · | on k. Hence these two metrics determine the same topoplogy on k, as in the preceding paragraph.

Let | · |_{1} and | · |_{2} be absolute value functions on k. If there is a positive real number a such that |x|_{2} = |x|_{1}^{a} (1.3.8)

for every x ∈ k, then | · |_{1} and | · |_{2} are said to be equivalent on k. In this case, the metrics associated to | · |_{1} and | · |_{2} determine the same topology on k, as in the previous paragraph. Conversely, if the metrics associated to | · |_{1} and | · |_{2} determine the same topology on k, then it is well known that | · |_{1} and | · |_{2} are equivalent on k, in the sense of (1.3.8).

Let | · | be an absolute value function on Q. A famous theorem of Ostrowski implies that | · | is either equivalent to the standard absolute value function on Q, or | · | is the trivial absolute value function on Q, or | · | is equivalent to the p-adic absolute value function on Q for some prime number p.

1.4 Completions

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces, and let E be a dense subset of X. Suppose that f is a uniformly continuous mapping from E into Y, with respect to the restriction of d_{X} to E. If Y is complete with respect to d_{Y}, then it is well known that there is a unique extension of f to a uniformly continuous mapping from X into Y. More precisely, uniqueness only uses continuity of the extension.

If X is not complete, then it is well known that one can pass to a completion, which is given by an isometric mapping from X onto a dense subset of a complete metric space. The completion is unique up to isometric equivalence, because of the extension theorem mentioned in the preceding paragraph.

Let X be a set with a semimetric d(x, y), and let E be a dense subset of X. If the restriction of d(x, y) to x, y ∈ E defines a semi-ultrametric on E, then one can check that d(x, y) is a semi-ultrametric on X. In particular, the completion of an ultrametric space is an ultrametric space too.

Let k be a field, and let | · | be an absolute value function on k. If k is not complete with respect to the metric associated to | · |, then one can pass to a completion. It is well known that the field operations on k can be extended to the completion, in such a way that the completion is also a field. The absolute value function on k can be extended to an absolute value function on the completion, which corresponds to the distance to 0 in the completion. The completion of k is unique, up to isometric isomorphic equivalence.

If |·| is an ultrametric absolute value function on k, then the extension of |·| to the completion of k is an ultrametric absolute value function as well. This is analogous to the earlier statement for ultrametric spaces, and can be obtained from that statement. Alternatively, let k_{1} be any field with an absolute value function |·|, and let k_{0} be a subfield of k_{1}. It is easy to see that |·| is archimedean on k_{1} if and only if the restriction of | · | to k_{0} is archimedean on k_{0}.

Let p be a prime number. The field Q_{p} of p-adic numbers is obtained by completing Q with respect to the p-adic absolute value function | · |_{p}. The corresponding extension of | · |_{p} to Q_{p} is also denoted | · |_{p}, and defines an ultrametric absolute value function on Q_{p}. If x ∈ Q_{p} and x ≠ 0, then one can check that |x|_{p} is an integer power of p.

Let k be a field with an absolute value function | · | again. If k has positive characteristic, then it is easy to see that | · | is non-archimedean on k. Suppose that | · | is archimedean on k, which implies that k has characteristic 0. If k is complete with respect to the metric associated to | · |, then another famous theorem of Ostrowski implies that k is isomorphic to R or C, in such a way that | · | corresponds to an absolute value function on R or C that is equivalent to the standard absolute value function.

1.5 Discreteness

Let k be a field, and let | · | be an absolute value function on k. Observe that

{|x| : x ∈ k, x ≠ 0} (1.5.1)

is a subgroup of the group R_{+} of positive real numbers with respect to multiplication. Of course, (1.5.1) is the trivial subgroup {1} of R_{+} exactly when | · | is the trivial absolute value function on k. If 1 is not a limit point of (1.5.1) with respect to the standard topology on R, then | · | is said to be discrete on k.

Put

ρ_{1} = sup{|x| : x ∈ k, |x| < 1}, (1.5.2)

so that 0 ≤ ρ_{1} ≤ 1. If |·| is the trivial absolute value function on k, then ρ_{1} = 0. Conversely, if | · | is not the trivial absolute value function on k, then there is a y ∈ k such that y ≠ 0 and |y| ≠ 1. This implies that there is an x ∈ k such that x ≠ 0 and |x| < 1, by taking x = y when |y| < 1 and x = 1/y when |y| > 1. Thus ρ_{1} > 0 when | · | is nontrivial on k.

If | · | is a discrete absolute value function on k, then ρ_{1} < 1. Conversely, if ρ_{1} < 1, then | · | is discrete on k. More precisely, the definition of ρ_{1} implies that there is no x ∈ k such that ρ_{1} < |x| < 1. If y ∈ k and |y| > 1, then we can apply the previous statement to x = 1/y, to get that 1/|y| ≤ ρ_{1}. It follows that 1 is not a limit point of (1.5.1) in R when ρ_{1} < 1, as desired.

Suppose that | · | is an archimedean absolute value function on k. This implies that k has characteristic 0, as in the previous section. Hence there is a natural embedding of Q into k. This leads to an absolute value function on Q, using | · | on k. It is easy to see that Q is archimedean with respect to this absolute value function, because k is archimedean with respect to | · |. Using this and Ostrowski’s classification of absolute value functions on Q mentioned in Section 1.3, we get that this absolute value function on Q is equivalent to the standard absolute value function. In particular, it follows that this absolute value function on Q is not discrete. This means that | · | is not discrete on k. If | · | is a discrete absolute value function on k, then | · | is non-archimedean on k, and hence | · | is an ultrametric absolute value function on k.

Suppose that |·| is a nontrivial discrete absolute value function on k, so that 0 < ρ_{1} < 1. If y, z ∈ k and |y| < |z|, then

|y| ≤ ρ_{1} |z|, (1.5.3)

because |y/z| = |y|/|z| < 1, and hence |y/z| ≤ ρ_{1}, by the definition (1.5.2) of ρ_{1}. One can check that the supremum is attained in (1.5.2), since otherwise there would be distinct elements of (1.5.1) close to ρ_{1}, whose quotient would be close to 1 but not equal to 1. Thus ρ_{1} is an element of (1.5.1), which implies that (1.5.1) contains all integer powers of ρ_{1}. In fact, one can verify that every element of (1.5.1) is an integer power of ρ_{1} in this case.

Suppose that | · | is an ultrametric absolute value function on k. If x, y ∈ k satisfy |x − y| < |y|, (1.5.4)

then |x| = |y|. (1.5.5)

More precisely, |x| ≤ max(|x − y|, |y|) = |y|, (1.5.6)

by the ultrametric version of the triangle inequality. Similarly,

|y| ≤ max(|x − y|, |x|), (1.5.7)

which implies that |y| ≤ |x| in this situation.

Let k_{0} be a subfield of k that is dense with respect to the ultrametric associated to | · |. The remarks in the preceding paragraph imply that

{|x| : x ∈ k_{0}, x ≠ 0} (1.5.8)

is the same as (1.5.1). In particular, if k is not already complete with respect to the ultrametric associated to | · |, then the nonzero values of the extension of | · | to the completion of k is the same as (1.5.1).

1.6 p-Adic integers

Let k be a field, let x be an element of k, and let n be a nonnegative integer. Observe that

(1 − x) ∑_{j=0}^{n} x^{j} = ∑_{j=0}^{n} x^{j} − ∑_{j=0}^{n} x^{j+1} = ∑_{j=0}^{n} x^{j} − ∑_{j=1}^{n+1} x^{j} = 1 − x^{n+1}, (1.6.1)

where x^{j} is interpreted as being the multiplicative identity element 1 in k when j = 0. If x ≠ 1, then it follows that

∑_{j=0}^{n} x^{j} = (1 − x^{n+1}) (1 − x)^{−1}. (1.6.2)

1.6. P -ADIC INTEGERS 9

Let | · | be an absolute value function on k, so that

|∑_{j=0}^{n} x^{j} − (1 − x)^{−1}| = |x^{n+1} (1 − x)^{−1}| = |x|^{n+1} |1 − x|^{−1}. (1.6.3)

If |x| < 1, then we get that

lim_{n→∞}|∑_{j=0}^{n} x^{j} − (1 − x)^{−1}| = 0. (1.6.4)

Let p be a prime number, and let y be an integer. Thus x = p y satisfies

|x|_{p} = p^{−1} |y|_{p} ≤ p^{−1} < 1, (1.6.5)

where | · |_{p} is the p-adic absolute value, as before. It follows that

lim_{n→∞}|∑_{j=0}^{n} p^{j} y^{j} − (1 − p y)^{−1}|_{p} = 0, (1.6.6)

as in (1.6.4). Note that ∑_{j=0}^{n} p^{j} y^{j} is an integer for each nonnegative integer n.

Suppose that z ∈ Q satisfies |z|_{p} ≤ 1. This means that z can be expressed as a/b, where a and b are integers, b ≠ 0, and b is not a multiple of p. Hence there is an integer c such that b c ≡ 1 modulo p, because the integers modulo p form a field. Thus z can be expressed as

z = (a c)/(b c) = a c (1 − p y)^{−1}, (1.6.7)

where y is an integer. This implies that z can be approximated by integers with respect to the p-adic metric, because of the analogous statement for (1 − p y)^{−1}, as in the preceding paragraph.

Put Z_{p} = {x ∈ Q_{p} : |x|_{p} ≤ 1}, (1.6.8)

which is the set of p-adic integers. Of course, the set Z of integers is contained in Z_{p}, by the definition of the p-adic absolute value. This implies that the closure of Z in Q_{p} with respect to the p-adic metric is contained in Z_{p}, because Z_{p} is a closed set in Q_{p}. Conversely, let x ∈ Z_{p} be given, and let us check that x is in the closure of Z in Q_{p}. Of course, Q is dense in Q_{p}, by construction. Thus x can be approximated by z ∈ Q with respect to the p-adic metric. In particular, if |x − z|_{p} ≤ 1, then |z|_{p} ≤ 1, by the ultrametric version of the triangle inequality. This means that x can be approximated by z ∈ Q with |z|_{p} ≤ 1 with respect to the p-adic metric. If z ∈ Q and |z|_{p} ≤ 1, then z can be approximated by integers with respect to the p-adic metric, as in the previous paragraph. This implies that x can be approximated by integers with respect to the p-adic metric, as desired.

1.7 Residue fields

Let k be a field, and suppose that | · | is an ultrametric absolute value function on k. It is easy to see that the open ball B(0, r) in k centered at 0 with radius r > 0 with respect to the ultrametric associated to | · | is a subgroup of k as a commutative group with respect to addition. Similarly, the closed ball \overline{B}(0, r) in k centered at 0 with radius r ≥ 0 is a subgroup of k with respect to addition. The closed unit ball \overline{B}(0, 1) is a subring of k, which contains the multiplicative identity element 1 in k in particular. Note that B(0, r) is an ideal in \overline{B}(0, 1) when 0 < r ≤ 1, and that \overline{B}(0, r) is an ideal in \overline{B}(0, 1) when 0 ≤ r ≤ 1.

Thus the quotient \overline{B}(0, 1)/B(0, r) (1.7.1)

can be defined as a commutative ring when 0 < r ≤ 1, and

\overline{B}(0, 1)/\overline{B}(0, r) (1.7.2)

can be defined as a commutative ring when 0 ≤ r ≤ 1. One can check that

\overline{B}(0, 1)/B(0, 1) (1.7.3)

is a field, which is the residue field associated to | · | on k. More precisely, a nonzero element of (1.7.3) comes from an element x of \overline{B}(0, 1) that is not in B(0, 1). This means that |x| = 1, so that 1/x is an element of \overline{B}(0, 1) too. The element of (1.7.3) corresponding to 1/x is the inverse of the given element of (1.7.3), as desired.

If | · | is the trivial absolute value function on k, then \overline{B}(0, 1) = k, B(0, 1) = {0}, and the residue field (1.7.3) reduces to k itself. If k has characteristic p > 0, and | · | is any ultrametric absolute value function on k, then it is easy to see that the reside field (1.7.3) has characteristic p as well.

Let k be any field with an ultrametric absolute value function | · | again, and let k_{0} be a subfield of k. The restriction of | · | to k_{0} defines an ultrametric absolute value function on k_{0}, and there is a natural embedding of the residue field associated to k_{0} into the residue field associated to k. If k_{0} is dense in k with respect to the ultrametric associated to | · |, then one can check that the embedding of the residue field associated to k_{0} into the residue field associated to k is surjective, so that the residue fields are isomorphic. In particular, if k is not complete with respect to the ultrametric associated to | · |, then the residue field associated to the completion of k is isomorphic to the residue field associated to k.

Let p be a prime number, and consider k = Q_{p} with the p-adic absolute value. In this case, \overline{B}(0, p^{−j}) = p^{j} Z_{p} (1.7.4)

for every j ∈ Z, where p^{j} Z_{p} is the set of p^{j} x, x ∈ Z_{p}. As before, Z_{p} is a subring of Q_{p}, p^{j} Z_{p} is an ideal in Z_{p} for each nonnegative integer j, and hence the quotient \overline{B}(0, 1)/\overline{B}(0, p^{−j}) = Z_{p}/(p^{j} Z_{p}) (1.7.5) is defined as a commutative ring when j ≥ 0. There is a natural ring homomorphism from Z into (1.7.5), which is the composiiton of the inclusion of Z in Z_{p} with the quotient mapping from Z_{p} onto (1.7.5). Observe that

Z ∩ (p^{j} Z_{p}) = p^{j} Z (1.7.6)

for every nonnegative integer j, which is the kernel of the homomorphism from Z into (1.7.5) just mentioned. Thus we get an injective ring homomorphism from Z/(p^{j} Z) (1.7.7)

into (1.7.6) for each nonnegative integer j. One can check that this homomorphism is surjective, because Z is dense in Z_{p}, as in the previous section. This shows that (1.7.5) is isomorphic to (1.7.7) as a ring for every nonnegative integer j.

1.8 Norms and ultranorms

Let k be a field with an absolute value function | · |, and let V be a vector space over k. A nonnegative real-valued function N on V is said to be a seminorm on V with respect to | · | if it satisfies the following two conditions. First,

N(t v) = |t| N(v) (1.8.1)

for every t ∈ k and v ∈ V. Second,

N(v + w) ≤ N(v) + N(w) (1.8.2)

for every v, w ∈ V. Note that (1.8.1) implies that N(0) = 0, by taking t = 0. If we also have that N(v) > 0 (1.8.3)

for every v ∈ V with v ≠ 0, then N is said to be a norm on V with respect to | · |. In particular, k may be considered as a one-dimensional vector space over itself, and | · | may be considered as a norm on k with respect to itself.

A nonnegative real-valued function N on V is said to be a semi-ultranorm on V with respect to | · | on k if it satisfies (1.8.1) and

N(v + w) ≤ max(N(v), N(w)) (1.8.4)

for every v, w ∈ V. If N also satisfies (1.8.3), then N is said to be an ultranorm on V with respect to | · |. Of course, (1.8.4) implies (1.8.2), so that semiultranorms and ultranorms are seminorms and ultranorms, respectively. If N is a semi-ultranorm on V with respect to |·| on k, and if N(v) > 0 for some v ∈ V, then one can check that | · | is an ultrametric absolute value function on k. If | · | is an ultrametric absolute value function on k, then | · | may be considered as an ultranorm on k as a one-dimensional vector space over itself.

Let | · | be any absolute value function on k again. If N is a seminorm on V with respect to | · |, then

d(v, w) = d_{N}(v, w) = N(v − w) (1.8.5)

defines a semimetric on V. If N is a norm on V, then (1.8.5) is a metric on V. If N is a semi-ultranorm on V, then (1.8.5) is a semi-ultrametric on V. Thus (1.8.5) is an ultrametric on V when N is an ultranorm on V.

Suppose for the moment that | · | is the trivial absolute value function on k. The trivial ultranorm is defined on V by putting N(v) equal to 1 when v ≠ 0, and equal to 0 when v = 0. It is easy to see that this defines an ultranorm on V, for which the corresponding ultrametric is the discrete metric.

Let | · | be any absolute value function on k, and let a be a positive real number with a ≤ 1. Remember that | · |^{a} defines an absolute value function on k too, as in Section 1.3. If N is a seminorm on V with respect to | · | on k, then N(v)^{a} is a seminorm on V with respect to | · |^{a} on k. This uses (1.3.1) to get the triangle inequality for N(v)^{a} from the one for N. If N is a norm on V with respect to | · | on k, then N(v)^{a} is a norm on V with respect to | · |^{a} on k.

If |·| is an ultrametric absolute value function on k, then |·|^{a} is an ultrametric absolute value function on k for every a > 0, as in Section 1.3. In this case, if N is a semi-ultranorm on V with respect to | · | on k, then N(v)^{a} is a semiultranorm on V with respect to | · |^{a} for every a > 0. If N is an ultranorm on V with respect to | · | on k, then N(v)^{a} is an ultranorm on V with respect to | · |^{a} on k for every a > 0.

Let N be a semi-ultranorm on V with respect to | · | on k. If v, w ∈ V satisfy

N(v − w) < N(w), (1.8.6)

then N(v) = N(w). (1.8.7)

This is analogous to the corresponding statement for ultrametric absolute value functions mentioned in Section 1.5. If N(v − w) ≤ N(w), then

N(v) ≤ max(N(v − w), N(w)) = N(w), (1.8.8)

by the semi-ultranorm version of the triangle inequality. We also have that

N(w) ≤ max(N(v − w), N(v)), (1.8.9)

which implies that N(w) ≤ N(v) when (1.8.6) holds.

Suppose that N is a norm on V with respect to an absolute value function |·| on k. If V is complete with respect to the metric associated to N, then V is said to be a Banach space with respect to N. Otherwise, one can pass to a completion of V. The vector space operations on V can be extended to the completion, so that the completion becomes a vector space over k. The extension of N to the completion corresponds to the distance to 0 on the completion, and defines a norm on the completion. If N is an ultranorm on V, then the extension of N to the completion of V is an ultranorm as well. The completion of V is unique, up to isometric isomorphic equivalence.

If N is a seminorm on V with respect to | · | on k, and if V_{0} is a linear subspace of V, then the restriction of N to V_{0} is a seminorm on V with respect to | · | on k. If V_{0} is dense in V with respect to the semimetric associated to N, and if the restriction of N to V_{0} is a semi-ultranorm on V_{0}, then it is easy to see that N is a semi-ultranorm on V. In particular, if N is an ultranorm on V, and if V is not already complete with respect to the ultrametric associated to N, then the extension of N to the completion of V is an ultranorm.

1.9 Bounded linear mappings

Let k be a field with an absolute value function | · |, and let V, W be vector spaces over k. Also let N_{V}, N_{W} be seminorms on V, W, respectively, with respect to | · | on k. A linear mapping T from V into W is said to be bounded with respect to N_{V} and N_{W} if there is a nonnegative real number C such that

N_{W} (T (v)) ≤ C N_{V} (v) (1.9.1)

for every v ∈ V. This implies that

N_{W} (T (v) − T (v′)) = N_{W} (T (v − v′)) ≤ C N_{V} (v − v′) (1.9.2)

for every v, v′ ∈ V, and in particular that T is continuous with respect to the semimetrics associated to N_{V} and N_{W} on V and W, respectively. Conversely, if a linear mapping T from V into W is continuous at 0 with respect to these semimetrics, and if | · | is not the trivial absolute value function on k, then one can check that T is bounded with respect to N_{V} and N_{W}.

Let BL(V, W) be the space of bounded linear mappings from V into W, with respect to N_{V} and N_{W}. If T ∈ BL(V, W), then put

(1.9.3) ∥T ∥_{op} = ∥T ∥_{op,V W} = inf{C ≥ 0 : (1.9.1) holds},

where more precisely the infimum is taken over all nonnegative real numbers C such that (1.9.1) holds for every v ∈ V. Note that the infimum is automatically attained in this situation, which is to say that (1.9.1) holds with C = ∥T ∥_{op}. One can verify that BL(V, W) is a vector space over k with respect to pointwise addition and scalar multiplication of mappings from V into W, and that (1.9.3) defines a seminorm on B(V, W) with respect to | · | on k. If N_{W} is a norm on W, then (1.9.3) is a norm on BL(V, W). If N_{W} is a semi-ultranorm on W, then (1.9.3) is a semi-ultranorm on BL(V, W). In particular, if N_{W} is an ultranorm on W, then (1.9.3) is an ultranorm on BL(V, W).

Let Z be another vector space over k, and let N_{Z} be a seminorm on Z with respect to | · | on k. Suppose that T_{1} is a bounded linear mapping from V into W with respect to N_{V} and N_{W}, and that T_{2} is a bounded linear mapping from W into Z with respect to N_{W} and N_{Z}. If v ∈ V, then

(1.9.4)  N_{Z}(T_{2}(T_{1}(v))) ≤ ∥T_{2}∥_{op,W Z} N_{W} (T_{1}(v))
≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z} N_{V} (v),

where the subscripts indicate the spaces involved in the corresponding operator seminorm. This implies that the composition T_{2} ∘ T_{1} is bounded as a linear mapping from V into Z, with

∥T_{2} ∘ T_{1}∥_{op,V Z} ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z}. (1.9.5)

Let us suppose from now on in this section that N_{W} is a norm on W, and that W is complete with respect to the metric associated associated to N_{W}. Under these conditions, one can check that BL(V, W) is complete with respect to the operator norm (1.9.3), using standard arguments. More precisely, if {T_{j}}_{j=1}^{∞} is a Cauchy sequence in BL(V, W) with respect to the metric associated to the operator norm, then {T_{j}(v)}_{j=1}^{∞} is a Cauchy sequence in W with respect to the metric associated to N_{W} for every v ∈ V. This implies that {T_{j}(v)}_{j=1}^{∞} converges to a unique element T (v) of W with respect to the metric associated to N_{W}, because W is supposed to be complete with respect to this metric. It is easy to see that T defines a linear mapping from V into W, because T_{j} is linear for each j. The Cauchy condition for {T_{j}}_{j=1}^{∞} with respect to the metric associated to the operator norm implies that the operator norms of the T_{j}’s are bounded, which can be used to get that T is a bounded linear mapping. One can use the Cauchy condition for {T_{j}}_{j=1}^{∞} again to obtain that this sequence converges to T with respect to the metric associated to the operator norm, as desired.

Suppose for convenience that N_{V} is a norm on V, although this is not really needed. Let V_{0} be a linear subspace of V that is dense in V with respect to the metric associated to N_{V}, and let T_{0} be a bounded linear mapping from V_{0} into W, with respect to the restriction of N_{V} to V_{0}. Note that T_{0} is uniformly continuous with respect to the metric on V_{0} associated to the restriction of N_{V} to V_{0}, and the metric on W associated to N_{W}, as in (1.9.2). It follows that there is a unique extension of T_{0} to a uniformly continuous mapping from V into W, with respect to the metrics associated to N_{V} and N_{W}, respectively, as mentioned at the beginning of Section 1.4. One can check that this extension is a bounded linear mapping from V into W with respect to N_{V} and N_{W}, with the same operator norm as T_{0} has on V_{0}.

1.10 Some norms on k^{n}

Let k be a field, and let n be a positive integer. The space k^{n} of n-tuples v = (v_{1}, …, v_{n}) of elements of k is a vector space over k with respect to coordinatewise addition and scalar multiplication. Let | · | be an absolute value function on k. It is easy to see that

∥v∥_{1} = ∑_{j=1}^{n} |v_{j}| (1.10.1)

and

∥v∥_{∞} = max_{1≤j≤n} |v_{j}| (1.10.2)

1.10. SOME NORMS ON K^{N} 15

are norms on k^{n} with respect to | · | on k. If | · | is an ultrametric absolute value function on k, then (1.10.2) is an ultranorm on k^{n}.

Observe that ∥v∥_{∞} ≤ ∥v∥_{1} ≤ n ∥v∥_{∞} (1.10.3)

for every v ∈ k^{n}. Let d_{1}(v, w) = ∥v − w∥_{1} (1.10.4)

and d_{∞}(v, w) = ∥v − w∥_{∞} (1.10.5)

be the metrics on k^{n} associated to (1.10.1) and (1.10.2), respectively. Thus

d_{∞}(v, w) ≤ d_{1}(v, w) ≤ n d_{∞}(v, w) (1.10.6)

for every v, w ∈ k^{n}, by (1.10.3). In particular, this implies that (1.10.4) and (1.10.5) determine the same topology on k^{n}. This is the same as the product topology on k^{n}, corresponding to the topology determined on k by the metric associated to | · |.

The standard basis vectors e_{1}, …, e_{n} in k^{n} are defined as usual by taking the jth coordinate of e_{l} to be equal to 1 when j = l and to 0 when j ≠ l, where 1 ≤ j, l ≤ n. Thus

v = ∑_{l=1}^{n} v_{l} e_{l} (1.10.7)

for every v ∈ k^{n}. Let W be a vector space over k, and let N_{W} be a seminorm on W with respect to | · | on k. If T is a linear mapping from k^{n} into W, then

T (v) = T ( ∑_{l=1}^{n} v_{l} e_{l}) = ∑_{l=1}^{n} v_{l} T (e_{l}) (1.10.8)

for every v ∈ k^{n}, and hence

N_{W} (T (v)) ≤ ∑_{l=1}^{n} |v_{l}| N_{W} (T (e_{l})). (1.10.9)

In particular,

N_{W} (T (v)) ≤ ( max_{1≤l≤n} N_{W} (T (e_{l}))) ∥v∥_{1} (1.10.10)

for every v ∈ k^{n}. This means that T is bounded as a linear mapping from k^{n} equipped with ∥v∥_{1} into W, with operator seminorm less than or equal to

max_{1≤l≤n} N_{W} (T (e_{l})). (1.10.11)

In fact, the operator seminorm of T is equal to (1.10.11) in this situation, because the operator seminorm of T is automatically greater than or equal to N_{W} (T (e_{l})) for each l = 1, …, n, since ∥e_{l}∥_{1} = 1. Similarly,

N_{W} (T (v)) ≤ ( ∑_{l=1}^{n} N_{W} (T (e_{l}))) ∥v∥_{∞} (1.10.12)

for every v ∈ k^{n}, by (1.10.9). This implies that T is bounded as a linear mapping from k^{n} equipped with ∥v∥_{∞} into W, with operator seminorm less than or equal to

∑_{l=1}^{n} N_{W} (T (e_{l})). (1.10.13)

Note that the operator seminorm of T is greater than or equal to (1.10.11), because ∥e_{l}∥_{∞} = 1 for each l = 1, …, n.

Suppose now that | · | is an ultrametric absolute value function on k, and that N_{W} is a semi-ultranorm on W with respect to | · | on k. Using (1.10.8), we get that

N_{W} (T (v)) ≤ max_{1≤l≤n}(|v_{l}| N_{W} (T (e_{l}))) ≤ ( max_{1≤l≤n} N_{W} (T (e_{l}))) ∥v∥_{∞} (1.10.14)

for every v ∈ k^{n}. This implies that T is bounded as a linear mapping from k^{n} equipped with ∥v∥_{∞} into W, with operator seminorm less than or equal to (1.10.11). The operator seminorm of T is also greater than or equal to (1.10.11), as in the preceding paragraph. Hence the operator seminorm of T with respect to ∥v∥_{∞} on k^{n} is equal to (1.10.11) in this case.

1.11 Inner products

Suppose for the moment that V and W are vector spaces over the field C of complex numbers, so that V and W may be considered as vector spaces over R as well. Let us say that a mapping T from V into W is real-linear if T is linear as a mapping from V into W as vector spaces over R, and that T is complex-linear if T is linear as a mapping from V into W as vector spaces over C. Thus a complex-linear mapping T from V into W is the same as a real-linear mapping that also satisfies T (i v) = i T (v) (1.11.1)

for every v ∈ V. A real-linear mapping T from V into W is said to be conjugatelinear if T (i v) = −i T (v) (1.11.2)

for every v ∈ V. This implies that

T (a v) = \overline{a} T (v) (1.11.3)

for every a ∈ C and v ∈ V, where \overline{a} is the usual complex-conjugate of a.

Suppose from now on in this section that k = R or C, with the standard absolute value function. Let V be a vector space over k, and let ⟨v, w⟩ be a k-valued function defined for v, w ∈ V. If the following three conditions are satisfied, then ⟨v, w⟩ is said to be an inner product on V. The first condition is that ⟨v, w⟩ be linear in v for each w ∈ V. The second condition is that

⟨w, v⟩ = ⟨v, w⟩ (1.11.4) for every v, w ∈ V in the real case, and that

⟨w, v⟩ = \overline{⟨v, w⟩} (1.11.5)

for every v, w ∈ V in the complex case. Note that ⟨v, w⟩ is linear in w for each v ∈ V in the real case, and conjugate-linear in w for each v ∈ V in the complex case. In the complex case, we also get that ⟨v, v⟩ is a real number for every v ∈ V, by (1.11.5). The third condition is that

⟨v, v⟩ > 0 (1.11.6)

for every v ∈ V with v ≠ 0. Of course, ⟨v, w⟩ = 0 when either v = 0 or w = 0, by the first two conditions. If⟨v, w⟩ is an inner product on V, then we put

∥v∥ = ⟨v, v⟩^{1/2} (1.11.7)

for every v ∈ V, using the nonnegative square root on the right side. It is well known that |⟨v, w⟩| ≤ ∥v∥ ∥w∥ (1.11.8)

for every v, w ∈ V, which is the Cauchy–Schwarz inequality. Using this, one can show that ∥ · ∥ defines a norm on V. If V is complete with respect to the metric associated to ∥ · ∥, then V is said to be a Hilbert space with respect to ⟨v, w⟩. Otherwise, one can pass to a completion, as usual.

Let n be a positive integer. The standard inner product on R^{n} is given by

⟨v, w⟩ = ⟨v, w⟩_{R^{n}} = ∑_{j=1}^{n} v_{j} w_{j}. (1.11.9)

Similarly, the standard inner product on C^{n} is given by

⟨v, w⟩ = ⟨v, w⟩_{C^{n}} = ∑_{j=1}^{n} v_{j} \overline{w_{j}}. (1.11.10)

In both cases, the corresponding norm is given by

∥v∥ = ∥v∥_{2} = ( ∑_{j=1}^{n} |v_{j}|^{2})^{1/2}. (1.11.11)

It is easy to see that ∥v∥_{∞} ≤ ∥v∥_{2} ≤ n^{1/2} ∥v∥_{∞} (1.11.12)

for every v ∈ R^{n} or C^{n}, where ∥v∥_{∞} is as in (1.10.2). One can also check that

∥v∥_{2} ≤ ∥v∥_{1} ≤ n^{1/2} ∥v∥_{2} (1.11.13)

for every v ∈ R^{n} or C^{n}, where ∥v∥_{1} is as in (1.10.1). More precisely, the first inequality in (1.11.13) can be verified using the first inequality in (1.10.3), and the second inequality in (1.11.13) can be obtained from the Cauchy–Schwarz inequality.

Let (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) be Hilbert spaces, both real or both complex, and let ∥ · ∥_{V} and ∥ · ∥_{W} be the corresponding norms on V and W, respectively. Also let T be a bounded linear mapping from V into W, with respect to ∥ · ∥_{V} and ∥ · ∥_{W}. It is well known that there is a unique bounded linear mapping T^{∗} from W into V such that

⟨T (v), w⟩_{W} = ⟨v, T^{∗}(w)⟩_{V} (1.11.14)

for every v ∈ V and w ∈ W. This mapping T^{∗} is called the adjoint of T. The adjoint (T^{∗})^{∗} of T^{∗} can be defined as a bounded linear mapping from V into W in the same way, and is equal to T. It is not difficult to show that

∥T^{∗}∥_{op,W V} = ∥T ∥_{op,V W} . (1.11.15)

Note that T ↦ T^{∗} is a linear mapping from BL(V, W) into BL(W, V) in the real case, and that this mapping is conjugate-linear in the complex case.

Let (Z, ⟨·, ·⟩_{Z}) be another Hilbert space, which is real when V and W are real, and complex when V and W are complex, and let ∥ · ∥_{Z} be the corresponding norm on Z. If T_{1} is a bounded linear mapping from V into W, and T_{2} is a bounded linear mapping from W into Z, then their composition T_{2} ∘ T_{1} is a bounded linear mapping from V into Z, as before. It is easy to see that

(T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗} (1.11.16)

as bounded linear mappings from Z into V.

1.12 Infinite series

Let k be a field with an absolute value function | · |, and let V be a vector space over k with a norm N with respect to | · | on k. An infinite series ∑_{j=1}^{∞} v_{j} with terms in V is said to converge in V if the corresponding sequence of partial sums ∑_{j=1}^{n} v_{j} converges to an element of V with respect to the metric associated to N. In this case, the value of the sum ∑_{j=1}^{∞} v_{j} is defined to be the limit of the sequence of partial sums. If ∑_{j=1}^{∞} v_{j} converges in V and t ∈ k, then it is easy to see that ∑_{j=1}^{∞} t v_{j} converges in V too, with

∑_{j=1}^{∞} t v_{j} = t ∑_{j=1}^{∞} v_{j}. (1.12.1)

Similarly, if ∑_{j=1}^{∞} w_{j} is another convergent series in V, then ∑_{j=1}^{∞}(v_{j} + w_{j}) converges in V as well, with

∑_{j=1}^{∞}(v_{j} + w_{j}) = ∑_{j=1}^{∞} v_{j} + ∑_{j=1}^{∞} w_{j}. (1.12.2)

A necessary condition for the convergence of an infinite series ∑_{j=1}^{∞} v_{j} with terms in V is that the corresponding sequence of partial sums be a Cauchy sequence with respect to the metric associated to N. This happens if and only if for every ϵ > 0 there is a positive integer L such that

N( ∑_{j=l}^{n} v_{j}) < ϵ (1.12.3)

for all l, n ∈ Z_{+} with n ≥ l ≥ L. In particular, this implies that

lim_{j→∞} N(v_{j}) = 0, (1.12.4)

by taking l = n in (1.12.3). Of course, if V is complete with respect to the metric associated to N, then the Cauchy condition (1.12.3) implies that ∑_{j=1}^{∞} v_{j} converges in V.

If ∑_{j=1}^{∞} N(v_{j}) converges as an infinite series of nonnegative real numbers, then ∑_{j=1}^{∞} v_{j} is said to converge absolutely with respect to N. Observe that

N( ∑_{j=l}^{n} v_{j}) ≤ ∑_{j=l}^{n} N(v_{j}) (1.12.5)

for every l, n ∈ Z_{+} with n ≥ l, by the triangle inequality for N. If ∑_{j=1}^{∞} v_{j} converges absolutely with respect to N, then it is easy to see that the Cauchy condition (1.12.3) holds, using (1.12.5). If V is complete with respect to the metric associated to N, then it follows that ∑_{j=1}^{∞} v_{j} converges in V. In this case, we also have that

N( ∑_{j=1}^{∞} v_{j}) ≤ ∑_{j=1}^{∞} N(v_{j}). (1.12.6)

Suppose for the moment that N is an ultranorm on V with respect to | · | on k, so that

N( ∑_{j=l}^{n} v_{j}) ≤ max_{l≤j≤n} N(v_{j}) (1.12.7)

for every n ≥ l ≥ 1. If (1.12.4) holds, then it follows that the Cauchy condition (1.12.3) holds too. If V is complete with respect to the ultrametric associated to N, then we get that ∑_{j=1}^{∞} v_{j} converges in V. Note that

N( ∑_{j=1}^{∞} v_{j}) ≤ max_{j≥1} N(v_{j}) (1.12.8)

in this situation. More precisely, the maximum on the right side of (1.12.8) is attained, because of (1.12.4).

Let us now take k = R or C, with the standard absolute value function. Let (V, ⟨·, ·⟩) be a real or complex inner product space, and let ∥ · ∥ be the corresponding norm on V, as in the previous section. Suppose that ∑_{j=1}^{∞} v_{j} is an infinite series of pairwise-orthogonal vectors in V, so that

⟨v_{j}, v_{l}⟩ = 0 (1.12.9)

when j ≠ l. This implies that

∥∑_{j=l}^{n} v_{j}∥^{2} = ∑_{j=l}^{n} ∥v_{j}∥^{2} (1.12.10)

for every n ≥ l ≥ 1. If ∑_{j=1}^{∞} ∥v_{j}∥^{2} converges as an infinite series of nonnegative real numbers, then the Cauchy condition (1.12.3) holds, with N = ∥ · ∥. Hence ∑_{j=1}^{∞} v_{j} converges in V when V is a Hilbert space, in which case we have that

∥∑_{j=1}^{∞} v_{j}∥^{2} = ∑_{j=1}^{∞} ∥v_{j}∥^{2}. (1.12.11)

Conversely, if the Cauchy condition (1.12.3) holds, then ∑_{j=1}^{∞} ∥v_{j}∥^{2} converges, because the partial sums are bounded.

1.13 Bounded bilinear mappings

Let k be a field, and let V, W, and Z be vector spaces over k. A mapping b from V × W into Z is said to be bilinear if b(v, w) is linear in v for each w ∈ W, and linear in w for each v ∈ V. Let | · | be an absolute value function on k, and let N_{V}, N_{W}, and N_{Z} be seminorms on V, W, and Z, respectively, with respect to | · | on k. If there is a nonnegative real number C such that

N_{Z}(b(v, w)) ≤ C N_{V} (v) N_{W} (w) (1.13.1)

for every v ∈ V and w ∈ W, then b is said to be bounded as a bilinear mapping from V × W into Z.

Let b be a bilinear mapping from V × W into Z that satisfies (1.13.1), and let v, v′ ∈ V and w, w′ ∈ W be given. Observe that

b(v, w) − b(v′, w′) = b(v − v′, w) + b(v′, w − w′), (1.13.2)

so that

N_{Z}(b(v, w) − b(v′, w′)) ≤ N_{Z}(b(v − v′, w)) + N_{Z}(b(v′, w − w′))
(1.13.3)  ≤ C ∥v − v′∥_{V} ∥w∥ + C ∥v′∥_{V} ∥w − w′∥_{W} .

One can use this to check that b is continuous with respect to the semimetrics associated to N_{V}, N_{W}, N_{Z} and the corresponding product topology on V × W.

Conversely, if a bilinear mapping b from V × W into Z is continuous at (0, 0) in V × W with respect to these semimetrics and the corresponding product topology on V × W, and if | · | is not the trivial absolute value function on k, then one can verify that b is bounded as a bilinear mapping.

Suppose for the moment that N_{V}, N_{W}, and N_{Z} are norms on V, W, and Z, respectively, and let V_{0}, W_{0} be dense linear subspaces of V and W with respect to the metrics associated to N_{V} and N_{W}. Let b_{0} be a bounded bilinear mapping from V_{0} × W_{0} into Z, using the restrictions of N_{V} and N_{W} to V_{0} and W_{0}, respectively. If Z is complete with respect to the metric associated to N_{Z}, then there is a unique extension of b_{0} to a bounded bilinear mapping from V × W into Z. More precisely, for each w ∈ W_{0}, one can first extend b_{0}(v, w) to a bounded linear mapping from V into Z, as a function of v. This defines a bounded bilinear mapping from V × W_{0} into Z, which can be extended to a bounded bilinear mapping from V × W into Z in the same way.

Suppose now that V = k^{n_{V}} and W = k^{n_{W}} for some positive integers n_{V} and n_{W}, and let N_{Z} be any seminorm on Z again. Also let e_{1}^{V} , …, e_{n_{V}}^{V} and e_{1}^{W} , …, e_{n_{W}}^{W} be the standard basis vectors in k^{n_{V}} and k^{n_{W}}, respectively, and let b be a bilinear mapping from k^{n_{V}} × k^{n_{W}} into Z. Observe that

b(v, w) = ∑_{j=1}^{n_{V}} ∑_{l=1}^{n_{W}} v_{j} w_{l} b(e_{j}^{n_{V}} , e_{l}^{n_{W}}) (1.13.4)

for every v ∈ V and w ∈ W, so that

N_{Z}(b(v, w)) ≤ ∑_{j=1}^{n_{V}} ∑_{l=1}^{n_{W}} |v_{j}| |w_{l}| N_{Z}(b(e_{j}^{n_{V}} , e_{l}^{n_{W}})). (1.13.5)

If we take N_{V} (v) to be

∥v∥_{1,n_{V}} = ∑_{j=1}^{n_{V}} |v_{j}|, (1.13.6)

and N_{W} (w) to be

∥w∥_{1,n_{W}} = ∑_{l=1}^{n_{W}} |w_{l}|, (1.13.7)

as in (1.10.1), then (1.13.5) implies that (1.13.1) holds with C equal to

max{N_{Z}(b(e_{j}^{n_{V}} , e_{l}^{n_{w}})) : 1 ≤ j ≤ n_{V} , 1 ≤ l ≤ n_{W} }. (1.13.8)

Similarly, if we take N_{V} (v) to be

∥v∥_{∞,n_{V}} = max_{1≤j≤n_{V}} |v_{j}|, (1.13.9)

and N_{W} (w) to be

∥w∥_{∞,n_{W}} = max_{1≤l≤n_{W}} |w_{l}|, (1.13.10)

as in (1.10.2), then (1.13.5) implies that (1.13.1) holds, with C equal to

∑_{j=1}^{n_{V}} ∑_{l=1}^{n_{W}} N_{Z}(b(e_{j}^{n_{V}} , e_{l}^{n_{W}})). (1.13.11)

If N_{Z} is a semi-ultranorm on Z, then we get that

N_{Z}(b(v, w)) (1.13.12)

≤ max{|v_{j}| |w_{l}| N(b(e_{j}^{n_{V}} , e_{l}^{n_{W}})) : 1 ≤ j ≤ n_{V} , 1 ≤ l ≤ n_{W} }

for every v ∈ k^{n_{V}} and k^{n_{W}}. In this case, if we take N_{V} (v) and N_{W} (w) to be as in (1.13.9) and (1.13.10), respectively, then (1.13.1) holds with C equal to (1.13.8).

1.14 Minkowski functionals

Let k be a field with an absolute value function | · |, and let V be a vector space over k. If t ∈ k and E ⊆ V, then we put

t E = {t v : v ∈ E}. (1.14.1)

Let us say that E is balanced in V if

t E ⊆ E (1.14.2)

for every t ∈ k with |t| ≤ 1. If |t| = 1, then it follows that

t E = E, (1.14.3)

by applying (1.14.2) to both t and 1/t. Note that a nonempty balanced subset of V contains 0.

Let us say that a balanced set A ⊆ V is absorbing if for every v ∈ V there is a t_{1} ∈ k such that t_{1} ≠ 0 and

v ∈ t_{1} A. (1.14.4)

This implies that v ∈ t A (1.14.5)

for every t ∈ k such that |t| ≥ |t_{1}|, because A is balanced. Equivalently, this means that t^{−1} v ∈ A (1.14.6)

when |t| ≥ |t_{1}|. Of course, if (1.14.4) holds with t_{1} = 0, then v = 0. We also have that 0 ∈ A, because A is balanced and nonempty, so that (1.14.5) holds for every t ∈ k. Clearly V is automatically balanced and absorbing as a subset of itself. If | · | is the trivial absolute value function on k, then V is the only balanced absorbing subset of itself.

1.14. MINKOWSKI FUNCTIONALS 23

Let N be a nonnegative real-valued function on V such that N(t v) = |t| N(v) (1.14.7) for every t ∈ k and v ∈ V. Put B_{N}(0, r) = {v ∈ V : N(v) < r} (1.14.8) for every positive real number r, and \overline{B}(0, r) = \{v ∈ V : N(v) ≤ r\} (1.14.9) for every nonnegative real number r. If t ∈ k and t ≠ 0, then t B_{N}(0, r) = B_{N}(0, |t| r) (1.14.10) for every r > 0, and t \overline{B}_{N}(0, r) = \overline{B}_{N}(0, |t| r) (1.14.11) for every r ≥ 0. In particular, B_{N}(0, r) is balanced in V for every r > 0, and \overline{B}_{N}(0, r) is balanced in V for every r ≥ 0. If | · | is not the trivial absolute value function on k, then B_{N}(0, r) and \overline{B}_{N}(0, r) are absorbing in V for every r > 0.

Let us suppose from now on in this section that |·| is not the trivial absolute value function on k. Let A be a balanced absorbing subset of V, and put

(1.14.12)  N_{A}(v) = inf{|t| : t ∈ k, t ≠ 0, v ∈ t A}
= inf{|t| : t ∈ k, t ≠ 0, t^{−1} v ∈ A}

for each v ∈ V. Note that N_{A}(0) = 0, so that we could have included the possibility of t = 0 in the first formulation of N_{A}(v). Of course, N_{A} is a nonnegative real-valued function on V, and one can check that N_{A}(t′ v) = |t′| N_{A}(v) (1.14.13) for every t′ ∈ k and v ∈ V. If v ∈ A, then N_{A}(v) ≤ 1, so that A ⊆ \overline{B}_{N_{A}}(0, 1). (1.14.14) If v ∈ V satisfies N_{A}(v) < 1, then there is a t ∈ k such that |t| < 1 and v ∈ t A. This implies that v ∈ A, because A is balanced in V, so that B_{N_{A}}(0, 1) ⊆ A. (1.14.15)

Suppose for the moment that | · | is discrete on k, as in Section 1.5. If v ∈ V and N_{A}(v) > 0, then it follows that the infimum in (1.14.12) is attained. If v ∈ V and N_{A}(v) ≤ 1, then there is a t ∈ k such that |t| ≤ 1 and v ∈ t A, because the infimum is attained when N_{A}(v) = 1. As before, this implies that v ∈ A, because A is balanced in V. Hence A = \overline{B}_{N_{A}}(0, 1) (1.14.16) in this situation.

1.15 Balanced subgroups

Let k be a field with an absolute value function | · | again, and let V be a vector space over k. Let us say that E ⊆ V is a balanced subgroup of V if E is balanced as a subset of V, as in the previous section, and E is a subgroup of V as a commutative group with respect to addition. If E ⊆ V is a nonempty balanced subset of V, then 0 ∈ E and −E = E. If we also have that

v + w ∈ E (1.15.1)

for every v, w ∈ E, then it follows that E is a balanced subgroup of V.

Let N be a semi-ultranorm on V with respect to |·| on k. Note that B_{N}(0, r) in (1.14.8) is the same as the open ball in V centered at 0 with radius r > 0 with respect to the semi-ultrametric associated to N, and that (1.14.9) is the same as the closed ball in V centered at 0 with radius r ≥ 0 with respect to the semi-ultrametric associated to N. It is easy to see that B_{N}(0, r) is a balanced subgroup in V for every r > 0, and that \overline{B}_{N}(0, r) is a balanced subgroup for every r ≥ 0. More precisely, (1.15.1) holds in both cases, by the semi-ultranorm version of the triangle inequality.

Of course, linear subspaces of V are balanced subgroups of V. If | · | is archimedean on k, then one can check that any balanced subgroup E of V is linear subspace of V. This uses the fact that for each v ∈ E and positive integer n, the sum of n v’s in V is an element of E. If | · | is the trivial absolute value function on k, then balanced subgroups of V are linear subspaces again.

If E is a balanced subgroup in V, and if the linear span of E in V is equal to V, then E is absorbing in V. More precisely, if | · | is the trivial absolute value function on E, then E is a linear subspace of V, and hence E = V. Otherwise, suppose that | · | is not the trivial absolute value function on k, and let v ∈ V be given. Thus v can be expressed as a linear combination of elements of E, by hypothesis. This implies that t v ∈ E when t ∈ k and |t| is sufficiently small, because E is a balanced subgroup of V.

Let us continue to suppose that | · | is not the trivial absolute value function on k. Let A is a balanced subgroup of V that is also absorbing in V, and let N_{A} be as in (1.14.12). We would like to check that N_{A} is a semi-ultranorm on V with respect to | · | on k. We have already seen that N_{A} satisfies the homogeneity condition (1.14.13), and so it is enough to show that N_{A} satisfies the semi-ultranorm version of the triangle inequality. Let v, w ∈ V be given, and let r be a positive real number such that

N_{A}(v), N_{A}(w) < r. (1.15.2)

This implies that there are nonzero elements t_{1}(v), t_{1}(w) of k such that

|t_{1}(v)|, |t_{1}(w)| < r (1.15.3)

and v ∈ t_{1}(v) A, w ∈ t_{1}(w) A, (1.15.4) by the definition (1.14.12) of N_{A}. Let us take t_{1} to be equal to t_{1}(v) or t_{1}(w), in such a way that |t_{1}| = max(|t_{1}(v)|, |t_{1}(w)|). (1.15.5)

Thus t_{1}(v) A, t_{1}(w) A ⊆ t_{1} A, because A is balanced. This implies that v and w are both elements of t_{1} A, by (1.15.4). It follows that

v + w ∈ t_{1} A, (1.15.6)

because A is a subgroup of V with respect to addition. This means that

N_{A}(v + w) ≤ |t_{1}|, (1.15.7)

by the definition (1.14.12) of N_{A}. Hence N_{A}(v+w) < r, by (1.15.3) and (1.15.5). This shows that N_{A}(v + w) ≤ max(N_{A}(v), N_{A}(w)), (1.15.8)

since r is any positive real number that satisfies (1.15.2).

Some basic notions related to Lie algebras

2.1 Modules and homomorphisms

Let k be a commutative ring with a multiplicative identity element 1 = 1_{k}, and let A be a commutative group, for which the group operations are expressed additively. Suppose that scalar multiplication on A by elements of k is defined, so that t a is defined as an element of A for every t ∈ k and a ∈ A. If scalar multiplication satisfies the usual compatibility conditions with the group operations on A and the ring operations on k, then A is said to be a module over k. If k is a field, then a module over k is the same as a vector space over k. Any abelian group may be considered as a module over Z, where n a is the sum of n a’s in A for each a ∈ A and n ∈ Z_{+}.

Let k be a commutative ring with a multiplicative identity element again, and let A be a module over k. A submodule of A is a subgroup A_{0} of A with respect to addition that is invariant under scalar multiplication by k. If k is a field, then this is the same as a linear subspace.

Let k be a field with an ultrametric absolute value function | · |, and let k_{1} be the closed unit ball in k with respect to | · |. Thus k_{1} is a subring of k that contains the multiplicative identity element in particular, as in Section 1.7. Let V be a vector space over k, which may be considered as a module over k_{1} as well. In this situation, a submodule of V as a module over k_{1} is the same as a balanced subgroup of V, as in Section 1.15.

Let k be a commutative ring with a multiplicative identity element, and let B be a module over k. If X is a nonempty set, then the space of all functions on X with values in B is a module over k, with respect to pointwise addition and scalar multiplication of the functions on X.

Let A be another module over k. A mapping ϕ from A into B is said to be a module homomorphism if ϕ is a group homomorphism with respect to addition that is also compatible with scalar multiplication by elements of k. One may say that ϕ is linear over k or k-linear in this case as well. If k is a field, then this is the same as a linear mapping between vector spaces.

The collection of all module homomorphisms from A into B may be denoted Hom(A, B), or Hom_{k}(A, B), to indicate the role of k. This is a module over k too, with respect to pointwise addition and scalar multiplication of mappings from A into B. More precisely, Hom(A, B) may be considered as a submodule of the module of all B-valued functions on A.

Let C be another module over k. If ϕ is a module homomorphism from A into B, and ψ is a module homomorphism from B into C, then their composition ψ ∘ ϕ defines a module homomorphism from A into C.

If ϕ is a one-to-one module homomorphism from A onto B, then the inverse mapping ϕ^{−1} is a module homomorphism from B onto A. In this case, ϕ is said to be a module isomorphism from A onto B. If ϕ is a module isomorphism from A onto B, and ψ is a module isomorphism from B onto C, then ψ ∘ ϕ is a module isomorphism from A onto C.

A mapping β from A × B into C is said to be bilinear over k if β(a, b) is linear over k in a as a mapping from A into C for every b ∈ B, and β(a, b) is linear over k in b as a mapping from B into C for every a ∈ A. If k is a field, then this is the same as the usual notion of bilinearity for a mapping from a product of vector spaces over k into another vector space over k.

In particular, we can take A = B, so that β is a bilinear mapping from A×A into C. If β(b, a) = β(a, b) (2.1.1)

for every a, b ∈ A, then β is said to be symmetric on A × A. Similarly, if

β(b, a) = −β(a, b) (2.1.2)

for every a, b ∈ A, then β is said to be antisymmetric on A × A. However, it is sometimes better to ask that β(a, a) = 0 (2.1.3)

for every a ∈ A. Of course,

β(a + b, a + b) = β(a, a) + β(a, b) + β(b, a) + β(b, b) (2.1.4)

for every a, b ∈ A, because of bilinearity. It is easy to see that (2.1.3) implies (2.1.2), using (2.1.4). If 1 + 1 has a multiplicative inverse in k, then (2.1.2) implies (2.1.3). If 1 + 1 = 0 in k, then (2.1.1) and (2.1.2) are the same.

2.2 Algebras

28 CHAPTER 2. SOME BASIC NOTIONS RELATED TO LIE ALGEBRAS

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. If A is equipped with a mapping from A × A into A that is bilinear over k, then A is said to be an algebra over k. In this case, we may also say that A is an algebra over k in the strict sense. The bilinear mapping may be expressed as (a, b) ↦ a b, (2.2.1) and we may refer to a b as the product of a, b ∈ A. If this bilinear mapping is symmetric, so that a b = b a (2.2.2)

for every a, b ∈ A, then we may say that A is commutative.

Let A be an algebra over k. If the associative law

(a b) c = a (b c) (2.2.3)

holds for every a, b, c ∈ A, then A is said to be an associative algebra over k. This is sometimes included in the definition of an algebra, and we do consider associativity to be part of the definition of a ring here.

Let A be an algebra over k in the strict sense. An element e of A is said to be the multiplicative identity element in A if

e a = a e = a (2.2.4)

for every a ∈ A. It is easy to see that this is unique when it exists.

Let A be any module over k, so that the space Hom_{k}(A, A) of module homomorphisms from A into itself is a module over k too. One can check that Hom_{k}(A, A) is an associative algebra over k, with composition of mappings as multiplication. The identity mapping on A is the multiplicative identity element in Hom_{k}(A, A).

Let A be an algebra over k in the strict sense, and let A_{0} be a submodule of A, as a module over k. If, for every a, b ∈ A_{0}, we have that a b ∈ A_{0}, then A_{0} is said to be a subalgebra of A. In this case, A_{0} is also an algebra over k in the strict sense, with respect to the restriction of multiplication on A to A_{0}. If A is an associative algebra, then A_{0} is associative as well.

Let A and B be algebras over k in the strict sense, so that A and B are modules over k in particular. Also let ϕ be a module homomorphism from A into B. If ϕ(a_{1} a_{2}) = ϕ(a_{1}) ϕ(a_{2}) (2.2.5)

for every a_{1}, a_{2} ∈ A, then one may say that ϕ is an algebra homomorphism from A into B. If A and B have multiplicative identity elements e_{A} and e_{B}, respectively, then one may require that

ϕ(e_{A}) = e_{B} (2.2.6)

too. If ϕ is a one-to-one algebra homomorphism from A onto B, then the inverse mapping ϕ^{−1} is an algebra homomorphism from B onto A, and ϕ is said to be an algebra isomorphism from A onto B. In this case, if A has a multiplicative identity element e_{A}, then ϕ(e_{A}) is the multiplicative identity element in B. An algebra isomorphism from A onto itself is called an algebra automorphism of A.

Let A be an algebra over k in the strict sense again. If a ∈ A, then put

M_{a}(x) = a x (2.2.7)

2.3. LIE ALGEBRAS 29 for every x ∈ A. Note that M_{a} defines a module homomorphism from A into itself for every a ∈ A, because of bilinearity of multiplication on A. Thus a ↦ M_{a} (2.2.8) may be considered as a mapping from A into the space Hom_{k}(A, A) of module homomorphisms from A into itself. More precisely, (2.2.8) is a module homomorphism from A into Hom_{k}(A, A) as modules over k, because multiplication on A is bilinear.

Suppose for the moment that A has a multiplicative identity element e. In this case, M_{e} is the identity mapping on A. We also get that M_{a}(e) = a e = a (2.2.9) for every a ∈ A, which implies that (2.2.8) is injective.

If a, b, x ∈ A, then M_{a}(M_{b}(x)) = M_{a}(b x) = a (b x) (2.2.10) and M_{a b}(x) = (a b) x. (2.2.11) If A is an associative algebra, then it follows that M_{a} ∘ M_{b} = M_{a b}, (2.2.12) as mappings from A into itself. This means that (2.2.8) is an algebra homomorphism from A into Hom_{k}(A, A), using composition as multiplication on the space Hom_{k}(A, A) of module homomorphisms from A into itself, as before.

2.3 Lie algebras

Let k be a commutative ring with a multiplicative identity element again, and let A be a module over k. Also let [x, y] be a mapping from A × A into A that is bilinear over k. Suppose that [x, x] = 0 (2.3.1) for every x ∈ A. This implies that [y, x] = −[x, y] (2.3.2) for every x, y ∈ A, as in Section 2.1. If 1 + 1 has a multiplicative inverse in k, then (2.3.2) implies (2.3.1), as before.

The Jacobi identity may be formulated as saying that [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 (2.3.3) for every x, y, z ∈ A. Alternatively, the Jacobi identity may be expressed as saying that [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (2.3.4)

for every x, y, z ∈ A. It is easy to see that (2.3.3) is equivalent to (2.3.4), if we have (2.3.2). If [x, y] is a bilinear mapping from A × A into A that satisfies (2.3.1) and either (2.3.3) or (2.3.4), then A is said to be a Lie algebra over k with respect to the Lie bracket [x, y]. In particular, A may be considered as an algebra over k in the strict sense, using [x, y] as multiplication on A.

If [x, y] = 0 (2.3.5)

for every x, y ∈ A, then A is said to be commutative as a Lie algebra. Commutativity of A as an algebra over k in the strict sense with respect to [x, y] means that [x, y] is symmetric in x and y, as in the previous section. If 1 + 1 is invertible in k, then commutativity of A as an algebra in the strict sense implies that A is commutative as a Lie algebra, because of (2.3.2). However, if 1+1 = 0 in k, then any Lie algebra over k is commutative as an algebra over k in the strict sense.

Let A be any algebra over k in the strict sense, where multiplication on A is expressed as in (2.2.1). Put

[x, y] = x y − y x (2.3.6)

for every x, y ∈ A, which defines a mapping from A × A into A. This mapping is bilinear over k, because multiplication on A is bilinear, by hypothesis. Of course, (2.3.6) satisfies (2.3.1) automatically. If A is associative with respect to the given operation of multiplication, then one can verify that (2.3.6) satisfies either of the Jacobi identities (2.3.3) or (2.3.4), so that A is a Lie algebra with respect to (2.3.6).

Let (A, [x, y]) be a Lie algebra over k, and let A_{0} be a submodule of A, as a module over k. As in the previous section, A_{0} is said to be a subalgebra of A if [x, y] ∈ A_{0} for every x, y ∈ A_{0}. In this situation, A_{0} is a Lie algebra over k with respect to the restriction of [x, y] to x, y ∈ A_{0}. One may also refer to A_{0} as a Lie subalgebra of A. If A is an associative algebra over k, and A_{0} is a subalgebra of A, then A_{0} is also a Lie subalgebra of A as a Lie algebra with respect to (2.3.6).

Let (A, [·, ·]_{A}) and (B, [·, ·]_{B}) be Lie algebras over k. Thus A and B are modules over k in particular, and we let ϕ be a module homomorphism from A into B. If ϕ([x, y]_{A}) = [ϕ(x), ϕ(y)]_{B} (2.3.7)

for every x, y ∈ A, then ϕ is said to be a Lie algebra homomorphism from A into B. Remember that A and B may be considered as algebras over k in the strict sense, using [·, ·]_{A} and [·, ·]_{B} as multiplication on A and B, respectively. A Lie algebra homomorphism from A into B is the same as an algebra homomorphism from A into B, as algebras over k in the strict sense, as in the previous section.

Let A and B be algebras over k in the strict sense, with multiplication expressed as in (2.2.1), and let ϕ be an algebra homomorphism from A into B. Also let [·, ·]_{A} and [·, ·]_{B} be the corresponding commutators on A and B, as in (2.3.6). If x, y ∈ A, then

ϕ([x, y]_{A}) = ϕ(x y − y x) = ϕ(x) ϕ(y) − ϕ(y) ϕ(x) = [ϕ(x), ϕ(y)]_{B}. (2.3.8)

This means that ϕ may be considered as an algebra homomorphism from A into B, using [·, ·]_{A} and [·, ·]_{B} as the algebra operations on A and B, respectively. If A and B are associative algebras with respect to their given operations of multiplication, then A and B are Lie algebras with respect to [·, ·]_{A} and [·, ·]_{B}, respectively, as before, and ϕ may be considered as a Lie algebra homomorphism from A into B with respect to these Lie brackets.

2.4 The adjoint representation

Let k be a commutative ring with a multiplicative identity element, and let (A, [x, y]_{A}) be a Lie algebra over k. If x ∈ A, then let ad x be the mapping from A into itself defined by (ad x)(y) = [x, y]_{A} (2.4.1)

for every y ∈ A. This is the same as the multiplication operator on A corresponding to x, as in Section 2.2, using the Lie bracket on A as multiplication. In particular, ad x is a module homomorphism from A into itself, as a module over k, because [x, y]_{A} is linear over k in y. Thus

x ↦ ad x (2.4.2)

defines a mapping from A into the space Hom_{k}(A, A) of module homomorphisms from A into itself. Remember that Hom_{k}(A, A) is a module over k with respect to pointwise addition and scalar multiplication of mappings from A into itself. As before, (2.4.2) is a module homomorphism from A into Hom_{k}(A, A) as modules over k, because [x, y]_{A} is linear in x over k.

If ϕ, ψ ∈ Hom_{k}(A, A), then put

[ϕ, ψ] = [ϕ, ψ]_{Hom_{k}(A,A)} = ϕ ∘ ψ − ψ ∘ ϕ, (2.4.3)

which is an element of Hom_{k}(A, A) too. Of course, Hom_{k}(A, A) is a Lie algebra over k with respect to (2.4.3), because Hom_{k}(A, A) is an associative algebra over k with respect to composition of mappings. It is well known that (2.4.2) is a Lie algebra homomorphism from A into Hom_{k}(A, A), with respect to (2.4.3) on Hom_{k}(A, A). To see this, let x, y, z ∈ A be given, and observe that

(2.4.4) ([ad x, ad y])(z) = (ad x)((ad y)(z)) − (ad y)((ad x)(z))
= (ad x)([y, z]_{A}) − (ad y)([x, z]_{A})
= [x, [y, z]_{A}]_{A} − [y, [x, z]_{A}]_{A}
= [x, [y, z]_{A}]_{A} + [y, [z, x]_{A}]_{A}.

This uses the fact that [x, z]_{A} = −[z, x]_{A}, as in (2.3.2), in the last step. We also have that (ad[x, y]_{A})(z) = [[x, y]_{A}, z]_{A} = −[z, [x, y]_{A}]_{A}, (2.4.5)

using (2.3.2) in the second step. The Jacobi identity (2.3.4) says exactly that the right sides of (2.4.4) and (2.4.5) are equal to each other. Thus

(ad[x, y]_{A})(z) = ([ad x, ad y])(z) (2.4.6)

for every z ∈ A, so that ad[x, y]_{A} = [ad x, ad y] (2.4.7)

as mappings from A into itself.

2.5 Derivations

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Also let δ be a module homomorphism from A into itself, as a module over k. If δ satisfies the product rule

δ(a b) = δ(a) b + a δ(b) (2.5.1)

for every a, b ∈ A, then δ is said to be a derivation on A. Let Der(A) be the collection of derivations on A. Remember that the space Hom_{k}(A, A) of all module homomorphisms from A into itself is a module over k with respect to pointwise addition and scalar multiplication of mappings on A. It is easy to see that Der(A) is a submodule of Hom_{k}(A, A), as a module over k. We also have that Hom_{k}(A, A) is an associative algebra over k, with respect to composition of mappings. This implies that Hom_{k}(A, A) is a Lie algebra over k with respect to the corresponding commutator bracket. It is well known that Der(A) is a Lie subalgebra of Hom_{k}(A, A) with respect to the commutator bracket.

More precisely, let δ, δ′ ∈ Der(A) and a, b ∈ A be given. Thus the commutator [δ, δ′] = δ ∘ δ′ − δ′ ∘ δ (2.5.2)

of δ and δ′ is defined as a module homomorphism from A into itself. Observe that

(2.5.3) ([δ, δ′])(a b) = δ(δ′(a b)) − δ′(δ(a b))
= δ(δ′(a) b + a δ′(b)) − δ′(δ(a) b + a δ(b))
= δ(δ′(a)) b + δ′(a) δ(b) + δ(a) δ′(b) + a δ(δ′(b))
−δ′(δ(a)) b − δ(a) δ′(b) − δ′(a) δ(b) − a δ′(δ(b)).

The middle pair of terms in the last two lines cancel each other, so that

(2.5.4) ([δ, δ′])(a b) = δ(δ′(a)) b + a δ(δ′(b)) − δ′(δ(a)) b − a δ′(δ(b))
= ([δ, δ′])(a) b + a ([δ, δ′])(b),

as desired.

Let a ∈ A be given, and put

δ_{a}(x) = a x − x a (2.5.5)

for every x ∈ A. Of course, the right side of (2.5.5) is the same as the commutator bracket corresponding to multiplication on A. Note that δ_{a} is a module homomorphism from A into itself, because of bilinearity over k of multiplication on A. Similarly, a ↦ δ_{a} (2.5.6)

is a module homomorphism from A into Hom_{k}(A, A), as modules over k. If A is an associative algebra over k and x, y ∈ A, then

(2.5.7)  δ_{a}(x y) = a (x y) − (x y) a
= (a x) y − x (y a)
= (a x) y − (x a) y + x (a y) − x (y a)
= δ_{a}(x) y + x δ_{a}(y),

so that δ_{a} ∈ Der(A).

Suppose now that (A, [·, ·]_{A}) is a Lie algebra over k. Let δ be a module homomorphism from A into itself, as a module over k. In this situation, the product rule says that

δ([a, b]_{A}) = [δ(a), b]_{A} + [a, δ(b)]_{A} (2.5.8)

for every a, b ∈ A. Thus δ ∈ Der(A) when this holds. Let x ∈ A be given, and let us verify that ad x = [x, ·]_{A} ∈ Der(A). If y, z ∈ A, then

(ad x)([y, z]_{A}) = [x, [y, z]_{A}]_{A} = −[y, [z, x]_{A}]_{A} − [z, [x, y]_{A}]_{A}, (2.5.9)

using the Jacobi identity in the second step. It follows that

(2.5.10)  (ad x)([y, z]_{A}) = [[x, y]_{A}, z]_{A} + [y, [x, z]_{A}]_{A}
= [(ad x)(y), z]_{A} + [y, (ad x)(z)]_{A},

as desired.

Let A be an algebra over k in the strict sense again, where the product of a, b ∈ A is denoted a b. If x, y ∈ A, then let [x, y] = x y − y x be the usual commutator bracket corresponding to multiplication in A. Let δ ∈ Der(A) be given, and observe that

(2.5.11) δ([x, y]) = δ(x y − y x) = δ(x) y + x δ(y) − δ(y) x − y δ(x)
= [δ(x), y] + [x, δ(y)]

for every x, y ∈ A. Of course, we can also consider A as an algebra over k in the strict sense with respect to [x, y]. It follows from (2.5.11) that δ is a derivation on A with respect to [x, y] as well.

2.6 Involutions

Let k be a commutative ring with a multiplicative identity element, and let A, B be algebras over k in the strict sense, where multiplication of x, y is expressed as x y. In particular, A and B are modules over k, and we let ϕ be a module homomorphism from A into B. If

ϕ(a_{1} a_{2}) = ϕ(a_{2}) ϕ(a_{1}) (2.6.1)

for every a_{1}, a_{2} ∈ A, then one may say that ϕ is an opposite algebra homomorphism from A into B. Of course, this is the same as an ordinary algebra homomorphism when A or B is commutative. If A and B have multiplicative identity elements e_{A} and e_{B}, respectively, then one may also ask that

ϕ(e_{A}) = e_{B}. (2.6.2)

If ϕ is a one-to-one opposite algebra homomorphism from A onto B, then ϕ^{−1} is an opposite algebra homomorphism from B onto A, and ϕ is said to be an opposite algebra isomorphism from A onto B. In this case, if A has a multiplicative identity element e_{A}, then it is easy to see that ϕ(e_{A}) is the multiplicative identity element in B. An opposite algebra automorphism on A is an oppposite algebra isomorphism from A onto itself.

Let [a_{1}, a_{2}]_{A} = a_{1} a_{2} −a_{2} a_{1} and [b_{1}, b_{2}]_{B} = b_{1} b_{2} −b_{2} b_{1} be the corresponding commutator brackets on A and B. If ϕ is an opposite algebra homomorphism from A into B and a_{1}, a_{2} ∈ A, then

(2.6.3) ϕ([a_{1}, a_{2}]_{A}) = ϕ(a_{1} a_{2} − a_{2} a_{1}) = ϕ(a_{2}) ϕ(a_{1}) − ϕ(a_{1}) ϕ(a_{2})
= −[ϕ(a_{1}), ϕ(a_{2})]_{B}.

An opposite algebra homomorphism x ↦ x^{∗} from A into itself is said to be an (algebra) involution on A if

(x^{∗})^{∗} = x (2.6.4)

for every x ∈ A. This implies that x ↦ x^{∗} is a one-to-one mapping from A onto itself, which is its own inverse mapping.

Let x ↦ x^{∗} be an opposite algebra automorphism on A. An element a of A is said to be self-adjoint with respect to x ↦ x^{∗} if

a^{∗} = a, (2.6.5)

and a is said to be anti-self-adjoint with respect to x ↦ x^{∗} if

a^{∗} = −a. (2.6.6)

The collections of self-adjoint and anti-self-ajoint elements of A are submodules of A, as a module over k. If 1 + 1 = 0 in k, then self-adjointness and anti-self-adjointness are the same. If 1+1 is invertible in k and a ∈ A is both self-adjoint and anti-self-adjoint, then a = 0.

Suppose that x ↦ x^{∗} is an algebra involution on A. If a is any element of A, then

a + a^{∗} (2.6.7)

is self-adjoint with respect to x ↦ x^{∗}, and

a − a^{∗} (2.6.8)

is anti-self-adjoint with respect to x ↦ x^{∗}. If 1 + 1 = 0 in k, then (2.6.7) and (2.6.8) are the same. If 1 + 1 is invertible in k, then every element of A can be expressed as the sum of self-adjoint and anti-self-adjoint elements of A, using (2.6.7) and (2.6.8). This expression is unique in this case, because 0 is the only element of A that is both self-adjoint and anti-self-adjoint.

Let a, b ∈ A be given, and let [a, b] = a b − b a be their usual commutator in A. If x ↦ x^{∗} is an opposite algebra automorphism on A, and a, b are anti-self-adjoint with respect to x ↦ x^{∗}, then

([a, b])^{∗} = −[a^{∗}, b^{∗}] = −[−a, −b] = −[a, b], (2.6.9)

using (2.6.3) in the first step. Thus [a, b] is anti-self-adjoint as well.

Suppose now that k is the field C of complex numbers, and that A and B are algebras in the strict sense over C. A conjugate-linear mapping ϕ from A into B is said to be a conjugate-linear algebra homomorphism if it preserves products as in (2.2.5), and ϕ is said to be a conjugate-linear opposite algebra homomorphism if it satisfies (2.6.1). If A and B are considered as algebras over the real numbers, then ϕ may be considered as a real-linear algebra homomorphism or opposite algebra homomorphism from A into B, as appropriate. If ϕ is a one-to-one conjugate-linear algebra homomorphism or opposite algebra homomorphism from A onto B, then ϕ^{−1} is a conjugate-linear algebra homomorphism or opposite algebra homomorphism from B onto A, as appropriate, and ϕ is said to be a conjugate-linear algebra isomorphism or opposite algebra isomorphism from A onto B, as appropriate. In particular, if A = B, then ϕ is said to be a conjugate-linear algebra automorphism or opposite algebra automorphism on A, as appropriate.

A conjugate-linear algebra homomorphism x ↦ x^{∗} from A into itself is said to be a conjugate-linear (algebra) involution on A if it satisfies (2.6.4) for every x ∈ A. In this case, x ↦ x^{∗} is a conjugate-linear opposite algebra automorphism on A, as before. Suppose that x ↦ x^{∗} is a conjugate-linear opposite algebra automorphism on A, which may be considered as a real-linear opposite algebra automorphism of A as an algebra over R. In particular, A may be considered as a vector space over R, and the collections of self-adjoint and anti-self-adjoint elements of A are real-linear subspaces of A, which is to say that they are linear subspaces of A as a vector space over R. In this situation, the anti-self-adjoint elements of A are exactly those that can be expressed as i times a self-adjoint element of A.

2.7 More on multiplication operators

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense again, where multiplication of a, b ∈ A is expressed as a b. If a ∈ A, then

M_{a}(x) = a x (2.7.1)

defines a module homomorphism from A into itself, as in Section 2.2. This is the operator of left multiplication by a on A. Similarly,

M̃_{a}(x) = x a (2.7.2)

defines a module homomorphism from A into itself, which is the operator of right multiplication by a on A. Of course, if A is commutative, then (2.7.1) and (2.7.2) are the same.

As before, a ↦ M̃_{a} (2.7.3)

defines a mapping from A into the space Hom_{k}(A, A) of all homomorphisms from A into itself, as a module over k. Bilinearity of multiplication on A implies that (2.7.3) is a module homomorphism from A into Hom_{k}(A, A), as modules over k. If A has a multiplicative identity element e, then M̃_{e} is the identity mapping on A. We also have that

M̃_{a}(e) = e a = a (2.7.4)

for every a ∈ A in this case, so that (2.7.3) is injective.

Observe that M̃_{a}(M̃_{b}(x)) = M̃_{a}(x b) = (x b) a (2.7.5)

and M̃_{b a}(x) = x (b a) (2.7.6)

for every a, b, x ∈ A. If A is an associative algebra, then we get that

M̃_{a} ∘ M̃_{b} = M̃_{b a} (2.7.7)

for every a, b ∈ A, as mappings from A into itself. This implies that (2.7.3) is an opposite algebra homomorphism from A into Hom_{k}(A, A), using composition of mappings as multiplication on Hom_{k}(A, A), as usual.

Let [x, y] = x y − y x be the usual commutator of x, y ∈ A. If A is an associative algebra, then it follows that

M̃_{[a,b]} = M̃_{a b−b a} = M̃_{a b} − M̃_{b a} = M̃_{b} ∘ M̃_{a} − M̃_{a} ∘ M̃_{b} (2.7.8)

for every a, b ∈ A. Similarly,

M_{[a,b]} = M_{a b−b a} = M_{a b} − M_{b a} = M_{a} ∘ M_{b} − M_{b} ∘ M_{a} (2.7.9) for every a, b ∈ A in this situation, using (2.2.12) in the third step. If a, b, x ∈ A, then M_{a}(M̃_{b}(x)) = M_{a}(x b) = a (x b) (2.7.10)

and M̃_{b}(M_{a}(x)) = M̃_{b}(a x) = (a x) b. (2.7.11)

These are the same when A is an associative algebra, in which case

M_{a} ∘ M̃_{b} = M̃_{b} ∘ M_{a} (2.7.12)

for every a, b ∈ A.

If a ∈ A, then let ad a be the mapping from A into itself defined by

(ad a)(x) = [a, x] = a x − x a = M_{a}(x) − M̃_{a}(x) (2.7.13)

for every x ∈ A. Equivalently,

ad a = M_{a} − M̃_{a}, (2.7.14)

which is a module homomorphism from A into itself, as a module over k. We also have that a ↦ ad a (2.7.15)

is a module homomorphism from A into Hom_{k}(A, A), as modules over k. If A is an associative algebra, then

(2.7.16) ad[a, b] = [ad a, ad b] = (ad a) ∘ (ad b) − (ad b) ∘ (ad a)

for every a, b ∈ A, as mappings from A into itself. More precisely, if A is an associative algebra over k, then A is also a Lie algebra over k with respect to the commutator bracket [x, y]. Thus (2.7.16) follows from the analogous statement for Lie algebras. Alternatively, one can use (2.7.14) to reduce to the properties (2.7.8), (2.7.9), and (2.7.12) of the multiplication operators M_{a} and M̃_{a}.

2.8 Matrices

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. If n ∈ Z_{+}, then we let M_{n}(A) be the space of n × n matrices with entries in A. An element of M_{n}(A) may be given as a = (a_{j,l}), where a_{j,l} ∈ A for every j, l = 1, …, n. It is easy to see that M_{n}(A) is also a module over k, with respect to entrywise addition and scalar multiplication.

Suppose that A is an algebra over k in the strict sense, where multiplication of x, y ∈ A is expressed as x y. If a, b ∈ M_{n}(A), then their product c = a b is defined as usual by

c_{j,r} = ∑_{l=1}^{n} a_{j,l} b_{l,r} (2.8.1)

for every j, r = 1, …, n. It is easy to see that this is bilinear in a and b over k, so that M_{n}(A) is an algebra in the strict sense over k with respect to matrix multiplication. If A is an associative algebra over k, then M_{n}(A) is associative with respect to matrix multiplication.

Suppose for the moment that A has a multiplicative identity element e. The identity matrix I = I_{n} in M_{n}(A) is the n × n matrix whose diagonal entries are equal to e, and whose other entries are equal to 0. This is the multiplicative identity element in M_{n}(A).

If a = (a_{j,l}) ∈ M_{n}(A), then the transpose a^{t} = (a_{j,l}^{t}) ∈ M_{n}(A) is defined as usual by a_{j,l}^{t} = a_{l,j} (2.8.2)

for every j, l = 1, …, n. Note that the mapping a ↦ a^{t} from a matrix to its transpose defines a module homomorphism from M_{n}(A) into itself, which is to say that it is linear over k. Let x ↦ x^{∗} be an opposite algebra automorphism on A, as in Section 2.6. If a = (a_{j,l}) ∈ M_{n}(A), then let a^{∗} = ((a^{∗})_{j,l}) ∈ M_{n}(A) be defined by (a^{∗})_{j,l} = (a_{l,j})^{∗} (2.8.3)

for every j, l = 1, …, n, which is to say that we apply x ↦ x^{∗} to the entries of the transpose a^{t} of a. One can check that this defines an opposite algebra automorphism on M_{n}(A), and an involution on M_{n}(A) when x ↦ x^{∗} is an involution on A.

If A is a commutative algebra, then the identity mapping on A defines an algebra involution on A. In this case, a^{∗} reduces to the transpose a^{t} of A.

Suppose now that k is the field C of complex numbers. Let x ↦ x^{∗} be a conjugate-linear opposite algebra automorphism on A, as in Section 2.6. In this situation, a ↦ a^{∗} is a conjugate-linear opposite algebra automorphism on M_{n}(A), and a conjugate-linear involution on M_{n}(A) when x ↦ x^{∗} is a conjugate-linear involution on A.

In particular, we can take A = C, as a commutative algebra over itself. Of course, complex-conjugation may be considered as a conjugate-linear involution on C. If a = (a_{j,l}) ∈ M_{n}(C), then let a^{∗} = ((a^{∗})_{j,l}) ∈ M_{n}(C) be defined by

(a^{∗})_{j,l} = \overline{a_{l,j}} (2.8.4)

for every j, l = 1, …, n, which is the same as the complex-conjugate of the entries of the transpose a^{t} of a. This defines a conjugate-linear involution on M_{n}(C), as before.

2.9 Traces of matrices

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. If A is a module over k, then M_{n}(A) is a module over k as well, with respect to entrywise addition and scalar multiplication, as in the previous section. If a = (a_{j,l}) ∈ M_{n}(A), then the trace of a is defined as an element of A as usual by

tr a = ∑_{j=1}^{n} a_{j,j}. (2.9.1)

It is easy to see that this defines a homomorphism from M_{n}(A) into A, as modules over k. We also have that

tr a^{t} = tr a (2.9.2)

for every a ∈ M_{n}(A), where the transpose a^{t} of a is defined as in the previous section.

Suppose that A is an algebra over k in the strict sense, where multiplication of x, y ∈ A is expressed as x y. If a, b ∈ M_{n}(A), then the products a b and b a are defined as elements of M_{n}(A) as in the previous section. Observe that

tr(a b) = ∑_{j=1}^{n} a_{j,l} b_{l,j} (2.9.3)

and

tr(b a) = ∑_{j=1}^{n} b_{l,j} a_{j,l}. (2.9.4)

If A is a commutative algebra over k, then we get that

tr(a b) = tr(b a) (2.9.5)

for every a, b ∈ M_{n}(A). Equivalently, this means that

tr(a b − b a) = 0 (2.9.6)

for every a, b ∈ M_{n}(A).

The nth general linear algebra gl_{n}(A) with entries in A is defined as an algebra over k in the following way. As a module over k, gl_{n}(A) is the same as M_{n}(A). We use the commutator [a, b] = a b − b a as the bilinear operation on gl_{n}(A), where the products a b and b a are as defined in the previous section. This makes gl_{n}(A) into an algebra over k in the strict sense. If A is an associative algebra over k, then M_{n}(A) is an associative algebra over k too, so that gl_{n}(A) is a Lie algebra over k.

Put sl_{n}(A) = {a ∈ gl_{n}(A) : tr a = 0}, (2.9.7) which defines a submodule of gl_{n}(A) as a module over k, or equivalently a submodule of M_{n}(A). If A is a commutative algebra over k, then

[a, b] ∈ sl_{n}(A) (2.9.8)

for every a, b ∈ gl_{n}(A), as in (2.9.6). In particular, this means that sl_{n}(A) is a subalgebra of gl_{n}(A) with respect to the commutator [a, b] when A is commutative. In this case, sl_{n}(A) is called the nth special linear algebra with entries in A. If A is a commutative associative algebra over k, then sl_{n}(A) is a Lie algebra over k with respect to [a, b].

2.10 Vector spaces and linear mappings

Let k be a field. If V and W are vector spaces over k, then the space L(V, W) of linear mappings from V into W is a vector space over k with respect to pointwise addition and scalar multiplication. This is the same as the space Hom_{k}(V, W) of module homomorphisms from V into W, where V and W are considered as modules over k. Similarly, if V is a vector space over k, then the space L(V) = L(V, V) of linear mappings from V into itself is an associative algebra over k with respect composition of mappings.

The general linear algebra gl(V) associated to a vector space V over k is defined as a Lie algebra over k in the following way. As a vector space over k, gl(V) is the same as L(V). If T_{1} and T_{2} are linear mappings from V into itself, then their commutator

[T_{1}, T_{2}] = T_{1} ∘ T_{2} − T_{2} ∘ T_{1} (2.10.1)

defines a linear mapping from V into itself as well. This defines a bilinear operation on gl(V), which we use to define the Lie bracket on gl(V). This satisfies the requirements of a Lie algebra, because L(V) is an associative algebra over k with respect to composition of linear mappings.

Suppose that V is a finite-dimensional vector space over k, with dimension n ≥ 1. Let v_{1}, …, v_{n} be a basis for V, as a vector space over k. Thus every v ∈ V can be expressed in a unique way as

v = ∑_{l=1}^{n} t_{l} v_{l}, (2.10.2)

where t_{1}, …, t_{n} ∈ k. Let a = (a_{j,l}) be an n × n matrix with entries in k. If v ∈ V is as in (2.10.2), then put

T_{a}(v) = ∑_{j=1}^{n} ( ∑_{l=1}^{n} a_{j,l} t_{l}) v_{j}, (2.10.3)

which defines an element of V. Of course, T_{a} is a linear mapping from V into itself, and a ↦ T_{a} (2.10.4)

is a linear mapping from M_{n}(k) into L(V). More precisely, (2.10.4) is a one-to-one mapping from M_{n}(k) onto L(V), and

T_{a} ∘ T_{b} = T_{a b} (2.10.5)

for every a, b ∈ M_{n}(k). This means that (2.10.4) is an algebra isomorphism from M_{n}(k) onto L(V), with respect to matrix multiplication on M_{n}(k), and composition of linear mappings on V. It follows that (2.10.4) is also a Lie algebra isomorphism from gl_{n}(k) onto gl(V), with respect to their corresponding commutator brackets.

If a = (a_{j,l}) ∈ M_{n}(k), then the trace of T_{a} is defined as an element of k by

tr T_{a} = tr a = ∑_{j=1}^{n} a_{j,j}, (2.10.6)

where tr a refers to the trace of a as a matrix, as in the previous section. This defines the trace tr T of every linear mapping T from V into itself, by the remarks in the preceding paragraph. Note that the trace is a linear mapping from L(V) into k. If T_{1}, T_{2} ∈ L(V), then

tr(T_{1} ∘ T_{2}) = tr(T_{2} ∘ T_{1}), (2.10.7)

by (2.9.5) and (2.10.5). It is well known that the trace of T ∈ L(V) does not depend on the choice of basis v_{1}, …, v_{n} for V, because of (2.9.5) or (2.10.7).

Put sl(V) = {T ∈ gl(V) : tr T = 0}, (2.10.8)

which is a linear subspace of gl(V), or equivalently of L(V). If T_{1}, T_{2} ∈ gl(V), then tr[T_{1}, T_{2}] = 0, (2.10.9)

by (2.10.7), and hence [T_{1}, T_{2}] ∈ sl(V). (2.10.10)

by (2.10.9). In particular, sl(V) is a subalgebra of gl(V) as a Lie algebra with respect to the commutator bracket. This is the special linear algebra associated to V. The mapping (2.10.4) defines a Lie algebra isomorphism from sl_{n}(k) onto sl(V).

2.11 Ideals and quotients

Let k be a commutative ring with a multiplicative identity element. If A and B are modules over k and ϕ is a module homomorphism from A into B, then the kernel of ϕ is the set of a ∈ A such that ϕ(a) = 0, as usual. Of course, this is a submodule of A.

If A is a module over k, and A_{0} is a submodule of A, then the quotient A/A_{0} can be defined as a module over k in the usual way. More precisely, one can consider the quotient A/A_{0} initially as a commutative group with respect to addition, and check that scalar multiplication on A/A_{0} by elements of k can be defined in a natural way. The corresponding quotient mapping is a module homomorphism from A onto A/A_{0}, with kernel equal to A_{0}.

Let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Also let A_{0} be a submodule of A, as a module over k. If

a b ∈ A_{0} (2.11.1)

for every a ∈ A and b ∈ A_{0}, then A_{0} is said to be a left ideal in A. Similarly, if (2.11.1) holds for every a ∈ A_{0} and b ∈ A, then A_{0} is said to be a right ideal in A. If A_{0} is both a left and right ideal in A, then A is said to be a two-sided ideal in A. Of course, if A is a commutative algebra, then left and right ideals in A are the same. If B is another algebra over k in the strict sense, and ϕ is an algebra homomorphism from A into B, then the kernel of ϕ is a two-sided ideal in A.

If A_{0} is a submodule of A, as a module over k, then the quotient A/A_{0} can be defined as a module over k too, as before. Let q_{0} be the corresponding quotient mapping from A onto A/A_{0}. Thus

(a, b) ↦ q_{0}(a b) (2.11.2)

is bilinear over k as a mapping from A × A into A/A_{0}. If A_{0} is a left ideal in A, then q_{0}(a b) = 0 (2.11.3) for every a ∈ A and b ∈ A_{0}, and (2.11.2) leads to a bilinear mapping from A × (A/A_{0}) into A/A_{0}. More precisely, if a, b ∈ A, then q_{0}(a b) only depends on a and q_{0}(b) in this case. Similarly, if A_{0} is a right ideal in A, then (2.11.3) holds for every a ∈ A_{0} and b ∈ A, and (2.11.2) leads to a bilinear mapping from (A/A_{0}) × A into A/A_{0}. If A_{0} is a two-sided ideal in A, then (2.11.3) holds when either a or b is in A_{0}, so that q_{0}(a b) only depends on q_{0}(a) and q_{0}(b). In this situation, (2.11.2) leads to a bilinear mapping from (A/A_{0}) × (A/A_{0}) into A/A_{0}, which makes A/A_{0} into an algebra over k in the strict sense, for which the quotient mapping q_{0} is an algebra homomorphism.

Suppose for the moment that A is an associative algebra over k. If A_{0} is a two-sided ideal in A, then A/A_{0} is an associative algebra over k as well. If A_{0} is a left ideal in A, then elements of A act on A/A_{0} by multiplication on the left, as in the preceding paragraph. Associativity of multiplication on A implies that the action on A/A_{0} by products of elements of A corresponds to the composition of the actions on A/A_{0} of the individual elements of A. Similarly, if A_{0} is a right ideal in A, then elements of A act on A/A_{0} by multiplication on the right, with the appropriate relationship between products of elements of A and their actions on A/A_{0}.

Suppose now that (A, [·, ·]_{A}) is a Lie algebra over k, and let A_{0} be a submodule of A as a module over k. If

[a, b]_{A} ∈ A_{0} (2.11.4)

for every a ∈ A and b ∈ A_{0}, then A_{0} is said to be an ideal in A as a Lie algebra. This is equivalent to saying that A_{0} is a left, right, or two-sided ideal in A, as an algebra over k in the strict sense. In this case, if q_{0} is the usual quotient mapping from A onto A_{0}, then q_{0}([a, b]_{A}) depends only on q_{0}(a) and q_{0}(b), as before. It is easy to see that A/A_{0} is also a Lie algebra over k with respect to the Lie bracket [·, ·]_{A/A_{0}} obtained from [·, ·]_{A} in this way.

Let A be an associative algebra over k again, where the product of a, b ∈ A is denoted a b. Let A_{0} be a two-sided ideal in A. Remember that A may also be considered as a Lie algebra over k with respect to the commutator bracket [a, b]_{A} = a b − b a. Under these conditions, A_{0} may be considered an an ideal in A as a Lie algebra with respect to [a, b]_{A} as well.

2.12 Bilinear forms

Let k be a commutative ring with a multiplicative identity element, and let A, C be modules over k. Also let β be a mapping from A×A into C that is bilinear over k. Of course, k may be considered as a module over itself as well, using multiplication on k as scalar multiplication. If C = k, as a module over itself, then β is said to be a bilinear form on A, as a module over k. As before, β is said to be symmetric on A × A when

β(b, a) = β(a, b) (2.12.1)

for every a, b ∈ A, and antisymmetric on A × A when

β(b, a) = −β(a, b) (2.12.2)

for every a, b ∈ A. It is sometimes better to ask that

β(a, a) = 0 (2.12.3)

for every a ∈ A, instead of (2.12.2). Remember that (2.12.3) implies (2.12.2), because of bilinearity, as in Section 2.1. If 1 + 1 = 0 in k, then (2.12.1) and (2.12.2) are the same. If 1 + 1 is invertible in k, then (2.12.2) implies (2.12.3).

If β is any bilinear mapping from A × A into C, then

β(a, b) + β(b, a) (2.12.4)

is symmetric on A × A, and

β(a, b) − β(b, a) (2.12.5)

is antisymmetric on A × A. If 1 + 1 = 0 in k, then (2.12.4) is the same as (2.12.5), and is equal to 0 when a = b. If 1 + 1 is invertible in k, then every bilinear mapping from A×A into C can be expressed as the sum of a symmetric bilinear mapping and an antisymmetric bilinear mapping. In this case, a bilinear mapping from A×A into C that is both symmetric and antisymmetric on A×A is identically 0 on A × A, which implies that the previous expression as a sum is unique.

Let β be a bilinear mapping from A × A into C, and let ϕ be a module homomorphism from A into itself. Let us say that ϕ is symmetric with respect to β on A if β(ϕ(a), b) = β(a, ϕ(b)) (2.12.6)

for every a, b ∈ A. Similarly, let us say that ϕ is antisymmetric with respect to β on A if β(ϕ(a), b) = −β(a, ϕ(b)) (2.12.7)

for every a, b ∈ A. The collections of module homomorphisms from A into itself that are symmetric or antisymmetric with respect to β are submodules of Hom_{k}(A, A). If 1+1 = 0 in k, then (2.12.6) and (2.12.7) are the same, as usual.

Let ϕ and ψ be module homomorphisms from A into itself, and let [ϕ, ψ] = ϕ ∘ ψ − ψ ∘ ϕ be their commutator with respect to composition. If ϕ and ψ are both symmetric on A with respect to β, then

β(([ϕ, ψ])(a), b) = β(ϕ(ψ(a)), b) − β(ψ(ϕ(a)), b)
(2.12.8)  = β(a, ψ(ϕ(b))) − β(a, ϕ(ψ(b))) = −β(a, ([ϕ, ψ])(b)))

for every a, b ∈ A. This also works when ϕ and ψ are both antisymmetric with respect to β on A, using antisymmetry twice in each term in the second step. In both cases, we get that [ϕ, ψ] is antisymmetric with respect to β on A. In particular, the collection of module homomorphisms from A into itself that are antisymmetric with respect to β is a Lie subalgebra of Hom_{k}(A, A), as a Lie algebra over k with respect to the commutator bracket.

2.13 Dual spaces and mappings

Let k be a field, and let V be a vector space over k. Remember that a linear functional on V is a linear mapping from V into k, where k is considered as a one-dimensional vector space over itself. The dual V′ of V is the space of all linear functionals on V, which is a vector space over k with respect to pointwise addition and scalar multiplication. If V has finite dimension, then it is well known that the dimension of V′ is the same as the dimension of V. This can be seen by expressing linear functionals on V in terms of a basis for V.

Let W be another vector space over k, and let T be a linear mapping from V into W. If μ is a linear functional on W, then

T′(μ) = μ ∘ T (2.13.1)

is a linear functional on V. This defines a linear mapping T′ from W′ into V′, which is the dual mapping associated to T. We also have that

T ↦ T′ (2.13.2)

is linear as a mapping from the space L(V, W) of linear mappings from V into W into the space L(W′, V′) of linear mappings from W′ into V′. The dual of the identity mapping I_{V} on V, as a linear mapping from V into itself, is the identity mapping I_{V′} on V′.

Let Z be a third vector space over k, let T_{1} be a linear mapping from V into W, and let T_{2} be a linear mapping from W into Z. Thus the composition T_{2} ∘T_{1} is a linear mapping from V into Z, whose dual maps Z′ into V′. If ν ∈ Z′, then

(T_{2} ∘ T_{1})′(ν) = ν ∘ (T_{2} ∘ T_{1}) = (ν ∘ T_{2}) ∘ T_{1} = T′_{1}(T′_{2}(ν)). (2.13.3)

This shows that (T_{2} ∘ T_{1})′ = T′_{1} ∘ T′_{2} (2.13.4)

as mappings from Z′ into V′.

Let V′′ = (V′)′ be the dual of V′. If v ∈ V and λ ∈ V′, then

L_{v}(λ) = λ(v) (2.13.5)

is an element of k. This defines L_{v} as a linear functional on V′, and

v ↦ L_{v} (2.13.6)

is a linear mapping from V into V′′. If v ∈ V and v ≠ 0, then one can find a λ ∈ V′ such that λ(v) ≠ 0, using a basis for V. This implies that (2.13.6) is injective as a mapping from V into V′′. If V has finite dimension, then V′ has the same dimension as V, and hence V′′ has the same dimension as well. In this case, (2.13.6) also maps V onto V′′.

Let λ_{1}, …, λ_{n} be n linear functionals on V for some positive integer n, and put Λ(v) = (λ_{1}(v), …, λ_{n}(v)) (2.13.7)

for each v ∈ V. This defines a linear mapping from V into the space k^{n} of n-tuples of elements of k, which is a vector space over k with respect to coordinatewise addition and scalar multiplication, as usual. Of course, the kernel of Λ is the same as the intersection of the kernels of λ_{1}, …, λ_{n}. If Λ is injective, then the dimension of V is less than or equal to n, by standard results in linear algebra. In particular, if Λ is injective and the dimension of V is equal to n, then Λ maps V onto k^{n}.

2.14 Nondegenerate bilinear forms

Let k be a field, and let V be a finite-dimensional vector space over k. Also let b(v, w) be a bilinear form on V. If w ∈ V, then

b_{w}(v) = b(v, w) (2.14.1)

defines a linear functional on V as a function of v, and w ↦ b_{w} defines a linear mapping from V into its dual space V′. The image

{b_{w} : w ∈ V } (2.14.2)

of this linear mapping is a linear subspace of V′. Note that (2.14.2) is equal to V′ exactly when w ↦ b_{w} is injective as a linear mapping from V into V′, because V and V′ have the same dimension.

If for every v ∈ V with v ≠ 0 there is a w ∈ V such that b(v, w) ≠ 0, then b is said to be nondegenerate on V. This is the same as saying that the intersections of the kernels of the b_{w}’s, w ∈ V, is the trivial subspace {0} of V. One can check that this happens exactly when (2.14.2) is equal to V′.

Suppose that b is a nondegenerate bilinear form on V, and let T be a linear mapping from V into itself. If w ∈ V, then b(T (v), w) defines a linear functional on V, as a function of v. This implies that there is a unique element T^{∗}(w) of V such that b(T (v), w) = b(v, T^{∗}(w)) (2.14.3)

for every v ∈ V, because b is nondegenerate on V. This defines a mapping T^{∗} from V into itself, which is the adjoint of T with respect to b. It is easy to see that T^{∗} is a linear mapping from V into itself, because T^{∗}(w) is uniquely determined by (2.14.3).

Remember that the space L(V) of linear mappings from V into itself is an algebra over k with respect to composition of mappings. One can check that T ↦ T^{∗} defines a linear mapping from L(V) into itself, because T^{∗} is uniquely determined by (2.14.3). Clearly I^{∗} = I, where I is the identity mapping on V.

If T is any linear mapping from V into itself, then T^{∗} = T exactly when T is symmetric with respect to b, as in Section 2.12. Similarly, T^{∗} = −T exactly when T is antisymmetric with respect to b.

If T is a linear mapping from V into itself and T^{∗} = 0 on V, then T = 0 on V, because of (2.14.3) and the nondegeneracy of b on V. This implies that T ↦ T^{∗} is a one-to-one mapping from L(V) onto itself, because L(V) is a finite-dimensional vector space over k.

Let T_{1} and T_{2} be linear mappings from V into itself. If v, w ∈ V, then

(2.14.4) b((T_{2} ∘ T_{1})(v), w) = b(T_{2}(T_{1}(v)), w) = b(T_{1}(v), T_{2}^{∗}(w))
= b(v, T_{1}^{∗}(T_{2}^{∗}(w))) = b(v, (T_{1}^{∗} ∘ T_{2}^{∗})(w)).

This implies that (T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗}, (2.14.5)

so that T ↦ T^{∗} is an opposite algebra automorphism on L(V), as in Section 2.6.

Let T be a linear mapping from V into itself again, so that T^{∗} and hence (T^{∗})^{∗} are defined as linear mappings from V into itself, as before. If b is symmetric on V, then

b(T (v), w) = b(T^{∗}(w), v) = b(w, (T^{∗})^{∗}(v)) = b((T^{∗})^{∗}(v), w) (2.14.6)

for every v, w ∈ V. Similarly, if b is antisymmetric on V, then

b(T (v), w) = −b(T^{∗}(w), v) = −b(w, (T^{∗})^{∗}(v)) = b((T^{∗})^{∗}(v), w) (2.14.7)

for every v, w ∈ V. In both cases, we get that

(T^{∗})^{∗} = T. (2.14.8)

It follows that T ↦ T^{∗} defines an involution on L(V), as an algebra over k with respect to composition, when b is symmetric or antisymmetric on V.

2.15 Sesquilinear forms

Let V be a vector space over the field C of complex numbers. A complex-valued function b on V ×V is said to be sesquilinear if b(v, w) is complex-linear in v for each w ∈ V, and b(v, w) is conjugate-linear in w for every v ∈ V. In particular, if we consider V and C as vector spaces over the real numbers, then it follows that b is bilinear over R. If we also have that

b(w, v) = \overline{b(v, w)} (2.15.1)

for every v, w ∈ V, then b is said to be Hermitian-symmetric on V, or equivalently b is a Hermitian form on V. The analogous Hermitian-antisymmetry condition b(w, v) = −\overline{b(v, w)} (2.15.2)

is the same as saying that i b(v, w) is a Hermitian form on V. If b is any sesquilinear form on V, then \overline{b(w, v)} is a sesquilinear form on V too,

b(v, w) + \overline{b(w, v)} (2.15.3)

is a Hermitian form on V, and

b(v, w) − \overline{b(w, v)} (2.15.4)

is Hermitian-antisymmetric on V. This permits us to express b(v, w) in a unique way as b_{1}(v, w) + i b_{2}(v, w), where b_{1}(v, w) and b_{2}(v, w) are Hermitian forms on V. Note that b(v, v) ∈ R for every v ∈ V when b is a Hermitian form on V.

Let b be a sesquilinear form on V, and let T be a linear mapping from V into itself. Let us say that T is self-adjoint with respect to b on V if

b(T (v), w) = b(v, T (w)) (2.15.5)

for every v, w ∈ V, and that T is anti-self-adjoint with respect to b on V if

b(T (v), w) = −b(v, T (w)) (2.15.6)

for every v, w ∈ V. One can check that T is anti-self-adjoint with respect to b on V if and only if i T is self-adjoint with respect to b on V. If we consider b as a real-bilinear mapping from V × V into C, then these self-adjointness and anti-self-adjointness conditions correspond exactly to the symmetry and antisymmetry conditions for module homomorphisms mentioned in Section 2.12. The space of self-adjoint linear mappings from V into itself with respect to b on V is a real-linear subspace of the space L(V) of linear mappings from V into itself, which is to say that it is a linear subspace of L(V) when L(V) is considered as a vector space over R. If T_{1}, T_{2} are self-adjoint linear mappings from V into itself with respect to b, then their commutator [T_{1}, T_{2}] = T_{1} ∘ T_{2} − T_{2} ∘ T_{1} with respect to composition is anti-self-adjoint with respect to b, as before. Similarly, if T_{1}, T_{2} are anti-self-adjoint linear mappings from V into itself with respect to b, then [T_{1}, T_{2}] is anti-self-adjoint with respect to b as well. It follows that the space of anti-self-adjoint linear mappings from V into itself with respect to b is a real Lie subalgebra of L(V), which is to say that it is a subalgebra of L(V) as a Lie algebra over R with respect to the commutator bracket.

Suppose from now on in this section that V has finite dimension as a complex vector space. If w ∈ V, then

b_{w}(v) = b(v, w) (2.15.7)

defines a linear functional on V as a function of v, as before. In this situation, w ↦ b_{w} is a conjugate-linear mapping from V into its dual space V′. The image

{b_{w} : w ∈ V } (2.15.8)

of this mapping is still a linear subspace of V′, as a complex vector space. One can check that (2.15.8) is equal to V′ exactly when w ↦ b_{w} is injective as a mapping from V into V′, because V and V′ have the same dimension as complex vector spaces.

If for every v ∈ V with v ≠ 0 there is a w ∈ V such that b(v, w) ≠ 0, then b is said to be nondegenerate as a sesquilinear form on V. This is the same as saying that the intersections of the kernels of the b_{w}’s, w ∈ V, is trivial, as before. This happens exactly when (2.15.8) is equal to V′, as in the previous section.

Let b be a nondegenerate sesquilinear form on V, and let T be a linear mapping from V into itself. Also let w ∈ V be given, so that b(T (v), w) defines a linear functional on V, as a function of v. It follows that there is a unique element T^{∗}(w) of V such that

b(T (v), w) = b(v, T^{∗}(w)) (2.15.9)

for every v ∈ V, because b is nondegenerate on V. The resulting mapping T^{∗} from V into itself is called the adjoint of T with respect to b. One can check that T^{∗} is a linear mapping from V into itself, using the sesquilinearity of b.

However, T ↦ T^{∗} is conjugate-linear as a mapping from L(V) into itself in this situation. We still have that I^{∗} = I, where I is the identity mapping on V. A linear mapping T from V into itself is self-adjoint with respect to b if and only if T^{∗} = T. Similarly, T is anti-self-adjoint with respect to b if and only if T^{∗} = −T.

If T is a linear mapping from V into itself and T^{∗} = 0 on V, then T = 0 on V, because of (2.15.9) and nondegeneracy of b on V, as before. If T_{1} and T_{2} are linear mappings from V into itself, then one can verify that

(T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗}, (2.15.10)

in the same way as before. It follows that T ↦ T^{∗} is a conjugate-linear opposite algebra automorphism on L(V), because L(V) is a finite-dimensional vector space over C.

Suppose now that b is also Hermitian-symmetric on V. Let T be a linear mapping from V into itself, so that T^{∗} and (T^{∗})^{∗} are defined as linear mappings from V into itself as well. If v, w ∈ V, then

b(T (v), w) = \overline{b(T^{∗}(w), v)} = \overline{b(w, (T^{∗})^{∗}(v))} = b((T^{∗})^{∗}(v), w). (2.15.11)

Thus (T^{∗})^{∗} = T, (2.15.12)

so that T ↦ T^{∗} is a conjugate-linear involution on L(V) in this case.

Submultiplicativity and invertibility

3.1 Invertibility

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e, where multiplication of a, b ∈ A is expressed as a b. An element a of A is said to be invertible in A if there is an element b of A such that

a b = b a = e. (3.1.1)

It is easy to see that b is unique when it exists, using associativity of multiplication on A. In this case, b is called the multiplicative inverse of a in A, and is denoted a^{−1}. Of course, e is its own inverse in A. If x and y are invertible elements of A, then x y is invertible in A too, with

(x y)^{−1} = y^{−1} x^{−1}. (3.1.2)

Thus the collection of invertible elements in A is a group.

Let x and y be commuting elements of A, so that x y = y x. If x is invertible in A, then x^{−1} commutes with y too. Suppose that w and z are commuting elements of A, and w z is invertible in A. Note that w z commutes with w and z, so that (w z)^{−1} commutes with w and z too. It follows that w and z are invertible in A, with w^{−1} = (w z)^{−1} z and z^{−1} = (w z)^{−1} w.

Let a ∈ A be given, and let n be a nonnegative integer. Using a standard computation, we get that

(e − a) ∑_{j=0}^{n} a^{j} = ( ∑_{j=0}^{n} a^{j}) (e − a) = e − a^{n+1}, (3.1.3)

where a^{j} is interpreted as being equal to e when j = 0. In particular, if a^{n+1} = 0, then it follows that e − a is invertible in A, with

(e − a)^{−1} = ∑_{j=0}^{n} a^{j}. (3.1.4)

If e − a^{n+1} is invertible in A, then (3.1.3) implies that e − a is invertible in A too, as in the previous paragraph.

Let n be a positive integer, and consider the algebra M_{n}(A) of n×n matrices with entries in A. The group of invertible elements of M_{n}(A) is denoted GL_{n}(A), and is called the nth general linear group with entries in A.

Suppose for the moment that A is also commutative, so that the determinant of a = (a_{j,l}) ∈ M_{n}(A) can be defined as an element of A in the usual way. If a ∈ GL_{n}(A), then det a is an invertible element of A. Conversely, if a ∈ M_{n}(A) and det a is an invertible element of A, then a ∈ GL_{n}(A), by Cramer’s rule. The nth special linear group SL_{n}(A) with entries in A consists of the a ∈ M_{n}(A) such that det a is the multiplicative identity element e in A. This is a normal subgroup of GL_{n}(A), because SL_{n}(A) is the kernel of the determinant as a group homomorphism from GL_{n}(A) into the multiplicative group of invertible elements in A.

Let k be a field, and let V be a vector space over k. Remember that the space L(V) of linear mappings from V into itself is an associative algebra with respect to composition of mappings, and with the identity mapping I = I_{V} on V as the multiplicative identity element in L(V). The group GL(V) of one-to-one linear mappings from V onto itself with respect to composition of mappings is the same as the group of invertible elements in L(V), and may be called the general linear group associated to V.

Suppose that V has finite dimension n ≥ 1, and let v_{1}, …, v_{n} be a basis for V. This leads to an algebra isomorphism from M_{n}(k) onto L(V), as in Section 2.10. The restriction of this mapping to GL_{n}(k) defines a group isomorphism from GL_{n}(k) onto GL(V).

The determinant of a linear mapping T from V into itself can be defined as an element of k as the determinant of the corresponding matrix in M_{n}(k). It is well known that this does not depend on the choice of basis v_{1}, …, v_{n} of V. Note that a ∈ M_{n}(k) is invertible exactly when det a ≠ 0, because k is a field. Thus T ∈ L(V) is invertible exactly when det T ≠ 0.

The special linear group SL(V) associated to V consists of the linear mappings T from V into itself such that det T = 1 in k. In particular, these linear mappings are invertible on V, and SL(V) is a normal subgroup of GL(V), because it is the kernel of the determinant as a group homomorphism from GL(V) into the multiplicative group of non-zero elements of k. The restriction of the algebra isomorphism from M_{n}(k) onto L(V) mentioned earlier to SL_{n}(k) defines a group isomorphism from SL_{n}(k) onto SL(V).

3.2 Submultiplicative seminorms

Let k be a field with an absolute value function | · |, and let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Also let N_{A} be a seminorm on A, as a vector space over k, and with respect to | · | on k. As in Section 1.13, multiplication on A is bounded as a bilinear mapping from A × A into A with respect to N_{A} on A if there is a nonnegative real number C such that

N_{A}(a b) ≤ C N_{A}(a) N_{A}(b) (3.2.1)

for every a, b ∈ A. If this holds with C = 1, then N_{A} is said to be submultiplicative on A. Similarly, if

N_{A}(a b) = N_{A}(a) N_{A}(b) (3.2.2)

for every a, b ∈ A, then N_{A} is said to be multiplicative on A.

Suppose for the moment that A has a multiplicative identity element e. If (3.2.1) holds for some C ≥ 0, then we get that

N_{A}(a) ≤ C N_{A}(a) N_{A}(e) (3.2.3)

for every a ∈ A. If N_{A}(a) > 0 for some a ∈ A, then it follows that

1 ≤ C N_{A}(e). (3.2.4)

Let V be a vector space over k, and let N_{V} be a seminorm on V with respect to | · | on k. Consider the space BL(V) = BL(V, V) of bounded linear mappings from V into itself, with respect to N_{V} on V. This is a subalgebra of the algebra L(V) of all linear mappings from V into itself, with composition of mappings as multiplication. Let ∥·∥_{op} = ∥·∥_{op,V V} be the operator seminorm on BL(V) corresponding to N_{V} on V, as in Section 1.9. This is a submultiplicative seminorm on BL(V) with respect to | · | on k, as before. It is easy to see that the identity mapping I = I_{V} on V is bounded with respect to N_{V}, with

∥I∥_{op} = 1 (3.2.5)

when N_{V} (v) > 0 for some v ∈ V, and ∥I∥_{op} = 0 otherwise.

Let A be an algebra over k in the strict sense again, and let N_{A} be a seminorm on A with respect to | · | on k that satisfies (3.2.1) for some C ≥ 0. If a ∈ A, then M_{a}(x) = a x (3.2.6)

defines a linear mapping from A into itself, as a vector space over k, as in Section 2.2. Using (3.2.1), we get that

N_{A}(M_{a}(x)) ≤ C N_{A}(a) N_{A}(x) (3.2.7)

for every x ∈ A, so that M_{a} is bounded as a linear mapping from A into itself with respect to N_{A}. More precisely, we have that

∥M_{a}∥_{op} ≤ C N_{A}(a) (3.2.8) for every a ∈ A, where ∥ · ∥_{op} = ∥ · ∥_{op,AA} is the operator seminorm on the space BL(A) of bounded linear mappings from A into itself with respect to N_{A}. If A has a multiplicative identity element e, then

N_{A}(a) = N_{A}(M_{a}(e)) ≤ ∥M_{a}∥_{op} N_{A}(e) (3.2.9)

for every a ∈ A.

Similarly, if a ∈ A, then M̃_{a}(x) = x a (3.2.10)

defines a linear mapping from A into itself, as in Section 2.7. As before, we can use (3.2.1) to get that

N_{A}(M̃_{a}(x)) ≤ C N_{A}(x) N_{A}(a) (3.2.11)

for every x ∈ A. This implies that M̃_{a} is bounded as a linear mapping from A into itself with respect to N_{A}, with

∥M̃_{a}∥_{op} ≤ C N_{A}(a) (3.2.12)

for every a ∈ A. If A has a multiplicative identity element e, then

N_{A}(a) = N_{A}(M̃_{a}(e)) ≤ ∥M̃_{a}∥_{op} N_{A}(e) (3.2.13)

for every a ∈ A.

Let α be a positive real number, and put

N̂_{A}(a) = α N_{A}(a) (3.2.14)

for every a ∈ A. This defines a seminorm on A as a vector space over k too, with respect to | · | on k. Using (3.2.1), we get that

N̂_{A}(a b) ≤ (C/α) N̂_{A}(a) N̂_{A}(b) (3.2.15)

for every a, b ∈ A. In particular, this means that N̂_{A} is submultiplicative on A when α ≥ C.

Alternatively, ∥M_{a}∥_{op} defines a seminorm on A as a vector space over k, with respect to | · | on k. Suppose that A is an associative algebra over k, so that a ↦ M_{a} is an algebra homomorphism from A into BL(A), as in Section 2.2. This implies that ∥M_{a}∥_{op} is submultiplicative as a seminorm on A, because ∥ · ∥_{op} is submultiplicative on BL(A). If A has a multiplicative identity element e, then M_{e} is the identity operator on A, as before. Hence ∥M_{e}∥_{op} is equal to 1 when N_{A}(a) > 0 for some a ∈ A, and is equal to 0 otherwise.

3.3 Some matrix seminorms

Let k be a field with an absolute value function |·|, and let A be an algebra over k in the strict sense again, where multiplication of a, b ∈ A is expressed as a b.

Also let n be a positive integer, and let M_{n}(A) be the space of n×n matrices with entries in A, which is an algebra over k in the strict sense with respect to matrix multiplication, as in Section 2.8. Suppose that N_{A} is a seminorm on A with respect to | · | on k that satisfies (3.2.1) for some C ≥ 0. If a = (a_{j,l}) ∈ M_{n}(A), then put

(3.3.1)  N_{∞}(a) = max_{1≤j,l≤n} N_{A}(a_{j,l}),
(3.3.2)  N_{1,∞}(a) = max_{1≤l≤n} ( ∑_{j=1}^{n} N_{A}(a_{j,l})),

and

N_{∞,1}(a) = max_{1≤j≤n} ( ∑_{l=1}^{n} N_{A}(a_{j,l})). (3.3.3)

It is easy to see that (3.3.1), (3.3.2), and (3.3.3) define seminorms on M_{n}(A), as a vector space over k, and with respect to | · | on k. If N_{A} is a norm on A, then (3.3.1), (3.3.2), and (3.3.3) are norms on M_{n}(A). Observe that

N_{∞}(a) ≤ N_{1,∞}(a) ≤ n N_{∞}(a) (3.3.4)

and N_{∞}(a) ≤ N_{∞,1}(a) ≤ n N_{∞}(a) (3.3.5)

for every a ∈ M_{n}(A). In addition,

N_{∞}(a^{t}) = N_{∞}(a) (3.3.6)

and N_{1,∞}(a^{t}) = N_{∞,1}(a) (3.3.7)

for every a ∈ M_{n}(A), where a^{t} ∈ M_{n}(A) is the transpose of a, as before.

Suppose for the moment that A has a multiplicative identity element e. Remember that the corresponding identity matrix I ∈ M_{n}(A) has diagonal entries equal to e, and all other entries equal to 0. Thus

N_{∞}(I) = N_{1,∞}(I) = N_{∞,1}(I) = N_{A}(e). (3.3.8)

Let a, b ∈ M_{n}(A) be given, and let c = a b be their product, so that

c_{j,r} = ∑_{l=1}^{n} a_{j,l} b_{l,r} (3.3.9)

for every j, r = 1, …, n. Observe that

N_{A}(c_{j,r}) ≤ ∑_{l=1}^{n} N_{A}(a_{j,l} b_{l,r}) ≤ C ∑_{l=1}^{n} N_{A}(a_{j,l}) N_{A}(b_{l,r}) (3.3.10)

3.4. CONTINUITY OF INVERSES 55

for every j, r = 1, …, r, using (3.2.1) in the second step. Thus

∑_{j=1}^{n} N_{A}(c_{j,r}) ≤ C ∑_{j=1}^{n} ∑_{l=1}^{n} N_{A}(a_{j,l}) N_{A}(b_{l,r})
(3.3.11)  = C ∑_{l=1}^{n} ∑_{j=1}^{n} N_{A}(a_{j,l}) N_{A}(b_{l,r})
≤ C N_{1,∞}(a) ∑_{l=1}^{n} N_{A}(b_{l,r}) ≤ C N_{1,∞}(a) N_{1,∞}(b)

for every l = 1, …, n, so that N_{1,∞}(c) ≤ C N_{1,∞}(a) N_{1,∞}(b). (3.3.12) Similarly,

∑_{r=1}^{n} N_{A}(c_{j,r}) ≤ C ∑_{r=1}^{n} ∑_{l=1}^{n} N_{A}(a_{j,l}) N_{A}(b_{l,r})
(3.3.13)  = C ∑_{l=1}^{n} ∑_{r=1}^{n} N_{A}(a_{j,l}) N_{A}(b_{l,r})
≤ C ∑_{l=1}^{n} N_{A}(a_{j,l}) N_{∞,1}(b) ≤ C N_{∞,1}(a) N_{∞,1}(b)

for every j = 1, …, n, so that N_{∞,1}(c) ≤ C N_{∞,1}(a) N_{∞,1}(b). (3.3.14)

Suppose now that N_{A} is a semi-ultranorm on A with respect to | · | on k. This implies that N_{∞} is a semi-ultranorm on M_{n}(A), as a vector space over k, and with respect to | · | on k. In this case, we have that

N_{A}(c_{j,r}) ≤ max_{1≤l≤n} N_{A}(a_{j,l} b_{l,r}) ≤ C max_{1≤l≤n}(N_{A}(a_{j,l}) N_{A}(b_{l,r})) (3.3.15)

for every j, r = 1, …, n, using (3.2.1) in the second step. It follows that N_{∞}(c) ≤ C N_{∞}(a) N_{∞}(b). (3.3.16)

3.4 Continuity of inverses

Let k be a field with an absolute value function | · |, and let A be an associative algebra over k with a multiplicative identity element e. Also let N_{A} be a seminorm on A as a vector space over k, with respect to | · | on k, and suppose that N_{A} satisfies the boundedness condition (3.2.1) with constant C ≥ 0. If a is an invertible element of A, then N_{A}(e) ≤ C N_{A}(a) N_{A}(a^{−1}). (3.4.1)

56 CHAPTER 3. SUBMULTIPLICATIVITY AND INVERTIBILITY

Let x, y be invertible elements of A, and observe that x^{−1} − y^{−1} = x^{−1} (y y^{−1}) − (x^{−1} x) y^{−1} = x^{−1} (y − x) y^{−1}. (3.4.2) It follows that N_{A}(x^{−1} − y^{−1}) ≤ C^{2} N_{A}(x^{−1}) N_{A}(y^{−1}) N_{A}(x − y). (3.4.3)

Hence

(3.4.4) N_{A}(y^{−1}) ≤ N_{A}(x^{−1}) + N_{A}(x^{−1} − y^{−1})
≤ N_{A}(x^{−1}) + C^{2} N_{A}(x^{−1}) N_{A}(y^{−1}) N_{A}(x − y),

so that (1 − C^{2} N_{A}(x^{−1}) N_{A}(x − y)) N_{A}(y^{−1}) ≤ N_{A}(x^{−1}). (3.4.5) If C^{2} N_{A}(x^{−1}) N_{A}(x − y) < 1, (3.4.6) then we get that N_{A}(y^{−1}) ≤ (1 − C^{2} N_{A}(x^{−1}) N_{A}(x − y))^{−1} N_{A}(x^{−1}). (3.4.7) Combining this with (3.4.3), we obtain that N_{A}(x^{−1} − y^{−1}) ≤ C^{2} (1 − C^{2} N_{A}(x^{−1}))^{−1} N_{A}(x^{−1})^{2} N_{A}(x − y) (3.4.8) when (3.4.6) holds.

Suppose for the moment that N_{A} is a semi-ultranorm on A, as a vector space over k. Let us check that N_{A}(x^{−1}) = N_{A}(y^{−1}) (3.4.9) when (3.4.6) holds. Of course, this is trivial when N_{A} ≡ 0 on A. Otherwise, if N_{A} ≢ 0 on A, then N_{A}(e) > 0, and hence N_{A}(y^{−1}) > 0, by (3.4.1). In this case, (3.4.6) implies that C^{2} N_{A}(x^{−1}) N_{A}(y^{−1}) N_{A}(x − y) < N_{A}(y^{−1}). (3.4.10) Combining this with (3.4.3), we get that N_{A}(x^{−1} − y^{−1}) < N_{A}(y^{−1}). (3.4.11) This implies (3.4.9) in this situation, as in (1.8.7). It follows that N_{A}(x^{−1} − y^{−1}) ≤ C^{2} N_{A}(x^{−1})^{2} N_{A}(x − y) (3.4.12) when (3.4.6) holds, by (3.4.3) and (3.4.9).

Let us now take x = e, for which there are some simplifications. If y is an invertible element of A, then e − y^{−1} = (y − e) y^{−1}, (3.4.13)

3.5. BANACH ALGEBRAS 57 so that N_{A}(e − y^{−1}) ≤ C N_{A}(y^{−1}) N_{A}(y − e). (3.4.14) This implies that N_{A}(y^{−1}) ≤ N_{A}(e) + N_{A}(e − y^{−1}) ≤ N_{A}(e) + C N_{A}(y^{−1}) N_{A}(y − e), (3.4.15) and hence (1 − C N_{A}(y − e)) N_{A}(y^{−1}) ≤ N_{A}(e). (3.4.16) If C N_{A}(y − e) < 1, (3.4.17) then it follows that N_{A}(y^{−1}) ≤ (1 − C N_{A}(y − e))^{−1} N_{A}(e). (3.4.18) This implies that N_{A}(e − y^{−1}) ≤ C (1 − C N_{A}(y − e))^{−1} N_{A}(e) N_{A}(y − e) (3.4.19) when (3.4.17) holds, because of (3.4.14).

Suppose now that N_{A} is a semi-ultranorm on A again, and let us verify that N_{A}(y^{−1}) = N_{A}(e) (3.4.20) when (3.4.17) holds. This is trivial when N_{A} ≡ 0 on A, as before. Suppose instead that N_{A} ≢ 0 on A, so that N_{A}(e) > 0, and thus N_{A}(y^{−1}) > 0. We can multiply both sides of (3.4.17) by N_{A}(y^{−1}), to get that C N_{A}(y^{−1}) N_{A}(y − e) < N_{A}(y^{−1}). (3.4.21) This implies that N_{A}(e − y^{−1}) < N_{A}(y^{−1}), (3.4.22) because of (3.4.14). This permits us to obtain (3.4.20) using (1.8.7), as before. Hence N_{A}(e − y^{−1}) ≤ C N_{A}(e) N_{A}(y − e) (3.4.23) when (3.4.17) holds, by (3.4.14) and (3.4.20).

3.5 Banach algebras

Let k be a field with an absolute value function | · |, and let A be an associative algebra over k. Also let ∥ · ∥ be a norm on A with respect to | · | on k such that ∥x y∥ ≤ C ∥x∥ ∥y∥ (3.5.1) for some C ≥ 0 and every x, y ∈ A. In this section, we ask that A be complete with respect to the metric associated to ∥ · ∥. Otherwise, one can pass to a completion of A, as usual. If (3.5.1) holds with C = 1, then A is said to be a Banach algebra with respect to ∥ · ∥.

Let us suppose too that A has a nonzero multiplicative identity element e, which is sometimes included in the definition of a Banach algebra. The condition

∥e∥ = 1 (3.5.2)

is sometimes included in the definition of a Banach algebra as well. Let a ∈ A be given, and remember that

(e − a) ∑_{j=0}^{n} a^{j} = ( ∑_{j=0}^{n} a^{j}) (e − a) = e − a^{n+1} (3.5.3)

for every nonnegative integer n, as in (3.1.3). Observe that

∥a^{j}∥ ≤ C^{j−1} ∥a∥^{j} (3.5.4)

for every positive integer j, by (3.5.1). Suppose that

C ∥a∥ < 1, (3.5.5)

so that

lim_{j→∞} ∥a^{j}∥ → 0, (3.5.6)

by (3.5.4). We also get that

∑_{j=0}^{∞} ∥a^{j}∥ (3.5.7)

converges as an infinite series of nonnegative real numbers, because

∑_{j=0}^{∞} C^{j} ∥a∥^{j} (3.5.8)

is a convergent geometric series. This means that ∑_{j=0}^{∞} a^{j} converges absolutely with respect to ∥ · ∥, and hence that ∑_{j=0}^{∞} a^{j} converges in A, because A is complete with respect to the metric associated to ∥ · ∥. The value of this sum satisfies

(e − a) ∑_{j=0}^{∞} a^{j} = ( ∑_{j=0}^{∞} a^{j}) (a − e) = e, (3.5.9)

by taking the limit as n → ∞ in (3.5.3). Thus e − a is invertible in A, with

(e − a)^{−1} = ∑_{j=0}^{∞} a^{j}. (3.5.10)

Let x be an invertible element of A, and let y be another element of A. Observe that y = x − (x − y) = x (e − x^{−1} (x − y)). (3.5.11)

Suppose that C^{2} ∥x^{−1}∥ ∥x − y∥ < 1, (3.5.12)

so that C ∥x^{−1} (x − y)∥ ≤ C^{2} ∥x^{−1}∥ ∥x − y∥ < 1. (3.5.13)

This implies that e−x^{−1} (x−y) is invertible in A, as in the preceding paragraph. It follows that y is invertible in A, by (3.5.11).

Let a be an element of A again, and let j_{0} be a positive integer. If e − a^{j_{0}} is invertible in A, then e − a is invertible in A, because of (3.5.3), with n = j_{0} − 1. In particular, this holds when

C ∥a^{j_{0}}∥ < 1, (3.5.14)

as before. Alternatively, one can use (3.5.1) to estimate ∥a^{j_{0} l+r}∥ in terms of C^{l−1} ∥a^{j_{0}}∥^{l} when l ≥ 1 and 0 ≤ r < j_{0}, to get that (3.5.6) holds and that (3.5.7) converges when (3.5.14) holds. This implies that ∑_{j=0}^{∞} a^{j} converges in A and satisfies (3.5.9) when (3.5.14) holds, as before.

3.6 Invertible linear mappings

Let k be a field with an absolute value function | · |, and let V, W be vector spaces over k with seminorms N_{V}, N_{W}, respectively, with respect to | · | on k. If T is a one-to-one linear mapping from V onto W, then the corresponding inverse mapping T^{−1} is a linear mapping from W onto V. As usual, T^{−1} is bounded with respect to N_{W}, N_{V} if there is a nonnegative real number C such that N_{V} (T^{−1}(w)) ≤ C N_{W} (w) (3.6.1)

for every w ∈ W. This is the same as saying that

N_{V} (v) ≤ C N_{W} (T (v)) (3.6.2)

for every v ∈ V.

Now let T be a linear mapping from V into W, and suppose that (3.6.2) holds for some C ≥ 0. If N_{V} is a norm on V, then it follows that T is injective on V. Let T_{1} be another linear mapping from V into W, and observe that

N_{V} (v) ≤ C N_{W} (T_{1}(v)) + C N_{W} (T_{1}(v) − T (v)) (3.6.3)

for every v ∈ V. If T_{1} − T is bounded as a linear mapping from V into W, then we get that

N_{V} (v) ≤ C N_{W} (T_{1}(v)) + C ∥T_{1} − T ∥_{op,V W} N_{V} (v) (3.6.4)

for every v ∈ V, where the operator seminorm ∥ · ∥_{op,V W} is as defined in Section 1.9. Thus (1 − C ∥T_{1} − T ∥_{op,V W}) N_{V} (v) ≤ C N_{W} (T_{1}(v)) (3.6.5) for every v ∈ V. If C ∥T_{1} − T ∥_{op,V W} < 1, (3.6.6)

then it follows that

N_{V} (v) ≤ C (1 − C ∥T_{1} − T ∥_{op,V W})^{−1} N_{W} (T_{1}(v)) (3.6.7)

for every v ∈ V.

Suppose that N_{W} is a semi-ultranorm on W with respect to | · | on k. In this case, we have that

N_{V} (v) ≤ C max(N_{W} (T_{1}(v)), N_{W} (T_{1}(v) − T (v))) (3.6.8)

for every v ∈ V, by (3.6.2). Suppose for the moment that

C N_{W} (T_{1}(v) − T (v)) < N_{V} (v) (3.6.9)

for every v ∈ V with N_{V} (v) > 0. It follows that

N_{V} (v) ≤ C N_{W} (T_{1}(v)) (3.6.10)

for every v ∈ V with N_{V} (v) > 0. Of course, (3.6.10) holds trivially when N_{V} (v) = 0, so that (3.6.10) holds for all v ∈ V. If T_{1} − T is bounded as a linear mapping from V into W, then

N_{W} (T_{1}(v) − T (v)) ≤ ∥T_{1} − T ∥_{op,V W} N_{V} (v) (3.6.11)

for every v ∈ V. If (3.6.6) holds, then we get that (3.6.9) holds when N_{V} (v) > 0.

A bounded linear mapping T from V onto W is said to be invertible as a bounded linear mapping if T is a one-to-one mapping from V onto W whose inverse T^{−1} is bounded with respect to N_{W}, N_{V}. The bounded linear mappings from V onto itself with bounded inverses are the same as the invertible elements of BL(V), as an algebra with respect to composition of mappings.

Suppose that N_{V} and N_{W} are norms on V and W, respectively, and that T is a bounded linear mapping from V into W that satisfies (3.6.2) for some C ≥ 0. If V is complete with respect to the metric associated to N_{V}, then it is easy to see that the image T (V) of V under T is complete with respect to the restriction of the metric on W associated to N_{W} to T (V). This implies that T (V) is a closed set in W with respect to the metric associated to N_{W}, by a standard argument.

3.7 Isometric linear mappings

Let k be a field with an absolute value function | · | again, and let V, W be vector spaces over k with seminorms N_{V}, N_{W}, respectively, with respect to | · | on k. A linear mapping T from V into W is said to be an isometry with respect to N_{V} and N_{W} if N_{W} (T (v)) = N_{V} (v) (3.7.1) for every v ∈ V. Of course, this is the same as saying that

N_{W} (T (v)) ≤ N_{V} (v) (3.7.2)

and N_{V} (v) ≤ N_{W} (T (v)) (3.7.3)

for every v ∈ V. The first condition (3.7.2) means that T is a bounded linear mapping from V into W with respect to N_{V} and N_{W}, with

∥T ∥_{op,V W} ≤ 1, (3.7.4)

where the operator seminorm is as defined in Section 1.9. The second condition (3.7.3) is the same as (3.6.2), with C = 1.

Let Z be another vector space over k with a seminorm N_{Z} with respect to |·| on k. Also let T_{1} be an isometric linear mapping from V into W with respect to N_{V} and N_{W}, and let T_{2} be an isometric linear mapping from W into Z with respect to N_{W} and N_{Z}. Observe that

N_{Z}((T_{2} ∘ T_{1})(v)) = N_{Z}(T_{2}(T_{1}(v))) = N_{W} (T_{1}(v)) = N_{V} (v) (3.7.5)

for every v ∈ V, so that T_{2} ∘ T_{1} is an isometric linear mapping from V into Z.

If T is a one-to-one linear mapping from V onto W, then (3.7.3) is equivalent to saying that T^{−1} is a bounded linear mapping from W into V with respect to N_{W} and N_{V}, with ∥T^{−1}∥_{op,W V} ≤ 1, (3.7.6)

as in the previous section. Thus T is an isometric linear mapping if and only T and T^{−1} are bounded linear mappings that satisfy (3.7.4) and (3.7.6). In particular, T is an isometric linear mapping if and only if T^{−1} is an isometric linear mapping. Of course, the identity mapping on V is an isometric linear mapping from V onto itself with respect to N_{V}. The collection of one-to-one isometric linear mappings from V onto itself is a group with respect to composition of mappings.

Suppose now that N_{W} is a semi-ultranorm on W with respect to | · | on k. Let T be a linear mapping from V into W that satisfies (3.7.3) for every v ∈ V. Let T_{1} be another linear mapping from V into W such that

N_{W} (T_{1}(v) − T (v)) < N_{V} (v) (3.7.7)

for every v ∈ V with N_{V} (v) > 0. Under these conditions, we have that

N_{V} (v) ≤ N_{W} (T_{1}(v)) (3.7.8)

for every v ∈ V, as in (3.6.10), with C = 1.

Let T be an isometric linear mapping from V into W, and let T_{1} be a bounded linear mapping from V into W. If

∥T_{1} − T ∥_{op,V W} ≤ 1, (3.7.9) then ∥T_{1}∥_{op,V W} ≤ 1, (3.7.10)

because of (3.7.4) and the hypothesis that N_{W} be a semi-ultranorm on W. If

∥T_{1} − T ∥_{op,V W} < 1, (3.7.11)

then (3.7.7) holds for every v ∈ V with N_{V} (v) > 0, so that T_{1} satisfies (3.7.8), as before. This shows that T_{1} is also an isometric linear mapping from V into W when (3.7.11) holds.

3.8 Hilbert space isometries

Let (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) be inner product spaces, both real or both complex, and let ∥ · ∥_{V} and ∥ · ∥_{W} be the corresponding norms on V and W, respectively, as in Section 1.11. If a linear mapping T from V into W satisfies

⟨T (u), T (v)⟩_{W} = ⟨u, v⟩_{V} (3.8.1)

for every u, v ∈ V, then it is easy to see that T is an isometry with respect to ∥·∥_{V} and ∥·∥_{W}, by taking u = v. Conversely, if T is an isometric linear mapping from V into W with respect to ∥ · ∥_{V} and ∥ · ∥_{W}, then one can check that T satisfies (3.8.1), using polarization identities. An isometric linear mapping from V onto W is also known as an orthogonal transformation in the real case, and a unitary transformation in the complex case. The orthogonal or unitary transformations from V onto itself form a group with respect to composition of mappings.

Suppose that V and W are Hilbert spaces, and that T is a bounded linear mapping from V into W. Let T^{∗} be the corresponding adjoint mapping from W into V, as in Section 1.11. Observe that

⟨T (u), T (v)⟩_{W} = ⟨u, T^{∗}(T (v))⟩_{V} (3.8.2)

for every u, v ∈ V. Thus T is an isometric linear mapping from V into W if and only if T^{∗} ∘ T = I_{V} , (3.8.3)

where I_{V} is the identity mapping on V. This is the same as saying that

T^{∗} = T^{−1} (3.8.4)

when T maps V onto W.

If T is any bounded linear mapping from V into W, then it is well known that ∥T^{∗} ∘ T ∥_{op,V V} = ∥T ∥_{V W}^{2} , (3.8.5)

where these operator norms are taken with respect to ∥ · ∥_{V} and ∥ · ∥_{W}, as appropriate. More precisely,

∥T^{∗} ∘ T ∥_{op,V V} ≤ ∥T ∥_{op,V W} ∥T^{∗}∥_{op,W V} = ∥T ∥_{op,V W}^{2} , (3.8.6) using (1.11.15) in the second step. We also have that

(3.8.7)  ∥T (v)∥_{W}^{2} = ⟨v, T^{∗}(T (v))⟩_{V} ≤ ∥v∥_{V} ∥T^{∗}(T (v))∥_{V}
≤ ∥T^{∗} ∘ T ∥_{op,V V} ∥v∥_{V}^{2}

for every v ∈ V, by taking u = v in (3.8.2) in the first step, and using the Cauchy–Schwarz inequality in the second step. This implies that

∥T ∥_{op,V W}^{2} ≤ ∥T^{∗} ∘ T ∥_{op,V V} , (3.8.8)

as desired.

Observe that w ∈ W satisfies T^{∗}(w) = 0 if and only if

⟨T (v), w⟩_{W} = 0 (3.8.9)

for every v ∈ V, by the definition of T^{∗}. If the image T (V) of V under T is dense in W with respect to the metric associated to ∥ · ∥_{W}, then it follows that T^{∗}(w) = 0. However, if T (V) is not dense in W, then there is a w ∈ W such that w ≠ 0 and (3.8.9) holds for every v ∈ V, by standard results about Hilbert spaces. Thus T (V) is dense in W if and only if the kernel of T^{∗} is trivial.

Suppose that ∥v∥_{V} ≤ C ∥T (v)∥_{W} (3.8.10)

for some nonnegative real number C and every v ∈ V. This implies that T (V) is a closed set in W with respect to the metric associated to ∥·∥_{W}, as in Section 3.6, because V is complete, by hypothesis. If T (V) is dense in W, then it follows that T (V) = W.

3.9 Preserving bilinear forms

Let k be a commutative ring with a multiplicative identity element, and let A_{1}, A_{2}, and C be modules over k. Also let β_{1}, β_{2} be bilinear mappings from A_{1} ×A_{1} and A_{2} × A_{2} into C, respectively, as in Section 2.12. Let us say that a module homomorphism ϕ from A_{1} into A_{2} preserves these bilinear mappings if

β_{2}(ϕ(a_{1}), ϕ(b_{1})) = β_{1}(a_{1}, b_{1}) (3.9.1)

for every a_{1}, b_{1} ∈ A_{1}. If ϕ is a one-to-one mapping from A_{1} onto A_{2}, then (3.9.1) is equivalent to asking that

β_{1}(ϕ^{−1}(a_{2}), ϕ^{−1}(b_{2})) = β_{2}(a_{2}, b_{2}) (3.9.2)

for every a_{2}, b_{2} ∈ A_{2}. Of course, this is the same as saying that ϕ^{−1} preserves β_{2}, β_{1}.

Let A_{3} be another module over k, and let β_{3} be a bilinear mapping from A_{3} × A_{3} into C. Suppose that ϕ_{1} is a module homomorphism from A_{1} into A_{2} that preserves β_{1}, β_{2}, and that ϕ_{2} is a module homomorphism from A_{2} into A_{3} that preserves β_{2}, β_{3}. This implies that

β_{3}(ϕ_{2}(ϕ_{1}(a_{1})), ϕ_{2}(ϕ_{1}(b_{1}))) = β_{2}(ϕ_{1}(a_{1}), ϕ_{1}(b_{1})) = β_{1}(a_{1}, b_{1}) (3.9.3)

for every a_{1}, b_{1} ∈ A_{1}, so that ϕ_{2} ∘ ϕ_{1} is a module homomorphism from A_{1} into A_{3} that preserves β_{1}, β_{3}.

Let ϕ be a module homomorphism from A_{1} into A_{2}, and suppose that

β_{2}(ϕ(a), ϕ(a)) = β_{1}(a, a) (3.9.4)

for every a ∈ A_{1}. If a, b ∈ A_{1}, then

β_{1}(a + b, a + b) = β_{1}(a, a) + β_{1}(a, b) + β_{1}(b, a) + β_{1}(b, b) (3.9.5)

and

(3.9.6) β_{2}(ϕ(a + b), ϕ(a + b)) = β_{2}(ϕ(a) + ϕ(b), ϕ(a) + ϕ(b))
= β_{2}(ϕ(a), ϕ(a)) + β_{2}(ϕ(a), ϕ(b))
+β_{2}(ϕ(b), ϕ(a)) + β_{2}(ϕ(b), ϕ(b)).

It follows that

β_{2}(ϕ(a), ϕ(b)) + β_{2}(ϕ(b), ϕ(a)) = β_{1}(a, b) + β_{1}(b, a) (3.9.7)

for every a, b ∈ A_{1}. If β_{1} and β_{2} are symmetric bilinear mappings, and if 1 + 1 is invertible in k, then we get that ϕ preserves β_{1}, β_{2}. Of course, (3.9.4) holds when ϕ preserves β_{1}, β_{2}.

Let A be a module over k, and remember that the space Hom_{k}(A, A) of module homomorphisms from A into itself is an associative algebra over k with respect to composition of mappings. If ϕ is a one-to-one module homomorphism from A onto itself, then ϕ^{−1} is a module homomorphism from A into itself as well. In this case, ϕ may be called an module automorphism of A. The module automorphisms of A are the same as the invertible elements of Hom_{k}(A, A), and form a group with respect to composition of mappings.

A bilinear mapping β from A × A into C is said to be invariant under an module automorphism ϕ on A if

β(ϕ(a), ϕ(b)) = β(a, b) (3.9.8)

for every a, b ∈ A, which is the same as saying that ϕ preserves β as a bilinear mapping from A × A into C for both the domain and range. The identity mapping on A obviously has this property. The collection of module automorphisms of A that preserve β is a subgroup of the group of all module automorphisms of A.

3.10 Preserving nondegenerate bilinear forms

Let k be a field, let V_{1}, V_{2} be vector spaces over k, and let b_{1}, b_{2} be bilinear forms on V_{1}, V_{2}, respectively. Suppose that V_{1} has finite dimension, and that b_{1} is nondegenerate on V_{1}, as in Section 2.14. Let T be a linear mapping from V_{1} into V_{2}. If w_{2} ∈ V_{2}, then b_{2}(T (v_{1}), w_{2}) defines a linear functional on V_{1}, as a function of v_{1}. This implies that there is a unique element T^{∗}(w_{2}) of V_{1} such that b_{2}(T (v_{1}), w_{2}) = b_{1}(v_{1}, T^{∗}(w_{2})) (3.10.1)

for every v_{1} ∈ V_{1}. This defines a linear mapping T^{∗} from V_{2} into V_{1}, which is the adjoint of T with respect to b_{1}, b_{2}. The mapping T ↦ T^{∗} is linear as a mapping from the space L(V_{1}, V_{2}) of linear mappings from V_{1} into V_{2} into the corresponding space L(V_{2}, V_{1}).

Let V_{3} be another vector space over k with a bilinear form b_{3}, and suppose that V_{2} also has finite dimension, and that b_{2} is nondegenerate on V_{2}. If T_{1} is a linear mapping from V_{1} into V_{2}, and T_{2} is a linear mapping from V_{2} into V_{3}, then their adjoints T_{1}^{∗} and T_{2}^{∗} can be defined as in the preceding paragraph. Similarly, T_{2} ∘ T_{1} is a linear mapping from V_{1} into V_{3}, whose adjoint can be defined as in the previous paragraph as well. Observe that

b_{3}((T_{2} ∘ T_{1})(v_{1}), w_{3}) = b_{3}(T_{2}(T_{1}(v_{1})), w_{3}) = b_{2}(T_{1}(v_{1}), T_{2}^{∗}(w_{3}))
(3.10.2)  = b_{1}(v_{1}, T_{1}^{∗}(T_{2}^{∗}(w_{3}))) = b_{1}(v_{1}, (T_{1}^{∗} ∘ T_{2}^{∗})(w_{3}))

for every v_{1} ∈ V_{1} and w_{3} ∈ V_{3}. This implies that

(T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗} (3.10.3)

as linear mappings from V_{3} into V_{1}.

Let us continue to suppose for the moment that V_{2} has finite dimension, and that b_{2} is nondegenerate on V_{2}. If T is a linear mapping from V_{1} into V_{2} and T^{∗} = 0 on V_{2}, then T = 0 on V_{1}, because of (3.10.1) and the nondegeneracy of b_{2} on V_{2}. This implies that T ↦ T^{∗} is a one-to-one mapping from L(V_{1}, V_{2}) onto L(V_{2}, V_{1}), because L(V_{1}, V_{2}) and L(V_{2}, V_{1}) are finite-dimensional vector spaces over k with the same dimension. Let T be a linear mapping from V_{1} into V_{2} again, and note that the adjoint (T^{∗})^{∗} of the adjoint T^{∗} of T can be defined as a linear mapping from V_{1} into V_{2} in the same way in this situation. If b_{1} and b_{2} are symmetric on V_{1} and V_{2}, respectively, then

(3.10.4)  b_{2}(T (v_{1}), w_{2}) = b_{1}(T^{∗}(w_{2}), v_{1}) = b_{2}(w_{2}, (T^{∗})^{∗}(v_{1}))
= b_{2}((T^{∗})^{∗}(v_{1}), w_{2})

for every v_{1} ∈ V_{1} and w_{2} ∈ V_{2}. Similarly, if b_{1} and b_{2} are antisymmetric on V_{1} and V_{2}, respectively, then

(3.10.5) b_{2}(T (v_{1}), w_{2}) = −b_{1}(T^{∗}(w_{2}), v_{1}) = −b_{2}(w_{2}, (T^{∗})^{∗}(v_{1}))
= b_{2}((T^{∗})^{∗}(v_{1}), w_{2})

66 CHAPTER 3. SUBMULTIPLICATIVITY AND INVERTIBILITY

for every v_{1} ∈ V_{1} and w_{2} ∈ V_{2}. In both cases, it follows that (T^{∗})^{∗} = T. (3.10.6) If one of b_{1} and b_{2} is symmetric, and the other is antisymmetric, then (T^{∗})^{∗} = −T, (3.10.7) by the analogous argument.

If T is any linear mapping from V_{1} into V_{2}, then b_{2}(T (v_{1}), T (w_{1})) = b_{1}(v_{1}, T^{∗}(T (w_{1}))) (3.10.8) for every v_{1}, w_{1} ∈ V_{1}, by the definition of T^{∗}. Thus T preserves b_{1}, b_{2}, as in the previous section, if and only if T^{∗} ∘ T = I_{V_{1}}, (3.10.9) where I_{V_{1}} is the identity mapping on V_{1}. In particular, this implies that T is injective, which could also be obtained more directly from the nondegeneracy of b_{1} on V_{1}. If V_{2} has the same dimension as V_{1}, then it follows that T maps V_{1} onto V_{2}. In this case, we get that T^{∗} = T^{−1}. (3.10.10)

3.11 Preserving sesquilinear forms

Let V_{1}, V_{2} be vector spaces over the complex numbers, and let b_{1}, b_{2} be sesquilinear forms on V_{1}, V_{2}, respectively. Let us say that a linear mapping T from V_{1} into V_{2} preserves b_{1}, b_{2} if b_{2}(T (v_{1}), T (w_{1})) = b_{1}(v_{1}, w_{1}) (3.11.1) for every v_{1}, w_{1} ∈ V_{1}. If T is a one-to-one linear mapping from V_{1} onto V_{2}, then this is the same as saying that b_{1}(T^{−1}(v_{2}), T^{−1}(w_{2})) = b_{2}(v_{2}, w_{2}) (3.11.2) for every v_{2}, w_{2} ∈ V_{2}, which means that T^{−1} preserves b_{2}, b_{1}.

Let V_{3} be another complex vector space with a sesquilinear form b_{3}. If T_{1} is a linear mapping from V_{1} into V_{2} that preserves b_{1}, b_{2}, and T_{2} is a linear mapping from V_{2} into V_{3} that preserves b_{2}, b_{3}, then it is easy to see that their composition T_{2} ∘ T_{1} preserves b_{1}, b_{3}.

Let T be a linear mapping from V_{1} into V_{2} that satisfies b_{2}(T (u_{1}), T (u_{1})) = b_{1}(u_{1}, u_{1}) (3.11.3) for every u_{1} ∈ V. One can check that T satisfies (3.11.1) for every v_{1}, w_{1} ∈ V_{1}, by applying (3.11.3) to u_{1} = v_{1} + w_{1} and to u_{1} = v_{1} + i w_{1}. Of course, (3.11.1) implies (3.11.3), by taking v_{1}, w_{1} = u_{1}.

Let V be a complex vector space, and let T be a one-to-one linear mapping from V onto itself. A sesquilinear form b on V is said to be invariant under T if b(T (v), T (w)) = b(v, w) (3.11.4)

for every v, w ∈ V, which is to say that T preserves b as a sesquilinear form on both the domain and range. The collection of one-to-one linear mappings from V onto itself that preserve b is a group with respect to composition.

Let V_{1}, V_{2} be complex vector spaces again, and let b_{1}, b_{2} be sesquilinear forms on them, respectively. Let us suppose for the rest of the section that V_{1} has finite dimension, and that b_{1} is nondegenerate on V_{1}, as in Section 2.15. Let T be a linear mapping from V_{1} into V_{2}, and let w_{2} ∈ V_{2} be given. Thus b_{2}(T (v_{1}), w_{2}) is a linear functional on V_{1}, as a function of v_{1}, so that there is a unique element T^{∗}(w_{2}) of V_{1} such that

b_{2}(T (v_{1}), w_{2}) = b_{1}(v_{1}, T^{∗}(w_{2})) (3.11.5)

for every v_{1} ∈ V_{1}. One can check that T^{∗} is a linear mapping from V_{2} into V_{1}, and that the mapping from T to its adjoint T^{∗} is conjugate-linear as a mapping from L(V_{1}, V_{2}) into L(V_{2}, V_{1}).

Let V_{3} be another complex vector space with a sesquilinear form b_{3}, and suppose that V_{2} has finite dimension, and that b_{2} is nondegenerate on b_{2}. If T_{1} is a linear mapping from V_{1} into V_{2}, and T_{2} is a linear mapping from V_{2} into V_{3}, then T_{2} ∘ T_{1} is a linear mapping from V_{1} into V_{3}, and the adjoints of T_{1}, T_{2}, and T_{3} can be defined as in the preceding paragraph. Under these conditions, one can verify that (T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗}, (3.11.6)

as linear mappings from V_{3} into V_{1}.

Let us continue to ask for the moment that V_{2} have finite dimension, and that b_{2} be nondegenerate on V_{2}. If T is a linear mapping from V_{1} into V_{2} such that T^{∗} = 0 on V_{2}, then T = 0 on V_{1}, because of (3.11.5) and the nondegeneracy of b_{2} on V_{2}. It follows that T ↦ T^{∗} is a one-to-one mapping from L(V_{1}, V_{2}) onto L(V_{2}, V_{1}), because L(V_{1}, V_{2}) and L(V_{2}, V_{1}) are finite-dimensional vector spaces over C with the same dimension. Let T be a linear mapping from V_{1} into V_{2} again, so that the adjoint (T^{∗})^{∗} of T^{∗} is defined as a linear mapping from V_{1} into V_{2}. Suppose that b_{1} and b_{2} are Hermitian-symmetric on V_{1} and V_{2}, respectively. Under these conditions, we have that

(3.11.7)  b_{2}(T (v_{1}), w_{2}) = \overline{b_{1}(T^{∗}(w_{2}), v_{1})} = \overline{b_{2}(w_{2}, (T^{∗})^{∗}(v_{1}))}
= b_{2}((T^{∗})^{∗}(v_{1}), w_{2})

for every v_{1} ∈ V_{1} and w_{2} ∈ V_{2}. This implies that

(T^{∗})^{∗} = T (3.11.8)

As before, b_{2}(T (v_{1}), T (w_{1})) = b_{1}(v_{1}, T^{∗}(T (w_{1}))) (3.11.9)

for every linear mapping T from V_{1} into V_{2} and v_{1}, w_{1} ∈ V_{1}. This implies that T preserves b_{1}, b_{2} if and only if

T^{∗} ∘ T = I_{V_{1}}, (3.11.10)

because b_{1} is nondegenerate on V_{1}. Note that T is injective in this case. If T maps V_{1} onto V_{2}, then (3.11.10) is the same as saying that

T^{∗} = T^{−1}. (3.11.11)

Of course, surjectivity of T follows from injectivity when V_{2} has the same dimension as V_{1}.

3.12 Bilinear forms and matrices

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. The space k^{n} of n-tuples of elements of k is a (free) module over k with respect to coordinatewise addition and scalar multiplication. Let C be another module over k, and let (β_{j,l}) be an n × n matrix with entries in C. Put

β(x, y) = ∑_{j=1}^{n} ∑_{l=1}^{n} β_{j,l} x_{l} y_{j} (3.12.1)

for every x, y ∈ k^{n}, where the terms of the sum are defined using multiplication on k and scalar multiplication on C. This defines a mapping from k^{n} × k^{n} into C that is bilinear over k, and it is easy to see that every bilinear mapping from k^{n} × k^{n} into C can be expressed as (3.12.1) in a unique way.

Observe that (3.12.1) is symmetric as a bilinear mapping from k^{n} × k^{n} into C if and only if (β_{j,l}) is symmetric as a matrix, which is to say that

β_{l,j} = β_{j,l} (3.12.2)

for every j, l = 1, …, n. Similarly, (3.12.1) is antisymmetric as a bilinear mapping from k^{n} × k^{n} into C if and only if (β_{j,l}) is antisymmetric as a matrix, in the sense that β_{l,j} = −β_{j,l} (3.12.3)

for every j, l = 1, …, n. Remember that (3.12.1) is antisymmetric as a bilinear mapping from k^{n} × k^{n} into C when

β(x, x) = 0 (3.12.4)

for every x ∈ k^{n}, as in Section 2.1. In this situation, one can check that (3.12.4) holds for every x ∈ k^{n} if and only if (β_{j,l}) is antisymmetric and

β_{j,j} = 0 (3.12.5) for every j = 1, …, n. If 1 + 1 is invertible in k, then (3.12.3) implies (3.12.5), by taking j = l.

Let a = (a_{j,l}) be an n × n matrix with entries in k. If x ∈ k^{n}, then let T_{a}(x) be the element of k^{n} whose jth coordinate is given by

(T_{a}(x))_{j} = ∑_{l=1}^{n} a_{j,l} x_{l} (3.12.6)

for each j = 1, …, n. This defines a module homomorphism from k^{n} into itself, and every module homomorphism from k^{n} into itself corresponds to a unique a ∈ M_{n}(k) in this way. More precisely, a ↦ T_{a} is an isomorphism from M_{n}(k) as an algebra over k with respect to matrix multiplication onto the algebra Hom_{k}(k^{n}, k^{n}) of module homomorphisms from k^{n} into itself with respect to composition of mappings. This corresponds to some of the remarks in Section 2.10 when k is a field.

Let a ∈ M_{n}(k) be given again, and observe that

β(T_{a}(x), y) = ∑_{j=1}^{n} ∑_{l=1}^{n} ∑_{r=1}^{n} β_{r,j} a_{j,l} x_{l} y_{r} (3.12.7)

for every x, y ∈ k^{n}, where the terms of the sum are again defined using multiplication on k and scalar multiplication on C. Similarly,

β(x, T_{a}(y)) = ∑_{j=1}^{n} ∑_{l=1}^{n} ∑_{r=1}^{n} β_{j,l} a_{j,r} x_{l} y_{r} (3.12.8)

for every x, y ∈ k^{n}. It follows that T_{a} is symmetric with respect to β, as in Section 2.12, if and only if

∑_{j=1}^{n} β_{r,j} a_{j,l} = ∑_{j=1}^{n} β_{j,l} a_{j,r} (3.12.9)

for every l, r = 1, …, n. Similarly, T_{a} is antisymmetric with respect to β if and only if

∑_{j=1}^{n} β_{r,j} a_{j,l} = − ∑_{j=1}^{n} β_{j,l} a_{j,r} (3.12.10)

for every l, r = 1, …, n.

We also have that

β(T_{a}(x), T_{a}(y)) = ∑_{j=1}^{n} ∑_{l=1}^{n} ∑_{m=1}^{n} ∑_{r=1}^{n} β_{m,j} a_{j,l} a_{m,r} x_{l} y_{r} (3.12.11)

for every x, y ∈ k^{n}, where the terms of the sum are defined using multiplication on k and scalar multiplication on C. Thus T_{a} preserves β, as in Section 3.9, if and only if

∑_{j=1}^{n} ∑_{m=1}^{n} β_{m,j} a_{j,l} a_{m,r} = β_{r,l} (3.12.12)

70 CHAPTER 3. SUBMULTIPLICATIVITY AND INVERTIBILITY for every l, r = 1, …, n. Note that T_{a} is a module automorphism of k^{n} exactly when a is invertible in M_{n}(k).

Let b ∈ M_{n}(k) be given, so that T_{b} can be defined as before, and

β(x, T_{b}(y)) = ∑_{j=1}^{n} ∑_{l=1}^{n} ∑_{r=1}^{n} β_{j,l} b_{j,r} x_{l} y_{r} (3.12.13)

for every x, y ∈ k^{n}, as in (3.12.8). Comparing this with (3.12.7), we get that β(T_{a}(x), y) = β(x, T_{b}(y)) (3.12.14) for every x, y ∈ k^{n} if and only if

∑_{j=1}^{n} β_{r,j} a_{j,l} = ∑_{j=1}^{n} β_{j,l}b_{j,r} (3.12.15)

for every l, r = 1, …, n.

The product of an n × n matrix with entries in C and an n × n matrix with entries in k, in either order, can be defined as an n × n matrix with entries in C in the usual way. Let us also use β to denote (β_{j,l}), as an element of M_{n}(C). Thus (3.12.9) is the same as saying that β a = a^{t} β (3.12.16) as elements of M_{n}(C), where a^{t} is the transpose of a, as in Section 2.8. Similarly, (3.12.10) is the same as saying that β a = −a^{t} β (3.12.17) as elements of M_{n}(C). We can reexpress (3.12.12) as a^{t} β a = β, (3.12.18) and (3.12.15) as β a = b^{t} β. (3.12.19)

Let us now take C = k, as a module over itself with respect to multiplication on k. If β is invertible in M_{n}(k), then (3.12.19) is the same as saying that b^{t} = β^{−1} a β. (3.12.20) If k is a field, then the invertibility of β in M_{n}(k) is equivalent to the nondegeneracy of the corresponding bilinear form (3.12.1) on k^{n} as a vector space over k, as in Section 2.14. In this case, (3.12.20) characterizes T_{b} as the adjoint of T_{a} with respect to (3.12.1).

3.13 Sesquilinear forms and matrices

Let n be a positive integer, so that the space C^{n} of n-tuples of complex numbers is a vector space over C with respect to coordinatewise addition and scalar multiplication, as usual. Also let (β_{j,l}) be an n × n matrix with entries in C, and put

β(z, w) = ∑_{j=1}^{n} ∑_{l=1}^{n} β_{j,l} z_{l} \overline{w_{j}} (3.13.1)

for every z, w ∈ C^{n}. This defines a sesquilinear form on C^{n}, and one can check that every sesquilinear form on C^{n} corresponds to a unique matrix (β_{j,l}) in this way. Note that (3.13.1) is Hermitian-symmetric on C^{n} if and only if

β_{l,j} = \overline{β_{j,l}} (3.13.2)

for every j, l = 1, …, n.

If a = (a_{j,l}) ∈ M_{n}(C) and z ∈ C^{n}, then let T_{a}(z) be the element of C^{n} whose jth coordinate is given by

(T_{a}(z))_{j} = ∑_{l=1}^{n} a_{j,l} z_{l} (3.13.3)

for each j = 1, …, n, as before. This defines a linear mapping from C^{n} into itself, and a ↦ T_{a} is an isomorphism from M_{n}(C) as an algebra over C with respect to matrix multiplication onto the algebra L(C^{n}) of linear mappings from C^{n} into itself with respect to composition of mappings, as in Section 2.10 and the previous section. Remember that a^{∗} = ((a^{∗})_{j,l}) ∈ M_{n}(C) is defined for each a ∈ M_{n}(C) by (a^{∗})_{j,l} = \overline{a_{l,j}}, (3.13.4)

as in Section 2.8, and that a ↦ a^{∗} is a conjugate-linear involution on M_{n}(C). Let a ∈ M_{n}(C) be given, and observe that

β(T_{a}(z), w) = ∑_{j=1}^{n} ∑_{l=1}^{n} ∑_{r=1}^{n} β_{r,j} a_{j,l} z_{l} \overline{w_{r}} (3.13.5)

for every z, w ∈ C^{n}. Similarly,

β(z, T_{a}(w)) = ∑_{j=1}^{n} ∑_{l=1}^{n} ∑_{r=1}^{n} β_{j,l} \overline{a_{j,r}} z_{l} \overline{w_{r}} (3.13.6)

for every z, w ∈ C^{n}. Thus T_{a} is self-adjoint with respect to β, as in Section 2.15, if and only if

∑_{j=1}^{n} β_{r,j} a_{j,l} = ∑_{j=1}^{n} β_{j,l} \overline{a_{j,r}} (3.13.7)

72 CHAPTER 3. SUBMULTIPLICATIVITY AND INVERTIBILITY for every l, r = 1, …, n. If we also use β to denote (β_{j,l}) as an element of M_{n}(C), then (3.13.7) is the same as saying that β a = a^{∗} β. (3.13.8)

Observe that

β(T_{a}(z), T_{a}(w)) = ∑_{j=1}^{n} ∑_{l=1}^{n} ∑_{m=1}^{n} ∑_{r=1}^{n} β_{m,j} a_{j,l} \overline{a_{m,r}} z_{l} \overline{w_{r}} (3.13.9)

for every z, w ∈ C^{n}. It follows that T_{a} preserves β(·, ·), as in Section 3.11, if and only if

∑_{j=1}^{n} ∑_{m=1}^{n} β_{m,j} a_{j,l} \overline{a_{m,r}} = β_{r,l} (3.13.10)

for every l, r = 1, …, n. This is the same as saying that a^{∗} β a = β (3.13.11) as elements of M_{n}(C).

If b ∈ M_{n}(C), then T_{b} can be defined as in (3.13.3), and

β(z, T_{b}(w)) = ∑_{j=1}^{n} ∑_{l=1}^{n} ∑_{r=1}^{n} β_{j,l} \overline{b_{j,r}} z_{l} \overline{w_{r}} (3.13.12)

for every z, w ∈ C^{n}, as before. Hence β(T_{a}(z), w) = β(z, T_{b}(w)) (3.13.13) for every z, w ∈ C^{n} if and only if

∑_{j=1}^{n} β_{r,j} a_{j,l} = ∑_{j=1}^{n} β_{j,l} \overline{b_{j,r}} (3.13.14)

for every l, r = 1, …, n. This is the same as saying that β a = b^{∗} β (3.13.15) as elements of M_{n}(C). Suppose that (β_{j,l}) is invertible as an element of M_{n}(C), which is equivalent to the nondegeneracy of β(·, ·) as a sesquilinear form on C^{n}, as in Section 2.15. In this situation, (3.13.15) is the same as saying that b^{∗} = β a β^{−1}, (3.13.16) which characterizes T_{b} as the adjoint of T_{a} with respect to β(·, ·).

3.14 Invertibility and involutions

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e. Also let x ↦ x^{∗} be an opposite algebra automorphism on A, as in Section 2.6. Thus e^{∗} = e, as before. If x is an invertible element of A, then

x^{∗} (x^{−1})^{∗} = (x^{−1} x)^{∗} = e^{∗} = e (3.14.1)

and (x^{−1})^{∗} x^{∗} = (x x^{−1})^{∗} = e^{∗} = e. (3.14.2)

This implies that x^{∗} is invertible in A, with

(x^{∗})^{−1} = (x^{−1})^{∗}. (3.14.3)

Let β be an element of A, and let us say that x ∈ A is self-adjoint with respect to β and the given opposite algebra automorphism on A if

β x = x^{∗} β. (3.14.4)

Similarly, let us say that x is anti-self-adjoint with respect to β and the given opposite algebra automorphism on A if

β x = −x^{∗} β. (3.14.5)

If β = e, then these reduce to the usual notions of self-adjointness and antiself-adjointness with respect to the given opposite algebra automorphism on A, as in Section 2.6. The collections of self-adjoint and anti-self-adjoint elements of A with respect to β and the given opposite algebra automorphism on A are submodules of A, as a module over k.

Suppose for the moment that x, y ∈ A are both anti-self-adjoint with respect to β and the given opposite algebra automorphism on A. This implies that

β x y = −x^{∗} β y = x^{∗} y^{∗} β = (y x)^{∗} β (3.14.6)

and β y x = −y^{∗} β x = y^{∗} x^{∗} β = (x y)^{∗} β. (3.14.7)

If [x, y] = x y − y x is the usual commutator of x and y in A, then we get that

β [x, y] = β x y − β y x = (y x)^{∗} β − (x y)^{∗} β = −([x, y])^{∗} β. (3.14.8)

Thus [x, y] is anti-self-adjoint with respect to β and the given opposite algebra automorphism on A too.

Suppose that x, y ∈ A satisfy

x^{∗} β x = β (3.14.9)

and y^{∗} β y = β. (3.14.10)

74 CHAPTER 3. SUBMULTIPLICATIVITY AND INVERTIBILITY This implies that (x y)^{∗} β x y = y^{∗} x^{∗} β x y = y^{∗} β y = β. (3.14.11) Note that (3.14.9) holds when x = e, because e^{∗} = e. If x is an invertible element of A that satisfies (3.14.9), then β = (x^{∗})^{−1} β x^{−1} = (x^{−1})^{∗} β x^{−1}, (3.14.12) using (3.14.3) in the second step. This shows that the collection of invertible elements x of A that satisfy (3.14.9) forms a group with respect to multiplication.

If β is an invertible element of A, then ϕ_{β}(x) = β^{−1} x^{∗} β (3.14.13) defines an opposite algebra automorphism on A. In this case, (3.14.4) is equivalent to ϕ_{β}(x) = x, (3.14.14) which means that x is self-adjoint with respect to ϕ_{β}. Similarly, (3.14.5) is equivalent to ϕ_{β}(x) = −x, (3.14.15) which means that x is anti-self-adjoint with respect to ϕ_{β}. We also have that (3.14.9) is equivalent to ϕ_{β}(x) x = e, (3.14.16) which is the same as saying that x^{−1} = ϕ_{β}(x) (3.14.17) when x is invertible in A. Of course, (3.14.13) reduces to the given opposite algebra automorphism x ↦ x^{∗} on A when β = e.

Suppose for the moment that x ↦ x^{∗} is an involution on A, as in Section 2.6. If x, β ∈ A, then (β x)^{∗} = x^{∗} β^{∗} (3.14.18) and (x^{∗} β)^{∗} = β^{∗} (x^{∗})^{∗} = β^{∗} x. (3.14.19) It follows that x is self-adjoint or anti-self-adjoint with respect to β and the given involution on A if and only if x is self-adjoint or anti-self-adjoint, respectively, with respect to β^{∗} and the given involution on A. We also have that (x^{∗} β x)^{∗} = x^{∗} β^{∗} (x^{∗})^{∗} = x^{∗} β x (3.14.20) for every x, β ∈ A, so that (3.14.9) is equivalent to x^{∗} β^{∗} x = β^{∗}. (3.14.21)

Let β be an invertible element of A again, so that ϕ_{β} can be defined on A as in (3.14.13). Observe that

(ϕ_{β}(x))^{∗} = (β^{−1} x^{∗} β)^{∗} = β^{∗} (x^{∗})^{∗} (β^{−1})^{∗} = β^{∗} x (β^{∗})^{−1} (3.14.22)

for every x ∈ A, using the hypothesis that x ↦ x^{∗} be an involution on A and (3.14.3) for β in the last step. Thus

ϕ_{β}(ϕ_{β}(x)) = β^{−1} β^{∗} x (β^{∗})^{−1} β (3.14.23)

for every x ∈ A. If β is either self-adjoint or anti-self-adjoint with respect to the given involution on A, then it follows that ϕ_{β} is an involution on A as well.

Suppose now that k is the field C of complex numbers, and that x ↦ x^{∗} is a conjugate-linear opposite algebra automorphism on A, as in Section 2.6. Thus x ↦ x^{∗} may be considered as a real-linear opposite algebra automorphism on A as an algebra over R, as before. In particular, if β ∈ A, then the collections of elements of A that are self-adjoint or anti-self-adjoint with respect to β and x ↦ x^{∗} are real-linear subspaces of A. In this situation, x ∈ A is anti-self-adjoint with respect to β if and only if i x is self-adjoint with respect to β.

3.15 Invertibility and seminorms

Let k be a field with an absolute value function | · |, and let A be an associative algebra over k with a submultiplicative seminorm N_{A} with respect to | · | on k. Suppose that A has a multiplicative identity element e such that

N_{A}(e) = 1. (3.15.1)

If x is an invertible element of A, then it follows that

N_{A}(x) N_{A}(x^{−1}) ≥ 1. (3.15.2)

In particular, if we have that

N_{A}(x), N_{A}(x^{−1}) ≤ 1, (3.15.3)

then N_{A}(x) = N_{A}(x^{−1}) = 1. (3.15.4)

Let y be another invertible element of A such that

N_{A}(y), N_{A}(y^{−1}) ≤ 1. (3.15.5)

Thus x y is an invertible element of A too,

N_{A}(x, y) ≤ N_{A}(x) N_{A}(y) ≤ 1, (3.15.6)

and N_{A}((x y)^{−1}) = N_{A}(y^{−1} x^{−1}) ≤ N_{A}(x^{−1}) N_{A}(y^{−1}) ≤ 1. (3.15.7)

This shows that the collection of invertible elements x of A that satisfy (3.15.3) forms a group with respect to multiplication.

Suppose that N_{A} is a semi-ultranorm on A, and that x is an invertible element of A that satisfies (3.15.3). Let y be another invertible element of A such that N_{A}(x − y) < 1. (3.15.8)

This implies that

N_{A}(y) ≤ max(N_{A}(x), N_{A}(x − y)) ≤ 1. (3.15.9)

We also have that N_{A}(y^{−1}) = N_{A}(x^{−1}) ≤ 1, (3.15.10)

as in (3.4.9). Thus y satisfies (3.15.5) under these conditions.

Let A be an algebra over k in the strict sense, and let N_{A} be a seminorm on A with respect to |·| on k. Also let x ↦ x^{∗} be an opposite algebra automorphism on A, as in Section 2.6. A basic compatibility condition between x ↦ x^{∗} and N_{A} is that there be a nonnegative real number C_{1} such that

N_{A}(x^{∗}) ≤ C_{1} N_{A}(x) (3.15.11)

for every x ∈ A. This is the same as saying that x ↦ x^{∗} is bounded as a linear mapping from A into itself, using N_{A} on the domain and range. Another compatibility condition is that there be a nonnegative real number C_{2} such that

N_{A}(x) ≤ C_{2} N_{A}(x^{∗}) (3.15.12)

for every x ∈ A. In particular, (3.15.11) and (3.15.12) hold with C_{1} = C_{2} = 1 if and only if N_{A}(x^{∗}) = N_{A}(x) (3.15.13)

for every x ∈ A, which is to say that x ↦ x^{∗} is an isometric linear mapping from A into itself with respect to N_{A}. If x ↦ x^{∗} is an involution on A, then (3.15.11) and (3.15.12) are equivalent, and with the same constant. If k is the field of complex numbers with the standard absolute value function, then one can consider conjugate-linear opposite algebra automorphisms on A as well.

Formal power series

4.1 Direct sums and products

Let k be a commutative ring with a multiplicative identity element, and let I be a nonempty set. Suppose that for each j ∈ I, A_{j} is a module over k. Under these conditions, the Cartesian product ∏_{j∈I} A_{j} of the A_{j}’s is a module over k too, with respect to coordinatewise addition and scalar multiplication. This is the direct product of the A_{j}’s, j ∈ I. If a ∈ ∏_{j∈I} A_{j} and l ∈ I, then we let a_{l} be the lth coordinate of a in A_{l}. Thus a ↦ a_{l} is the standard coordinate projection from ∏_{j∈I} A_{j} onto A_{l}. Of course, this mapping is linear over k.

Let ⨁_{j∈I} A_{j} be the set of a ∈ ∏_{j∈I} A_{j} such that a_{l} = 0 for all but finitely many l ∈ I. This is the direct sum of A_{j}, j ∈ I. Note that ⨁_{j∈I} A_{j} is a submodule of ∏_{j∈I} A_{j}, as a module over k. If I has only finitely many elements, then ⨁_{j∈I} A_{j} is the same as ∏_{j∈I} A_{j}. If I = {1, …, n} for some n ∈ Z_{+}, then we may use the notation ⨁_{j=1}^{n} A_{j} or ∏_{j=1}^{n} A_{j}.

If A_{j} is an algebra over k in the strict sense for each j ∈ I, then ∏_{j∈I} A_{j} is an algebra in the strict sense over k with respect to coordinatewise multiplication, and ⨁_{j∈I} A_{j} is a two-sided ideal in ∏_{j∈I} A_{j}. If A_{j} is commutative for every j ∈ I, then ∏_{j∈I} A_{j} is commutative as well. Similarly, if A_{j} is associative for every j ∈ I, then ∏_{j∈I} A_{j} is associative. If A_{j} has a multiplicative identity element for every j ∈ I, then we get a multiplicative identity element in ∏_{j∈I} A_{j}. If A_{j} is a Lie algebra for every j ∈ I, then ∏_{j∈I} A_{j} is a Lie algebra.

Now let k be a field with an absolute value function | · |, and let I be a nonempty set again. Suppose that V_{j} is a vector space over k for each j ∈ I, so that the direct product ∏_{j∈I} V_{j} is a vector space over k too. If N_{l} is a seminorm on V_{l} with respect to | · | on k for some l ∈ I, then it is easy to see that

Ñ_{l}(v) = N_{l}(v_{l}) (4.1.1)

defines a seminorm on ∏_{j∈I} V_{j} with respect to |·| on k. If N_{l} is a semi-ultranorm on V_{l}, then (4.1.1) is a semi-ultranorm on ∏_{j∈I} V_{j}.

As before, the direct sum ⨁_{j∈I} V_{j} is a linear subspace of ∏_{j∈I} V_{j}. Suppose that N_{j} is a seminorm on V_{j} for each j ∈ I, and put

Ñ^{1}(v) = ∑_{j∈I} N_{j}(v_{j}) (4.1.2)

for every v ∈ ⨁_{j∈I} V_{j}. More precisely, if v ∈ ⨁_{j∈I} V_{j}, then v_{j} = 0 for all but finitely many j ∈ I, so that the sum on the right side of (4.1.2) reduces to a finite sum of nonnegative real numbers. One can check that (4.1.2) defines a seminorm on ⨁_{j∈I} V_{j} with respect to | · | on k, which is a norm when N_{j} is a norm on V_{j} for every j ∈ I.

Similarly, put

Ñ^{∞}(v) = max_{j∈I} N_{j}(v_{j}) (4.1.3)

for every v ∈ ⨁_{j∈I} V_{j}, which reduces to the maximum of finitely many nonnegative real numbers. One can verify that this defines a seminorm on ⨁_{j∈I} V_{j}, which is a norm when N_{j} is a norm on V_{j} for every j ∈ I. If N_{j} is a semiultranorm on V_{j} for each j ∈ I, then (4.1.3) is a semi-ultranorm on ⨁_{j∈I} V_{j}. Observe that Ñ^{∞}(v) ≤ Ñ^{1}(v) (4.1.4)

for every v ∈ ⨁_{j∈I} V_{j}. If I has only finitely many elements, then

Ñ^{1}(v) ≤ (#I) Ñ^{∞}(v) (4.1.5)

for every v ∈ ⨁_{j∈I} V_{j}, where #I is the number of elements in I.

Let a_{j} be a positive real number for each j ∈ I. As before,

Ñ_{a}^{1}(v) = ∑_{j∈I} a_{j} N_{j}(v_{j}) (4.1.6)

defines a seminorm on ⨁_{j∈I} V_{j} with respect to | · | on k. Similarly,

Ñ_{a}^{∞}(v) = max_{j∈I} (a_{j} N_{j}(v_{j})) (4.1.7)

defines a seminorm on ⨁_{j∈I} V_{j} with respect to | · | on k, which is a semiultranorm when N_{j} is a semi-ultranorm on V_{j} for each j ∈ I. Clearly

Ñ_{a}^{∞}(v) ≤ Ñ_{a}^{1}(v) (4.1.8)

for every v ∈ ⨁_{j∈I} V_{j}. If I has only finitely many elements, then

Ñ_{a}^{1}(v) ≤ ( ∑_{j∈I} a_{j}) Ñ^{∞}(v) (4.1.9)

for every v ∈ ⨁_{j∈I} V_{j}. If I has infinitely many elements, then ∑_{j∈I} a_{j} can be defined as an extended real number, as the supremum of the corresponding finite subsums. If the supremum is finite, then (4.1.9) still holds and is nontrivial.

4.2 Bilinear mappings and Cauchy products

Let k be a commutative ring with a multiplicative identity element, let A, B, and C be modules over k, and let β be a mapping from A × B into C that is bilinear over k. Also let ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} be infinite series with terms in A and B, respectively, considered formally for the moment. Put

c_{n} = ∑_{j=0}^{n} β(a_{j}, b_{n−j}) (4.2.1)

for each nonnegative integer n. It is easy to see that

∑_{n=0}^{∞} c_{n} = β( ∑_{j=0}^{∞} a_{j}, ∑_{l=0}^{∞} b_{l}), (4.2.2)

at least formally. More precisely, suppose for the moment that there are nonnegative integers J, L such that a_{j} = 0 when j > J and b_{l} = 0 when l > L. If n > J +L, then it follows that c_{n} = 0. Thus the infinite series ∑_{j=0}^{∞} a_{j}, ∑_{l=0}^{∞} b_{l}, and ∑_{n=0}^{∞} c_{n} reduce to the finite sums ∑_{j=0}^{J} a_{j}, ∑_{l=0}^{L} b_{l}, and ∑_{n=0}^{J+L} c_{n}, respectively, and the formal argument for (4.2.2) works in this case.

In particular, if A is an algebra over k in the strict sense, then we can take β to be the corresponding mapping from A × A into A. Let ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} be infinite series with terms in A, and let us express multiplication of a, b ∈ A as a b. The Cauchy product of these series is defined to be the series ∑_{n=0}^{∞} c_{n}, where

c_{n} = ∑_{j=0}^{n} a_{j} b_{n−j}, (4.2.3)

as in (4.2.1). Thus

∑_{n=0}^{∞} c_{n} = ( ∑_{j=0}^{∞} a_{j}) ( ∑_{l=0}^{∞} b_{l}), (4.2.4)

at least formally, as before.

Suppose for the moment that k = R with the standard absolute value function, and that a_{j}, b_{l} are nonnegative real numbers for every j, l ≥ 0. If c_{n} is as in (4.2.3), then c_{n} is a nonnegative real number for every n ≥ 0. If J, L are nonnegative integers, then one can verify that

( ∑_{j=0}^{J} a_{j}) ( ∑_{l=0}^{L} b_{l}) ≤ ∑_{n=0}^{J+L} c_{n}. (4.2.5)

Similarly, if N is a nonnegative integer, then

∑_{n=0}^{N} c_{n} ≤ ( ∑_{j=0}^{N} a_{j}) ( ∑_{l=0}^{N} b_{l}). (4.2.6)

If ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converge, then it follows that ∑_{n=0}^{∞} c_{n} converges, and that the sums satisfy (4.2.4).

Let k be any field with an absolute value function | · |, and let A, B, and C be vector spaces over k with norms N_{A}, N_{B}, and N_{C}, respectively, with respect to | · | on k. Also let β be a bounded bilinear mapping from A × B into C with respect to these norms, so that there is a nonnegative real number C(β) such that N_{C}(β(a, b)) ≤ C(β) N_{A}(a) N_{B}(b) (4.2.7)

for every a ∈ A and b ∈ B. Suppose that A, B, and C are complete with respect to the metrics associated to N_{A}, N_{B}, and N_{C}, respectively. Let ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} be infinite series with terms in A and B that converge absolutely with respect to N_{A} and N_{B}, respectively, so that ∑_{j=0}^{∞} N_{A}(a_{j}) and ∑_{l=0}^{∞} N_{B}(b_{l}) converge as infinite series of nonnegative real numbers. If c_{n} is as in (4.2.1), then

N_{C}(c_{n}) ≤ ∑_{j=0}^{n} N_{C}(β(a_{j}, b_{n−j})) ≤ C(β) ∑_{j=0}^{n} N_{A}(a_{j}) N_{B}(b_{n−j}) (4.2.8)

for every n ≥ 0. The sum on the right side of (4.2.8) is the same as the nth term of the Cauchy product of ∑_{j=0}^{∞} N_{A}(a_{j}) and ∑_{l=0}^{∞} N_{B}(b_{l}). It follows that ∑_{n=0}^{∞} N_{C}(c_{n}) converges as an infinite series of nonnegative real numbers, with

∑_{n=0}^{∞} N_{C}(c_{n}) ≤ C(β) ( ∑_{j=0}^{∞} N_{A}(a_{j})) ( ∑_{l=0}^{∞} N_{B}(b_{l})). (4.2.9)

Thus ∑_{n=0}^{∞} c_{n} converges absolutely with respect to N_{C}, and ∑_{j=0}^{∞} a_{j}, ∑_{l=0}^{∞} b_{l}, and ∑_{n=0}^{∞} c_{n} converge in A, B, and C, respectively, by completeness. One can check that (4.2.2) holds in this situation, by approximating these sums by finite sums.

Suppose now that N_{A}, N_{B}, and N_{C} are ultranorms on A, B, and C, respectively, and let ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} be infinite series with terms in A and B, respectively, such that

lim_{j→∞} N_{A}(a_{j}) = lim_{l→∞} N_{B}(b_{l}) = 0. (4.2.10)

If c_{n} is as in (4.2.1) again, then

N_{C}(c_{n}) ≤ max_{0≤j≤n} N_{C}(β(a_{j}, b_{n−j})) ≤ C(β) max_{0≤j≤n}(N_{A}(a_{j}) N_{B}(b_{l})) (4.2.11)

for every n ≥ 0. One can verify that

lim_{n→∞} N_{C}(c_{n}) = 0, (4.2.12)

using (4.2.10) and (4.2.11). It follows that ∑_{j=0}^{∞} a_{j}, ∑_{l=0}^{∞} b_{l}, and ∑_{n=0}^{∞} c_{n} converge in A, B, and C, respectively, because of completeness, as in Section 1.12. One can check that (4.2.2) holds in this situation too, by approximating these sums by finite sums, as before.

4.3 Formal power series and modules

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, and let T be an indeterminate. As in [4, 11], we shall try to use upper-case letters like T for indeterminates, and lower-case letters for elements of k or A. A formal power series in T with coefficients in A can be expressed as

f(T) = ∑_{j=0}^{∞} f_{j} T^{j}, (4.3.1)

where f_{j} is an element of A for each nonnegative integer j. The space A[[T]] of these formal power series can be defined as the space of functions on the set Z_{+} ∪ {0} of nonnegative integers with values in A, where (4.3.1) corresponds to j ↦ f_{j} as an A-valued function on Z_{+} ∪ {0}. This is a module over k with respect to pointwise addition and scalar multiplication of A-valued functions on Z_{+} ∪ {0}, which corresponds to termwise addition and scalar multiplication of formal power series as in (4.3.1).

Similarly, a formal polynomial in T with coefficients in A can be expressed as

f(T) = ∑_{j=0}^{n} f_{j} T^{j} (4.3.2)

for some nonnegative integer n, where f_{j} ∈ A for each j = 0, …, n. This may be considered as a formal power series in T too, with f_{j} = 0 for j > n. The space A[T] of these formal polynomials can be defined as the space of A-valued functions on Z_{+} ∪ {0} that are equal to 0 at all but finitely many nonnegative integers. Of course, A[T] is a submodule of A[[T]], as a module over k. Observe that A[[T]] corresponds to the direct product of copies of A indexed by the set Z_{+} ∪{0} of nonnegative integers, as a module over k, and that A[T] corresponds to the analogous direct sum.

There is a natural mapping from A into A[T], which sends a ∈ A to the formal polynomial f(T) with f_{0} = a and f_{j} = 0 when j ≥ 1. This is an injective module homomorphism from A into A[T], and it is sometimes convenient to think of A as a submodule of A[T] in this way. Note that the mapping

f(T) ↦ f_{0} (4.3.3)

is a module homomorphism from A[[T]] onto A.

If f(T) ∈ A[[T]] and l is a nonnegative integer, then

f(T) T^{l} = ∑_{j=0}^{∞} f_{j} T^{j+l} = ∑_{j=l}^{∞} f_{j−l} T^{j} (4.3.4)

defines an element of A[[T]], which is the same as f(T) when l = 0. Of course, (4.3.4) is in A[T] when f(T) ∈ A[T]. The mapping

f(T) ↦ f(T) T^{l} (4.3.5) is an injective module homomorphism from A[[T]] into itself for every l ≥ 0.

Let us say that f(T) ∈ A[[T]] vanishes to order n for some nonnegative integer n if f_{j} = 0 for every j = 0, …, n. Equivalently, this means that f(T) = g(T) T^{n+1} for some g(T) ∈ A[[T]]. The collection of f(T) ∈ A[[T]] that vanishes to order n is a submodule of A[[T]]. This submodule is the same as the kernel of (4.3.3) when n = 0.

4.4 Sums and extensions

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, and let T be an indeterminate again. Also let f_{l}(T) = ∑_{j=0}^{∞} f_{l,j} T^{j} be an element of A[[T]] for each l ∈ Z_{+}, and let f(T) be another element of A[[T]]. Let us say that {f_{l}(T)}_{l=1}^{∞} eventually agrees with f(T) termwise if for each nonnegative integer j there is a positive integer L_{j} such that f_{l,j} = f_{j} (4.4.1)

for every l ≥ L_{j}. This implies that f_{l}(T) − f(T) vanishes to order n ≥ 0 when l ≥ max(L_{0}, …, L_{n}). As in the previous section, A[[T]] can be defined as the space of A-valued functions on Z_{+} ∪ {0}, which is the same as the Cartesian product of the family of copies of A indexed by Z_{+} ∪ {0}. Consider the product topology on A[[T]] as a Cartesian product, using the discrete topology on A in each factor. The condition that {f_{l}(T)}_{l=1}^{∞} eventually agree termwise with f(T) is equivalent to the convergence of {f_{l}(T)}_{l=1}^{∞} to f(T) with respect to this product topology.

Similarly, let us say that {f_{l}(T)}_{l=1}^{∞} is termwise eventually constant if for each nonnegative integer j there is an L_{j} ∈ Z_{+} such that f_{l,j} does not depend on l when l ≥ L_{j}. In this case, we can define f_{j} ∈ A for each j ≥ 0 by putting f_{j} = f_{l,j} when l ≥ L_{j}. This defines f(T) = ∑_{j=0}^{∞} f_{j} T^{j} as an element of A[[T]], and we have that {f_{l}(T)}_{l=1}^{∞} eventually agrees termwise with f(T). Conversely, if {f_{l}(T)}_{l=1}^{∞} eventually agrees termwise with some f(T) ∈ A[[T]], then {f_{l}(T)}_{l=1}^{∞} is termwise eventually constant.

If {f_{l}(T)}_{l=1}^{∞} eventually agrees termwise with some f(T) ∈ A[[T]] and α ∈ k, then {α f_{l}(T)}_{l=1}^{∞} eventually agrees termwise with α f(T). In this situation, we also have that {f_{l}(T) T^{r}}_{l=1}^{∞} eventually agrees termwise with f(T) T^{r} for every nonnegative integer r. If {g_{l}(T)}_{l=1}^{∞} is another sequence of elements of A[[T]] that eventually agrees termwise with g(T) ∈ A[[T]], then {f_{l}(T) + g_{l}(T)}_{l=1}^{∞} eventually agrees termwise with f(T) + g(T).

Let a_{l}(T) = ∑_{j=0}^{∞} a_{l,j} T^{j} be a formal power series in T with coefficients in A for each l ∈ Z_{+}. Suppose that {a_{l}(T)}_{l=1}^{∞} eventually agrees termwise with 0, so that for each j ≥ 0 there is an L_{j} ∈ Z_{+} such that a_{l,j} = 0 for every l ≥ L_{j} It follows that the coefficient of T^{j} in

∑_{l=1}^{n} a_{l}(T) (4.4.2)

does not depend on n when n ≥ L_{j} − 1, so that the sequence of these sums is termwise eventually constant. Under these conditions, we can define

∑_{l=1}^{∞} a_{l}(T) (4.4.3)

as a formal power series in T with coefficients in A, by taking the coefficient of T^{j} in (4.4.3) to be the coefficient of T^{j} in (4.4.2) when l ≥ L(n) − 1, as before. By construction, the sequence of partial sums (4.4.2) eventually agrees termwise with (4.4.3).

If α ∈ k, then {α a_{l}(T)}_{l=1}^{∞} eventually agrees termwise with 0 too, and

∑_{l=1}^{∞} α a_{l}(T) = α ∑_{l=1}^{∞} a_{l}(T). (4.4.4)

If r is a nonnegative integer, then {a_{l}(T) T^{r}}_{l=1}^{∞} eventually agrees termwise with 0 as well, and

∑_{l=1}^{∞} a_{l}(T) T^{r} = ( ∑_{l=1}^{∞} a_{l}(T)) T^{r}. (4.4.5)

If {b_{l}(T)}_{l=1}^{∞} is another sequence of elements of A[[T]] that eventually agrees termwise with 0, then {a_{l}(T)+b_{l}(T)}_{l=1}^{∞} eventually agrees termwise with 0, and

∑_{l=1}^{∞}(a_{l}(T) + b_{l}(T)) = ∑_{l=1}^{∞} a_{l}(T) + ∑_{l=1}^{∞} b_{l}(T). (4.4.6)

Of course, one can deal with sequences and series that start with l = 0 in the same way.

Let B be another module over k, so that B[[T]] is a module over k too, as before. Also let ϕ be a module homomorphism from A into B[[T]]. Note that a module homomorphism from A into B may be considered as a module homomorphism from A into B[[T]], by considering B as a submodule of B[[T]]. If f(T) = ∑_{l=0}^{∞} f_{l} T^{l} ∈ A[[T]], then ϕ(f_{l}) ∈ B[[T]] for each l ≥ 0, and ϕ(f_{l}) T^{l} automatically vanishes to order l−1 for every l ≥ 1. In particular, {ϕ(f_{l}) T^{l}}_{l=0}^{∞} eventually agrees termwise with 0, and we put

ϕ(f(T)) = ∑_{l=0}^{∞} ϕ(f_{l}) T^{l}, (4.4.7)

where the sum is defined as an element of B[[T]] as in (4.4.3). This defines a module homomorphism from A[[T]] into B[[T]], which agrees with the initial homomorphism from A into B[[T]] when A is considered as a submodule of A[[T]]. If we start with a module homomorphism ϕ from A into B[T], and if f(T) ∈ A[T], then (4.4.7) reduces to a finite sum in B[T].

If f(T) ∈ A[[T]] vanishes to order n for some nonnegative integer n, then it is easy to see that ϕ(f(T)) vanishes to order n as well. More precisely, one can check that ϕ(f(T) T^{r}) = ϕ(f(T)) T^{r} (4.4.8) for every f(T) ∈ A[[T]] and nonnegative integer r. If f(T) = ∑_{l=0}^{∞} f_{l} T^{l} ∈ A[[T]] and j is a nonnegative integer, then the total coefficient of T^{j} in (4.4.7) is the sum of the coefficients of T^{j−l} in ϕ(f_{l}) for l = 0, …, j, and in particular depends only on ϕ(f_{l}) for l ≤ j. If {f_{r}(T)}_{r=1}^{∞} is a sequence of elements of A[[T]] that eventually agrees termwise with f(T), then it follows that {ϕ(f_{r}(T))}_{r=1}^{∞} eventually agrees termwise with ϕ(f(T)).

4.5 Extending bilinear mappings

Let k be a commutative ring with a multiplicative identity element, let A, B, and C be modules over k, and let T be an indeterminate. As before, C[[T]] is a module over k, and we let β be a mapping from A×B into C[[T]] that is bilinear over k. There is a natural way to extend β to a mapping from A[[T]] × B[[T]] into C[[T]], as follows. Let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} and g(T) = ∑_{l=0}^{∞} g_{l} T^{l} be formal power series in T with coefficients in A and B, respectively. Put

h_{n}(T) = ∑_{j=0}^{n} β(f_{j}, g_{n−j}) (4.5.1)

for each nonnegative integer n, which is an element of C[[T]]. Thus h_{n}(T) T^{n} automatically vanishes to order n − 1 for every n ≥ 1, so that

h(T) = ∑_{n=0}^{∞} h_{n}(T) T^{n} (4.5.2)

defines an element of C[[T]] too, as in the previous section. Put

β(f(T), g(T)) = h(T). (4.5.3)

This defines a mapping from A[[T]] × B[[T]] into C[[T]] that is bilinear over k and agrees with the initial mapping from A × B into C[[T]], with A and B considered as submodules of A[[T]] and B[[T]], respectively.

If f(T) ∈ A[T] and g(T) ∈ B[T], then (4.5.1) is equal to 0 for all but finitely many n, so that (4.5.2) reduces to a finite sum. If the inital mapping β sends A × B into C[T], then (4.5.1) is in C[T] for every n ≥ 0. In this case, it follows that (4.5.2) is an element of C[T] when f(T) ∈ A[T] and g(T) ∈ B[T].

If f(T) ∈ A[[T]] and g(T) ∈ B[[T]] vanish to order r_{1} and r_{2}, respectively, for some nonnegative integers r_{1}, r_{2}, then (4.5.1) is equal to 0 when n ≤ r_{1} +r_{2}, so that (4.5.2) vanishes to order r_{1} + r_{2}. More precisely, one can verify that

β(f(T) T^{r}, g(T) T^{m}) = β(f(T), g(T)) T^{r+m} (4.5.4)

for every f(T) ∈ A[[T]], g(T) ∈ B[[T]], and nonnegative integers r, m. If f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ A[[T]], g(T) = ∑_{l=0}^{∞} g_{l} T^{l} ∈ B[[T]], and r is a nonnegative integer, then the total coefficient of T^{r} in (4.5.2) is the sum of the coefficients of T^{r−n} in (4.5.1) for n = 0, …, r. In particular, this only involves (4.5.1) for n ≤ r, and hence depends only on f_{j} and g_{l} for j, l ≤ r. If {f_{m}(T)}_{m=1}^{∞} and {g_{m}(T)}_{m=1}^{∞} are sequences of elements of A[[T]] and B[[T]] that eventually agree termwise with f(T) and g(T), respectively, then it follows that {β(f_{m}(T), g_{m}(T))}_{m=1}^{∞} eventually agrees termwise with β(f(T), g(T)).

Let {a_{m}(T)}_{m=0}^{∞} and {b_{r}(T)}_{r=0}^{∞} be sequences of elements of A[[T]] and B[[T]], respectively, that eventually agree termwise with 0. Thus ∑_{m=0}^{∞} a_{m}(T) and ∑_{r=0}^{∞} b_{r}(T) can be defined as elements of A[[T]] and B[[T]], respectively, as in the previous section. Using the extension of β to A[[T]] × B[[T]] defined earlier, we get that β(a_{m}(T), b_{r}(T)) is defined as an element of C[[T]] for all m, r ≥ 0. Put

c_{N}(T) = ∑_{m=0}^{N} β(a_{m}(T), b_{N−m}(T)) (4.5.5)

for every nonnegative integer N, which is an element of C[[T]]. Note that β(a_{m}(T), b_{r}(T)) vanishes to arbitrarily large order when m or r is sufficiently large in this situation, by the remarks in the preceding paragraph. This implies that {c_{N}(T)}_{N=0}^{∞} eventually agrees termwise with 0, so that ∑_{N=0}^{∞} c_{N}(T) defines an element of C[[T]], as in the previous section. One can check that

∑_{N=0}^{∞} c_{N}(T) = β( ∑_{m=0}^{∞} a_{m}(T), ∑_{r=0}^{∞} b_{r}(T)) (4.5.6)

as elements of C[[T]], as in Section 4.2. More precisely, this means that for each nonnegative integer j, the coefficients of T^{j} on both sides of (4.5.6) are the same. This reduces to an analogous statement for finite sums for each j ≥ 0 in this situation.

Observe that (4.5.1) is the same as

∑_{l=0}^{n} β(f_{n−l}, g_{l}). (4.5.7)

Suppose now that A = B. If the initial mapping β from A × A into C[[T]] is symmetric or antisymmetric, then it is easy to see that the extension of β to A[[T]] × A[[T]] has the same property, using the fact that (4.5.1) is the same as (4.5.7). Suppose that the initial mapping β on A × A satisfies

β(a, a) = 0 (4.5.8)

for every a ∈ A, and let us check that the extension of β to A[[T]] × A[[T]] satisfies β(f(T), f(T)) = 0 (4.5.9)

for every f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ A[[T]]. To do this, it suffices to verify that

∑_{j=0}^{n} β(f_{j}, f_{n−j}) = 0 (4.5.10)

for every nonnegative integer n. Remember that (4.5.8) implies that β is antisymmetric on A × A, as in Section 2.1. If n is odd, then (4.5.10) reduces to the antisymmetry of β on A × A. If n is even, then one can use the antisymmetry of β on A × A to reduce (4.5.10) to the condition that β(f_{n/2}, f_{n/2}) = 0, which follows from (4.5.8).

4.6 Formal power series and algebras

Let k be a commutative ring with a multiplicative identity element again, and let T be an indeterminate. Suppose that A is an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} and g(T) = ∑_{l=0}^{∞} g_{l} T^{l} be formal power series in T with coefficients in A, and put

h_{n} = ∑_{j=0}^{n} f_{j} g_{n−j} (4.6.1)

for each nonnegative integer n. Thus h(T) = ∑_{n=0}^{∞} h_{n} T^{n} is a formal power series in T with coefficients in A too, and we put

f(T) g(T) = h(T). (4.6.2)

This defines a mapping from A[[T]] × A[[T]] into A[[T]], which is the same as the mapping obtained from multiplication on A as in the previous section. Using this definition of multiplication on A[[T]], we get that A[[T]] is an algebra over k in the strict sense. Note that A[T] is a subalgebra of A[[T]], and that A corresponds to a subalgebra of A[T], using the identification mentioned in Section 4.3. The mapping f(T) ↦ f_{0} mentioned in Section 4.3 defines an algebra homomorphism from A[[T]] onto A. If multiplication on A is commutative, then multiplication on A[[T]] is commutative as well, as in the remark about symmetry of β in the previous section. Similarly, if multiplication on A is associative, then one can check that multiplication on A[[T]] is associative too. If A has a multiplicative identity element e, then the corresponding formal polynomial in T is the multiplicative identity element in A[[T]].

In particular, we can take A = k, considered as an algebra over itself. Thus k[[T]] is a commutative associative algebra over k with a multiplicative identity element, and k[T] is a subalgebra of k[[T]]. We can identify k with a subalgebra of k[T], where the multiplicative identity element in k corresponds to the multiplicative identity element in k[[T]].

If A is any module over k, then A[[T]] may be considered as a module over k[[T]]. More precisely, if f(T) ∈ k[[T]] and g(T) ∈ A[[T]], then f(T) g(T) can be defined as a formal power series in T with coefficients in A as in (4.6.2), where the terms in the sum on the right side of (4.6.1) are defined using scalar multiplication on A. Equivalently, scalar multiplication on A corresponds to a mapping from k × A into A that is bilinear over k, which can be extended to a mapping from k[[T]] × A[[T]] into A[[T]] as in the previous section. Similarly, A[T] may be considered as a module over k[T].

4.7. INVERTIBILITY IN A[[T]] 87

Let A and B be modules over k, and let ϕ be a module homomorphism from A into B[[T]], as modules over k. As in Section 4.4, there is a natural way to extend ϕ to a mapping from A[[T]] into B[[T]]. It is easy to see that this mapping is a module homomorphism from A[[T]] into B[[T]], as modules over k[[T]]. If the initial mapping sends A into B[T], then the restricton to A[T] of the extension to A[[T]] is a module homomorphism into B[T], as modules over k[T]. Let C be another module over k, and let β be a mapping from A × B into C[[T]] that is bilinear over k. One can check that the extension of β to a mapping from A[[T]] × B[[T]] into C[[T]] defined in the previous section is bilinear over k[[T]]. If the initial mapping sends A × B into C[T], then the restriction to A[T] × B[T] of the extension to A[[T]] × B[[T]] is bilinear over k[T] as a mapping into C[T]. If A is an algebra over k in the strict sense, then A[[T]] may be considered as an algebra over k[[T]] in the strict sense, and A[T] may be considered as an algebra over k[T] in the strict sense.

Suppose that (A, [·, ·]_{A}) is a Lie algebra over k, and let f(T) and g(T) be formal power series in T with coefficients in A again. In this situation, (4.6.1) should be expressed as

h_{n} = ∑_{j=0}^{n}[f_{j}, g_{n−j}]_{A} (4.6.3)

for each n ≥ 0, and we put

[f(T), g(T)]_{A[[T]]} = h(T) = ∑_{n=0}^{∞} h_{n} T^{n}. (4.6.4)

One can verify that A[[T]] is a Lie algebra over k with respect to (4.6.4). More precisely, one can use the fact that [a, a]_{A} = 0 for every a ∈ A to get that [f(T), f(T)]_{A[[T]]} = 0 for every f(T) ∈ A[[T]], as in the previous section. One can also get the Jacobi identity for [·, ·]_{A[[T]]} from the Jacobi identity for [·, ·]_{A}. As before, A[T] is a Lie subalgebra of A[[T]], as a Lie algebra over k. One can consider A[[T]] as a Lie algebra over k[[T]], and A[T] as a Lie algebra over k[T].

4.7 Invertibility in A[[T]]

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e. Also let T be an indeterminate, so that A[[T]] may be considered as an associative algebra over k[[T]] as in the previous section. Let us identify e with the corresponding formal power series in T with coefficients in A, which is the multiplicative identity element in A[[T]]. If a(T) ∈ A[[T]], then a(T)^{l} can be defined as an element of A[[T]] for every l ∈ Z_{+} using multiplication on A[[T]], and we interpret a(T)^{l} as being equal to e when l = 0. Observe that

(e − a(T)) ∑_{l=0}^{n} a(T)^{l} = ( ∑_{l=0}^{n} a(t)^{l}) (e − a(T)) = e − a(T)^{n+1} (4.7.1)

for every nonnegative integer n, by a standard computation.

Suppose that a(T) vanishes to order 0, as in Section 4.3, so that the coefficient of T^{0} in a(T) is equal to 0. This implies that a(T)^{l} vanishes to order l − 1 for every l ∈ Z_{+}. It follows that the coefficient of T^{j} in

∑_{l=0}^{n} a(T)^{l} (4.7.2)

is the same for n ≥ j. As in Section 4.4, we define

∑_{l=0}^{∞} a(T)^{l} (4.7.3)

as a formal power series in T with coefficients in A by taking the coefficient of T^{j} in (4.7.3) to be the same as the coefficient of T^{j} in (4.7.2) when n ≥ j. In particular, {a(T)^{l}}_{l=0}^{∞} eventually agrees termwise with 0, as in Section 4.4, and the sequence of partial sums (4.7.2) eventually agrees termwise with (4.7.3).

Using (4.7.1), one can check that

(e − a(T)) ∑_{l=0}^{∞} a(T)^{l} = ( ∑_{l=0}^{∞} a(T)^{l}) (e − a(T)) = e. (4.7.4)

More precisely, for each nonnegative integer j, the coefficient of T^{j} in each of the three expressions in (4.7.4) is the same as in the corresponding expression in (4.7.1) when n ≥ j. It follows that (4.7.3) is the multiplicative inverse of e − a(T) in A[[T]].

Let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} be a formal power series in T with coefficients in A. If f_{0} is invertible as an element of A, then f(T) can be expressed as f_{0} (e−a(T)), where a(T) ∈ A[[T]] vanishes to order 0. This implies that f(T) is invertible in A[[T]], because e − a(T) is invertible in A[[T]], as in the previous paragraph. Conversely, if f(T) is invertible in A[[T]], then f_{0} is invertible in A, because f(T) ↦ f_{0} is an algebra homomorphism from A[[T]] onto A.

Of course, the collection of invertible elements of A[[T]] is a group with respect to multiplication of formal power series. The collection of f(T) ∈ A[[T]] with f_{0} = e is a subgroup of this group.

4.8 Homomorphisms over k[T]

Let k be a commutative ring with a multiplicative identity element, let A, B be modules over k, and let T be an indeterminate. Also let ϕ be a homomorphism from A[T] into B[T], as modules over k. Suppose that

ϕ(f(T) T) = ϕ(f(T)) T (4.8.1)

for every f(T) ∈ A[T]. This implies that

ϕ(f(T) T^{r}) = ϕ(f(T)) T^{r} (4.8.2)

4.8. HOMOMORPHISMS OVER K[T] 89

for every f(T) ∈ A[T] and r ∈ Z_{+}, by applying (4.8.1) repeatedly. Of course, (4.8.2) holds trivially when r = 0. It follows that ϕ is a homomorphism from A[T] into B[T] as modules over k[T] under these conditions. Conversely, if ϕ is a homomorphism from A[T] into B[T] as modules over k[T], then ϕ is a homomorphism from A[T] into B[T] as modules over k that satisfies (4.8.1).

Let ϕ be a homomorphism from A[T] into B[T] as modules over k[T] again. It is easy to see that ϕ is uniquely determined on A[T] by its restriction to A, considered as a submodule of A[T] as a module over k, as in Section 4.3. Remember that every homomorphism from A into B[T], as modules over k, can be extended to a homomorphism from A[T] into B[T] as modules over k[T], as in Sections 4.4 and 4.6. This gives a natural isomorphism between Hom_{k}(A, B[T]) and Hom_{k[T]}(A[T], B[T]), as modules over k.

In fact, Hom_{k}(A, B[T]) may be considered as a module over k[T]. More precisely, Hom_{k}(A, B[T]) is a submodule of the space of all B[T]-valued functions on A, as a module over k[T]. The isomorphism between Hom_{k}(A, B[T]) and Hom_{k[T]}(A[T], B[T]) mentioned in the preceding paragraph is linear over k[T].

Now let ϕ be a homomorphism from A[[T]] into B[[T]], as modules over k, that satisfies (4.8.1) for every f(T) ∈ A[[T]]. This implies that (4.8.2) holds for every f(T) ∈ A[[T]] and nonnegative integer r, as before. If f(T) ∈ A[[T]] vanishes to order n for some nonnegative integer n, then we get that ϕ(f(T)) vanishes to order n as well, by expressing f(T) as an element of A[[T]] times T^{n+1}. Using this, one can check that ϕ is a homomorphism from A[[T]] into B[[T]], as modules over k[[T]]. Conversely, if ϕ is a homomorphism from A[[T]] into B[[T]] as modules over k[[T]], then ϕ is a homomorphism from A[[T]] into B[[T]] as modules over k[T], and hence ϕ is a homomorphism from A[[T]] into B[[T]] as modules over k that satisfies (4.8.1) for every f(T) ∈ A[[T]].

Let ϕ be a homomorphism from A[[T]] into B[[T]] as modules over k that satisfies (4.8.1) for every f(T) ∈ A[[T]] again. If f(T) = ∑_{l=0}^{∞} f_{l} T^{l} ∈ A[[T]] and j is a nonnegative integer, then

ϕ( ∑_{l=j+1}^{∞} f_{l} T^{l}) (4.8.3)

vanishes to order j, as before. This implies that the total coefficient of T^{j} in ϕ(f(T)) is the same as for

ϕ( ∑_{l=0}^{j} f_{l} T^{l}) = ∑_{l=0}^{j} ϕ(f_{l}) T^{l}. (4.8.4)

It follows that ϕ is uniquely determined on A[[T]] by its restriction to A, considered as a submodule of A[[T]] as a module over k. We have also seen that every homomorphism from A into B[[T]], as modules over k, can be extended to a homomorphism from A[[T]] into B[[T]] as modules over k[[T]], as in Sections 4.4 and 4.6.

This gives a natural isomorphism between Hom_{k}(A, B[[T]]) and

Hom_{k[[T]]}(A[[T]], B[[T]]), (4.8.5) as modules over k. We may also consider Hom_{k}(A, B[[T]]) as a module over k[[T]], because it is a submodule of the space of all B[[T]]-valued functions on A, as a module over k[[T]]. It is easy to see that the isomorphism between Hom_{k}(A, B[[T]]) and (4.8.5) is linear over k[[T]].

Let C be another module over k, and let β be a mapping from A[T] × B[T] into C[T]. If β is bilinear over k and

β(f(T) T, g(T)) = β(f(T), g(T) T) = β(f(T), g(T)) T (4.8.6)

for every f(T) ∈ A[T] and g(T) ∈ B[T], then

β(f(T) T^{r}, g(T) T^{m}) = β(f(T), g(T)) T^{r+m} (4.8.7)

for all f(T) ∈ A[T], g(T) ∈ B[T], and nonnegative integers r, m, and hence β is bilinear over k[T]. Conversely, if β is bilinear over k[T], then β is bilinear over k and satisfies (4.8.6). It is easy to see that β is uniquely determined on A[T] × B[T] by its restriction to A × B in this situation. We have seen that every mapping from A × B into C[T] that is bilinear over k can be extended to a mapping from A[T] × B[T] into C[T] that is bilinear over k[T], as in Sections 4.5 and 4.6.

Let β be a mapping from A[[T]]×B[[T]] into C[[T]]. Suppose that β is bilinear over k and satisfies (4.8.6) for every f(T) ∈ A[[T]] and g(T) ∈ B[[T]], which implies that (4.8.7) holds for all f(T) ∈ A[[T]], g(T) ∈ B[[T]], and nonnegative integers r, m. If f(T) ∈ A[[T]], g(T) ∈ B[[T]] vanish to orders r_{1}, r_{2} for some nonnegative integers r_{1}, r_{2}, respectively, then it follows that β(f(T), g(T)) vanishes to order r_{1} + r_{2}. One can use this to verify that β is bilinear over k[[T]]. Conversely, if β is bilinear over k[[T]], then β is bilinear over k[T], and hence β is bilinear over k and satisfies (4.8.6) for every f(T) ∈ A[[T]] and g(T) ∈ B[[T]].

Suppose that β is bilinear over k and satisfies (4.8.6) for every f(T) ∈ A[[T]] and g(T) ∈ B[[T]] again. Let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ A[[T]], and g(T) = ∑_{l=0}^{∞} g_{l} T^{l} ∈ B[[T]] be given. If r is any nonnegative integer, then the total coefficient of T^{r} in β(f(T), g(T)) is the same as for

β( ∑_{j=0}^{r} f_{j} T^{j}, ∑_{l=0}^{r} g_{l} T^{l}). (4.8.8)

This implies that β is uniquely determined on A[[T]] × B[[T]] by its restriction to A × B. We have seen that every mapping from A × B into C[[T]] that is bilinear over k can be extended to a mapping from A[[T]] × B[[T]] into C[[T]] that is bilinear over k[[T]], as in Sections 4.5 and 4.6.

4.9 Formal power series and homomorphisms

Let k be a commutative ring with a multiplicative identity element again, let A, B be modules over k, and let T be an indeterminate. Remember that the space Hom_{k}(A, B) of module homomorphisms from A into B is a module over k with respect to pointwise addition and scalar multiplication, as in Section 2.1. Thus the corresponding spaces (Hom_{k}(A, B))[T] and (Hom_{k}(A, B))[[T]] of formal polynomials and power series in T with coefficients in Hom_{k}(A, B) can be defined as in Section 4.3. More precisely, (Hom_{k}(A, B))[T] and (Hom_{k}(A, B))[[T]] may be considered as modules over k[T] and k[[T]], respectively, as in Section 4.6. Let us see how elements of these modules are related to homomorphisms from A into B[T] and B[[T]], respectively.

Let

ϕ(T) = ∑_{j=0}^{∞} ϕ_{j} T^{j} (4.9.1)

be a formal power series in T with coefficients in Hom_{k}(A, B), so that ϕ_{j} is a homomorphism from A into B for every j ≥ 0. If a ∈ A, then

(ϕ(T))(a) = ∑_{j=0}^{∞} ϕ_{j}(a) T^{j} (4.9.2)

defines a formal power series in T with coefficients in B, and the mapping from a ∈ A to (4.9.2) is a homomorphism from A into B[[T]], as modules over k. Conversely, every homomorphism from A into B[[T]] as modules over k corresponds to a sequence {ϕ_{j}}_{j=0}^{∞} of homomorphisms from A into B in this way, and hence to an element of (Hom_{k}(A, B))[[T]]. This defines a natural isomorphism between Hom_{k}(A, B[[T]]) and (Hom_{k}(A, B))[[T]], as modules over k. This isomorphism is linear over k[[T]] as well, where Hom_{k}(A, B[[T]]) is considered as a module over k[[T]], as in the previous section.

Similarly, if

ϕ(T) = ∑_{j=0}^{n} ϕ_{j} T^{j} (4.9.3)

is a formal polynomial in T with coefficients in Hom_{k}(A, B), then

(ϕ(T))(a) = ∑_{j=0}^{n} ϕ_{j}(a) T^{j} (4.9.4)

is a formal polynomial in T with coefficients in B for every a ∈ A, and the mapping from a ∈ A to (4.9.4) is a homomorphism from A into B[T], as modules over k. In the other direction, a homomorphism from A into B[T] as modules over k may be considered as a homomorphism from A into B[[T]], and corresponds to a sequence {ϕ_{j}}_{j=0}^{∞} of homomorphisms from A into B as in (4.9.2). The condition that this mapping sends A into B[T] means that for each a ∈ A, we have that ϕ_{j}(a) = 0 for all but finitely many j. In particular, this holds when ϕ_{j} = 0 for all but finitely many j, as mappings from A into B. If for every a ∈ A we have that ϕ_{j}(a) = 0 for all but finitely many j, and if A is finitely generated as a module over k, then ϕ_{j} = 0 for all but finitely many j.

Consider the mapping from (4.9.3) to (4.9.4), as a module homomorphism from A into B[T]. This defines a natural homomorphism from (Hom_{k}(A, B))[T] into Hom_{k}(A, B[T]), as modules over k. It is easy to see that this mapping is injective. This mapping is also linear over k[T], where Hom_{k}(A, B[T]) is considered as a module over k[T], as in the previous section. If A is finitely generated as a module over k, then this mapping is surjective, as in the preceding paragraph.

4.10 Extensions and compositions

Let us continue with the same notation and hypotheses as in the preceding section. Let a(T) = ∑_{m=0}^{∞} a_{m} T^{m} be a formal power series in T with coefficients in A. If ϕ(T) is a formal power series in T with coefficients in Hom_{k}(A, B) as in (4.9.1), then

(ϕ(T))(a_{m}) = ∑_{j=0}^{∞} ϕ_{j}(a_{m}) T^{j} (4.10.1)

defines a formal power series in T with coefficients in B for every nonnegative integer m, as in (4.9.2). Using this, we can define

(ϕ(T))(a(T)) = ∑_{m=0}^{∞}(ϕ(T))(a_{m}) T^{m} (4.10.2)

as a formal power series in T with coefficients in B, as in Section 4.4. This corresponds to extending the homomorphism from A into B[[T]] associated to ϕ(T) to a homomorphism from A[[T]] into B[[T]], as modules over k[[T]], as before. Of course, if a(T) and ϕ(T) are formal polynomials in T with coefficients in A and Hom_{k}(A, B), respectively, then (4.10.1) is a formal polynomial in T with coefficients in B for each m ≥ 0, and (4.10.2) is a formal polynomial in T with coefficients in B too. This corresponds to extending the homomorphism from A into B[T] associated to ϕ(T) to a homomorphism from A[T] into B[T], as modules over k[T].

Alternatively, put E(ϕ, a) = ϕ(a) (4.10.3)

for every ϕ ∈ Hom_{k}(A, B) and a ∈ A, which is the natural evaluation mapping

(4.10.4) from Hom_{k}(A, B) × A into B.

This mapping is bilinear over k, and can be extended to a mapping

(4.10.5) from (Hom_{k}(A, B))[[T]] × A[[T]] into B[[T]],

as in Section 4.5. More precisely, let ϕ(T) ∈ (Hom_{k}(A, B))[[T]] and a(T) in A[[T]] be given as before, and put

E_{n}(ϕ(T), a(T)) = ∑_{j=0}^{n} ϕ_{j}(a_{n−j}) (4.10.6)

for each nonnegative integer n. This is an element of B for every n ≥ 0, so that

E(ϕ(T), a(T)) = ∑_{n=0}^{∞} E_{n}(ϕ(T), a(T)) T^{n} (4.10.7)

is an element of B[[T]]. It is easy to see that this is the same as (4.10.2). If ϕ(T) ∈ (Hom_{k}(A, B))[T] and a(T) ∈ A[T], then (4.10.6) is equal to 0 for all but finitely many n ≥ 0, so that (4.10.6) is an element of B[T]. This corresponds to extending (4.10.3) to a mapping

(4.10.8) from (Hom_{k}(A, B))[T] × A[T] into B[T],

as before.

Let C be another module over k, let ϕ(T) be a formal power series in T with coefficients in Hom_{k}(A, B) as in (4.9.1), and let

ψ(T) = ∑_{l=0}^{∞} ψ_{l} T^{l} (4.10.9)

be a formal power series in T with coefficients in Hom_{k}(B, C). Note that the composition ψ_{l} ∘ ϕ_{j} of ϕ_{j} and ψ_{l} defines a module homomorphism from A into C for all j, l ≥ 0. Thus

(ψ(T) ∘ ϕ(T))_{n} = ∑_{l=0}^{n} ψ_{l} ∘ ϕ_{n−l} (4.10.10)

is an element of Hom_{k}(A, C) for every nonnegative integer n. Put

ψ(T) ∘ ϕ(T) = ∑_{n=0}^{∞}(ψ(T) ∘ ϕ(T))_{n} T^{n}, (4.10.11)

which defines a formal power series in T with coefficients in Hom_{k}(A, C). If ϕ(T) and ψ(T) are formal polynomials in T, then (4.10.10) is equal to 0 for all but finitely many n ≥ 0, so that (4.10.11) is a formal polynomial in T as well.

The composition of module homomorphisms from A into B and from B into C defines a mapping

(4.10.12) from Hom_{k}(A, B) × Hom_{k}(B, C) into Hom_{k}(A, C)

that is bilinear over k. The definition of (4.10.11) corresponds to the extension of this bilinear mapping to mappings

(4.10.13)  from  (Hom_{k}(A, B))[[T]] × (Hom_{k}(B, C))[[T]]
into  (Hom_{k}(A, C))[[T]]

and

(4.10.14)  from  (Hom_{k}(A, B))[T] × (Hom_{k}(B, C))[T]
into  (Hom_{k}(A, C))[T],

as in Section 4.5.

As before, ϕ(T) ∈ (Hom_{k}(A, B))[[T]] corresponds to a homomorphism from A into B[[T]] as modules over k, which can be extended to a homomorphism from A[[T]] into B[[T]] as modules over k[[T]]. Similarly, ψ(T) corresponds to a homomorphism from B[[T]] into C[[T]] as modules over k[[T]], and (4.10.11) corresponds to a homomorphism from A[[T]] into C[[T]] as modules over k[[T]]. One can check that the homomorphism corresponding to (4.10.11) is the same as the composition of the homomorphisms corresponding to ϕ(T) and ψ(T). If ϕ(T) ∈ (Hom_{k}(A, B))[T] and ψ(T) ∈ (Hom_{k}(B, C))[T], then (4.10.11) is an element of (Hom_{k}(A, C))[T], as before. In this case, (4.10.11) corresponds to a homomorphism from A[T] into C[T], as modules over k[T], which is the composition of the homomorphisms from A[T] into B[T] and from B[T] into C[T] corresponding to ϕ(T) and ψ(T), respectively.

4.11 Two-step extensions

Let k be a commutative ring with a multiplicative identity element, let A, B, C be modules over k, and let T be an indeterminate. If β is a mapping from A×B into C[[T]] that is bilinear over k, then β can be extended to a mapping from A[[T]] × B[[T]] into C[[T]] that is bilinear over k[[T]], as in Sections 4.5 and 4.6. One can also look at this in terms of extending module homomorphisms, as in Sections 4.4 and 4.6, in two steps. More precisely, if a ∈ A, then β(a, b) defines a mapping from B into C[[T]], as a function of b, that is linear over k. This can be extended to a mapping from B[[T]] into C[[T]] that is linear over k[[T]], as before. If b(T) ∈ B[[T]], then we can use the extension just mentioned to get β(a, b(T)) as a function of a ∈ A with values in C[[T]] that is linear over k. This can be extended to a mapping from A[[T]] into C[[T]] that is linear over k[[T]], which extends β to a mapping from A[[T]] × B[[T]] into C[[T]] that is bilinear over k[[T]]. Similarly, if β is a mapping from A × B into C[T] that is bilinear over k, then β can be extended to a mapping from A[T] × B[T] into C[T] that is bilinear over k[T] in two steps.

Let β be a mapping from A×B into C[[T]] that is bilinear over k again, and put ρ_{a}(b) = β(a, b) (4.11.1)

for every a ∈ A and b ∈ B. If a ∈ A, then ρ_{a} defines a mapping from B into C[[T]] that is linear over k, which is to say that ρ_{a} ∈ Hom_{k}(B, C[[T]]). Thus

a ↦ ρ_{a} (4.11.2)

defines a mapping from A into Hom_{k}(B, C[[T]]), which is linear over k. Remember that there are natural isomorphisms between Hom_{k}(B, C[[T]]) and each of Hom_{k[[T]]}(B[[T]], C[[T]]) and (Hom_{k}(B, C))[[T]], as modules over k[[T]], as in Sections 4.8 and 4.9. The isomorphism with Hom_{k[[T]]}(B[[T]], C[[T]]) gives the extension of ρ_{a} to B[[T]], and the isomorphism with (Hom_{k}(B, C))[[T]] can be used to extend (4.11.2) to A[[T]].

Now let β be a mapping from A×B into C[T] that is bilinear over k. If a ∈ A, then (4.11.1) defines ρ_{a} as a mapping from B into C[T] that is linear over k, and hence an element of Hom_{k}(B, C[T]). Similarly, (4.11.2) defines a mapping from A into Hom_{k}(B, C[T]) that is linear over k. There is a natural isomorphism between Hom_{k}(B, C[T]) and Hom_{k[T]}(B[T], C[T]), as modules over k[T], as in Section 4.8. This permits one to identify ρ_{a} with a homomorphism from B[T] into C[T], as modules over k[T].

As in Section 4.9, there is a natural injection from (Hom_{k}(B, C))[T] into Hom_{k}(B, C[T]) that is linear over k[T]. If (4.11.2) corresponds to a mapping from A into (Hom_{k}(B, C))[T] that is linear over k, then this mapping can be extended to one from A[T] into (Hom_{k}(B, C))[T] that is linear over k[T] in the usual way. Otherwise, (4.11.2) can still be extended to a mapping from A[T] into Hom_{k}(B, C[T]) or equivalently Hom_{k[T]}(B[T], C[T]) that is linear over k[T]. This basically just uses the fact that Hom_{k}(B, C[T]) or equivalently Hom_{k[T]}(B[T], C[T]) is a module over k[T]. Of course, one can also look at this in terms of extending (4.11.1) in a for b fixed, as mentioned at the beginning of the section.

4.12 Extending algebra homomorphisms

Let k be a commutative ring with a multiplicative identity element, let T be an indeterminate, and let A, B be algebras over k in the strict sense. As in Section 4.6, multiplication on A and B can be extended to A[[T]] and B[[T]], respectively, so that they become algebras over k[[T]] in the strict sense. In particular, A[[T]] and B[[T]] may be considered as algebras over k.

Let ϕ be a homomorphism from A into B[[T]], as modules over k. Thus ϕ can be expressed as

ϕ(a) = ∑_{j=0}^{∞} ϕ_{j}(a) T^{j} (4.12.1)

for each a ∈ A, where ϕ_{j} is a homomorphism from A into B, as modules over k, for every nonnegative integer j. In order for ϕ to be a homomorphism from A into B[[T]], as algebras over k, we need to have that

ϕ(a a′) = ϕ(a) ϕ(a′) (4.12.2)

for every a, a′ ∈ A. Of course,

ϕ(a a′) = ∑_{n=0}^{∞} ϕ_{n}(a a′) T^{n}, (4.12.3)

and

(4.12.4)  ϕ(a) ϕ(a′) = ( ∑_{j=0}^{∞} ϕ_{j}(a) T^{j}) ( ∑_{l=0}^{∞} ϕ_{l}(a′) T^{l})

= ∑_{n=0}^{∞} ( ∑_{j=0}^{n} ϕ_{j}(a) ϕ_{n−j}(a′)) T^{n}.

It follows that (4.12.2) holds if and only if

ϕ_{n}(a a′) = ∑_{j=0}^{n} ϕ_{j}(a) ϕ_{n−j}(a′) (4.12.5)

for every nonnegative integer n.

As in Section 4.4, we can extend ϕ to a module homomorphism from A[[T]] into B[[T]] by putting

ϕ(f(T)) = ∑_{j=0}^{∞} ϕ(f_{j}) T^{j} (4.12.6)

for every f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ A[[T]]. If g(T) = ∑_{l=0}^{∞} g_{l} T^{l} is another element of A[[T]], then h(T) = f(T) g(T) is defined by putting h(T) = ∑_{n=0}^{∞} h_{n} T^{n}, where

h_{n} = ∑_{j=0}^{n} f_{j} g_{n−j} (4.12.7)

for every n ≥ 0, as in Section 4.6. Thus

ϕ(f(T) g(T)) = ϕ(h(T)) = ∑_{n=0}^{∞} ϕ(h_{n}) T^{n}, (4.12.8)

where the sum on the right is defined as an element of B[[T]] as in Section 4.4 again. If ϕ is an algebra homomorphism from A into B[[T]], then

ϕ(h_{n}) = ∑_{j=0}^{n} ϕ(f_{j} g_{n−j}) = ∑_{j=0}^{n} ϕ(f_{j}) ϕ(g_{n−j}) (4.12.9)

for each n ≥ 0. This implies that

ϕ(f(T) g(T)) = ∑_{n=0}^{∞} ( ∑_{j=0}^{n} ϕ(f_{j}) ϕ(g_{n−j})) T^{n} = ϕ(f(T)) ϕ(g(T)). (4.12.10)

More precisely, this follows from the definition of multiplication on B[[T]] when ϕ maps A into B. Otherwise, if ϕ(f_{j}) and ϕ(g_{l}) are elements of B[[T]], then the second step in (4.12.10) can be obtained as in (4.5.6). This shows that the extension of ϕ to a module homomorphism from A[[T]] into B[[T]] defined in (4.12.6) is an algebra homomorphism in this case.

Similarly, multiplication on A and B can be extended to A[T] and B[T], respectively, so that they become algebras in the strict sense over k[T], as in Section 4.6. If ϕ is a homomorphism from A into B[T], as modules over k, then we can extend ϕ to a module homomorphism from A[T] into B[T] as in (4.12.6). If ϕ is an algebra homomorphism from A into B[T], then this extension is an algebra homomorphism from A[T] into B[T], as before. Of course, there are analogous statements for opposite algebra homomorphisms.

4.13 Some remarks about commutators

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e. Also let T be an indeterminate, and let us identify e with the corresponding formal power series in T with coefficients in A in the usual way, which is the multiplicative identity element in A[[T]]. Suppose that a(T) = ∑_{j=0}^{∞} a_{j} T^{j} and b(T) = ∑_{j=0}^{∞} b_{j} T^{j} are elements of A[[T]] with a_{0} = b_{0} = e. Note that a(T) and b(T) are invertible in A[[T]], as in Section 4.7.

Put α(T) = a(T) − e = ∑_{j=1}^{∞} a_{j} T^{j} and β(T) = b(T) − e = ∑_{j=1}^{∞} b_{j} T^{j}, so that a(T) = e + α(T) and b(T) = e + β(T). Thus

a(T) b(T) = e + α(T) + β(T) + α(T) β(T). (4.13.1)

As in Section 4.7,

a(T)^{−1} = ∑_{l=0}^{∞}(−α(T))^{l} = e + ∑_{l=1}^{∞}(−α(T))^{l} (4.13.2)

and

b(T)^{−1} = ∑_{l=0}^{∞}(−β(T))^{l} = e + ∑_{l=1}^{∞}(−β(T))^{l}. (4.13.3)

If n ∈ Z_{+}, then we let O(T^{n}) refer to any element of A[[T]] that vanishes to order n − 1, which means that it can be expressed as an element of A[[T]] times T^{n}. Observe that

a(T) b(T) a(T)^{−1} b(T)^{−1} = e + O(T), (4.13.4)

because a(T), b(T), a(T)^{−1}, b(T)^{−1} = e + O(T). More precisely,

a(T) b(T) a(T)^{−1} b(T)^{−1} = e + O(T^{2}), (4.13.5)

because a(T) = e + a_{1} T + O(T^{2}), b(T) = e + b_{1} T + O(T^{2}), (4.13.6)

and a(T)^{−1} = e − a_{1} T + O(T^{2}), b(T)^{−1} = e − b_{1} T + O(T^{2}). (4.13.7)

We also have that

a(T) = e + a_{1} T + a_{2} T^{2} + O(T^{3}), b(T) = e + b_{1} T + b_{2} T^{2} + O(T^{3}), (4.13.8)

and

a(T) b(T) = e + a_{1} T + b_{1} T + a_{2} T^{2} + b_{2} T^{2} + a_{1} b_{1} T^{2} + O(T^{3}). (4.13.9)

Using (4.13.2), we get that

(4.13.10)  a(T)^{−1} = e − α(T) + α(T)^{2} + O(T^{3})
= e − a_{1} T − a_{2} T^{2} + a_{1}^{2} T^{2} + O(T^{3}).

Similarly, b(T)^{−1} = e − b_{1} T − b_{2} T^{2} + b_{1}^{2} T^{2} + O(T^{3}). (4.13.11)

It follows that

(4.13.12) a(T)^{−1} b(T)^{−1} = e − a_{1} T − b_{1} T − a_{2} T^{2} − b_{2} T^{2}
+a_{1}^{2} T^{2} + b_{2}^{2} T^{2} + a_{1} b_{1} T^{2} + O(T^{3}).

Combining (4.13.9) and (4.13.12), it is not difficult to verify that

a(T) b(T) a(T)^{−1} b(T)^{−1} = e + a_{1} b_{1} T^{2} − b_{1} a_{1} T^{2} + O(T^{3})
(4.13.13)  = e + α(T) β(T) − β(T) α(T) + O(T^{3}).

4.14 Formal power series and involutions

Let k be a commutative ring with a multiplicative identity element, and let T be an indeterminate. Also let A be an algebra over k in the strict sense, and let x ↦ x^{∗} be an opposite algebra automorphism on A. If k = C, then we may wish to consider opposite algebra automorphisms that are conjugate-linear, as usual. If a(T) = ∑_{j=0}^{∞} a_{j} T^{j} ∈ A[[T]], then

a(T)^{∗} = ∑_{j=0}^{∞} a_{j}^{∗} T^{j} (4.14.1)

defines an element of A[[T]], which is in A[T] when a(T) ∈ A[T]. Of course,

a(T) ↦ a(T)^{∗} (4.14.2)

defines an opposite algebra automorphism on A[[T]]. Note that a(T) is selfadjoint with respect to (4.14.2) if and only if a_{j} is self-adjoint for every j ≥ 0, and similarly a(T) is anti-self-adjoint with respect to (4.14.2) if and only if a_{j} is anti-self-adjoint for every j ≥ 0. If x ↦ x^{∗} is an involution on A, then (4.14.2) is an involution on A[[T]].

Suppose now that A is an associative algebra over k with a multiplicative identity element e. Remember that e^{∗} = e, as in Section 2.6. Let y be an element of A, and put y_{0} = y − e, so that y = e + y_{0}, y^{∗} = e + y_{0}^{∗}, and

y^{∗} y = (e + y_{0}^{∗}) (e + y_{0}) = e + y_{0} + y_{0}^{∗} + y_{0}^{∗} y_{0}. (4.14.3)

Suppose that y_{1}, y_{2} ∈ A satisfy

y_{0} = y_{1} + y_{2}, y_{1}^{∗} = y_{1}, y_{2}^{∗} = −y_{2}, and y_{1} y_{2} = y_{2} y_{1}. (4.14.4)

Under these conditions, y_{0}^{∗} = y_{1} − y_{2}, and

y^{∗} y = e + 2 · y_{1} + (y_{1} − y_{2}) (y_{1} + y_{2}) = e + 2 · y_{1} + y_{1}^{2} − y_{2}^{2}. (4.14.5)

Let a(T) = ∑_{j=0}^{∞} a_{j} T^{j} be an element of A[[T]] with a_{0} = e, and put α(T) = a(T) − e. As in (4.14.3), we have that

a(T)^{∗} a(T) = e + α(T) + α(T)^{∗} + α(T)^{∗} α(T). (4.14.6)

In particular, a necessary condition for

a(T)^{∗} a(T) = e (4.14.7)

to hold is that a_{1}^{∗} = −a_{1}. (4.14.8)

More precisely, (4.14.8) is equivalent to a(T)^{∗} a(T) = e+O(T^{2}), in the notation of the previous section. Of course, (4.14.7) is the same as saying that

a(T)^{∗} = a(T)^{−1}, (4.14.9)

which also implies that a(T) commutes with a(T)^{∗}.

Suppose that β(T) = ∑_{j=0}^{∞} β_{j} T^{j}, γ(T) = ∑_{j=0}^{∞} γ_{j} T^{j} ∈ A[[T]] satisfy β_{0} = γ_{0} = 0, α(T) = β(T) + γ(T),

β(T)^{∗} = β(T), γ(T)^{∗} = −γ(T), and β(T) γ(T) = γ(T) β(T). (4.14.10)

Under these conditions, we have that

a(T)^{∗} a(T) = e + 2 · β(T) + β(T)^{2} − γ(T)^{2}, (4.14.11)

as in (4.14.5). In this situation, (4.14.7) holds exactly when

2 · β(T) = −β(T)^{2} + γ(T)^{2}. (4.14.12)

Let us suppose from now on in this section that 1+1 has a multiplicative inverse in k. It is easy to see that β(T) is uniquely determined by γ(T) and (4.14.12).

More precisely, if γ(T) ∈ A[[T]] satisfies γ_{0} = 0, then there is a unique β(T) ∈ A[[T]] that satisfies β_{0} = 0 and (4.14.12). Indeed, for each j ∈ Z_{+}, one can use (4.14.12) to get β_{j} from β_{l} with l < j and the coefficients of γ(T). One can in fact approximate β in terms of polynomials in γ(T)^{2} with coefficients in k, using (4.14.12). This implies that β(T) commutes with γ(T), which could also be verified more directly. If γ(T)^{∗} = −γ(T), as in (4.14.10), then

(γ(T)^{2})^{∗} = (γ(T)^{∗})^{2} = (−γ(T))^{2} = γ(T)^{2}. (4.14.13)

In this case, one can check that β(T)^{∗} = β(T). If we take α(T) = β(T) + γ(T) and a(T) = e + α(T), as before, then (4.14.11) holds, which implies (4.14.7), by (4.14.12).

Some related notions

5.1 Adjoining nilpotent elements

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. If n ∈ Z_{+}, then we would like to define A[ϵ_{n}] as a module over k, where ϵ_{n} is an additional element that is considered to satisfy

ϵ_{n}^{n+1} = 0. (5.1.1)

The elements of A[ϵ_{n}] may be expressed as formal sums of the form

a = a_{0} + a_{1} ϵ_{n} + ⋯ + a_{n} ϵ_{n}^{n}, (5.1.2)

where a_{0}, a_{1}, …, a_{n} ∈ A. Addition and scalar multiplication on A[ϵ_{n}] are defined termwise, so that A[ϵ_{n}] becomes a module over k that contains A as a submodule. More precisely, A[ϵ_{n}] is isomorphic to the direct sum of n+1 copies of A, as a module over k. If a ∈ A[ϵ_{n}] is as in (5.1.2), then

a ϵ_{n} = a_{0} ϵ_{n} + ⋯ + a_{n−1} ϵ_{n}^{n} (5.1.3)

defines an element of A[ϵ_{n}] as well. This defines

a ↦ a ϵ_{n} (5.1.4)

as a module homomorphism from A[ϵ_{n}] into itself. It is sometimes convenient to consider the a_{0} term on the right side of (5.1.2) as being a_{0} ϵ_{n}^{0}, so that multiplication by ϵ_{n}^{0} corresponds to the identity mapping on A[ϵ_{n}].

Let B be another module over k, so that B[ϵ_{n}] can be defined as in the previous paragraph. Also let ϕ be a homomorphism from A into B[ϵ_{n}], as modules over k. If a ∈ A[ϵ_{n}] is as in (5.1.2), then

ϕ(a) = ϕ(a_{0}) + ϕ(a_{1}) ϵ_{n} + ⋯ + ϕ(a_{n}) ϵ_{n}^{n} (5.1.5)

defines an element of B[ϵ_{n}]. This defines an extension of ϕ to a homomorphism from A[ϵ_{n}] into B[ϵ_{n}], as modules over k. It is easy to see that

ϕ(a ϵ_{n}) = ϕ(a) ϵ_{n} (5.1.6) for every a ∈ A[ϵ_{n}], and that this extension of ϕ to A[ϵ_{n}] is uniquely determined by these properties.

Let C be another module over k, so that C[ϵ_{n}] can be defined as before. Also let β be a mapping from A × B into C[ϵ_{n}] that is bilinear over k. We can extend β to a mapping from A[ϵ_{n}] × B[ϵ_{n}] into C[ϵ_{n}], as follows. Let a ∈ A[ϵ_{n}] be as in (5.1.2), and let

b = b_{0} + b_{1} ϵ_{n} + ⋯ + b_{n} ϵ_{n}^{n} (5.1.7)

be an element of B[ϵ_{n}], with b_{0}, b_{1}, …, b_{n} ∈ B. Thus

β(a, b) = ∑_{j=0}^{n} ∑_{l=0}^{n} β(a_{j}, b_{l}) ϵ_{n}^{j+l} (5.1.8)

defines an element of C[ϵ_{n}], which extends β to a mapping from A[ϵ_{n}] × B[ϵ_{n}] into C[ϵ_{n}] that is bilinear over k. One can check that

β(a ϵ_{n}, b) = β(a, b ϵ_{n}) = β(a, b) ϵ_{n} (5.1.9)

for every a ∈ A[ϵ_{n}] and b ∈ B[ϵ_{n}], and that this extension of β to A[ϵ_{n}] × B[ϵ_{n}] is uniquely determined by the properties. One can also look at this in terms of extending β in each variable separately, as in the previous paragraph.

Suppose now that A = B, so that we start with a mapping β from A × A into C[ϵ_{n}]. If β is symmetric or antisymmetric on A×A, then the extension of β to A[ϵ_{n}]×A[ϵ_{n}] into C[ϵ_{n}] as in the preceding paragraph has the same property. Similarly, if β(a, a) = 0 (5.1.10)

for every a ∈ A, then (5.1.10) holds for every a ∈ A[ϵ_{n}]. To see this, let a ∈ A[ϵ_{n}] be given as in (5.1.2), so that

β(a, a) = ∑_{j=0}^{n} ∑_{l=0}^{n} β(a_{j}, a_{l}) ϵ_{n}^{j+l}, (5.1.11)

as in (5.1.8). By hypothesis, β(a_{j}, a_{j}) = 0 for each j. Remember that β is antisymmetric on A × A, as in Section 2.1. This implies that

β(a_{j}, a_{l}) + β(a_{l}, a_{j}) = 0 (5.1.12)

when j ≠ l, which can be used to get that (5.1.11) is equal to 0.

5.2 Adjoining nilpotent elements to algebras

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Also let n be a positive integer, and let ϵ_{n} be as in the previous section, so that A[ϵ_{n}] can be defined as a module over k as before. If a, b ∈ A[ϵ_{n}] are as in (5.1.2) and (5.1.7), respectively, then

a b = ∑_{j=0}^{n} ∑_{l=0}^{n} a_{j} b_{l} ϵ_{n}^{j+l} (5.2.1)

defines an element of A[ϵ_{n}] too. This extends multiplication on A to a mapping from A[ϵ_{n}] × A[ϵ_{n}] into A[ϵ_{n}] that is bilinear over k, as in the previous section. Thus A[ϵ_{n}] becomes an algebra over k in the strict sense as well.

If multiplication on A is commutative, then this extension of multiplication to A[ϵ_{n}] is commutative too. If multiplication on A is associative, then one can verify that multiplication on A[ϵ_{n}] is associative as well. If A has a multiplicative identity element e, then e is also the multiplicative identity element in A[ϵ_{n}]. We can apply this to A = k, to get k[ϵ_{n}] as a commutative associative algebra over k with a multiplicative identity element.

Let A be a module over k, so that A[ϵ_{k}] can be defined as a module over k as in the previous section. In fact, A[ϵ_{n}] may be considered as a module over k[ϵ_{n}]. More precisely, let a ∈ k[ϵ_{n}] be as in (5.1.2), with a_{0}, a_{1}, …, a_{n} ∈ k, and let b ∈ A[ϵ_{n}] be as in (5.1.7), with b_{0}, b_{1}, …, b_{n} ∈ A. Under these conditions, a b can be defined as an element of A[ϵ_{n}] as in (5.2.1), where a_{j} b_{l} is defined as an element of A using scalar multiplication with respect to k. This is the same as extending scalar multiplication on A as a bilinear mapping from k ×A into A to a bilinear mapping from k[ϵ_{n}] × A[ϵ_{n}] into A[ϵ_{n}], as in the previous section.

Let B be another module over k, so that B[ϵ_{n}] can be defined as before. It is easy to see that a mapping ϕ from A[ϵ_{n}] into B[ϵ_{n}] is linear over k[ϵ_{n}] if and only if ϕ is linear over k and (5.1.6) holds for every a ∈ A[ϵ_{n}]. Similarly, let C be a module over k, and let C[ϵ_{n}] be as before. One can check that a mapping β from A[ϵ_{n}] × B[ϵ_{n}] into C[ϵ_{n}] is bilinear over k[ϵ_{n}] if and only if β is bilinear over k and satisfies (5.1.9) for ever a ∈ A[ϵ_{n}] and b ∈ B[ϵ_{n}]. If A is an algebra over k in the strict sense, then A[ϵ_{n}] may be considered as an algebra over k[ϵ_{n}] in the strict sense.

If (A, [·, ·]_{A}) is a Lie algebra over k, then [·, ·]_{A} can be extended to A[ϵ_{n}] as before. More precisely, if a, b ∈ A[ϵ_{n}] are as in (5.1.2) and (5.1.7), respectively, then

[a, b]_{A[ϵ_{n}]} = ∑_{j=0}^{n} ∑_{l=0}^{n}[a_{j}, b_{l}]_{A} ϵ_{n}^{j+l} (5.2.2)

defines an element of A[ϵ_{n}] as well. As in the previous section, [a, a]_{A[ϵ_{n}]} = 0 for every a ∈ A[ϵ_{n}], because of the analogous property of [·, ·]_{A} on A. One can verify that (5.2.2) satisfies the Jacobi identity on A[ϵ_{n}], using the Jacobi identity for [·, ·]_{A} on A. Thus A[ϵ_{n}] is a Lie algebra over k with respect to (5.2.2), which may be considered as a Lie algebra over k[ϵ_{n}].

Let A be a module over k again, and let T be an indeterminate, so that the space A[[T]] of formal power series in T with coefficients in A can be defined as in Section 4.3. If f(T) = ∑_{j=0}^{∞} f_{j} T^{j} is an element of A[[T]], then

f_{0} + f_{1} ϵ_{n} + ⋯ + f_{n} ϵ_{n}^{n} (5.2.3)

5.3. INVERTIBILITY IN A[ϵ_{N}] 103

defines an element of A[ϵ]. This defines a homomorphism from A[[T]] onto A[ϵ_{n}], as modules over k. This homomorphism also maps the space A[T] of formal polynomials in T with coefficients in A onto A[ϵ_{n}]. Note that multiplication by T on A[[T]] corresponds to multiplication by ϵ_{n} on A[ϵ_{n}] with respect to this homomorphism.

If A is an algebra over k in the strict sense, then one can check that the mapping from f(T) in A[[T]] to (5.2.3) defines a homomorphism from A[[T]] onto A[ϵ_{n}] as algebras over k. In particular, we can apply this to A = k.

5.3 Invertibility in A[ϵ_{n}]

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e. Also let n be a positive integer, and let A[ϵ_{n}] be as in the previous two sections. Remember that a ∈ A[ϵ_{n}] can be expressed as

a = a_{0} + a_{1} ϵ_{n} + ⋯ + a_{n} ϵ_{n}^{n}, (5.3.1)

where a_{0}, a_{1}, …, a_{n} ∈ A. The mapping

a ↦ a_{0} (5.3.2)

defines an algebra homomorphism from A[ϵ_{n}] onto A. If a is invertible in A[ϵ_{n}], then it follows that a_{0} is invertible in A.

Note that

(e − a) ∑_{l=0}^{n} a^{l} = ( ∑_{l=0}^{n} a^{l}) (e − a) = e − a^{n+1} (5.3.3)

for every a ∈ A[ϵ_{n}], where a^{l} is interpreted as being equal to e when l = 0, as usual. If a_{0} = 0, then a^{n+1} = 0, so that (5.3.3) reduces to

(e − a) ∑_{l=0}^{n} a^{l} = ( ∑_{l=0}^{n} a^{l}) (e − a) = e. (5.3.4)

This means that e − a is invertible in A[ϵ_{n}], with

(e − a)^{−1} = ∑_{l=0}^{n} a^{l}. (5.3.5)

If b = b_{0} + b_{1} ϵ_{n} + ⋯ + b_{n} ϵ_{n} (5.3.6)

is an element of A[ϵ_{n}], where b_{0} is an invertible element of A, then b can be expressed as b_{0} (e − a), where a ∈ A[ϵ_{n}] is as in (5.3.1), with a_{0} = 0. It follows that b is invertible in A[ϵ_{n}], because e − a is invertible.

104 CHAPTER 5. SOME RELATED NOTIONS

Suppose now that a, b ∈ A[ϵ_{n}] are as in (5.3.1) and (5.3.6), respectively, with a_{0} = b_{0} = e. Put α = a − e and β = b − e, so that a = e + α, b = e + β, a b = e + α + β + α β, (5.3.7) and α, β are multiples of ϵ_{n}. As in the preceding paragraph, a and b are invertible elements of A[ϵ_{n}], with

a^{−1} = ∑_{l=0}^{n}(−α)^{l} = e + ∑_{l=1}^{n}(−α)^{l} (5.3.8)

and

b^{−1} = ∑_{l=0}^{n}(−β)^{l} = e + ∑_{l=1}^{n}(−β)^{l}. (5.3.9)

We also have that

(5.3.10)  a b a^{−1} b^{−1} = e + a_{1} b_{1} ϵ_{n}^{2} − b_{1} a_{1} ϵ_{n}^{2} + O(ϵ_{n}^{3})
= e + α β − β α + O(ϵ_{n}^{3}),

where O(ϵ_{n}^{3}) refers to any element of A[ϵ_{n}] that is a multiple of ϵ_{n}^{3}. This can be verified in the same way as in Section 4.13, or reduced to that situation, using the homomorphism from A[[T]] onto A[[ϵ_{n}]] mentioned in the previous section.

If n = 1, then α = a_{1} ϵ_{1}, β = b_{1} ϵ_{1}, and (5.3.7) reduces to a b = e + α + β = e + a_{1} ϵ_{1} + b_{1} ϵ_{1}. (5.3.11) In this case, (5.3.8) and (5.3.9) reduce to a^{−1} = e − α = e − a_{1} ϵ_{1} (5.3.12) and b^{−1} = e − β = e − b_{1} ϵ_{1}. (5.3.13) Note that a and b commute in this situation, so that a b a^{−1} b^{−1} = e.

If n = 2, then α = a_{1} ϵ_{2} + a_{2} ϵ_{2}^{2}, β = b_{1} ϵ_{2} + b_{2} ϵ_{2}^{2}, and (5.3.7) reduces to a b = e + a_{1} ϵ_{2} + a_{2} ϵ_{2}^{2} + b_{1} ϵ_{2} + b_{2} ϵ_{2}^{2} + a_{1} b_{1} ϵ_{2}^{2}. (5.3.14) Similarly, (5.3.10) reduces to a b a^{−1} b^{−1} = e + a_{1} b_{1} ϵ_{2}^{2} − b_{1} a_{1} ϵ_{2}^{2}. (5.3.15)

5.4 Adjoining two nilpotent elements

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. We would like to define A[ϵ_{1}, η_{1}] as a module over k, where ϵ_{1} and η_{1} are additional elements that are considered to satisfy ϵ_{1}^{2} = η_{1}^{2} = 0, ϵ_{1} η_{1} = η_{1} ϵ_{1}. (5.4.1)

The elements of A[ϵ_{1}, η_{1}] can be expressed as formal sums of the form

a = a_{0,0} + a_{1,0} ϵ_{1} + a_{0,1} η_{1} + a_{1,1} ϵ_{1} η_{1}, (5.4.2)

where a_{0,0}, a_{1,0}, a_{0,1}, a_{1,1} ∈ A. Addition and scalar multiplication on A[ϵ_{1}, η_{1}] are defined termwise, so that A[ϵ_{1}, η_{1}] becomes a module over k, which is isomorphic to the direct sum of four copies of A. By construction, A[ϵ_{1}, η_{1}] contains a copy of A, and in fact A[ϵ_{1}, η_{1}] contains copies of A[ϵ_{1}] and A[η_{1}], which are defined as in Section 5.1. If a ∈ A[ϵ_{1}, η_{1}] is as in (5.4.2), then

a ϵ_{1} = a_{0,0} ϵ_{1} + a_{0,1} ϵ_{1} η_{1}, a η_{1} = a_{0,0} η_{1} + a_{1,0} ϵ_{1} η_{1} (5.4.3)

define elements of A[ϵ_{1}, η_{1}]. This defines

a ↦ a ϵ_{1}, a ↦ a η_{1} (5.4.4)

as module homomorphisms from A[ϵ_{1}, η_{1}] into itself. One can also look at A[ϵ_{1}, η_{1}] in terms of adjoining ϵ_{1} and η_{1} separately, as before.

Let B be another module over k, and let ϕ be a homomorphism from A into B[ϵ_{1}, η_{1}], as modules over k, where B[ϵ_{1}, η_{1}] is defined as in the previous paragraph. If a ∈ A[ϵ_{1}, η_{1}] is as in (5.4.2), then

ϕ(a) = ϕ(a_{0,0}) + ϕ(a_{1,0}) ϵ_{1} + ϕ(a_{0,1}) η_{1} + ϕ(a_{1,1}) ϵ_{1} η_{1} (5.4.5)

defines an element of B[ϵ_{1}, η_{1}]. This extends ϕ to a homomorphism from A[ϵ_{1}, η_{1}] into B[ϵ_{1}, η_{1}], as modules over k. This extension satisfies

ϕ(a ϵ_{1}) = ϕ(a) ϵ_{1}, ϕ(a η_{1}) = ϕ(a) η_{1} (5.4.6)

for every a ∈ A[ϵ_{1}, η_{1}], and is uniquely determined by these properties.

Similarly, let C be another module over k, and let β be a mapping from A × B into C[ϵ_{1}, η_{1}] that is bilinear over k, where C[ϵ_{1}, η_{1}] is defined as before. One can extend β to a mapping from A[ϵ_{1}, η_{1}] × B[ϵ_{1}, η_{1}] into C[ϵ_{1}, η_{1}] that is bilinear over k and satisfies

β(a ϵ_{1}, b) = β(a, b ϵ_{1}) = β(a, b) ϵ_{1} (5.4.7)

and β(a η_{1}, b) = β(a, b η_{1}) = β(a, b) η_{1} (5.4.8)

for every a ∈ A[ϵ_{1}, η_{1}] and b ∈ B[ϵ_{1}, η_{1}].

Now let A be an algebra over k in the strict sense. Multiplication on A can be extended to a mapping from A[ϵ_{1}, η_{1}]×A[ϵ_{1}, η_{1}] into A[ϵ_{1}, η_{1}] that is bilinear over k, as in the preceding paragraph, so that A[ϵ_{1}, η_{1}] becomes an algebra over k in the strict sense too. If multiplication on A is commutative, then multiplication on A[ϵ_{1}, η_{1}] is commutative as well. If multiplication on A is associative, then one can check that multiplication on A[ϵ_{1}, η_{1}] is associative. If A has a multiplicative identity element e, then e is the multiplicative identity element in A[ϵ_{1}, η_{1}] too. One can look at A[ϵ_{1}, η_{1}] as an algebra in terms adjoining ϵ_{1} and η_{1} separately, as before. Thus these and other properties of A[ϵ_{1}, η_{1}] can be obtained from the remarks in Sections 5.1 and 5.2.

5.5 Invertibility in A[ϵ_{1}, η_{1}]

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e. Thus A[ϵ_{1}, η_{1}] is an associative algebra that contains A as a subalgebra, as in the previous section, and e is the multiplicative identity element of A[ϵ_{1}, η_{1}]. As before, a ∈ A[ϵ_{1}, η_{1}] as in (5.4.2), with a_{0,0}, a_{1,0}, a_{0,1}, a_{1,1} ∈ A. The mapping

a ↦ a_{0,0} (5.5.1)

defines an algebra homomorphism from A[ϵ_{1}, η_{1}] onto A. If a is invertible in A[ϵ_{1}, η_{1}], then a_{0,0} is invertible in A.

As usual,

(e − a) ∑_{l=0}^{2} a^{l} = ( ∑_{l=0}^{2} a^{l}) (e − a) = e − a^{3} (5.5.2)

for every a ∈ A[ϵ_{1}, η_{1}], where a^{l} is interpreted as being e when l = 0. If a is as in (5.4.2), with a_{0,0} = 0, then a^{3} = 0, and hence

(e − a) ∑_{l=0}^{2} a^{l} = ( ∑_{l=0}^{2} a^{l}) (e − a) = e. (5.5.3)

It follows that e − a is invertible in A[ϵ_{1}, η_{1}], with

(e − a)^{−1} = ∑_{l=0}^{2} a^{l} = e + a + a^{2}. (5.5.4)

If b = b_{0,0} + b_{1,0} ϵ_{1} + b_{0,1} η_{1} + b_{1,1} ϵ_{1} η_{1} (5.5.5)

is an element of A[ϵ_{1}, η_{1}], where b_{0,0} is an invertible element of A, then b can be expressed as b_{0,0} (e − a), where a ∈ A[ϵ_{1}, η_{1}] is as in (5.4.2), with a_{0,0} = 0. This implies that b is invertible in A[ϵ_{1}, η_{1}], because e − a is invertible.

Suppose that a, b ∈ A[ϵ_{1}, η_{1}] are as in (5.4.2) and (5.5.5), respectively, with a_{0,0} = b_{0,0} = e. Put

α = a − e = a_{1,0} ϵ_{1} + a_{0,1} η_{1} + a_{1,1} ϵ_{1} η_{1} (5.5.6)

and β = b − e = b_{1,0} ϵ_{1} + b_{0,1} η_{1} + b_{1,1} ϵ_{1} η_{1}. (5.5.7)

Thus a = e + α, b = e + β, and

a b = e + α + β + α β = e + α + β + a_{1,0} b_{0,1} ϵ_{1} η_{1} + a_{0,1} b_{1,0} ϵ_{1} η_{1}. (5.5.8)

We also have that a and b are invertible elements of A[ϵ_{1}, η_{1}], as in the preceding paragraph, with

a^{−1} = ∑_{l=0}^{2}(−α)^{l} = e − α + α^{2} = e − α + 2 · a_{1,0} a_{0,1} ϵ_{1} η_{1} (5.5.9)

and

b^{−1} = ∑_{l=0}^{2}(−β)^{l} = e − β + β^{2} = e − β + 2 · b_{1,0} b_{0,1} ϵ_{1} η_{1}. (5.5.10)

Here 2 · x denotes x + x for any element x of A or A[ϵ_{1}, η_{1}].

Suppose now that a_{0,1} = a_{1,1} = b_{1,0} = b_{1,1} = 0, so that

a = e + a_{1,0} ϵ_{1}, b = e + b_{0,1} η_{1} (5.5.11)

and α = a_{1,0} ϵ_{1}, β = b_{0,1} η_{1}. In this case, (5.5.8) reduces to

a b = e + a_{1,0} ϵ_{1} + b_{0,1} η_{1} + a_{1,0} b_{0,1} ϵ_{1} η_{1}. (5.5.12)

Similarly, (5.5.9) and (5.5.10) reduce to

a^{−1} = e − a_{1,0} ϵ_{1}, b^{−1} = e − b_{0,1} η_{1}. (5.5.13)

Thus a^{−1} b^{−1} = e − a_{1,0} ϵ_{1} − b_{0,1} η_{1} + a_{1,0} b_{0,1} ϵ_{1} η_{1}. (5.5.14)

Combining (5.5.12) and (5.5.14), one can verify that

a b a^{−1} b^{−1} = e + a_{1,0} b_{0,1} ϵ_{1} η_{1} − b_{0,1} a_{1,0} ϵ_{1} η_{1}. (5.5.15)

5.6 Differentiation

Let k be a commutative ring with a multiplicative identity element, and let T be an indeterminate. Also let A be a module over k, so that the spaces A[T] and A[[T]] of formal polynomials and power series in T with coefficients in A can be defined as in Section 4.3. If f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ A[[T]], then the formal derivative of f(T) is defined by

f′(T) = ∑_{j=1}^{∞} j · f_{j} T^{j−1} = ∑_{j=0}^{∞}(j + 1) · f_{j+1} T^{j}, (5.6.1)

where j · a is the sum of j a’s in A for every j ∈ Z_{+} and a ∈ A. Thus f′(T) ∈ A[[T]] too, and f(T) ↦ f′(T) (5.6.2)

is a homomorphism from A[[T]] into itself, as a module over k. If f(T) ∈ A[T], then f′(T) ∈ A[T] as well.

Let B and C be two more modules over k, and let β be a mapping from A × B into C that is bilinear over k. Let f(T) ∈ A[[T]] be given as before, as well as g(T) = ∑_{l=0}^{∞} g_{l} T^{l} ∈ B[[T]]. Put

h_{n} = ∑_{j=0}^{n} β(f_{j}, g_{n−j}) (5.6.3)

for every nonnegative integer n, and

β(f(T), g(T)) = h(T) = ∑_{n=0}^{∞} h_{n} T^{n}, (5.6.4)

as in Section 4.5. We would like to verify that

h′(T) = β(f′(T), g(T)) + β(f(T), g′(T)). (5.6.5)

This is the same as saying that

(5.6.6)  (n + 1) · h_{n+1} = ∑_{j=0}^{n} β((j + 1) · f_{j+1}, g_{n−j})

+ ∑_{j=0}^{n} β(f_{j}, (n − j + 1) · g_{n−j+1})

for every n ≥ 0. By the definition (5.6.3) of h_{n}, we have that

(n + 1) · h_{n+1} = ∑_{j=0}^{n+1}(n + 1) · β(f_{j}, g_{n+1−j})

(5.6.7)  = ∑_{j=0}^{n+1} j · β(f_{j}, g_{n+1−j}) + ∑_{j=0}^{n+1}(n + 1 − j) · β(f_{j}, g_{n+1−j})

= ∑_{j=1}^{n+1} j · β(f_{j}, g_{n+1−j}) + ∑_{j=0}^{n}(n + 1 − j) · β(f_{j}, g_{n+1−j})

= ∑_{j=0}^{n}(j + 1) · β(f_{j+1}, g_{n−j}) + ∑_{j=0}^{n}(n − j + 1) · β(f_{j}, g_{n−j+1})

for each n ≥ 0. This implies (5.6.6), as desired.

Let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. As in Section 4.6, multiplication on A can be extended to A[[T]], so that A[[T]] becomes an algebra over k in the strict sense too. If f(T) and g(T) are elements of A[[T]] and h(T) = f(T) g(T), then

h′(T) = f′(T) g(T) + f(T) g′(T), (5.6.8)

as in (5.6.5). Thus (5.6.2) defines a derivation on A[[T]], as an algebra over k. Similarly, the restriction of (5.6.2) to A[T] defines a derivation on A[T], as an algebra over k.

Let A be a module over k again, and remember that A[[T]] may be considered as a module over k[[T]], as in Section 4.6. Let f(T) ∈ k[[T]] and g(T) ∈ A[[T]] be given, so that h(T) = f(T) g(T) defines an element of A[[T]] as well. Under these conditions, f′(T) ∈ k[[T]], g′(T), h′(T) ∈ A[[T]], and (5.6.8) holds in A[[T]]. This may be considered as another instance of (5.6.5). More precisely, this uses scalar multiplication on A as a bilinear mapping from k × A into A, and its extension to k[[T]] × A[[T]].

5.7 Polynomial functions

Let k be a commutative ring with multiplicative identity element, let A be a module over k, and let T be an indeterminate. Also let f(T) = ∑_{j=0}^{n} f_{j} T^{j} be a formal polynomial in T with coefficients in A, as in Section 4.3. If t ∈ k, then

f(t) = ∑_{j=0}^{n} f_{j} t^{j} (5.7.1)

defines an element of A, where f_{j} t^{j} is defined using scalar multiplication on A, and t^{j} is interpreted as being the multiplicative identity element 1 in k when j = 0. The mapping f(T) ↦ f(t) (5.7.2)

defines a homomorphism from A[T] into A, as modules over k.

Let B and C be modules over k as well, and let β be a mapping from A × B into C that is bilinear over k. If f(T) ∈ A[T] and g(T) ∈ B[T], then

h(T) = β(f(T), g(T)) (5.7.3)

can be defined as an element of C[T] as in Section 4.5. Under these conditions, one can check that h(t) = β(f(t), g(t)) (5.7.4)

for every t ∈ k.

Now let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Remember that multiplication on A can be extended to A[T], so that A[T] becomes an algebra over k in the strict sense, as in Section 4.6. If f(T), g(T) ∈ A[T] and h(T) = f(T) g(T), then

h(t) = f(t) g(t) (5.7.5)

for every t ∈ k, as in (5.7.4). Thus (5.7.2) defines a homomorphism from A[T] into A, as algebras over k.

Let A be a module over k again, so that A[T] may be considered as a module over k[T], as in Section 4.6. Let f(T) ∈ k[T] and g(T) ∈ A[T] be given, and let h(T) = f(T) g(T) be their product in A[T]. If t ∈ k, then f(t) is defined as an element of k, g(t) and h(t) are defined as elements of A, and (5.7.5) holds, as in (5.7.4).

Let t ∈ k be given, and suppose that ϵ ∈ k satisfies

ϵ^{2} = 0. (5.7.6)

Noe that (t + ϵ)^{j} = t^{j} + j · t^{j−1} ϵ (5.7.7) for every positive integer j. If f(T) = ∑_{j=0}^{n} f_{j} T^{j} ∈ A[T], then

(5.7.8)  f(t + ϵ) = ∑_{j=0}^{n} f_{j} (t + ϵ)^{j}

= ∑_{j=0}^{n} f_{j} t^{j} + ∑_{j=1}^{n} j · f_{j} t^{j−1} ϵ = f(t) + f′(t) ϵ.

Here f′(T) ∈ A[T] is as defined in the previous section, so that f′(t) is defined as an element of A as before.

Let A be an associative algebra over k with a multiplicative identity element e. If f(T) = ∑_{j=0}^{n} f_{j} t^{j} ∈ k[T] and a ∈ A, then

f(a) = ∑_{j=0}^{n} f_{j} a^{j} (5.7.9)

is defined as an element of A, where a^{j} is interpreted as being equal to e when j = 0. Let g(T) be another element of k[T], so that h(T) = f(T) g(T) is defined as an element of k[T] too. It is easy to see that

h(a) = f(a) g(a), (5.7.10)

so that f(T) ↦ f(a) (5.7.11)

defines a homomorphism from k[T] into A, as algebras over k.

Suppose that a, ϵ ∈ A satisfy

a ϵ = ϵ a (5.7.12)

and ϵ^{2} = 0. As in (5.7.7), we have that

(a + ϵ)^{j} = a^{j} + j · a^{j−1} ϵ (5.7.13)

for every j ∈ Z_{+}. Using this, we get that

f(a + ϵ) = f(a) + f′(a) ϵ, (5.7.14)

as in (5.7.8).

5.8 Several commuting indeterminates

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. Also let n be a positive integer, and let T_{1}, …, T_{n} be n commuting indeterminates. As usual, a multi-index of length n is an n-tuple α = (α_{1}, …, α_{n}) of nonnegative integers, and we put

|α| = α_{1} + ⋯ + α_{n}. (5.8.1)

The corresponding formal monomial

T^{α} = T_{1}^{α_{1}} ⋯ T_{n}^{α_{n}} (5.8.2)

in T_{1}, …, T_{n} has degree |α|. A formal power series in T_{1}, …, T_{n} with coefficients in A can be expressed as

f(T) = f(T_{1}, …, T_{n}) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} T^{α}, (5.8.3)

where f_{α} ∈ A for every multi-index α. The space A[[T_{1}, …, T_{n}]] of all such formal power series can be defined as the space of all A-valued functions on the set (Z_{+} ∪ {0})^{n} of all multi-indices of length n. This is a module over k with respect to pointwise addition and scalar multiplication of A-valued functions on (Z_{+}∪{0})^{n}, which corresponds to termwise addition and scalar multiplication of formal power series as in (5.8.3). As a module over k, A[[T_{1}, …, T_{n}]] corresponds to the direct product of copies of A indexed by (Z_{+} ∪ {0})^{n}.

A formal polynomial in T_{1}, …, T_{n} with coefficients in A can be expressed as

f(T) = f(T_{1}, …, T_{n}) = ∑_{|α|≤N} f_{α} T^{α}, (5.8.4)

where the sum is taken over multi-indices α with |α| ≤ N for some nonnegative integer N, and f_{α} ∈ A for each such α. Of course, we can take f_{α} = 0 when |α| > N, so that (5.8.4) may be considered as a formal power series in T_{1}, …, T_{n} with coefficients in A. The space A[T_{1}, …, T_{n}] of all such formal polynomials can be defined as the space of all A-valued functions on (Z_{+} ∪ {0})^{n} that are equal to 0 for all but finitely many α, which is a submodule of A[[T_{1}, …, T_{n}]]. As a module over k, A[T_{1}, …, T_{n}] corresponds to the direct sum of copies of A indexed by (Z_{+} ∪{0})^{n}. We can identify A with the submodule of A[T_{1}, …, T_{n}] consisting of f(T) as in (5.8.4) with N = 0.

Suppose that A is an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Let f(T) ∈ A[[T_{1}, …, T_{n}]] be as in (5.8.3), and let

g(T) = ∑_{β∈(Z_{+}∪{0})^{n}} g_{β} T^{β} (5.8.5)

be another element of A[[T_{1}, …, T_{n}]]. If α and β are multi-indices of length n, then α + β can be defined as a multi-index of length n by coordinatewise addition, and we have that

|α + β| = |α| + |β|. (5.8.6)

Put

h_{γ} = ∑_{α+β=γ} f_{α} g_{β} (5.8.7)

for every multi-index γ of length n, where the sum is taken over all pairs of multi-indices α, β such that α + β = γ. There are only finitely many such pairs, so that the sum on the right side of (5.8.7) is a finite sum of elements of A. Thus

h(T) = ∑_{γ∈(Z_{+}∪{0})^{n}} h_{γ} T^{γ} (5.8.8)

defines an element of A[[T_{1}, …, T_{n}]], and we put

f(T) g(T) = h(T). (5.8.9)

This extends multiplication on A to A[[T_{1}, …, T_{n}]], so that the latter becomes an algebra over k in the strict sense too. It is easy to see that A[T_{1}, …, T_{n}] is a subalgebra of A[[T_{1}, …, T_{n}]] with respect to this definition of multiplication. If multiplication on A is commutative or associative, then one can check that multiplication on A[[T_{1}, …, T_{n}]] has the same property. If A has a multiplicative identity element e, then the corresponding formal polynomial in T_{1}, …, T_{n} is the multiplicative identity element in A[[T_{1}, …, T_{n}]]. In particular, k[[T_{1}, …, T_{n}]] is a commutative associative algebra over k.

Let A be a module over k again, let f(T) ∈ k[[T_{1}, …, T_{n}]] be as in (5.8.3), and let g(T) ∈ A[[T_{1}, …, T_{n}]] be as in(5.8.5). Thus f_{α} g_{β} is defined as an element of A for all multi-indices α, β, using scalar multiplication on A. If γ is a multiindex of length n, then h_{γ} can be defined as an element of A as in (5.8.7). This permits us to define h(T) as an element of A[[T_{1}, …, T_{n}]] as in (5.8.8), which can be used to define f(T) g(T). One can verify that A[[T_{1}, …, T_{n}]] is a module over k[[T_{1}, …, T_{n}]] in this way. Similarly, if f(T) ∈ k[T_{1}, …, T_{n}] and g(T) ∈ A[T_{1}, …, T_{n}], then h(T) ∈ A[T_{1}, …, T_{n}]. Using this definition of scalar multiplication, A[T_{1}, …, T_{n}] becomes a module over k[T_{1}, …, T_{n}].

Let l and m be positive integers, and let X_{1}, …, X_{l}, Y_{1}, …, Y_{m} be commuting indeterminates. If β and γ are multi-indices of length l and m, respectively, then let us identify (β, γ) with a multi-index of length l + m. This corresponds to identifying (Z_{+} ∪ {0})^{l} × (Z_{+} ∪ {0})^{m} with (Z_{+} ∪ {0})^{l+m}. We may consider

X^{β} Y^{γ} = X_{1}^{β_{1}} ⋯ X_{l}^{β_{l}} Y_{1}^{γ_{1}} ⋯ Y_{m}^{γ_{m}} (5.8.10)

as a formal monomial in the variables X_{1}, …, X_{l}, Y_{1}, …, Y_{m} of degree |β| + |γ|. As before, A[[X_{1}, …, X_{l}]] is a module over k, so that

(A[[X_{1}, …, X_{l}]])[[Y_{1}, …, Y_{m}]] (5.8.11)

can be defined as a module over k as well. There is a simple one-to-one correspondence between the elements of (5.8.11) and

A[[X_{1}, …, X_{l}, Y_{1}, …, Y_{m}]], (5.8.12)

which defines an isomorphism between these modules over k. This correspondence takes (A[X_{1}, …, X_{l}])[Y_{1}, …, Y_{m}] (5.8.13)

onto A[X_{1}, …, X_{l}, Y_{1}, …, Y_{m}]. (5.8.14)

If A is an algebra over k in the strict sense, then we get an isomorphism between (5.8.11) and (5.8.12) as algebras over k. In particular, we get an isomorphism between (k[[X_{1}, …, X_{l}]])[[Y_{1}, …, Y_{m}]] (5.8.15)

and k[[X_{1}, …, X_{l}, Y_{1}, …, Y_{m}]] (5.8.16)

as algebras over k. Similarly, if A is a module over k, then scalar multiplication on (5.8.11) by elements of (5.8.15) corresponds to scalar multiplication on (5.8.12) by elements of (5.8.16).

5.9 Polynomial functions in several variables

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. As usual, we let k^{n} be the space of n-tuples of elements of k. If t = (t_{1}, …, t_{n}) ∈ k^{n} and α is a multi-index of length n, then t^{α} is defined as an element of k by t^{α} = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}}. (5.9.1)

Here t_{j}^{α_{j}} is interpreted as being the multiplicative identity element 1 in k when α_{j} = 0, as before. If β is another multi-index of length n, then

t^{α+β} = t^{α} t^{β}. (5.9.2)

Let A be a module over k, and let T_{1}, …, T_{n} be n commuting indeterminates. Also let f(T) be a formal polynomial in T_{1}, …, T_{n} with coefficients in A, as in (5.8.4). If t ∈ k^{n}, then

f(t) = ∑_{|α|≤N} f_{α} t^{α} (5.9.3)

defines an element of A, where f_{α} t^{α} is defined using scalar multiplication on A for each multi-index α. The mapping

f(T) ↦ f(t) (5.9.4)

defines a homomorphism from A[T_{1}, …, T_{n}] into A, as modules over k.

Let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Remember that multiplication on A can be extended to A[T_{1}, …, T_{n}], as in the previous section. Let f(T), g(T) ∈ A[T_{1}, …, T_{n}] be given, and put h(T) = f(T) g(T). If t ∈ k^{n}, then one can check that

h(t) = f(t) g(t). (5.9.5)

This means that (5.9.4) defines a homomorphism from A[T_{1}, …, T_{n}] into A, as algebras over k.

Let A be a module over k again, and remember that A[T_{1}, …, T_{n}] may be considered as a module over k[T_{1}, …, T_{n}], as in the previous section. Let f(T) in k[T_{1}, …, T_{n}] and g(T) ∈ A[T_{1}, …, T_{n}] be given, so that h(T) = f(T) g(T) is defined as an element of A[T_{1}, …, T_{n}] as well. If t ∈ k^{n}, then f(t) ∈ k, g(t) and h(t) are elements of A, and one can verify that (5.9.5) holds.

Let A be an associative algebra over k with a multiplicative identity element e, and let A^{n} be the space of n-tuples of elements of A. Suppose that a = (a_{1}, …, a_{n}) ∈ A^{n} has commuting coordinates, so that

a_{j} a_{l} = a_{l} a_{j} (5.9.6)

for all j, l = 1, …, n. Of course, this condition holds trivially when n = 1. If α is a multi-index of length n, then a^{α} is defined as an element of A by

a^{α} = a_{1}^{α_{1}} ⋯ a_{n}^{α_{n}}, (5.9.7)

where a_{j}^{α_{j}} is interpreted as being equal to e when α_{j} = 0. Note that

a^{α+β} = a^{α} a^{β} (5.9.8)

for all multi-indices α, β under these conditions. Let f(T) be a formal polynomial in T_{1}, …, T_{n} with coefficients in k, as in (5.8.4) again. As before, f(a) is defined as an element of A by

f(a) = ∑_{|α|≤N} f_{α} a^{α}, (5.9.9)

where f_{α} a^{α} is defined using scalar multiplication on A. If g(T) ∈ k[T_{1}, …, T_{n}] too and h(T) = f(T) g(T), then one can verify that

h(a) = f(a) g(a). (5.9.10)

It follows that f(T) ↦ f(a) is a homomorphism from k[T_{1}, …, T_{n}] into A, as algebras over k, since this mapping is clearly linear over k.

5.10 Partial derivatives

Let n be a positive integer, let α be a multi-index of length n, and let l be a positive integer with l ≤ n. The multi-index α(l) of length n is defined by

α_{j}(l) = α_{j} when j ≠ l (5.10.1)

and

(5.10.2)  α_{l}(l) = α_{l} − 1 when α_{l} ≥ 1
= 0  when α_{l} = 0.

Similarly, let α^{+}(l) be the multi-index of length n defined by

(5.10.3)  α_{j}^{+}(l) = α_{j}  when j ≠ l
= α_{l} + 1 when j = l.

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, and let T_{1}, …, T_{n} be n commuting indeterminates. Also let f(T) be a formal power series in T_{1}, …, T_{n} with coefficients in A, as in (5.8.3). The formal partial derivative of f(T) in T_{l} can be defined as a formal power series in T_{1}, …, T_{n} with coefficients in A by

∂_{l}f(T) = \frac{∂}{∂T_{l}} f(T) = ∑_{α∈(Z_{+}∪\{0\})^{n}}(α_{l} + 1) · f_{α^{+}(l)} T^{α}. (5.10.4)

This is basically the same as

∑_{α∈(Z_{+}∪{0})^{n}} α_{l} · f_{α} T^{α(l)} = ∑_{α_{l}≥1} α_{l} · f_{α} T^{α(l)}, (5.10.5)

where the second sum is taken over all multi-indices α with α_{l} ≥ 1. Note that

f(T) ↦ ∂_{l}f(T) (5.10.6)

defines a homomorphism from A[[T_{1}, …, T_{n}]] into itself, as a module over k. Of course, if f(T) ∈ A[T_{1}, …, T_{n}], then the previous sums reduce to finite sums, and ∂_{l}f(T) ∈ A[T_{1}, …, T_{n}]. One can check that

∂_{l}(∂_{m}f(T)) = ∂_{m}(∂_{l}f(T)) (5.10.7)

for every l, m = 1, …, n and f(T) ∈ A[[T_{1}, …, T_{n}]].

If n = 1, then (5.10.4) reduces to the definition of the derivative in Section 5.6. If n > 1, then we can identify f(T) ∈ A[[T_{1}, …, T_{n}]] with a formal power series in T_{l} whose coefficients are formal power series in the other variables T_{j}, j ≠ l, with coefficients in A, as in Section 5.8. The derivative of this formal power series in T_{l} can be defined as in Section 5.6, as a formal power series in T_{l} whose coefficients are formal power series in the other variables. This differentiated formal power series can be identified with a formal power series in T_{1}, …, T_{n}, as before, which is the same as (5.10.4).

Suppose that A is an algebra over k in the strict sense, so that A[[T_{1}, …, T_{n}]] is an algebra over k in the strict sense as well, as in Section 5.8. Under these conditions, (5.10.6) defines a derivation on A[[T_{1}, …, T_{n}]]. This can be reduced to the analogous statement for polynomials in one variable in Section 5.6, as in the preceding paragraph, or verified directly as in the n = 1 case.

Let A be a module over k again, and remember that A[[T_{1}, …, T_{n}]] may be considered as a module over k[[T_{1}, …, T_{n}]], as in Section 5.8. If f(T) is an element of k[[T_{1}, …, T_{n}]] and g(T) ∈ A[[T_{1}, …, T_{n}]], then

∂_{l}(f(T) g(T)) = (∂_{l}f(T)) g(T) + f(T) (∂_{l}g(T)), (5.10.8)

as elements of A[[T_{1}, …, T_{n}]]. This can be reduced to the analogous statement for polynomials in one variable in Section 5.6, as before, or verified directly in a similar way.

Let t ∈ k^{n} be given, and suppose that u ∈ k^{n} satisfies

u_{j} u_{l} = 0 (5.10.9)

for all j, l = 1, …, n. Of course, t + u is defined as an element of k^{n}, using coordinatewise addition. Let α be a multi-index of length n, so that t^{α} and (t + u)^{α} are defined as elements of k, as in (5.9.1). As in (5.7.7),

(t_{l} + u_{l})^{α_{l}} = t_{l}^{α_{l}} + α_{l} · t_{l}^{α_{l}−1} u_{l} (5.10.10)

for each l = 1, …, n when α_{l} ≥ 1, because u_{l}^{2} = 0. If α_{l} = 0, then (t_{l} + u_{l})^{α_{l}} = t_{l}^{α_{l}} = 1. Thus (t_{l} + u_{l})^{α_{l}} = t_{l}^{α_{l}} + α_{l} · t_{l}^{α_{l}(l)} u_{l} (5.10.11)

for every l = 1, …, n. Using this, one can check that

(t + u)^{α} = t^{α} + ∑_{l=1}^{n} α_{l} · t^{α(l)} u_{l}. (5.10.12)

If f(T) ∈ A[T_{1}, …, T_{n}], then it follows that

f(t + u) = f(t) + ∑_{l=1}^{n}(∂_{l}f)(t) u_{l}. (5.10.13)

Let A be an associative algebra over k with a multiplicative identity element e, and suppose that a ∈ A^{n} has commuting coordinates, as in the previous section. Also let u be an element of A^{n} that satisfies (5.10.9), and whose coordinates commute with the coordinates of a, so that

a_{j} u_{l} = u_{l} a_{j} (5.10.14)

for all j, l = 1, …, n. Note that u has commuting coordinates, and hence a + u has commuting coordinates. Let α be a multi-index of length n again, so that a^{α} and (a + u)^{α} are defined as elements of A as before. As in (5.10.11),

(a_{l} + u_{l})^{α_{l}} = a_{l}^{α_{l}} + α_{l} · a_{l}^{α_{l}(l)} u_{l} (5.10.15)

for every l = 1, …, n. This implies that

(a + u)^{α} = a^{α} + ∑_{l=1}^{n} α_{l} · a^{α(l)} u_{l}, (5.10.16)

as before. If f(T) ∈ k[T_{1}, …, T_{n}], then we get that

f(a + u) = f(a) + ∑_{l=1}^{n}(∂_{l}f)(a) u_{l}. (5.10.17)

5.11 Formal differential operators

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Also let ∂_{1}, …, ∂_{n} be commuting formal symbols, which may be used to represent partial derivatives, as in the previous section. If α is a multi-index of length n, then let

∂^{α} = ∂_{1}^{α_{1}} ⋯ ∂_{n}^{α_{n}} (5.11.1)

be the corresponding formal product of ∂_{l}’s.

Let T_{1}, …, T_{n} be n commuting indeterminates, as before. A formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] can be expressed as

∑_{|α|≤N} a^{α}(T) ∂^{α}, (5.11.2)

where N is a nonnegative integer, the sum is taken over all multi-indices α of length n with |α| ≤ N, and a^{α}(T) ∈ k[[T_{1}, …, T_{n}]] for each such α. As usual, we can take a^{α}(T) = 0 when |α| > N, so that a^{α}(T) is defined for every multiindex α. The space of these formal differential operators can be defined as the space of functions α ↦ a^{α}(T) from (Z_{+} ∪ {0})^{n} into k[[T_{1}, …, T_{n}]] such that a^{α}(T) = 0 for all but finitely many α. This is a module over k with respect to pointwise addition and scalar multiplication, which corresponds to termwise addition and scalar multiplication of sums as in (5.11.2). As a module over k, this corresponds to the direct sum of copies of k[[T_{1}, …, T_{n}]] indexed by (Z_{+} ∪ {0})^{n}. We can identify elements of k[[T_{1}, …, T_{n}]] with sums of the form (5.11.2) with N = 0.

Multiplication on k[[T_{1}, …, T_{n}]] can be extended to these formal differential operators, with

∂_{l}(b^{β}(T) ∂^{β}) = (∂_{l} b^{β}(T)) ∂^{β} + b^{β}(T) ∂_{l} ∂^{β} (5.11.3)

for every l = 1, …, n, multi-index β, and b^{β}(T) ∈ k[[T_{1}, …, T_{n}]]. Note that

∂_{l} ∂^{β} = ∂^{β^{+}(l)}, (5.11.4)

in the notation of the previous section. The space of these formal differential operators is an associative algebra over k in this way, which contains k[[T_{1}, …, T_{n}]] as a subalgebra. The multiplicative identity element of k is also the multiplicative identity element in the space of these formal differential operators, when considered as an element of k[[T_{1}, …, T_{n}]] and hence a formal differential operator, as before.

A formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] can be expressed as in (5.11.2), with a^{α}(T) ∈ k[T_{1}, …, T_{n}] for each α. The space of these formal differential operators is a subalgebra of the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]]. The space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] contains k[T_{1}, …, T_{n}] as a subalgebra, as before.

Let A be a module over k. If α is a multi-index of length n and f(T) is a formal power series in T_{1}, …, T_{n} with coefficients in A, then

∂^{α}f(T) = ∂_{1}^{α_{1}} ⋯ ∂_{n}^{α_{n}}f(T) (5.11.5)

defines an element of A[[T_{1}, …, T_{n}]] as well, where partial derivatives are defined on A[[T_{1}, …, T_{n}]] as in the previous section. This is interpreted as being equal to f(T) when α = 0. Similarly, if (5.11.2) is a formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]], then

( ∑_{|α|≤N} a^{α}(T) ∂^{α})f(T) = ∑_{|α|≤N} a^{α}(T) ∂^{α}f(T) (5.11.6)

defines an element of A[[T_{1}, …, T_{n}]]. Thus (5.11.2) induces a mapping from A[[T_{1}, …, T_{n}]] into itself, which is linear over k. If the coefficients a^{α}(T) of (5.11.2) are elements of k[T_{1}, …, T_{n}] and f(T) ∈ A[T_{1}, …, T_{n}], then (5.11.6) is in A[T_{1}, …, T_{n}] too.

Remember that the space

Hom_{k}(A[[T_{1}, …, T_{n}]], A[[T_{1}, …, T_{n}]]) (5.11.7)

of homomorphisms from A[[T_{1}, …, T_{n}]] into itself, as a module over k, is an associative algebra over k with respect to composition of mappings. The remarks in the preceding paragraph define a mapping from the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] into (5.11.7). One can check that this mapping is an algebra homomorphism, with respect to multiplication of formal differential operators, as mentioned earlier.

Suppose now that A = k, as a module over itself. Let us also suppose for the moment that if m is a positive integer, t ∈ k, and m · t = 0 in k, then t = 0. In particular, this holds when k = Z, or k is a field of characteristic 0, or at least an algebra over Q. In this case, one can verify that a formal differential operator (5.11.2) in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] is uniquely determined by the corresponding mapping from k[[T_{1}, …, T_{n}]] into itself. More precisely, (5.11.2) is uniquely determined by the restriction of this mapping to k[T_{1}, …, T_{n}]. This uses the fact that

∂^{α}T^{β} = 0 (5.11.8)

when α, β are multi-indices such that α_{j} > β_{j} for some j = 1, …, n. This also uses the fact that

∂^{α} T^{α} = ∏_{j=1}^{n}(α_{j}!) (5.11.9)

for every multi-index α.

5.12 First-order differential operators

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let ∂_{1}, …, ∂_{n} be commuting formal symbols, as in the previous section. Also let T_{1}, …, T_{n} be commuting indeterminates, and let

a(T) = (a^{1}(T), …, a^{n}(T)) (5.12.1)

be an n-tuple of formal power series in T_{1}, …, T_{n} with coefficients in k. Put

D_{a(T)} = ∑_{j=1}^{n} a^{j}(T) ∂_{j}, (5.12.2)

which defines a formal partial differential operator in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]], as in the previous section.

Let b(T) = (b^{1}(T), …, b^{n}(T)) be another n-tuple of formal power series in T_{1}, …, T_{n} with coefficients in k, so that D_{b(T)} can be defined as before. The products D_{a(T)} D_{b(T)} and D_{b(T)} D_{a(T)} can be defined as formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] as well, as in the previous section. It is easy to see that

D_{a(T)} D_{b(T)} − D_{b(T)} D_{a(T)} (5.12.3)

= ∑_{j=1}^{n} ∑_{l=1}^{n}(a^{j}(T) ∂_{j}b^{l}(T) − b^{j}(T) ∂_{j}a^{l}(T)) ∂_{l},

using (5.11.3). Put c(T) = (c^{1}(T), …, c^{n}(T)), where

c^{l}(T) = ∑_{j=1}^{n}(a^{j}(T) ∂_{j}b^{l}(T) − b^{j}(T) ∂_{j}a^{l}(T)) (5.12.4)

for each l = 1, …, n. Thus c(T) is another n-tuple of elements of k[[T_{1}, …, T_{n}]], and D_{a(T)} D_{b(T)} − D_{b(T)} D_{a(T)} = D_{c(T)}. (5.12.5)

If f(T) ∈ k[[T_{1}, …, T_{n}]], then

D_{a(T)}f(T) = ∑_{j=1}^{n} a^{j}(T) ∂_{j}f(T) (5.12.6)

defines an element of k[[T_{1}, …, T_{n}]] too, as in the previous section. One can check that this defines a derivation on k[[T_{1}, …, T_{n}]], because partial derivatives define derivations on k[[T_{1}, …, T_{n}]].

If a^{j}(T) ∈ k[T_{1}, …, T_{n}] for each j = 1, …, n, and f(T) ∈ k[T_{1}, …, T_{n}], then (5.12.6) defines an element of k[T_{1}, …, T_{n}], as before. In this case, we get a derivation on k[T_{1}, …, T_{n}]. If b^{j}(T) ∈ k[T_{1}, …, T_{n}] for each j = 1, …, n as well, then (5.12.4) is an element of k[T_{1}, …, T_{n}] for every l = 1, …, n.

Let δ be any derivation on k[T_{1}, …, T_{n}], as an algebra over k. Note that T_{j} may be considered as an element of k[T_{1}, …, T_{n}] for each j = 1, …, n, where more precisely the coefficient of T_{j} is the multiplicative identity element in k. Thus a^{j}(T) = δ(T_{j}) (5.12.7) defines an element of k[T_{1}, …, T_{n}] for each j = 1, …, n. This permits us to define a(T) as an n-tuple of elements of k[T_{1}, …, T_{n}] as in (5.12.1), so that D_{a(T)} can be defined as in (5.12.2). If f(T) ∈ k[T_{1}, …, T_{n}], then one can verify that δ(f(T)) = D_{a(T)}f(T), (5.12.8)

where the right side is defined as in (5.12.6).

Now let δ be a derivation on k[[T_{1}, …, T_{n}]], as an algebra over k. In this case, (5.12.7) defines an element of k[[T_{1}, …, T_{n}]] for each j = 1, …, n, so that we can define a(T) as an n-tuple of elements of k[[T_{1}, …, T_{n}]] as in (5.12.1). Thus D_{a(T)} can be defined as in (5.12.2), and (5.12.8) holds for every f(T) ∈ k[T_{1}, …, T_{n}], as in the preceding paragraph. Of course, we would like to extend this to f(T) ∈ k[[T_{1}, …, T_{n}]].

Let us say that f(T) ∈ ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} T^{α} ∈ k[[T_{1}, …, T_{n}]] vanishes to order L for some nonnegative integer L if f_{α} = 0 for every multi-index α with |α| ≤ L. If g(T) ∈ k[[T_{1}, …, T_{n}]], then

g(T) T^{β} (5.12.9)

vanishes to order |β| − 1 for every nonzero multi-index β. If f(T) vanishes to order L for some L ≥ 0, then f(T) can be expressed as a finite sum of elements of k[[T_{1}, …, T_{n}]] of the form (5.12.9), with |β| = L + 1.

If f(T) vanishes to order L for some L ≥ 1, then δ(f(T)) vanishes to order L − 1. This can be verified directly when f(T) is of the form (5.12.9) with |β| = L+1, and otherwise one can reduce to that case, as in the preceding paragraph. One can use this to get that (5.12.8) holds for every f(T) ∈ k[[T_{1}, …, T_{n}]], as desired.

5.13 Homogeneous formal polynomials

Let k be a commutative ring with a multiplicative identity element, let T_{1}, …, T_{n} be n commuting indeterminates for some positive integer n, and let A be a module over k. A formal polynomial f(T) in T_{1}, …, T_{n} with coefficients in A is said to be homogeneous of degree d for some nonnegative integer d if f(T) can be expressed as

f(T) = ∑_{|α|=d} f_{α} T^{α}, (5.13.1)

where the sum is taken over all multi-indices α of length n with |α| = d, and f_{α} is an element of A for all such α. Equivalently, this means that the coefficient f_{α} of T^{α} in f(T) is equal to 0 when |α| ≠ d. The space A_{d}[T_{1}, …, T_{n}] of these formal polynomials is a submodule of A[T_{1}, …, T_{n}], as a module over k. Note that A_{d}[T_{1}, …, T_{n}] corresponds to a direct sum of copies of A indexed by multiindices α with |α| = d, and that A[T_{1}, …, T_{n}] can be viewed as the direct sum of A_{d}[T_{1}, …, T_{n}] over all nonnegative integers d. Similarly, A[[T_{1}, …, T_{n}]] can be viewed as the direct product of A_{d}[T_{1}, …, T_{n}] over all nonnegative integers d. If f(T) ∈ A_{d}[T_{1}, …, T_{n}] for some d ≥ 1, then it is easy to see that ∂_{l}f(T) is homogeneous of degree d − 1 for every l = 1, …, n.

Suppose that A is an algebra over k in the strict sense. If f(T), g(T) are homogeneous formal polynomials in T_{1}, …, T_{n} with coefficients in A for some nonnegative integers d_{1}, d_{2}, respectively, then one can check that f(T) g(T) is homogeneous of degree d_{1} + d_{2}.

Let us now take A = k, and let

a^{j}(T) = ∑_{l=1}^{n} a_{l}^{j} T_{l} (5.13.2)

be elements of k_{1}[T_{1}, …, T_{n}] for j = 1, …, n, so that a_{l}^{j} ∈ k for all j, l = 1, …, n. Thus ∂_{l}a^{j}(T) = a_{l}^{j} for every j, l = 1, …, n. Put a(T) = (a^{1}(T), …, a^{n}(T)), and let ∂_{1}, …, ∂_{n} be commuting formal symbols, as in the previous two sections. Consider the corresponding formal differential operator D_{a(T)} in ∂_{1}, …, ∂_{n}, as before. If f(T) is a formal polynomial in T_{1}, …, T_{n} with coefficients in k, then D_{a(T)}f(T) is defined as an element of k[T_{1}, …, T_{n}] too. More precisely, if f(T) is homogeneous of degree d, then D_{a(T)}f(T) is homogeneous of degree d as well. In particular, if

f(T) = ∑_{j=1}^{n} f_{j} T_{j} (5.13.3)

is homogeneous of degree 1, so that f_{j} ∈ k for j = 1, …, n, then

D_{a(T)}f(T) = ∑_{j=1}^{n} a^{j}(T) ∂_{j}f(T) = ∑_{j=1}^{n} ∑_{l=1}^{n} a_{l}^{j} f_{j} T_{l}. (5.13.4)

Let

b^{j}(T) = ∑_{l=1}^{n} b_{l}^{j} T_{l} (5.13.5)

be elements of k_{1}[T_{1}, …, T_{n}] for j = 1, …, n, and put b(T) = (b^{1}(T), …, b^{n}(T)). Put

c^{j}(T) = ∑_{l=1}^{n}(a^{l}(T) b_{l}^{j} − b^{l}(T) a_{l}^{j}) = ∑_{l=1}^{n} ∑_{m=1}^{n}(a_{m}^{l} b_{l}^{j} − b_{m}^{l} a_{l}^{j}) T_{m} (5.13.6)

for each j = 1, …, n, and c(T) = (c^{1}(T), …, c^{n}(T)). Note that c^{j}(T) is an element of k_{1}[T_{1}, …, T_{n}] for j = 1, …, n. If D_{b(T)} and D_{c(T)} are the formal differential operators in ∂_{1}, …, ∂_{n} corresponding to b(T) and c(T), respectively, then D_{a(T)} D_{b(T)} − D_{b(T)} D_{a(T)} = D_{c(T)}, (5.13.7)

as in (5.12.5).

5.14 Homogeneous differential operators

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Also let T_{1}, …, T_{n} be commuting indeterminates, and let ∂_{1}, …, ∂_{n} be commuting formal symbols, as in Section 5.11. Consider a formal differential operator

L = ∑_{|α|≤N} a^{α}(T) ∂^{α} (5.14.1)

in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}]. Let us say that L is homogeneous of degree d for some integer d if the following conditions hold. If α is a multiindex such that d ≥ −|α|, then a^{α}(T) should be homogeneous of degree d + |α| as a formal polynomial in T_{1}, …, T_{n} with coefficients in k. Otherwise, if d is strictly less than −|α|, then a^{α}(T) = 0. The space of these formal differential operators that are homogeneous of degree d is a submodule of the space of all formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}], as a module over k.

Let L_{1} and L_{2} be formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}], so that their product L_{1} L_{2} can be defined as a formal differential operator with coefficients in k[T_{1}, …, T_{n}] as well. If L_{1}, L_{2} are homogeneous of degrees d_{1}, d_{2} ∈ Z, respectively, then one can check that L_{1} L_{2} is homogeneous of degree d_{1} + d_{2}. More precisely, one can start with the case where L_{1} = ∂_{j} for some j = 1, …, n, so that d_{1} = −1. Using this, one can obtain the analogous statement for L_{1} = ∂^{α} for some multi-index α, so that d_{1} = −|α|. One can use this to obtain the analogous statement when L_{1} is homogeneous of any degree d_{1}.

Let L_{1} be a formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] again, and let A be a module over k. Also let f(T) be a formal polynomial in T_{1}, …, T_{n} with coefficients in A, so that L_{1}f(T) is defined as an element of A[T_{1}, …, T_{n}] too. Suppose that L_{1} is homogeneous of degree d_{1} ∈ Z, and that f(T) is homogeneous of degree d for some nonnegative integer d. One can verify that L_{1}f(T) is homogeneous of degree d_{1} + d when d_{1} ≥ −d, and that L_{1}f(T) = 0 otherwise. Indeed, if L_{1} = ∂_{j} for some j = 1, …, n, then this was mentioned in the previous section. As before, one can use this to obtain the analogous statement when L_{1} = ∂^{α} for some multi-index α. The analogous statement for any L_{1} follows easily from this.

Remember that the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] is an algebra over k. The collection of such formal differential operators that are homogeneous of degree 0 is a subalgebra of this algebra. This subalgebra is generated as an an algebra over k by homogeneous differential operators of degree 0 as in (5.14.1) with N = 1. To see this, one can start with a homogeneous differential operator of degree 0 of the form T^{α} ∂^{β}, where α, β are multi-indices with |α| = |β| ≥ 2. This can be approximated by a product of |α| = |β| operators of the form T_{j} ∂_{l} for some j, l = 1, …, n. More precisely, one can choose the approximation so that the difference is a formal differential operator of lower order. One can repeat the process to express any homogeneous differential operator of degree 0 as a finite sum of products of homogeneous differential operators of degree 0 and order less than or equal to 1, as desired.

5.15 Some algebras of differential operators

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, let T_{1}, …, T_{n} be commuting indeterminates, and let ∂_{1}, …, ∂_{n} be commuting formal symbols, as before. Remember that

(5.15.1) the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]]

is an associative algebra over k, as in Section 5.11. Similarly,

(5.15.2) the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}]

is a subalgebra of (5.15.1). We also have that

(5.15.3) the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] that are homogeneous of degree 0

is a subalgebra of (5.15.2), as in the previous section.

If a(T) = (a^{1}(T), …, a^{n}(T)) is an element of the space (k[[T_{1}, …, T_{n}]])^{n} of n-tuples of formal power series in T_{1}, …, T_{n} with coefficients in k, then put D_{a(T)} = ∑_{j=1}^{n} a^{j}(T) ∂_{j}, as in Section 5.12. Note that

{D_{a(T)} : a(T) ∈ (k[[T_{1}, …, T_{n}]])^{n}} (5.15.4)

is a submodule of (5.15.1), as a module over k. Remember that the commutator of two elements of (5.15.4) is an element of (5.15.4) too, as in Section 5.12. Thus (5.15.4) is a Lie subalgebra of (5.15.1), as a Lie algebra over k with respect to the commutator bracket. Similarly,

{D_{a(T)} : a(T) ∈ (k[T_{1}, …, T_{n}])^{n}} (5.15.5)

is a submodule of (5.15.2), as a module over k, and a Lie subalgebra of (5.15.2), as a Lie algebra over k with respect to the commutator bracket.

Remember that k_{1}[T_{1}, …, T_{n}] is the space of homogeneous formal polynomials in T_{1}, …, T_{n} of degree one with coefficients in k, as in Section 5.13. Put

g_{n}(k) = {D_{a(T)} : a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n}}, (5.15.6)

which is a submodule of (5.15.3), as a module over k. In fact, (5.15.6) is a Lie subalgebra of (5.15.3), as a Lie algebra over k with respect to the commutator bracket.

Of course, a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n} can be expressed as a^{j}(T) = ∑_{l=1}^{n} a_{l}^{j} T_{l} for j = 1, …, n, where a_{l}^{j} ∈ k for every j, l = 1, …, n. Thus the elements of (k_{1}[T_{1}, …, T_{n}])^{n} correspond to n × n matrices with entries in k in an obvious way. Using this, it is easy to see that g_{n}(k) is isomorphic to gl_{n}(k) as a Lie algebra over k, as in Section 5.13.

Put

s_{n}(k) = {D_{a(T)} : a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n}, ∑_{j=1}^{n} a_{j}^{j} = 0}, (5.15.7)

where a_{l}^{j}, 1 ≤ j, l ≤ n, corresponds to a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n} as in the preceding paragraph. This is a Lie subalgebra of g_{n}(k), as a Lie algebra over k, which corresponds to sl_{n}(k) under the isomorphism between g_{n}(k) and gl_{n}(k) mentioned before. More precisely,

[g_{n}(k), g_{n}(k)] ⊆ s_{n}(k), (5.15.8)

as in Section 2.9.

If a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n}, then D_{a(T)} defines a mapping from k_{1}[T_{1}, …, T_{n}] into itself that is linear over k, as in the previous two sections. Multiplication of differential operators corresponds to composition of the associated linear mappings on k_{1}[T_{1}, …, T_{n}], so that commutators of differential operators correspond to commutators of the associated linear mappings on k_{1}[T_{1}, …, T_{n}]. Of course, k_{1}[T_{1}, …, T_{n}] is freely generated by T_{1}, …, T_{n}, as a module over k. The linear mapping on k_{1}[T_{1}, …, T_{n}] associated to D_{a(T)} corresponds to an n × n matrix whose entries are determined by the coefficients of the components of a(T), as before.

Remember that formal polynomials in T_{1}, …, T_{n} with coefficients in k determine k-valued polynomial functions on k^{n}, as in Section 5.9. In particular, elements of k_{1}[T_{1}, …, T_{n}] correspond exactly to mappings from k^{n} into k that are linear over k. Similarly, elements of (k_{1}[T_{1}, …, T_{n}])^{n} correspond exactly to mappings from k^{n} into itself that are linear over k.

Of course, mappings from k^{n} into itself that are linear over k correspond to n × n matrices with entries in k in the usual way. Using this, matrix multiplication corresponds to composition of mappings on k^{n}. Thus commutators of matrices correspond to commutators of linear mappings on k^{n}.

If a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n}, then we get a linear mapping from k^{n} into itself, and an n×n matrix with entries in k, as in the previous two paragraphs. One can check that the n×n matrix associated to D_{a(T)} as mentioned earlier is the same as the transpose of the matrix associated to the linear mapping from k^{n} into itself. This means that commutators of these differential operators correspond to −1 times the commutators of the linear mappings on k^{n} associated to the same elements of (k_{1}[T_{1}, …, T_{n}])^{n}.

Bilinear actions and representations

6.1 Bilinear actions

Let k be a commutative ring with a multiplicative identity element, and let A, V be modules over k. Also let β be a mapping from A × V into V that is bilinear over k. This may be described as a bilinear action of A on V over k, or more precisely as a bilinear action of A on V on the left. It is sometimes convenient to consider a mapping from V × A into V that is bilinear over k as a bilinear action of A on V on the right. If A is an associative algebra over k, or a Lie algebra over k, then we may be interested in bilinear actions that satisfy additional conditions, as in Sections 6.4 and 6.5.

Alternatively, we may use the notation

ρ_{a}(v) = β(a, v), (6.1.1)

where a ∈ A and v ∈ V. The bilinearity of β means that ρ_{a} is a module homomorphism from V into itself for each a ∈ A, and that a ↦ ρ_{a} defines a module homomorphism from A into the space Hom_{k}(V, V) of module homomorphisms from V into itself. We may use ρ to denote a module homomorphism from A into Hom_{k}(V, V) in this way, which defines a bilinear action of A on V as in (6.1.1). We may also use the notation

a · v = β(a, v) (6.1.2)

for a ∈ A and v ∈ V. A bilinear action of A on V on the right may be expressed by v · a for a ∈ A and v ∈ V.

Let T be an indeterminate, and remember that k[T], A[T], and V [T] are the corresponding spaces of formal polynomials in T with coefficients in k, A, and V, respectively. A bilinear action of A on V over k can be extended to a bilinear action of A[T] on V [T] over k[T]. More precisely, one could start with a mapping from A × V into V [T] that is bilinear over k. This can be extended to a mapping from A[T] × V [T] into V [T] that is bilinear over k[T], as in Sections 4.5 and 4.6.

Remember that k[[T]], A[[T]], and V [[T]] are the corresponding spaces of formal power series in T with coefficients in k, A, and T, respectively. A bilinear action of A on V over k can be extended to a bilinear action of A[[T]] on V [[T]] over k[[T]]. As before, one could start with a mapping from A × V into V [[T]] that is bilinear over k, which can be extended to a mapping from A[[T]]×V [[T]] into V [[T]] that is bilinear over k[[T]].

Suppose now that k is a field with an absolute value function |·|, and that A, V are vector spaces over k with seminorms N_{A}, N_{V}, respectively, with respect to | · | on V. Remember that a bilinear mapping β from A × V into V is said to be bounded with respect to these seminorms if there is a nonnegative real number C such that

N_{V} (β(a, v)) ≤ C N_{A}(a) N_{V} (v) (6.1.3)

for every a ∈ A and v ∈ V, as in Section 1.13. Let ρ_{a} be as in (6.1.1), so that (6.1.3) can be reformulated as saying that

N_{V} (ρ_{a}(v)) ≤ C N_{A}(a) N_{V} (v) (6.1.4)

for every a ∈ A and v ∈ V. This is the same as saying that for each a ∈ A, ρ_{a} is bounded as a linear mapping from V into itself with respect to N_{V}, with

∥ρ_{a}∥_{op,V V} ≤ C N_{A}(a). (6.1.5)

This can also be reformulated as saying that a ↦ ρ_{a} is bounded as a linear mapping from A into the space BL(V) of bounded linear mappings from V into itself with respect to N_{V}, with the corresponding operator seminorm of this mapping being less than or equal to C.

6.2 Subactions and homomorphisms

Let k be a commutative ring with a multiplicative identity element, let A and V be modules over k, and let β be a mapping from A×V into V that is bilinear over k. Also let W be a submodule of V, as a module over k, and suppose that

β(a, w) ∈ W (6.2.1)

for every a ∈ A and w ∈ W. This means that the restriction of β to A × W defines a mapping into W that is bilinear over k, and hence a bilinear action of A on W. Equivalently, if ρ_{a} is as in (6.1.1), then (6.2.1) says that

ρ_{a}(W) ⊆ W (6.2.2)

for every a ∈ A. Thus the restriction of ρ_{a} to W defines a module homomorphism from W into itself for every a ∈ A, and the mapping from a ∈ A to the restriction of ρ_{a} to W defines a module homomorphism from A into Hom_{k}(W, W). If the bilinear action of A on V is expressed as in (6.1.2), then (6.2.1) can be reformulated as saying that

a · w ∈ W (6.2.3)

for every a ∈ A and w ∈ W. If A acts on V on the right, then the corresponding condition is that w · a ∈ W (6.2.4)

for every a ∈ A and w ∈ W. In this case, A acts on W on the right, as before.

As a basic class of examples, suppose that A is an algebra over k in the strict sense, where multiplication of a b ∈ A is expressed as a b. We can use multiplication on A to define bilinear actions of A on itself, on the left and on the right. Let A_{0} be a submodule of A, as a module over k. The condition that A_{0} be a left ideal in A says exactly that the action of A on itself on the left maps A_{0} into itself. Similarly, the condition that A_{0} be a right ideal in A says that the action of A on itself on the right maps A_{0} into itself.

Let A and V be modules over k again, and let β^{V} be a mapping from A × V into V that is bilinear over k. Let Z be another module over k, let β^{Z} be a mapping from A×Z into Z that is bilinear over k, and let ϕ be a homomorphism from V into Z, as modules over k. If

ϕ(β^{V} (a, v)) = β^{Z}(a, ϕ(v)) (6.2.5)

for every a ∈ A and v ∈ V, then we say that ϕ intertwines the actions of A on V and Z. Equivalently, if a ∈ A, then let ρ_{a}^{V} and ρ_{a}^{Z} be the module homomorphisms from V and Z into themselves, respectively, associated to β^{V} and β^{Z} as in (6.1.1). It is easy to see that (6.2.5) is the same as saying that

ϕ ∘ ρ_{a}^{V} = ρ_{a}^{Z} ∘ ϕ (6.2.6)

for every a ∈ A, as mappings from V into Z. If these bilinear actions of A on V and Z are expressed as in (6.1.2), then (6.2.5) can be reexpressed as

ϕ(a · v) = a · ϕ(v) (6.2.7)

for every a ∈ A and v ∈ V. If A acts on V and Z on the right, then ϕ intertwines these actions when ϕ(v · a) = ϕ(v) · a (6.2.8)

for every a ∈ A and v ∈ V.

6.3 Quotient actions

Let k be a commutative ring with a multiplicative identity element, let V be a module over k, and let W be a submodule of V. Remember that the quotient V/W can be defined as a module over k too, as in Section 2.11. Let q be the corresponding quotient mapping from V onto V/W.

Suppose that ϕ is a homomorphism from V into itself, as a module over k, such that ϕ(W) ⊆ W. (6.3.1)

Thus q ∘ ϕ is a homomorphism from V into V/W, as modules over k, whose kernel contains W. Under these conditions, there is a unique mapping ψ from V/W into itself such that ψ ∘ q = q ∘ ϕ (6.3.2)

as mappings from V into W, by standard arguments. Equivalently, this means that ψ(q(v)) = q(ϕ(v)) (6.3.3)

for every v ∈ V. Of course, ψ is a homomorphism from V/W into itself, as a module over k.

Let A be another module over k, and let β^{V} be a mapping from A × V into V that is bilinear over k. Suppose that the action of A on V maps W into itself, as in (6.2.1), with β = β^{V}. This implies that q(β^{V} (a, w)) = 0 for every a ∈ A and w ∈ W. It follows that for a ∈ A and v ∈ V, q(β^{V} (a, v)) actually depends only on a and q(v). This permits us to define a mapping β^{V/W} from A×(V/W) into V/W that is bilinear over k and satisfies

β^{V/W} (a, q(v)) = q(β^{V} (a, v)) (6.3.4)

for every a ∈ A and v ∈ V. Equivalently, if a ∈ A, then let ρ_{a}^{V} be the module homomorphism from V into itself associated to β^{V} as in (6.1.1). There is a unique mapping ρ_{a}^{V/W} from V/W into itself such that

ρ_{a}^{V/W} ∘ q = q ∘ ρ_{a}^{V} , (6.3.5)

as in (6.3.2). More precisely, ρ_{a}^{V W} is a module homomorphism from V/W into itself, as before, and one can check that a ↦ ρ_{a}^{V/W} is a module homomorphism from A into Hom_{k}(V/W, V/W). If the bilinear action of A on V is expressed as in (6.1.2), then the induced bilinear action of A on V/W can be expressed in the same way, with a · q(v) = q(a · v) (6.3.6)

for every a ∈ A and v ∈ V. Similarly, if A acts on V on the right, then we get a bilinear action of A on V/W on the right, with

q(v) · a = q(v · a) (6.3.7)

for every a ∈ A and v ∈ V. Note that q intertwines the actions of A on V and V/W, by construction.

Let A be an algebra over k in the strict sense, so that multiplication on A defines bilinear actions of A on itself, on the left and on the right, as in the previous section. If A_{0} is a left ideal in A, then the action of A on itself on the left maps A_{0} into itself, and we get a bilinear action of A on the quotient A/A_{0} on the left, as in the preceding paragraph. Similarly, if A_{0} is a right ideal in A, then the action of A on itself on the right maps A_{0} into itself, and we get a bilinear action of A on A/A_{0} on the right. This was mentioned earlier in Section 2.11, in terms of bilinear mappings.

6.4 Representations of associative algebras

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k, and let V be a module over k. Remember that the space Hom_{k}(V, V) of module homomorphisms from V into itself is an associative algebra over k with respect to composition of mappings. A representation of A on V is an algebra homomorphism from A into Hom_{k}(V, V). If A has a multiplicative identity element e, then one may also require that the representation send e to the identity mapping on V. In this case, if a ∈ A has a multiplicative inverse in A, then the representation sends a to an invertible mapping on V.

A representation of A on V may be denoted ρ, where for each a ∈ A, ρ_{a} denotes the corresponding module homomorphism from V into itself. Thus ρ_{a}(v) is the image of v ∈ V under ρ_{a}. Note that ρ_{a}(v) is linear over k in a and v, because the representation is linear over k as a mapping from A into Hom_{k}(V, V), and elements of Hom_{k}(V, V) are linear over k by definition. The multiplicative property of an algebra homomorphism can be expressed as

ρ_{a} ∘ ρ_{b} = ρ_{a b} (6.4.1)

for every a, b ∈ A, which is the same as saying that

ρ_{a}(ρ_{b}(v)) = ρ_{a b}(v) (6.4.2)

for every a, b ∈ A and v ∈ V. If A has a multiplicative identity element e, then one may require that ρ_{e} be the identity mapping on V, as before, so that

ρ_{e}(v) = v (6.4.3)

for every v ∈ V.

It is sometimes convenient to express a representation ρ of A on V by

ρ_{a}(v) = a · v (6.4.4)

for every a ∈ A and v ∈ V. As before, a · v should be linear over k in a and v, so that a · v corresponds to a mapping from A × V into V that is bilinear over k. The multiplicative property (6.4.2) can be reexpressed in this notation as

a · (b · v) = (a b) · v (6.4.5)

for every a, b ∈ A and v ∈ V. If A has a multiplicative identity element e, then (6.4.3) can be reexpressed as e · v = v (6.4.6) for every v ∈ V. We may also call V a (left) module over A, as an associative algebra over k, with respect to this representation.

Suppose now that we have an action of A on V on the right, so that for each a ∈ A and v ∈ V, v · a is defined as an element of V. Suppose that v · a is linear over k in a and v, so that v · a corresponds to a mapping from V × A into V that is bilinear over k. If we also have that

(v · a) · b = v · (a b) (6.4.7)

for every a, b ∈ A and v ∈ V, then V is said to be a right module over A, as an associative algebra over k. If A has a multiplicative identity element e, then one may require that v · e = v (6.4.8)

for every v ∈ V, as usual.

If V is a right module over A, as in the preceding paragraph, then

ρ_{a}(v) = v · a (6.4.9)

defines a module homomorphism from V into itself for each a ∈ A, because v · a is linear over k in v. The mapping from a ∈ A to ρ_{a} ∈ Hom_{k}(V, V) is linear over k, because v · a is linear over k in a. The multiplicativity condition (6.4.7) is the same as saying that

ρ_{b}(ρ_{a}(v)) = ρ_{a b}(v) (6.4.10)

for every a, b ∈ A and v ∈ V, which means that

ρ_{b} ∘ ρ_{a} = ρ_{a b} (6.4.11)

for every a, b ∈ A. Thus a ↦ ρ_{a} is an opposite algebra homomorphism from A into Hom_{k}(V, V). If A has a multiplicative identity element e, then (6.4.8) says that ρ_{e} is the identity mapping on V.

Remember that A is a module over k in particular. We may also consider A as both a right and left module over itself, where the actions of A on itself as a module over k on the left and the right are given by multiplication on A. The linearity conditions for these actions correspond to the definition of an algebra over k in the strict sense. Similarly, the conditions (6.4.5) and (6.4.7) correspond in this situation to associativity of multiplication on A. If A has a multiplicative identity element e, then (6.4.6) and (6.4.8) hold automatically.

Equivalently, the representation of A on itself as in (6.4.4) corresponds to the multiplication operators discussed in Section 2.2. Similarly, if V = A, then (6.4.9) corresponds to the right multiplication operators discussed in Section 2.7.

6.5 Representations of Lie algebras

Let k be a commutative ring with a multiplicative identity element, let (A, [·, ·]_{A}) be a Lie algebra over k, and let V be a module over k. The space Hom_{k}(V, V) of module homomorphisms from V into itself is an associative algebra over k with respect to compositions of mappings, and hence a Lie algebra over k with respect to the corresponding commutator bracket. A Lie algebra homomorphism from A into Hom_{k}(V, V), as a Lie algebra over k, is also known as a Lie algebra representation of A on V.

As before, a Lie algebra representation of A on V may be denoted ρ, where ρ_{a} is the module homomorphism from V into itself corresponding to a ∈ A, and ρ_{a}(v) is the image of v ∈ V under ρ_{a}. Thus ρ_{a}(v) is linear over k in a and v, and

ρ_{[a,b]_{A}} = ρ_{a} ∘ ρ_{b} − ρ_{b} ∘ ρ_{a} (6.5.1)

for every a, b ∈ A. Equivalently, this means that

ρ_{[a,b]_{A}}(v) = ρ_{a}(ρ_{b}(v)) − ρ_{b}(ρ_{a}(v)) (6.5.2)

for every a, b ∈ A and v ∈ V.

Let ρ be a Lie algebra representation of A on V, and put

ρ_{a}(v) = a · v (6.5.3)

for every a ∈ A and v ∈ V. This is linear over k in a and v, so that it corresponds to a mapping from A × V into V that is bilinear over k. Using this notation, (6.5.2) can be reexpressed as saying that

([a, b]_{A}) · v = a · (b · v) − b · (a · v) (6.5.4)

for every a, b ∈ A and v ∈ V. We may also call V a module over A, as a Lie algebra over k, with respect to this representation.

Suppose for the moment that A is an associative algebra over k, where multiplication of a, b ∈ A is expressed as a b. Thus A may be considered as a Lie algebra over k with respect to the corresponding commutator bracket [a, b] = a b − b a. If ρ is a representation of A on V, where A is considered as an associative algebra over k, then ρ is a Lie algebra representation of A on V too. Equivalently, if V is a left module over A as an associative algebra over k, then V is a module over A as a Lie algebra over k as well. Suppose now that V is a right module over A as an associative algebra over k, and let ρ be as in (6.4.9). This means that a ↦ ρ_{a} is an opposite algebra homomorphism from A into Hom_{k}(V, V), as before. If a, b ∈ A, then

ρ_{[a,b]} = ρ_{a b} − ρ_{b a} = ρ_{b} ∘ ρ_{a} − ρ_{a} ∘ ρ_{b}. (6.5.5)

It follows that −ρ_{a} defines a Lie algebra representation of A on V in this case.

Let A be any Lie algebra over k, and remember that the corresponding adjoint representation was defined in Section 2.4. This defines a representation of A on itself, as a module over k.

6.6 Subrepresentations

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k. Also let ρ be a representation of A on a module V over k. Suppose that W is a submodule of V such that

ρ_{a}(W) ⊆ W (6.6.1)

for every a ∈ A. Thus, for each a ∈ A, the restriction of ρ_{a} to W defines a module homomorphism from W into itself. This defines a representation of A on W, which is a subrepresentation of ρ on V.

As before, V may be considered as a left module over A, with a · v = ρ_{a}(v) for every a ∈ A and v ∈ V. Using this notation, (6.6.1) is the same as saying that a · w ∈ W (6.6.2)

for every a ∈ A and w ∈ W. Under these conditions, we may also say that W is a (left) submodule of V, as a left module over A.

Similarly, suppose that V is a right module over A. If W is a submodule of V as a module over k, and if w · a ∈ W (6.6.3)

for every a ∈ A and w ∈ W, then we say that W is a (right) submodule of V, as a right module over A.

As a basic class of examples, let V be a module over k, and let A be a subalgebra of Hom_{k}(V, V), as an associative algebra over k with respect to composition of mappings. There is an obvious representation of A on V, because the elements of A are already module homomorphisms from V into itself. Let W be a submodule of V, as a module over k, such that

a(W) ⊆ W (6.6.4)

for every a ∈ A. The restrictions of the elements of A to W defines a subrepresentation of the obvious representation of A on V just mentioned. Equivalently, V is a left module over A in an obvious way, and W is a left submodule of V as a left module over A.

As another basic class of examples, let A be any associative algebra over k. We may consider A as both a left and right module over itself, using multiplication on the left and on the right, as in Section 6.4. A left ideal in A is the same as a left submodule of A as a left module over itself, and similarly a right ideal in A is the same as a right submodule of A as a right module over itself.

Now let (A, [·, ·]_{A}) be a Lie algebra over k, and let ρ be a representation of A as a Lie algebra on a module V over k. Suppose that W is a submodule of V, as a module over k, such that (6.6.1) holds for every a ∈ A. Hence the restriction of ρ_{a} to W defines a homomorphism from W into itself, as a module over k, for every a ∈ A. This defines a representation of A as a Lie algebra on W, which is a subrepresentation of ρ on V.

As usual, V may be considered as a module over A as a Lie algebra, with a · v = ρ_{a}(v) for every a ∈ A and v ∈ V. The condition (6.6.1) on W can be reexpressed in this notation as (6.6.2), as before. In this case, W may be called a submodule of V, as a module over A, as a Lie algebra over k.

Let V be a module over k, and let A be a Lie subalgebra of Hom_{k}(V, V), as a Lie algebra with respect to the usual commutator bracket. As before, there is an obvious representation of A as a Lie algebra over k on V, because the elements of A are already homomorphisms from V into itself, as a module over k. If W is a submodule of V, as a module over k, that satisfies (6.6.4) for every a ∈ A, then the restrictions of the elements of A to W defines a subrepresentation of this representation of A as a Lie algebra over k on V. This is the same as saying that V is a module over A as a Lie algebra over k in an obvious way, and that W is a submodule of V as a module over A.

If A is any Lie algebra over k, then subrepresentations of the adjoint representation of A correspond exactly to ideals in A.

6.7 Homomorphisms between representations

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k again. Also let V, W be modules over k, and let ρ^{V}, ρ^{W} be representations of A on V, W, respectively. Suppose that ϕ is a homomorphism from V into W, as modules over k, such that

ϕ ∘ ρ_{a}^{V} = ρ_{a}^{W} ∘ ϕ (6.7.1)

for every a ∈ A, which is the same as saying that

ϕ(ρ_{a}^{V} (v)) = ρ_{a}^{W} (ϕ(v)) (6.7.2)

for every a ∈ A and v ∈ V. In this case, we say that ϕ intertwines the representations ρ^{V} and ρ^{W}, or that ϕ is a homomorphism between these representations. If ϕ is a one-to-one mapping from V onto W, then ϕ^{−1} intertwines ρ^{W} and ρ^{V} as a mapping from W onto V, and we say that ϕ defines an isomorphism between ρ^{V} and ρ^{W}.

Let us consider V and W as left modules over A, with

a · v = ρ_{a}^{V} (v), a · w = ρ_{a}^{W} (w) (6.7.3)

for every a ∈ A, v ∈ V, and w ∈ W. Thus (6.7.2) may be reexpressed as

ϕ(a · v) = a · ϕ(v) (6.7.4)

for every a ∈ A and v ∈ V. We may also say that ϕ defines a homomorphism from V into W as left modules over A in this situation. If ϕ is a one-to-one mapping from V onto W, then it follows that ϕ^{−1} is a homomorphism from W into V, as left modules over A. Under these conditions, ϕ is said to be an isomorphism from V onto W, as left modules over A.

Suppose now that V and W are right modules over A, so that v · a and w · a are defined as elements of V and W, respectively, for every a ∈ A, v ∈ V, and w ∈ W. Let ϕ be a homomorphism from V into W, as modules over k. If

ϕ(v · a) = ϕ(v) · a (6.7.5)

for every a ∈ A and v ∈ V, then ϕ is said to be a homomorphism from V into W, as right modules over A. If ϕ is also a one-to-one mapping from V onto W, then ϕ^{−1} is a homomorphism from W into V, as right modules over A. As before, ϕ is said to be an isomorphism from V onto W, as right modules over A, under these conditions.

Let (A, [·, ·]_{A}) be a Lie algebra over k, and let ρ^{V}, ρ^{W} be representations of A as a Lie algebra on modules V, W over k, respectively. Also let ϕ be a homomorphism from V into W, as modules over k. If ϕ satisfies (6.7.1) for every a ∈ A, then ϕ is said to intertwine ρ^{V} and ρ^{W}, or equivalently be a homomorphism between these representations. We may consider V and W as modules over A as a Lie algebra over k, as in (6.7.3). Using this notation, we can reexpress (6.7.2) as (6.7.4), and we say that ϕ is a homomorphism from V into W as modules over A, as a Lie algebra over k. If ϕ is a one-to-one mapping from V onto W, then ϕ^{−1} is a homomorphism from W into V, as modules over A. In this case, ϕ is an isomorphism between these representations of A, or equivalently an isomorphism from V onto W, as modules over A.

6.8 Quotient representations

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k. Also let V be a module over k, and let ρ^{V} be a representation of A on V. Suppose that

ρ_{a}^{V} (W) ⊆ W (6.8.1)

for every a ∈ A, as in Section 6.6. This implies that for each a ∈ A there is a unique mapping ρ_{a}^{V/W} from V/W into itself such that

ρ_{a}^{V/W} ∘ q = q ∘ ρ_{a}^{V} (6.8.2)

as mappings from V into V/W, as in Section 6.3. One can check that this defines ρ^{V/W} as a representation of A on V/W.

Equivalently, suppose that V is a left module over A, and that W is a left submodule of V. If a ∈ A and v ∈ V, then q(a · v) defines an element of V/W that is equal to 0 when v ∈ W. This permits us to define an action of A on V/W on the left, with a · q(v) = q(a · v) (6.8.3)

for every a ∈ A and v ∈ V. This defines V/W as a left module over A, as before. If A_{0} is a left ideal in A, then note that the quotient A/A_{0} is a left module over A.

Similarly, suppose that V is a right module over A, and that W is a right submodule of V. If a ∈ A and v ∈ V, then q(v · a) is an element of V/W that is equal to 0 when v ∈ W. Using this, we can define an action of A on V/W on the right, with q(v) · a = q(v · a) (6.8.4)

for every a ∈ A and v ∈ V. One can verify that this defines V/W as a right module over A. If A_{0} is a right ideal in A, then the quotient A/A_{0} is a right module over A.

Suppose now that A is a Lie algebra over k, and that ρ^{V} is a representation of A on V. If W satisfies (6.8.1) for every a ∈ A, then one can define ρ^{V/W} on V/W as in (6.8.2). One can check that ρ^{V/W} is a representation of A on V/W. Equivalently, if V is a module over A, and if W is a submodule of V as a module over A, then we can define the action of A on V/W as in (6.8.3). This makes V/W a module over A, as a Lie algebra over k, as before.

In each of these situations, the quotient mapping q intertwines the actions of A on V and V/W, by construction.

6.9 Sums of representations

Let k be a commutative ring with a multiplicative identity element, and let I be a nonempty set. Also let V_{j} be a module over k for every j ∈ I. Thus the direct product ∏_{j∈I} V_{j} of the V_{j}’s defines a module over k, and the direct sum ⨁_{j∈I} V_{j} is a submodule of ∏_{j∈I} V_{j}, as in Section 4.1. If v ∈ ∏_{j∈I} V_{j} and j ∈ I, then v_{j} denotes the jth coordinate of v in V_{j}, as before.

Let A be a module over k, and suppose that for each j ∈ I, β^{V_{j}} is a mapping from A × V_{j} into V_{j} that is bilinear over k. If a ∈ A and v ∈ ∏_{j∈I} V_{j}, then β(a, v) can be defined as an element of ∏_{j∈I} V_{j} by putting

(β(a, v))_{j} = β^{V_{j}}(a, v_{j}) (6.9.1)

for every j ∈ I. It is easy to see that this defines β as a mapping from A×∏_{j∈I} V_{j} into ∏_{j∈I} V_{j} that is bilinear over k. If v ∈ ⨁_{j∈I} V_{j}, then

β(a, v) ∈ ⨁_{j∈I} V_{j} (6.9.2)

for every a ∈ A.

Equivalently, if the bilinear action of A on V_{j} is given by ρ^{V_{j}} for each j ∈ I, as in Section 6.1, then the corresponding bilinear action ρ of A on ∏_{j∈I} V_{j} is defined by (ρ_{a}(v))_{j} = ρ_{a}^{V_{j}}(v_{j}). (6.9.3)

Similarly, if the bilinear action of A on V_{j} is expressed as a · v_{j} for every a ∈ A, v_{j} ∈ V_{j}, and j ∈ I, then the corresponding bilinear action of A on ∏_{j∈I} V_{j} can be expressed by a · v for every a ∈ A and v ∈ ∏_{j∈I} V_{j}, where

(a · v)_{j} = a · v_{j} (6.9.4) for each j ∈ I. If A acts on V_{j} on the right for each j ∈ I, so that the bilinear action is expressed as v_{j} · a for every a ∈ A, v_{j} ∈ V_{j}, and j ∈ I, then the corresponding bilinear action of A on ∏_{j∈I} V_{j} on the right can be expressed as v · a for every a ∈ A and v ∈ ∏_{j∈I} V_{j}, where

(v · a)_{j} = v_{j} · a (6.9.5)

for each j ∈ I.

Let A be an associative algebra over k. If ρ^{V_{j}} is a representation of A on V_{j} for each j ∈ I, then one can check that (6.9.3) defines ρ as a representation of A on ∏_{j∈I} V_{j}. Equivalently, if V_{j} is a left module over A for every j ∈ I, then ∏_{j∈I} V_{j} is a left module over A with respect to (6.9.4). Note that ⨁_{j∈I} V_{j} is a left submodule of ∏_{j∈I} V_{j}, as a left module over A, as in (6.9.2). Similarly, if V_{j} is a right module over A for every j ∈ I, then ∏_{j∈I} V_{j} a right module over A with respect to (6.9.5), and ⨁_{j∈I} V_{j} is a right submodule of ∏_{j∈I} V_{j}, as a right module over A.

Now let A be a Lie algebra over k. If ρ^{V_{j}} is a representation of A on V_{j} for every j ∈ I, then one can verify that (6.9.3) defines ρ as a representation of A on ∏_{j∈I} V_{j}. Equivalently, if V_{j} is a module over A for every j ∈ I, then ∏_{j∈I} V_{j} is a module over A with respect to (6.9.4). As before, ⨁_{j∈I} V_{j} is a submodule of ∏_{j∈I} V_{j}, as a module over A.

Let V be a module over k, and let V_{1}, V_{2} be submodules of V. Observe that

V_{1} + V_{2} = {v_{1} + v_{2} : v_{1} ∈ V_{1}, v_{2} ∈ V_{2}} (6.9.6)

is a submodule of V too. The direct sum V_{1} ⊕ V_{2} of V_{1} and V_{2} can be defined as a module over k as in Section 4.1, with I = {1, 2}. More precisely, V_{1} ⊕ V_{2} can be defined as a set as the Cartesian product V_{1} × V_{2} of V_{1} and V_{2}, consisting of all ordered pairs (v_{1}, v_{2}) with v_{1} ∈ V_{1} and v_{2} ∈ V_{2}. Addition and scalar multiplication can be defined on V_{1} ⊕ V_{2} coordinatewise, as usual. Observe that

(v_{1}, v_{2}) ↦ v_{1} + v_{2} (6.9.7)

defines a homomorphism from V_{1} ⊕ V_{2} onto V_{1} + V_{2}, as modules over k. If

V_{1} ∩ V_{2} = {0}, (6.9.8)

then (6.9.7) is injective as a mapping from V_{1} ⊕ V_{2} into V.

Let A be a module over k again, and suppose that we have a bilinear action of A on V. If this action maps V_{1} and V_{2} into themselves, then it maps V_{1} + V_{2} into itself as well. In this case, we also get a corresponding bilinear action of A on V_{1} ⊕ V_{2}, as before. Of course, (6.9.7) intertwines the bilinear actions of A on V_{1} ⊕ V_{2} and V_{1} + V_{2}.

6.10 Compatible bilinear mappings

Let k be a commutative ring with a multiplicative identity element, and let A and V be modules over k. Also let ρ be a bilinear action of A on V, so that ρ_{a} is a module homomorphism from V into itself for every a ∈ A, and a ↦ ρ_{a} is a module homomorphism from A into Hom_{k}(V, V), as in Section 6.1. Let W be another module over k, and let μ be a mapping from V × V into W that is bilinear over k. Suppose that for every a ∈ A there is an ã ∈ A such that

μ(ρ_{a}(u), v) = μ(u, ρ_{a}(v)) (6.10.1)

for every u, v ∈ V. If k = C, then one may consider a sesquilinear form μ on V. Let V_{0} be a submodule of V, and put

(6.10.2) (V_{0})^{⊥} = (V_{0})^{⊥,μ} = {u ∈ V : for every v ∈ V_{0}, μ(u, v) = 0}.

Suppose that ρ satisfies the compatibility condition with μ in the preceding paragraph, and that ρ_{a}(V_{0}) ⊆ V_{0} (6.10.3)

for every a ∈ A. Let a ∈ A, u ∈ (V_{0})^{⊥}, and v ∈ V_{0} be given, and let ã ∈ A be as in (6.10.1). Observe that

μ(ρ_{a}(u), v) = μ(u, ρ_{a}(v)) = 0, (6.10.4)

because ρ_{a}(v) ∈ V_{0}, by (6.10.3). This means that ρ_{a}(u) ∈ (V_{0})^{⊥}, so that

ρ_{a}((V_{0})^{⊥}) ⊆ (V_{0})^{⊥}. (6.10.5)

Now let A be an associative algebra over k, and suppose that ρ is a representation of A on V. Also let a ↦ a^{∗} be an opposite algebra automorphism on A. One way that (6.10.1) can hold is with ã = a^{∗}, so that

μ(ρ_{a}(u), v) = μ(u, ρ_{a^{∗}}(v)) (6.10.6)

for every a ∈ A and u, v ∈ V. If k = C and μ is a sesquilinear form on V, then one may consider a conjugate-linear opposite algebra automorphism on A.

If ρ satisfies (6.10.6), then

μ(ρ_{a}(u), ρ_{a}(v)) = μ(u, ρ_{a^{∗}}(ρ_{a}(v))) = μ(u, ρ_{a^{∗} a}(v)) (6.10.7)

for every a ∈ A and u, v ∈ V. Suppose that A has a multiplicative identity element e, and that ρ_{e} is the identity mapping on V. If a ∈ A satisfies a^{∗} a = e, then (6.10.7) implies that ρ_{a} preserves μ.

In some situations there may be an opposite algebra automorphism T ↦ T^{⋆} on the algebra Hom_{k}(V, V) of module homomorphisms from V into itself such that μ(T (u), v) = μ(u, T^{⋆}(v)) (6.10.8)

for every u, v ∈ V and T ∈ Hom_{k}(V, V). In this case, one can ask that ρ be compatible with these opposite algebra automorphisms on A and Hom_{k}(V, V), in the sense that (ρ_{a})^{⋆} = ρ_{a^{∗}} (6.10.9) for every a ∈ A. Note that (6.10.6) follows from (6.10.8) and (6.10.9). More precisely, one might have an opposite algebra automorphism on a subalgebra of Hom_{k}(V, V) that satisfies (6.10.8), and a representation ρ of A on V with values in this subalgebra of Hom_{k}(V, V). If k = C, then one may consider conjugate-linear opposite algebra automorphisms and a sesquilinear form on V again.

Suppose now that (A, [·, ·]_{A}) is a Lie algebra over k, and that ρ is a representation of A as a Lie algebra on V. A natural compatibility condition for ρ with μ is that ρ_{a} be antisymmetric with respect to μ for every a ∈ A, so that

μ(ρ_{a}(u), v) = −μ(u, ρ_{a}(v)) (6.10.10)

for every u, v ∈ V. This means that (6.10.1) holds with ã = −a.

Let us take V = A, as a module over k, and ρ to be the adjoint representation on A. Thus, for each x ∈ A, ρ_{x} = ad x = ad_{x} is the module homomorphism from A into itself defined by

ad_{x}(z) = [x, z]_{A}, (6.10.11)

as in Section 2.4. In this situation, (6.10.10) is the same as saying that

μ(ad_{w}(x), y) = −μ(x, ad_{w}(y)) (6.10.12)

for every w, x, y ∈ A. Equivalently, this means that

μ([w, x]_{A}, y) = −μ(x, [w, y]_{A}) (6.10.13)

for every w, x, y ∈ A, which can also be expressed as

μ([x, w]_{A}, y) = μ(x, [w, y]_{A}). (6.10.14)

This property is sometimes described by saying that μ is associative on A × A, as on p21 of [14].

6.11 Representations and formal power series

Let k be a commutative ring with a multiplicative identity element, let V be a module over k, and let T be an indeterminate. Remember that there are natural isomorphisms between Hom_{k}(V, V [[T]]),

Hom_{k[[T]]}(V [[T]], V [[T]]), (6.11.1)

and (Hom_{k}(V, V))[[T]], (6.11.2)

as modules over k[[T]], as in Sections 4.8 and 4.9. Of course, Hom_{k}(V, V) and (6.11.1) are associative algebras over k and k[[T]], respectively, with composition of mappings as multiplication. Similarly, (6.11.2) is an associative algebra over k[[T]], as in Section 4.6. In fact, the natural isomorphism between (6.11.2) and (6.11.1) preserves multiplication, as in Section 4.10.

As before, there is also a natural isomorphism between Hom_{k}(V, V [T]) and

Hom_{k[T]}(V [T], V [T]), (6.11.3)

and a natural injective homomorphism from

(Hom_{k}(V, V))[T] (6.11.4)

into (6.11.3), as modules over k[T]. Note that (6.11.3) is an associative algebra over k[T] with respect to composition of mappings, and that (6.11.4) is an associative algebra over k[T] too, as in Section 4.6. The natural injective homomorphism from (6.11.4) into (6.11.3) preserves multiplication, as in Section 4.10.

Let A be an associative algebra over k, so that A[[T]] and A[T] are associative algebras over k[[T]] and k[T], respectively, as in Section 4.6. Remember that a representation of A on V corresponds to an algebra homomorphism from A into Hom_{k}(V, V), as in Section 6.4. This can be extended to an algebra homomorphism from A[[T]] into (6.11.2), as algebras over k[[T]], as in Section 4.12. This corresponds to an algebra homomorphism from A[[T]] into (6.11.1), which is to say a representation of A[[T]] on V [[T]]. Similarly, we get an algebra homomorphism from A[T] into (6.11.4), as algebras over k[T], as in Section 4.12. This leads to an algebra homomorphism from A[T] into (6.11.3), which gives a representation of A[T] on V [T]. If V is a right module over A, then we can get V [[T]] as a right module over A[[T]], and V [T] as a right module over A[T], in the same way.

Remember that an associative algebra is automatically a Lie algebra with respect to the corresponding commutator bracket. The natural algebra isomorphism between (6.11.2) and (6.11.1) mentioned earlier automatically preserves commutator brackets. Similarly, the natural injective algebra homomorphism from (6.11.4) into (6.11.3) automatically preserves commutator brackets. Let A be a Lie algebra over k, so that A[[T]] and A[T] are Lie algebras over k[[T]] and k[T], as in Section 4.6. A representation of A on V is the same as a homomorphism from A into Hom_{k}(V, V), as a Lie algebra over k with respect to the commutator bracket. This can be extended to a Lie algebra homomorphism from A[[T]] into (6.11.2), as in Section 4.12. This corresponds to a Lie algebra homomorphism from A[[T]] into (6.11.1), which is a representation of A[[T]] as a Lie algebra over k[[T]] on V [[T]]. We also get a Lie algebra homomorphism from A[T] into (6.11.4), as in Section 4.12. This leads to a Lie algebra homomorphism from A[T] into (6.11.3), and hence a representation of A[T] as a Lie algebra over k[T] on V [T].

6.12 Opposite algebras

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. The corresponding opposite algebra A^{op} is defined as an algebra over k in the strict sense as follows. As a module over k, A^{op} is the same as A. The product of a, b ∈ A^{op} is defined to be the product b a of b and a in A. Thus multiplication in A^{op} is the same as multiplication in A exactly when A is commutative. If A has a multiplicative identity element e, then e is also the multiplicative identity element in A^{op}. If A is associative, then it is easy to see that A^{op} is associative as well. By construction, the identity mapping on A is an opposite algebra isomorphism between A and A^{op}.

Let B be another algebra over k in the strict sense. An algebra homomorphism from A into B may also be considered as an algebra homomorphism from A^{op} into B^{op}. An opposite algebra homomorphism from A into B corresponds to an algebra homomorphism from A^{op} into B, or from A into B^{op}.

Suppose that A is an associative algebra over k, and let V be a module over k. A bilinear action of A on V makes V into a left module over A exactly when it makes V into a right module over A^{op}. Similarly, a bilinear action of A on V makes V into a right module over A exactly when it makes V into a left module over A^{op}. Equivalently, this corresponds to a representation of A^{op} on V.

Let n be a positive integer, and let M_{n}(A) and M_{n}(A^{op}) be the corresponding spaces of n × n matrices with entries in A and A^{op}, respectively, as in Section 2.8. Thus M_{n}(A) and M_{n}(A^{op}) are the same as modules over k, using entrywise addition and scalar multiplication. Remember that M_{n}(A) and M_{n}(A^{op}) are algebras over k with respect to matrix multiplication. Let a, b be n×n matrices with entries in A, which can also be considered as n × n matrices with entries in A^{op}. The product c of a and b in M_{n}(A) is given by

c_{j,r} = ∑_{l=1}^{n} a_{j,l} b_{l,r} (6.12.1)

for every j, r = 1, …, n, as usual. Let c̃ be the product of a and b in M_{n}(A^{op}). This means that

c̃_{j,r} = ∑_{l=1}^{n} b_{l,r} a_{j,l} (6.12.2)

for every j, r = 1, …, n, where the terms in the sum on the right use multiplication in A.

Let a^{t}, b^{t}, and c^{t} be the transposes of a, b, and c, respectively, as in Section 2.8. Thus

c_{j,r}^{t} = c_{r,j} = ∑_{l=1}^{n} a_{r,l} b_{l,j} (6.12.3)

for every j, r = 1, …, n. This is the same as the product of b^{t} and a^{t} in M_{n}(A^{op}). This means that a ↦ a^{t} (6.12.4)

is an opposite algebra homomorphism from M_{n}(A) into M_{n}(A^{op}). More precisely, (6.12.4) is an opposite algebra isomorphism from M_{n}(A) onto M_{n}(A^{op}), because (6.12.4) is a one-to-one mapping from M_{n}(A) onto M_{n}(A^{op}).

6.13 Matrices and associative algebras

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k. Also let n be a positive integer, and let A^{n} be the space of n-tuples of elements of A, as usual. Of course, A^{n} is a module over k with respect to coordinatewise addition and scalar multiplication. Similarly, A^{n} may be considered as both a left and right module over A, with respect to coordinatewise multiplication. More precisely, if a ∈ A and x = (x_{1}, …, x_{n}) is an element of A^{n}, then a · x and x · a are defined as elements of A^{n} by

a · x = (a x_{1}, …, a x_{n}) (6.13.1)

and x · a = (x_{1} a, …, x_{n} a), (6.13.2)

respectively. If A is commutative, then (6.13.1) and (6.13.2) are the same. Note that (a · x) · b = a · (x · b) (6.13.3)

for every a, b ∈ A and x ∈ A^{n}.

Let α = (α_{j,l}) be an n × n matrix with entries in A, which is to say an element of M_{n}(A). If x ∈ A^{n}, then let T_{α}^{L}(x) be the element of A^{n} whose jth coordinate is given by

(T_{α}^{L}(x))_{j} = ∑_{l=1}^{n} α_{j,l} x_{l} (6.13.4)

for every j = 1, …, n. Similarly, let T_{α}^{R}(x) be the element of A^{n} whose jth coordinate is given by

(T_{α}^{R}(x))_{j} = ∑_{l=1}^{n} x_{l} α_{j,l} (6.13.5)

for every j = 1, …, n. If A is commutative, then (6.13.4) and (6.13.5) are the same. It is easy to see that T_{α}^{L} and T_{α}^{R} are homomorphisms from A^{n} into itself, as a module over k. Observe that

(T_{α}^{L}(x)) · a = T_{α}^{L}(x · a) (6.13.6)

and a · (T_{α}^{R}(x)) = T_{α}^{R}(a · x) (6.13.7)

for every a ∈ A and x ∈ A^{n}. Thus T_{α}^{L} is a homomorphism from A^{n} into itself as a right module over A, and T_{α}^{R} is a homomorphism from A^{n} into itself as a left module over A.

If t ∈ k, then t α = (t α_{j,l}) ∈ M_{n}(A), as in Section 2.8. Clearly

T_{t α}^{L}(x) = t T_{α}^{L}(x) (6.13.8)

and T_{t α}^{R}(x) = t T_{α}^{R}(x) (6.13.9) for every x ∈ A^{n}. Let β = (β_{j,l}) be another n × n matrix with entries in A, so that α + β ∈ M_{n}(A) too. Of course,

T_{α+β}^{L}(x) = T_{α}^{L}(x) + T_{β}^{L}(x) (6.13.10)

and T_{α+β}^{R}(x) = T_{α}^{R}(x) + T_{β}^{R}(x) (6.13.11)

for every x ∈ A^{n}. One can check that

T_{α}^{L}(T_{β}^{L}(x)) = T_{α β}^{L}(x) (6.13.12)

for every x ∈ A^{n}, where α β ∈ M_{n}(A) is defined using matrix multiplication, as in Section 2.8. Let γ be the product of α and β as elements of M_{n}(A^{op}), where A^{op} is the opposite algebra associated to A, as in the previous section. One can verify that T_{α}^{R}(T_{β}^{R}(x)) = T_{γ}^{R}(x) (6.13.13)

for every x ∈ A^{n}.

Suppose that A has a multiplicative identity element e. Remember that the corresponding identity matrix in M_{n}(A) has diagonal entries equal to e and all other entries equal to 0, as in Section 2.8. If α is the identity matrix, then T_{α}^{L} and T_{α}^{R} are equal to the identity mapping on A^{n}. Let u^{1}, …, u^{n} be the elements of A^{n} with u_{j}^{l} = e when j = l and u_{j}^{l} = 0 when j ≠ l. Thus

x = ∑_{l=1}^{n} x_{l} · u^{l} = ∑_{l=1}^{n} u^{l} · x_{l} (6.13.14)

for every x ∈ A^{n}.

If T is any homomorphism from A^{n} into itself, as a right module over A, then

T (x) = T ( ∑_{l=1}^{n} u^{l} · x_{l}) = ∑_{l=1}^{n} T (u^{l}) · x_{l} (6.13.15)

for every x ∈ A^{n}. This means that T can be represented in a unique way as T_{α}^{L}, with α ∈ M_{n}(A). More precisely, α_{j,l} is the jth coordinate of T (u^{l}) for each j, l = 1, …, n. Similarly, if T is a homomorphism from A^{n} into itself, as a left module over A, then

T (x) = T ( ∑_{l=1}^{n} x_{l} · u^{l}) = ∑_{l=1}^{n} x_{l} · T (u^{l}) (6.13.16)

for every x ∈ A^{n}. This implies that T can be represented in a unique way as T_{α}^{R}, where α_{j,l} is the jth coordinate of T (u^{l}) for every j, l = 1, …, n.

6.14 Irreducibility

Let k be a commutative ring with a multiplicative identity element, let A, V be modules over k, and let ρ be a bilinear action of A on V. Thus ρ_{a} is a homomorphism from V into itself, as a module over k, for every a ∈ A, and a ↦ ρ_{a} is a homomorphism from A into the space Hom_{k}(V, V) of module homomorphisms from V into itself, as modules over k, as in Section 6.1. Suppose that there is no submodule W of V, as a module over k, such that W ≠ {0}, V and ρ_{a}(W) ⊆ W (6.14.1)

for every a ∈ A. In this case, one may say that ρ is irreducible on V, or equivalently that V is simple with respect to the action of ρ. The condition that V ≠ {0} is typically included in the definition of irreducibility or simplicity as well.

Let V_{1}, V_{2} be modules over k, and let ρ^{1}, ρ^{2} be bilinear actions of A on V_{1}, V_{2}, respectively. Suppose that ϕ is a homomorphism from V_{1} into V_{2}, as modules over k, that intertwines ρ^{1} and ρ^{2}, as in Section 6.2. If v_{1} ∈ V_{1} is in the kernel of ϕ and a ∈ A, then

ϕ(ρ_{a}^{1}(v_{1})) = ρ_{a}^{2}(ϕ(v_{1})) = 0, (6.14.2)

so that ρ_{a}^{1}(v_{1}) is in the kernel of ϕ too. If ρ^{1} is irreducible on V_{1}, then it follows that the kernel of ϕ is either trivial or equal to V_{1}, so that ϕ is either injective or equal to 0 on V_{1}. This is part of Schur’s lemma.

Similarly, ρ_{a}^{2}(ϕ(V_{1})) = ϕ(ρ_{a}^{1}(V_{1})) ⊆ ϕ(V_{1}) (6.14.3)

for every a ∈ A. If ρ^{2} is irreducible on V_{2}, then it follows that ϕ(V_{1}) = {0} or V_{2}, so that either ϕ = 0 on V_{1} or ϕ is surjective. This is another part of Schur’s lemma.. If ρ^{1} and ρ^{2} are both irreducible, then either ϕ = 0 or ϕ is a bijection.

Let V be a module over k again, and let ρ be a bilinear action of A on V. Remember that Hom_{k}(V, V) is an associative algebra over k with respect to composition of mappings. Consider the space Hom_{k,ρ}(V, V) of ϕ ∈ Hom_{k}(V, V) that intertwine ρ. It is easy to see that Hom_{k,ρ}(V, V) is a subalgebra of Hom_{k}(V, V), and that Hom_{k,ρ}(V, V) contains the identity mapping on V. If ρ is irreducible, then every nonzero element of Hom_{k,ρ}(V, V) is invertible, as in the preceding paragraph.

Suppose that k is an algebraically closed field, and that V has positive finite dimension as a vector space over k. If ϕ is any linear mapping from V into itself, then it is well known that there is a λ ∈ k such that ϕ has a nonzero eigenvector in V with eigenvalue λ. Let E_{λ} be the corresponding eigenspace of eigenvectors of ϕ in V with eigenvalue λ. Let ρ be a bilinear action of A on V again, and suppose that ϕ intertwines ρ. If a ∈ A, then ρ_{a} maps E_{λ} into itself, by a standard argument. If ρ is irreducible on V, then it follows that E_{λ} = V. This is another part of Schur’s lemma.

6.15 Representations and differential operators

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, and let n be a positive integer. Also let T_{1}, …, T_{n} be n commuting indeterminates, and let let ∂_{1}, …, ∂_{n} be n commuting formal symbols, which may be used to represent partial derivatives, as before. Remember that the space A[[T_{1}, …, T_{n}]] of formal power series in T_{1}, …, T_{n} with coefficients in A is a module over k, with respect to termwise addition and scalar multiplication, as in Section 5.8.

The space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] is an associative algebra over k, as in Section 5.11. These formal differential operators determine mappings from A[[T_{1}, …, T_{n}]] into itself that are linear over k, as before. This defines a representation of the algebra of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] on A[[T_{1}, …, T_{n}]].

The space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] is a subalgebra of the algebra of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]]. Thus A[[T_{1}, …, T_{n}]] may be considered as a left module over the algebra of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}]. The mappings from A[[T_{1}, …, T_{n}]] into itself corresponding to formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] map the space A[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in A into itself, as in Section 5.11. This means that

(6.15.1) A[T_{1}, …, T_{n}] is a submodule of A[[T_{1}, …, T_{n}]],

as a left module over the algebra of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}].

Similarly, the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] that are homogeneous of degree 0 is a subalgebra of the algebra of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}], as in Section 5.14. This permits us to consider A[T_{1}, …, T_{n}] as a left module over the algebra of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] that are homogeneous of degree 0. If d is a nonnegative integer, then the space A_{d}[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in A that are homogeneous of degree d is a submodule of A[T_{1}, …, T_{n}], as a module over k, as in Section 5.13. In fact,

(6.15.2) A_{d}[T_{1}, …, T_{n}] is a submodule of A[T_{1}, …, T_{n}],

as a left module over the algebra of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] that are homogeneous of degree 0, as in Section 5.14.

If a(T) = (a^{1}(T), …, a^{n}(T)) is in the space (k[[T_{1}, …, T_{n}]])^{n} of n-tuples of elements of k[[T_{1}, …, T_{n}]], then put D_{a(T)} = ∑_{j=1}^{n} a^{j}(T) ∂_{j}, as in Section 5.12. The space of these formal differential operators is a Lie subalgebra of the space of all formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]], as a Lie algebra over k with respect to the commutator bracket, as in Section 5.15. Thus we may consider A[[T_{1}, …, T_{n}]] as a module over the Lie algebra consisting of D_{a(T)}, a(T) ∈ (k[[T_{1}, …, T_{n}]])^{n}.

Similarly, the space of D_{a(T)}, a(T) ∈ (k[T_{1}, …, T_{n}])^{n}, is a Lie subalgebra of the Lie algebra consisting of D_{a(T)}, a(T) ∈ (k[[T_{1}, …, T_{n}]])^{n}. This permits us to consider A[[T_{1}, …, T_{n}]] as a module over the Lie algebra consisting of D_{a(T)}, a(T) ∈ (k[T_{1}, …, T_{n}])^{n}. We may also consider A[T_{1}, …, T_{n}] as a submodule of A[[T_{1}, …, T_{n}]], as a module over the Lie algebra consisting of D_{a(T)}, with a(T) ∈ (k[T_{1}, …, T_{n}])^{n}.

The space g_{n}(k) consisting of D_{a(T)}, a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n}, is a Lie subalgebra of the Lie algebra of D_{a(T)}, a(T) ∈ (k[T_{1}, …, T_{n}]), as in Section 5.15. Thus (6.15.3) we may consider A[T_{1}, …, T_{n}] as a module over g_{n}(k),

as a Lie algebra over k. If d is a nonnegative integer, then

(6.15.4) A_{d}[T_{1}, …, T_{n}] is a submodule of A[T_{1}, …, T_{n}], as a module over g_{n}(k).

Remember that s_{n}(k) is the Lie subalgebra of g_{n}(k) consisting of D_{a(T)} where the matrix of coefficients of the components of a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n} has trace 0. We may consider A_{d}[T_{1}, …, T_{n}] as a module over s_{n}(k) too, as a Lie algebra over k, for each d ≥ 0.

Representations and multilinear mappings

7.1 Some remarks about subalgebras

Let k be a commutative ring with a multiplicative identity element, and let A_{1}, A_{2} be algebras over k in the strict sense. The direct sum A_{1} ⊕ A_{2} of A_{1} and A_{2} can be defined as an algebra over k in the strict sense as in Section 4.1, with I = {1, 2}. More precisely, A_{1} ⊕ A_{2} can be defined as a set as the Cartesian product of A_{1} and A_{2}, consisting of all ordered pairs (a_{1}, a_{2}) with a_{1} ∈ A_{1} and a_{2} ∈ A_{2}. Addition, scalar multiplication, and multiplication on A_{1} ⊕ A_{2} are defined coordinatewise, as usual. In particular, if multiplication of a_{1}, b_{1} ∈ A_{1} and a_{2}, b_{2} ∈ A_{2} are expressed as a_{1} b_{1} and a_{2} b_{2}, respectively, then

(a_{1}, 0) (0, a_{2}) = 0 (7.1.1)

in A_{1} ⊕ A_{2} for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}.

If A is a module over k and A_{1}, A_{2} ⊆ A are submodules of A, then

A_{1} + A_{2} = {a_{1} + a_{2} : a_{1} ∈ A_{1}, a_{2} ∈ A_{2}} (7.1.2)

is a submodule of A, as before. Suppose that A is an algebra over k in the strict sense. If A_{1}, A_{2} are left ideals in A, then (7.1.2) is a left ideal in A. Similarly, if A_{1}, A_{2} are right ideals in A, then (7.1.2) is a right ideal in A. If A_{1}, A_{2} are two-sided ideals in A, then it follows that (7.1.2) is a two-sided ideal in A.

Let A_{1}, A_{2} be subalgebras of A. If

a_{1} a_{2} = a_{2} a_{1} = 0 (7.1.3)

for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}, then it is easy to see that (7.1.2) is a subalgebra of A. In this situation, (a_{1}, a_{2}) ↦ a_{1} + a_{2} (7.1.4) defines a homomorphism from A_{1} ⊕ A_{2} into A, as algebras over k in the strict sense. If (A, [·, ·]_{A}) is a Lie algebra over k, then (7.1.3) is the same as saying that [a_{1}, a_{2}]_{A} = 0 (7.1.5)

for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}.

Let A and B be algebras over k in the strict sense, and let ϕ, ψ be algebra homomorphisms from A into B. Suppose that

ϕ(x) ψ(y) = ψ(y) ϕ(x) = 0 (7.1.6)

for every x, y ∈ A. This implies that

(ϕ(x) + ψ(x)) (ϕ(y) + ψ(y)) (7.1.7)

= ϕ(x) ϕ(y) + ϕ(x) ψ(y) + ψ(x) ϕ(y) + ψ(x) ψ(y)
= ϕ(x) ϕ(y) + ψ(x) ψ(y) = ϕ(x y) + ψ(x y)

for every x, y ∈ A, so that ϕ + ψ defines an algebra homomorphism from A into B as well. If (B, [·, ·]_{B}) is a Lie algebra over k, then (7.1.6) is the same as saying that [ϕ(x), ψ(y)]_{B} = 0 (7.1.8)

for every x, y ∈ A.

Let A be a Lie algebra over k, and let ρ^{1}, ρ^{2} be Lie algebra representations of A on a module V over k. Let us say that that ρ^{1} and ρ^{2} are commuting representations of A on V if

ρ_{a}^{1} ∘ ρ_{b}^{2} = ρ_{b}^{2} ∘ ρ_{a}^{1} (7.1.9)

for every a, b ∈ A. Under these conditions,

(ρ^{1} + ρ^{2})_{a} = ρ_{a}^{1} + ρ_{a}^{2} (7.1.10)

defines a Lie algebra representation of A on V too, as in the preceding paragraph.

7.2 Representations on linear mappings

Let k be a commutative ring with a multiplicative identity element, and let V, W be modules over k. Remember that the space Hom_{k}(V, W) of module homomorphisms from V into W is a module over k too, with respect to pointwise addition and scalar multiplication. Let T be a module homomorphism from V into itself, and let ϕ be a module homomorphism from V into W. Thus

T̃(ϕ) = ϕ ∘ T (7.2.1)

defines another module homomorphism from V into W. This defines T̃ as a homomorphism from Hom_{k}(V, W) into itself, as a module over k.

148 CHAPTER 7. REPRESENTATIONS AND MULTILINEAR MAPPINGS

Let R be another module homomorphism from V into itself, so that the composition T ∘ R is a module homomorphism from V into itself as well. In particular, R̃ and ˜ (T ∘ R) can be defined as homomorphisms from Hom_{k}(V, W), as a module over k, as in the preceding paragraph. If ϕ is a module homomorphism from V into W, then

˜ (T ∘ R)(ϕ) = ϕ ∘ (T ∘ R) = (ϕ ∘ T) ∘ R = R̃(T̃(ϕ)). (7.2.2)

This means that ˜ (T ∘ R) = R̃ ∘ T̃ (7.2.3)

as mappings from Hom_{k}(V, W). Note that

T ↦ T̃ (7.2.4)

is linear over k, as a mapping from Hom_{k}(V, V) into the space of mappings from Hom_{k}(V, W) into itself. More precisely, this defines an opposite algebra homomorphism from Hom_{k}(V, V) into the algebra of module homomorphisms from Hom_{k}(V, W) into itself. If T is the identity mapping on V, then T̃ is the identity mapping on Hom_{k}(V, W).

Let A be an associative algebra over k, and let ρ be a representation of A on V. If a ∈ A, then ρ_{a} is a module homomorphism from V into itself, so that ρ̃_{a} can be defined as before, as a module homomorphism from Hom_{k}(V, W) into itself. By hypothesis, a ↦ ρ_{a} is an algebra homomorphism from A into Hom_{k}(V, V), which implies that ρ̃_{a} is an opposite algebra homomorphism from A into the algebra of module homomorphisms from Hom_{k}(V, W) into itself. Similarly, if a ↦ ρ_{a} is an opposite algebra homomorphism from A into Hom_{k}(V, V), then a ↦ ρ̃_{a} is an algebra homomorphism from A into the algebra of module homomorphisms from Hom_{k}(V, W) into itself. Equivalently, if V is a left module over A, then Hom_{k}(V, W) becomes a right module over A in this way, and if V is a right module over A, then Hom_{k}(V, W) becomes a left module over A.

Now let A be a Lie algebra over k, and let ρ be a representation of A on V. As before, ρ̃_{a} is defined as a module homomorphism from V into W for every a ∈ A. Under these conditions, one can check that a ↦ −ρ̃_{a} defines a representation of A as a Lie algebra on Hom_{k}(V, W).

7.3 Multilinear mappings

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Also let V_{1}, …, V_{n} be n modules over k, and let W be another module over k. A mapping μ from V_{1} × ⋯ × V_{n} into W is said to be multilinear over k if μ is linear over k in each variable separately. This reduces to ordinary linearity over k when n = 1, and to bilinearity over k when n = 2.

The space of mappings μ from ∏_{j=1}^{n} V_{j} into W that are multilinear over k may be denoted L(V_{1}, …, V_{n}; W), or L_{k}(V_{1}, …, V_{n}; W) to indicate the role of k. It is easy to see that L_{k}(V_{1}, …, V_{n}; W) is a module over k with respect to pointwise addition and scalar multiplication of mappings from ∏_{j=1}^{n} V_{j} into W. More precisely, L_{k}(V_{1}, …, V_{n}; W) may be considered as a submodule of the module of all W -valued functions on ∏_{j=1}^{n} V_{j}. Of course, L_{k}(V_{1}, …, V_{n}; W) reduces to Hom_{k}(V_{1}, W) when n = 1.

Suppose for the moment that V_{1}, …, V_{n} are the same module V over k, and that n ≥ 2. Let us say that μ ∈ L_{k}(V_{1}, …, V_{n}; W) is symmetric if μ(v_{1}, …, v_{n}) is invariant under permutations of the variables v_{1}, …, v_{n}. This reduces to the earlier notion of symmetry for bilinear mappings when n = 2. Similarly, let us say that μ ∈ L_{k}(V_{1}, …, V_{n}; W) is antisymmetric if

μ(v_{1}, …, v_{j−1}, v_{l}, v_{j+1}, …, v_{l−1}, v_{j}, v_{l+1}, …, v_{n}) (7.3.1)

= −μ(v_{1}, …, v_{j−1}, v_{j}, v_{j+1}, …, v_{l−1}, v_{l}, v_{l+1}, …, v_{n})

for every v_{1}, …, v_{n} ∈ V and 1 ≤ j < l ≤ n, which is to say that interchanging two of the variables corresponds to taking the additive inverse of the value of μ. This reduces to the earlier notion of antisymmetry for bilinear mappings when n = 2. As usual, it is sometimes better to ask that

μ(v_{1}, …, v_{n}) = 0 (7.3.2)

whenever v_{j} = v_{l} for some j ≠ l. This implies that μ is antisymmetric, by the same type of argument as for bilinear mappings. If 1 + 1 is invertible in k, then this condition holds when μ is antisymmetric, as before.

Let V_{1}, …, V_{n} be n modules over k for some n ≥ 1 again, and let j be an integer with 1 ≤ j ≤ n. Also let A_{j} be a module homomorphism from V_{j} into itself, and let μ be a mapping from ∏_{j=1}^{n} V_{j} into W that is multilinear over k. If v_{1} ∈ V_{1}, …, v_{n} ∈ V_{n}, then put

(Ã_{j}(μ))(v_{1}, …, v_{n}) = μ(v_{1}, …, v_{j−1}, A_{j}(v_{j}), v_{j+1}, …, v_{n}). (7.3.3)

This defines a mapping Ã_{j}(μ) from ∏_{j=1}^{n} V_{j} into W, which corresponds to composing μ with A_{j} in the jth variable. Note that Ã_{j}(μ) is multilinear over k, because A_{j} is linear over k. It is easy to see that Ã_{j} is linear over k, as a mapping from L(V_{1}, …, V_{n}; W) into itself. The mapping

A_{j} ↦ Ã_{j} (7.3.4)

is linear over k as well, as a mapping from Hom_{k}(V_{j}, V_{j}) into the space of module homomorphisms from L(V_{1}, …, V_{n}; W) into itself.

Let B_{j} be another module homomorphism from V_{j} into itself, and let μ be a mapping from ∏_{j=1}^{n} V_{j} into W that is multilinear over k again. The composition B_{j} ∘A_{j} of A_{j} and B_{j} is a module homomorphism from V_{j} into itself too, so that ˜ (B_{j} ∘ A_{j})(μ) can be defined as a multilinear mapping from ∏_{j=1}^{n} V_{j} into W as before. More precisely, if v_{1} ∈ V_{1}, …, v_{n} ∈ V_{n}, then

( ˜ (B_{j} ∘ A_{j})(μ))(v_{1}, …, v_{n}) (7.3.5)

= μ(v_{1}, …, v_{j−1}, (B_{j} ∘ A_{j})(v_{j}), v_{j+1}, …, v_{n})
= μ(v_{1}, …, v_{j−1}, B_{j}(A_{j}(v_{j})), v_{j+1}, …, v_{n}).

This is the same as

(B̃_{j}(μ))(v_{1}, …, v_{j−1}, A_{j}(v_{j}), v_{j+1}, …, v_{n}) (7.3.6)

= (Ã_{j}(B̃_{j}(μ)))(v_{1}, …, v_{n}),

so that ˜ (B_{j} ∘ A_{j}) = Ã_{j} ∘ B̃_{j} (7.3.7)

as mappings from L(V_{1}, …, V_{n}; W) into itself. Of course, this was mentioned in Section 7.2 when n = 1.

7.4 Boundedness and multilinearity

Let k be a field with an absolute value function | · |, let n be a positive integer, and let V_{1}, …, V_{n} and W be vector spaces over k. Also let N_{V_{1}}, …, N_{V_{n}} and N_{W} be seminorms on V_{1}, …, V_{n} and W, respectively, and with respect to | · | on k. A multilinear mapping μ from V_{1} × ⋯ × V_{n} into W is said to be bounded with respect to N_{V_{1}}, …, N_{V_{n}} and N_{W} if there is a nonnegative real number C such that N_{W} (μ(v_{1}, …, v_{n})) ≤ C N_{V_{1}}(v_{1}) ⋯ N_{V_{n}}(v_{n}) (7.4.1)

for every v_{1} ∈ V_{1}, …, v_{n} ∈ V_{n}. This reduces to the earlier definitions of boundedness for linear and bilinear mappings when n = 1 and n = 2, respectively, as in Sections 1.9 and 1.13.

Let BL(V_{1}, …, V_{n}; W) be the space of bounded multilinear mappings from ∏_{j=1}^{n} V_{j} into W, with respect to N_{V_{1}}, …, N_{V_{n}} and N_{W}. It is easy to see that this is a linear subspace of the space of all multilinear mappings from ∏_{j=1}^{n} V_{j} into W. If μ ∈ BL(V_{1}, …, V_{n}; W), then put

(7.4.2) ∥μ∥ = ∥μ∥_{V_{1},...,V_{n};W} = inf{C ≥ 0 : (7.4.1) holds},

where more precisely the infimum is taken over all nonnegative real numbers C such that (7.4.1) holds for every v_{1} ∈ V_{1}, …, v_{n} ∈ V_{n}. This reduces to the operator seminorm of a bounded linear mapping when n = 1, as in (1.9.3). As before, the infimum in (7.4.2) is automatically attained, so that (7.4.1) holds with C = ∥μ∥. One can check that (7.4.2) defines a seminorm on BL(V_{1}, …, V_{n}; W) with respect to | · |, and that (7.4.2) is a norm on BL(V_{1}, …, V_{n}; W) when N_{W} is a norm on W. Similarly, if N_{W} is a semi-ultranorm on W, then (7.4.2) is a semi-ultranorm on BL(V_{1}, …, V_{n}; W).

Suppose for the moment that n ≥ 2, and let μ be a multilinear mapping from ∏_{j=1}^{n} V_{j} into W. If v_{n} ∈ V_{n}, then

μ_{v_{n}}(v_{1}, …, v_{n−1}) = μ(v_{1}, …, v_{n−1}, v_{n}) (7.4.3)

defines a multilinear mapping from V_{1} × ⋯ × V_{n−1} into W. In addition,

T_{μ}(v_{n}) = μ_{v_{n}} (7.4.4) defines a linear mapping from V_{n} into L(V_{1}, …, V_{n−1}; W). Note that every linear mapping from V_{n} into L(V_{1}, …, V_{n−1}; W) corresponds to a multilinear mapping from ∏_{j=1}^{n} V_{j} into W in this way. If μ is also bounded as a multilinear mapping, then

N_{W} (μ_{v_{n}}(v_{1}, …, v_{n−1})) = N_{W} (μ(v_{1}, …, v_{n}))
(7.4.5)  ≤ ∥μ∥_{V_{1},...,V_{n};W} N_{V_{1}}(v_{1}) ⋯ N_{V_{n−1}}(v_{n−1}) N_{V_{n}}(v_{n})

for every v_{1} ∈ V_{1}, …, v_{n−1} ∈ V_{n−1}, v_{n} ∈ V_{n}. This implies that μ_{v_{n}} is bounded as a multilinear mapping for each v_{n} ∈ V_{n}, with

∥μ_{v_{n}}∥_{V_{1},...,V_{n−1};W} ≤ ∥μ∥_{V_{1},...,V_{n};W} N_{V_{n}}(v_{n}). (7.4.6)

Using ∥ · ∥_{V_{1},...,V_{n−1};W} on BL(V_{1}, …, V_{n−1}; W), we get that (7.4.4) is bounded as a linear mapping from V_{n} into BL(V_{1}, …, V_{n−1}; W), with

∥T_{μ}∥_{op} ≤ ∥μ∥_{V_{1},...,V_{n};W} . (7.4.7)

Conversely, suppose that (7.4.4) is a bounded linear mapping from V_{n} into BL(V_{1}, …, V_{n−1}; W). If v_{n} ∈ V, then μ_{v_{n}} ∈ BL(V_{1}, …, V_{n−1}; W), with

∥μ_{v_{n}}∥_{V_{1},...,V_{n−1};W} ≤ ∥T_{μ}∥_{op} N_{V_{n}}(v_{n}). (7.4.8)

If v_{1} ∈ V_{1}, …, v_{n−1} ∈ V_{n−1}, then we get that

N_{W} (μ(v_{1}, …, v_{n−1}, v_{n})) = N_{W} (μ_{v_{n}}(v_{1}, …, v_{n−1}))
(7.4.9)  ≤ ∥μ_{v_{n}}∥_{V_{1},...,V_{n−1};W} N_{V_{1}}(v_{1}) ⋯ N_{V_{n−1}}(v_{n−1})
≤ ∥T_{μ}∥_{op} N_{V_{1}}(v_{1}) ⋯ N_{V_{n−1}}(v_{n−1}) N_{V_{n}}(v_{n}).

This implies that μ ∈ BL(V_{1}, …, V_{n}; W), with

∥μ∥_{V_{1},...,V_{n};W} ≤ ∥T_{μ}∥_{op}. (7.4.10)

It follows that ∥T_{μ}∥_{op} = ∥μ∥_{V_{1},...,V_{n};W} , (7.4.11)

by (7.4.7).

Suppose now that n ≥ 1, 1 ≤ j ≤ n, and that A_{j} is a bounded linear mapping from V_{j} into itself, with respect to N_{V_{j}} on V_{j}. Let μ be a bounded multilinear mapping from ∏_{j=1}^{n} V_{j} into W, and let Ã_{j}(μ) be the multilinear mapping from ∏_{j=1}^{n} V_{j} into W corresponding to μ and A_{j} as in the previous section. If v_{1} ∈ V_{1}, …, v_{n} ∈ V_{n}, then

N_{W} ((Ã_{j}(μ))(v_{1}, …, v_{n})) (7.4.12)

= N_{W} (μ(v_{1}, …, v_{j−1}, A_{j}(v_{j}), v_{j+1}, v_{n}))
≤ ∥μ∥_{V_{1},...,V_{n};W} N_{V_{1}}(v_{1}) ⋯ N_{V_{j−1}}(v_{j−1}) N_{V_{j}}(A_{j}(v_{j}))
N_{V_{j+1}}(v_{j+1}) ⋯ N_{V_{n}}(v_{n})
≤ ∥A_{j}∥_{op,V_{j}V_{j}} ∥μ∥_{V_{1},...,V_{n};W} N_{V_{1}}(v_{1}) ⋯ N_{V_{n}}(v_{n}),

where ∥A_{j}∥_{op,V_{j}V_{j}} is the usual operator seminorm of A_{j} associated to N_{V_{j}} on V_{j}. This shows that Ã_{j}(μ) is also bounded as a multilinear mapping, with

∥Ã_{j}(μ)∥_{V_{1},...,V_{n};W} ≤ ∥A_{j}∥_{op,V_{j}V_{j}} ∥μ∥_{V_{1},...,V_{n};W} . (7.4.13)

Similarly, let A be a bounded linear mapping from W into itself, with respect to N_{W} on W. It is easy to see that the composition A∘μ of μ with A is multilinear over k, as a mapping from ∏_{j=1}^{n} V_{j} into W. If v_{1} ∈ V_{1}, …, v_{n} ∈ V_{n}, then

N_{W} (A(μ(v_{1}, …, v_{n}))) ≤ ∥A∥_{op,W W} N_{W} (μ(v_{1}, …, v_{n}))
(7.4.14)  ≤ ∥A∥_{op,W W} ∥μ∥_{V_{1},...,V_{n};W} N_{V_{1}}(v_{1}) ⋯ N_{V_{n}}(v_{n}),

where ∥A∥_{op,W W} is the operator seminorm of A associated to N_{W} on W. Thus A ∘ μ is bounded as a multilinear mapping too, with

∥A ∘ μ∥_{V_{1},...,V_{n};W} ≤ ∥A∥_{op,W W} ∥μ∥_{V_{1},...,V_{n};W} . (7.4.15)

7.5 Representations on multilinear mappings

Let k be a commutative ring with a multiplicative identity element, and let V_{1}, …, V_{n} and W be modules over k. The space L_{k}(V_{1}, …, V_{n}; W) of mappings from ∏_{j=1}^{n} V_{j} into W that are multilinear over k is a module over k with respect to pointwise addition and scalar multiplication of mappings, as in Section 7.3. Let A be an associative algebra over k, and suppose for the moment that W is a left module over A. If a ∈ A and μ ∈ L(V_{1}, …, V_{n}; W), then

a · (μ(v_{1}, …, v_{n})) (7.5.1)

defines a multilinear mapping from ∏_{j=1}^{n} V_{j} into W as well. It is easy to see that this makes L(V_{1}, …, V_{n}; W) into a left module over A.

Suppose that V_{j} is a left module over A for some j, 1 ≤ j ≤ n. If a ∈ A and μ ∈ L(V_{1}, …, V_{n}; W), then

μ(v_{1}, …, v_{j−1}, a · v_{j}, v_{j+1}, …, v_{n}) (7.5.2)

defines another multilinear mapping from ∏_{j=1}^{n} V_{j} into W. This defines an opposite algebra homomorphism from A into the algebra of module homomorphisms from L(V_{1}, …, V_{n}; W) into itself, as in Section 7.3. Equivalently, L(V_{1}, …, v_{n}; W) may be considered as a right module over A in this way. This was mentioned in Section 7.2 when n = 1.

Suppose from now on in this section that A is a Lie algebra over k. If W is a module over A as a Lie algebra over k, a ∈ A, and μ ∈ L(V_{1}, …, V_{n}; W), then (7.5.1) defines an element of L(V_{1}, …, V_{n}; W), as before. This makes L(V_{1}, …, V_{n}; W) into a module over A as a Lie algebra over k. Similarly, if V_{j} is a module over A as a Lie algebra over k for some j, a ∈ A, and μ is in L(V_{1}, …, V_{n}; W), then (7.5.2) is an element of L(V_{1}, …, V_{n}; W), and hence

−μ(v_{1}, …, v_{j−1}, a · v_{j}, v_{j+1}, …, v_{n}) (7.5.3) is an element of L(V_{1}, …, V_{n}; W). One can check that L(V_{1}, …, V_{n}; W) is a module over A as a Lie algebra over k with respect to (7.5.3), which was mentioned in Section 7.2 when n = 1.

Suppose that V_{1}, …, V_{n} and W are all modules over A as a Lie algebra. Let a ∈ A and μ ∈ L_{k}(V_{1}, …, V_{n}; W) be given, and let us define a · μ as a mapping from ∏_{j=1}^{n} V_{j} into W, as follows. If v_{j} ∈ V_{j} for each j = 1, …, n, then we put

(a · μ)(v_{1}, …, v_{n}) = a · (μ(v_{1}, …, v_{n}))

+ ∑_{j=1}^{n}(−μ(v_{1}, …, v_{j−1}, a · v_{j}, v_{j+1}, …, v_{n})). (7.5.4)

It is easy to see that (7.5.4) is multilinear over k as a mapping from ∏_{j=1}^{n} V_{j} into W, and that (7.5.4) is linear in a over k. One can verify that this makes L(V_{1}, …, V_{n}; W) into a module over A as a Lie algebra over k, using the remarks in the preceding paragraph. More precisely, each term on the right side of (7.5.4) defines a Lie algebra representation of A on L(V_{1}, …, V_{n}; W), as before. One can check directly that these n + 1 representations of A on L(V_{1}, …, V_{n}; W) commute with each other. Hence their sum defines a Lie algebra representation of A on L(V_{1}, …, V_{n}; W) too, as in Section 6.5.

Suppose that V_{1}, …, V_{n} are the same module V over k, with the same Lie algebra representation of A. Let a ∈ A and μ ∈ L(V_{1}, …, V_{n}; W) be given again. If μ is a symmetric multilinear mapping, then (7.5.4) is symmetric as well. If μ is antisymmetric, then one can verify that (7.5.4) is antisymmetric too. If μ(v_{1}, …, v_{n}) = 0 (7.5.5)

whenever v_{j} = v_{l} for some j ≠ l, then one can check that (7.5.4) satisfies the same condition. More precisely, if v_{j} = v_{l} for some j ≠ l, then this uses the antisymmetry of μ for the two terms on the right side of (7.5.4) that involve a · v_{j} and a · v_{l}. Otherwise, one can apply the hypothesis on μ directly to the other terms on the right side of (7.5.4).

7.6 Centralizers and invariant elements

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is denoted a b. The centralizer of a set E ⊆ A in A is the set of a ∈ A that commute with every x ∈ E, which is to say that

a x = x a (7.6.1)

for every x ∈ E. This is a submodule of A as a module over k, and a subalgebra of A when A is associative. The center of A is the centralizer of A in itself.

Now let (A, [·, ·]_{A}) be a Lie algebra over k. The centralizer of a set E ⊆ A in A as a Lie algebra is given by

(7.6.2) C_{A}(E) = {a ∈ A : [a, x]_{A} = 0 for every x ∈ E}.

It is easy to see that this is a Lie subalgebra of A, using the Jacobi identity. The center of A as a Lie algebra is given by

(7.6.3) Z(A) = C_{A}(A) = {a ∈ A : [a, x]_{A} = 0 for every x ∈ A},

which is automatically an ideal in A.

Note that (7.6.2) is contained in the centralizer of A as an algebra in the strict sense, and that (7.6.3) is contained in the center of A as an algebra in the strict sense. If 1 + 1 is invertible in k, then (7.6.2) is the same as the centralizer of E in A as an algebra in the strict sense, and (7.6.3) is the same as the center of A as an algebra in the strict sense.

If A is an associative algebra over k, then A is a Lie algebra over k with respect to the corresponding commutator bracket [x, y] = x y − y x. Of course, (7.6.1) is the same as saying that [a, x] = 0. In this case, the centralizer of E ⊆ A in A as an associative algebra is the same as the centralizer of E in A as a Lie algebra with respect to the commutator bracket. In particular, the center of A as an associative algebra is the same as the center of A as a Lie algebra.

Let (A, [·, ·]_{A}) be a Lie algebra over k again, and let ρ be a representation of A as a Lie algebra over k on a module V over k. An element v of V is said to be invariant under ρ if ρ_{a}(v) = 0 (7.6.4)

for every a ∈ A, as on p31 of [25]. As usual, V may be considered as a module over A as a Lie algebra over k, with a · v = ρ_{a}(v) for every a ∈ A and v ∈ V. Thus (7.6.4) can be reexpressed as saying that

a · v = 0 (7.6.5)

for every a ∈ A.

If every v ∈ V is invariant under ρ, then ρ is said to act trivially on V. It is easy to see that the collection of v ∈ V that are invariant under ρ is a submodule of V, as a module over k. This defines a subrepresentation of ρ on V, on which ρ acts trivially.

If A_{0} is any Lie subalgebra of A, then the restriction of ρ_{a} to a ∈ A_{0} defines a representation of A_{0} on V, as a Lie algebra over k. If v is any element of V, then the collection of a ∈ A such that (7.6.4) or equivalently (7.6.5) holds is a Lie subalgebra of A.

Remember that the adjoint representation of A is a representation of A as a Lie algebra on itself, as a module over k. The collection of elements of A that are invariant under the adjoint representation is the same as the center of A as a Lie algebra.

7.7 Invariant multilinear mappings

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let V, W be modules over k, and let ρ^{V}, ρ^{W} be representations of A as a Lie algebra over k on V, W, respectively.

Remember that the space Hom_{k}(V, W) of homomorphisms from V into W as modules over k is a module over k too, with respect to pointwise addition and scalar multiplication of mappings. If a ∈ A, then ρ_{a}^{V} and ρ_{a}^{W} are homomorphisms from V and W into themselves, as modules over k. If ϕ ∈ Hom_{k}(V, W), then it follows that ϕ ∘ ρ_{a}^{V} and ρ_{a}^{W} ∘ ϕ are homomorphisms from V into W as well, as modules over k. The mappings

ϕ ↦ ρ_{a}^{W} ∘ ϕ (7.7.1)

and ϕ ↦ −ϕ ∘ ρ_{a}^{V} (7.7.2)

define homomorphisms from Hom_{k}(V, W) into itself, as a module over k. These define representations of A as a Lie algebra over k on Hom_{k}(V, W), as in Section 7.5. We also saw that ρ_{a}(ϕ) = ρ_{a}^{W} ∘ ϕ − ϕ ∘ ρ_{a}^{V} (7.7.3)

defines a representation of A as a Lie algebra over k on Hom_{k}(V, W). Remember that ϕ ∈ Hom_{k}(V, W) is said to be invariant under ρ when (7.7.3) is equal to 0 for every a ∈ A, as in the previous section. This happens exactly when ϕ intertwines the representations ρ^{V}, ρ^{W} of A on V, W, respectively, as in Section 6.7.

Let V, W be modules over k again, and let ρ^{V} be a respresentation of A as a Lie algebra over k on V. Remember that the space L_{k}(V, V; W) of mappings from V × V into W that are bilinear over k is a module over k with respect to pointwise addition and scalar multiplication. If a ∈ A and β ∈ L_{k}(V, V; W), then ρ_{a}^{V} is a homomorphism from V into itself, as a module over k, and

β(ρ_{a}^{V} (u), v), β(u, ρ_{a}^{V} (v)) (7.7.4)

define elements of L_{k}(V, V; W). The mappings

β(u, v) ↦ −β(ρ_{a}^{V} (u), v) (7.7.5)

and β(u, v) ↦ −β(u, ρ_{a}^{V} (v)) (7.7.6)

define homomorphisms from L_{k}(V, V; W) into itself, as a module over k. These define representations of A as a Lie algebra over k on L_{k}(V, V; W), as in Section 7.5, and (ρ_{a}(β))(u, v) = −β(ρ_{a}^{V} (u), v) − β(u, ρ_{a}^{V} (v)) (7.7.7)

defines a representation of A as a Lie algebra over k on L_{k}(V, V; W) too. The condition that ρ_{a}(β) = 0 (7.7.8)

as an element of L_{k}(V, V; W) is the same as saying that (7.7.7) is equal to 0 for every u, v ∈ V, which means that ρ_{a}^{V} is antisymmetric on V with respect to β. Thus β is invariant under the representation (7.7.7) of A on L_{k}(V, V; W) exactly when ρ_{a}^{V} is antisymmetric on V with respect to β for every a ∈ A.

Let V_{1}, …, V_{n} and W be modules over k, and let μ be a mapping from ∏_{j=1}^{n} V_{j} into W that is multilinear over k. Also let l ∈ {1, …, n} be given, and let V_{l}^{μ} be the set of v_{l} ∈ V_{l} such that

μ(v_{1}, …, v_{l−1}, v_{l}, v_{l+1}, …, v_{n}) = 0 (7.7.9)

for every v_{j} ∈ V_{j} with 1 ≤ j ≤ n and j ≠ l. Note that V_{l}^{μ} is a submodule of V_{l}, as a module over k. Let A be a Lie algebra over k, and suppose that V_{1}, …, V_{n} and W are modules over A. Thus, for each a ∈ A, a · μ can be defined as a mapping from ∏_{j=1}^{n} V_{j} into W that is multilinear over k, as in Section 7.5. If a · μ = 0 as a mapping on ∏_{j=1}^{n} V_{j}, and if v_{l} ∈ V_{l}^{μ}, then it is easy to see that a · v_{l} ∈ V_{l}^{μ} too. This means that V_{l}^{μ} is a submodule of V_{l}, as a module over A, when a · μ = 0 for every a ∈ A.

Let V be a module over k, and let β be a mapping from V × V into V that is bilinear over k. Thus V is an algebra in the strict sense over k, with respect to β. Let A be a Lie algebra over k, and suppose that V is a module over A. If a ∈ A, then a · β is defined as a mapping from V × V into V that is bilinear over k by

(a · β)(v, w) = a · (β(v, w)) − β(a · v, w) − β(v, a · w) (7.7.10)

for every v, w ∈ V, as in Section 7.5. Observe that a · β = 0 as a mapping on V × V exactly when δ_{a}(v) = a · v defines a derivation on V with respect to β.

7.8 Traces of linear mappings

Let k be a commutative ring with a multiplicative identity element, let A_{0} be a commutative associative algebra over k, and let n be a positive integer. Remember that the space M_{n}(A_{0}) of n × n matrices with entries in A_{0} is an associative algebra over k, using entrywise addition and scalar multiplication, and matrix multiplication. The trace of an element of M_{n}(A_{0}) defines a homomorphism from M_{n}(A_{0}) into A_{0} as modules over k, which satisfies

tr(a b) = tr(b a) (7.8.1)

for every a, b ∈ M_{n}(A_{0}). Put

B_{0}(a, b) = tr(a b) (7.8.2)

for every a, b ∈ M_{n}(A_{0}), which defines a mapping from M_{n}(A_{0}) × M_{n}(A_{0}) into A_{0}. This mapping is bilinear over k, and symmetric in a, b.

Let a, b, x ∈ M_{n}(A_{0}) be given, and observe that

B_{0}(a x, b) = tr((a x) b) = tr(a (x b)) = B_{0}(a, x b). (7.8.3)

We also have that

(7.8.4) B_{0}(x a, b) = tr((x a) b) = tr(x (a b))
= tr((a b) x) = tr(a (b x)) = B_{0}(a, b x).

Of course, a ↦ a x, a ↦ x a (7.8.5)

define homomorphisms from M_{n}(A_{0}) into itself, as a module over k. Similarly,

C_{x}(a) = [x, a] = x a − a x (7.8.6)

defines C_{x} as a homomorphism from M_{n}(A_{0}) into itself, as a module over k. It is easy to see that B_{0}(C_{x}(a), b) = −B_{0}(a, C_{x}(b)), (7.8.7)

using (7.8.3) and (7.8.4).

Let k^{n} be the space of n-tuples of elements of k, which is a (free) module over k with respect to coordinatewise addition and scalar multiplication. If a = (a_{j,l}) is an n × n matrix with entries in k, then

(T_{a}(v))_{j} = ∑_{l=1}^{n} a_{j,l} v_{l} (7.8.8)

defines a module homomorphism from k^{n} into itself, as usual. The mapping a ↦ T_{a} defines an isomorphism from M_{n}(k) onto Hom_{k}(k^{n}, k^{n}), as associative algebras over k. The trace of T_{a} is defined as an element of k to be the trace of a, which defines the trace as a homomorphism from Hom_{k}(k^{n}, k^{n}) into k, as modules over k. If R, T ∈ Hom_{k}(k^{n}, k^{n}), then we have that

tr(R ∘ T) = tr(T ∘ R), (7.8.9)

as in (7.8.1).

Let V be a module over k that is isomorphic to k^{n} as a module over k, so that Hom_{k}(V, V) is isomorphic to Hom_{k}(k^{n}, k^{n}) as associative algebras over k. The trace can be defined as a homomorphism from Hom_{k}(V, V) into k, as modules over k, as before. One can check that this definition of the trace does not depend on the module isomorphism between V and k^{n}, because of (7.8.9). Of course, if k is a field, then an n-dimensional vector space V over k is isomorphic to k^{n} as a vector space over k. This corresponds to choosing a basis for V, and the trace of a linear mapping from V into itself does not depend on the choice of the basis.

Put B(T_{1}, T_{2}) = tr(T_{1} ∘ T_{2}) (7.8.10)

for every T_{1}, T_{2} ∈ Hom_{k}(V, V), which defines a symmetric bilinear form on Hom_{k}(V, V). If R ∈ Hom_{k}(V, V), then

C_{R}(T) = [R, T] = R ∘ T − T ∘ R (7.8.11)

defines a homomorphism from Hom_{k}(V, V) into itself, as a module over k. As in (7.8.7), we have that

B(C_{R}(T_{1}), T_{2}) = −B(T_{1}, C_{R}(T_{2})) (7.8.12)

for every T_{1}, T_{2} ∈ Hom_{k}(V, V), so that C_{R} is antisymmetric on Hom_{k}(V, V) with respect to (7.8.10).

7.9 The Killing form

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let V be a module over k, and let ρ be a representation of A as a Lie algebra on k. Suppose that V is isomorphic to k^{n} as a module over k for some positive integer n. If T is a homomorphism from V into itself, as a module over k, then the trace tr T = tr_{V} T of T can be defined as an element of k as in the previous section. Put

B_{ρ}(x, y) = tr_{V} (ρ_{x} ∘ ρ_{y}) (7.9.1)

for every x, y ∈ A, which is the trace of ρ_{x} ∘ ρ_{y} as a module homomorphism from V into itself. This defines a mapping from A × A into k that is bilinear over k. Note that (7.9.1) is symmetric in x and y.

Let w, x, y ∈ A be given, and observe that

B_{ρ}([w, x]_{A}, y) = tr_{V} (ρ_{[w,x]_{A}} ∘ ρ_{y}) = tr_{V} ([ρ_{w}, ρ_{x}] ∘ ρ_{y}). (7.9.2)

The right side is equal to −tr_{V} (ρ_{x} ∘ [ρ_{w}, ρ_{y}]), (7.9.3)

as in the previous section. It follows that

B_{ρ}([w, x]_{A}, y) = −B_{ρ}(x, [w, y]_{A}). (7.9.4)

If x ∈ A, then ad x = ad_{x} is defined as a module homomorphism from A into itself by ad_{x}(z) = [x, z]_{A} (7.9.5)

for every z ∈ A, as in Section 2.4. Thus (7.9.4) can be reformulated as saying that B_{ρ}(ad_{w}(x), y) = −B_{ρ}(x, ad_{w}(y)) (7.9.6)

for every w, x, y ∈ A.

Remember that the space of bilinear forms on A may be considered as a module over A, with respect to the adjoint representation on A and the trivial representation of A on k, as in Section 7.5. Using this, (7.9.6) is the same as saying that (7.9.1) is invariant under this action of A on bilinear forms on A, as in Section 7.7. This corresponds to Proposition 1.1 on p32 of [25], which was formulated for a field k. Equivalently, (7.9.4) means that (7.9.1) is associative as a bilinear mapping on A × A, as in Section 6.10. See also p27 of [14].

If A is isomorphic to k^{n} as a module over k for some positive integer n, then we can take V = A and ρ_{x} = ad_{x} in the previous paragraphs. In this case, (7.9.1) becomes b(x, y) = tr_{A}(ad_{x} ∘ ad_{y}) (7.9.7)

for x, y ∈ A. This is known as the Killing form on A, as on p21 of [14], and Definition 1.2 on p32 of [25].

7.10 Invariant subspaces and traces

Let k be a commutative ring with a multiplicative identity element, and let V be a module over k. Remember that the collection Hom_{k}(V, V) of all homomorphisms from V into itself, as a module over k, is an associative algebra over k with respect to composition of mappings. Let W be a submodule of V, and let A_{W} be the collection of all T ∈ Hom_{k}(V, V) such that

T (W) ⊆ W. (7.10.1)

Note that A_{W} is a subalgebra of Hom_{k}(V, V). If T ∈ A_{W}, then let T_{W} be the restriction of T to W, which defines a module homomorphism from W into itself. Of course, T ↦ T_{W} (7.10.2)

defines an algebra homomorphism from A_{W} into the algebra Hom_{k}(W, W) of all module homomorphisms from W into itself. Let q be the canonical quotient mapping from V onto the quotient module V/W. If T ∈ A_{W}, then there is a unique module homomorphism T_{V/W} from V/W into itself such that

T_{V/W} ∘ q = q ∘ T. (7.10.3)

It is easy to see that T ↦ T_{V/W} (7.10.4)

defines an algebra homomorphism from A_{W} into the algebra Hom_{k}(V/W, V/W) of all module homomorphisms from V/W into itself.

Suppose that V/W is isomorphic to k^{n} as a module over k for some positive integer n, so that V/W is a free module over k of rank n. This means that there are n elements z_{1}, …, z_{n} of V such that every element of V/W can be expressed in a unique way as a linear combination of q(z_{1}), …, q(z_{n}) with coefficients in k. Let Z be the submodule of V consisting of linear combinations of z_{1}, …, z_{n} with coefficients in k. More precisely, every element of Z can be expressed in a unique way as a linear combination of z_{1}, …, z_{n} with coefficients in k, because of the analogous property of q(z_{1}), …, q(z_{n}) in V/W. Note that the restriction of q to Z defines an isomorphism from Z onto V/W, as modules over k. It is easy to see that every element of V can be expressed in a unique way as the sum of elements of W and Z, so that V may be identified with the direct sum of W and Z, as a module over k. Suppose that W is isomorphic to k^{m} as a module over k for some positive integer m as well. Under these conditions, V is isomorphic to k^{m+n} as a module over k.

If T ∈ A_{W}, then the traces of T, T_{W}, and T_{V/W} on V, W, and V/W, respectively, can be defined as elements of k, as in Section 7.8. Observe that

tr_{V} T = tr_{W} T_{W} + tr_{V/W} T_{V/W} , (7.10.5)

where the subscripts indicate the spaces on which the traces are taken. In particular, if T (V) ⊆ W, (7.10.6) then T_{V/W} = 0, and tr_{V} T = tr_{W} T_{W} . (7.10.7)

If T_{1}, T_{2} ∈ Hom_{k}(V, V), then put

B_{V} (T_{1}, T_{2}) = tr_{V} (T_{1} ∘ T_{2}), (7.10.8)

as in Section 7.8. Let B_{W} (·, ·) and B_{V/W} (·, ·) be the analogous bilinear forms on Hom_{k}(W, W) and Hom_{k}(V/W, V/W), respectively. Suppose that T_{1}, T_{2} ∈ A_{W}, and let T_{1,W} , T_{2,W} ∈ Hom_{k}(W, W) and T_{1,V/W} , T_{2,V/W} ∈ Hom_{k}(V/W, V/W) be as before. Note that

(T_{1} ∘ T_{2})_{W} = T_{1,W} ∘ T_{2,W} , (T_{1} ∘ T_{2})_{V/W} = T_{1,V/W} ∘ T_{2,V/W} , (7.10.9)

because (7.10.2) and (7.10.4) are algebra homomorphisms. It follows that

B_{V} (T_{1}, T_{2}) = B_{W} (T_{1,W} , T_{2,W}) + B_{V/W} (T_{1,V/W} , T_{2,V/W}), (7.10.10)

by applying (7.10.5) to T = T_{1} ∘ T_{2}.

Let (A, [·, ·]) be a Lie algebra over k, and let B be an ideal in A, so that the quotient A/B is defined as a Lie algebra over k too. If x ∈ A, then ad_{x} = ad_{A,x} is defined as a module homomorphism from A into itself by

ad_{x}(z) = ad_{A,x}(z) = [x, z] (7.10.11)

for every z ∈ A, as in Section 2.4. Let (ad_{x})_{B} be the restriction of ad_{x} to B, which maps B into itself, because B is an ideal in A. Similarly, let (ad_{x})_{A/B} be the mapping from A/B into itself which is induced by ad_{x} on A. Suppose that B and A/B are isomorphic as modules over k to k^{m} and k^{n}, respectively, for some positive integers m and n. This implies that A is isomorphic to k^{m+n} as a module over k, as before. If x, y ∈ A, then

(7.10.12)  tr_{A}(ad_{x} ∘ ad_{y}) = tr_{B}((ad_{x})_{B} ∘ (ad_{y})_{B})
+tr_{A/B}((ad_{x})_{A/B} ∘ (ad_{y})_{A/B}),

as in (7.10.10).

7.11 Radicals of bilinear mappings

Let k be a commutative ring with a multiplicative identity element, and let V, W be modules over k. Also let β be a mapping from V × V into W that is bilinear over k. Note that

(7.11.1) V^{β} = {u ∈ V : β(u, v) = 0 for every v ∈ V }

is a submodule of V. This may be called the radical of β in V, as on p22 of [14].

Let (A, [·, ·]) be a Lie algebra over k, and let β be a mapping from A × A into W that is bilinear over k. Thus we take V = A in the preceding paragraph, as a module over k. If x ∈ A, then ad_{x} = ad_{A,x} is the module homomorphism from A into itself defined by (7.10.11), as before. Suppose that

β(ad_{w}(x), y) = −β(x, ad_{w}(y)) (7.11.2)

for every w, x, y ∈ A, which is the same as saying that

β([x, w], y) = β(x, [w, y]) (7.11.3)

for every w, x, y ∈ A, as in Section 6.10. In this case, we may say that β is associative as a bilinear form on A, as in Section 6.10. Equivalently, β is invariant with respect to the representation on the space of bilinear mappings from A × A into W corresponding to the adjoint representation on A and the trivial representation on W, as in Section 7.7. It is easy that the radical A^{β} of β in A is an ideal in A as a Lie algebra over k, as on p22 of [14], and p44 of [25]. This may be considered as a particular case of statements in Sections 6.10 and 7.7.

Suppose that A is isomorphic to k^{r} as a module over k for some positive integer r. If x, y ∈ A, then

β(x, y) = tr_{A}(ad_{x} ∘ ad_{y}) (7.11.4)

is defined as an element of k, as in Section 7.8. This defines a symmetric bilinear mapping from A × A into k that satisfies (7.11.2), as in Section 7.9. Thus the radical A^{β} of (7.11.4) in A is an ideal in A as a Lie algebra over k, as in the preceding paragraph.

Let B be an ideal in A as a Lie algebra over k, so that the quotient A/B is a Lie algebra over k as well. Suppose that B and A/B are isomorphic to k^{m} and k^{n}, respectively, as modules over k for some positive integers m and n. This implies that A is isomorphic to k^{m+n} as a module over k, as in the previous section. If x, y ∈ A, then let (ad_{x})_{B}, (ad_{y})_{B} be the restrictions of ad_{x}, ad_{y} to B, and let (ad_{x})_{A/B}, (ad_{y})_{A/B} be the mappings from A/B into itself induced by ad_{x}, ad_{y}, as before. If x ∈ B, then ad_{x} maps A into B, so that the induced mapping (ad_{x})_{A/B} is equal to 0. This implies that the second term on the right side of (7.10.12) is equal to 0 for every y ∈ A. It follows that

(7.11.5) tr_{A}(ad_{x} ∘ ad_{y}) = tr_{B}((ad_{x})_{B} ∘ (ad_{y})_{B})

for every x ∈ B and y ∈ A.

If x ∈ B and B is commutative as a Lie algebra, then (ad_{x})_{B} is equal to 0. This implies that the right side of (7.11.5) is equal to 0 for every y ∈ A. Under these conditions, we get that

β(x, y) = tr_{A}(ad_{x} ∘ ad_{y}) = 0 (7.11.6)

for every x ∈ B and y ∈ A. This means that

B ⊆ A^{β} (7.11.7)

7.12 Tensor products

Let k be a commutative ring with a multiplicative identity element, let n ≥ 2 be an integer, and let V_{1}, …, V_{n} be n modules over k. The tensor product

⨂_{j=1}^{n} V_{j} = V_{1} ⨂ ⋯ ⨂ V_{n} (7.12.1)

of these modules over k is a module over k with the following two properties. First, the tensor product comes equipped with a mapping from ∏_{j=1}^{n} V_{j} into ⨂_{j=1}^{n} V_{j} that is multilinear over k. The image of (v_{1}, …, v_{n}) ∈ ∏_{j=1}^{n} V_{j} in ⨂_{j=1}^{n} V_{j} under this mapping is often expressed as v_{1} ⊗ ⋯ ⊗ v_{n}. Second, let W be any module over k, and let μ be any mapping from ∏_{j=1}^{n} V_{j} into W that is multilinear over k. Under these conditions, μ can be expressed in a unique way as the composition of the mapping from ∏_{j=1}^{n} V_{j} into ⨂_{j=1}^{n} V_{j} just mentioned with a homomorphism from ⨂_{j=1}^{n} V_{j} into W, as modules over k. Equivalently, this means that there is a unique module homomorphism μ̃ from ⨂_{j=1}^{n} V_{j} into W such that μ̃(v_{1} ⊗ ⋯ ⊗ v_{n}) = μ(v_{1}, …, v_{n}) (7.12.2)

for every (v_{1}, …, v_{n}) ∈ ∏_{j=1}^{n} V_{j}. The tensor product is unique up to a suitable isomorphic equivalence.

Note that ⨂_{j=1}^{n} V_{j} is generated as a module over k by the associated image of ∏_{j=1}^{n} V_{j}. This means that every element of ⨂_{j=1}^{n} V_{j} can be expressed as a finite sum of terms of the form v_{1} ⊗⋯⊗v_{n}, where v_{j} ∈ V_{j} for each j = 1, …, n. This is clear from the standard construction of the tensor product, and it can also be obtained from the uniqueness of the tensor product.

Let W_{1}, …, W_{n} another collection of n modules over k, and suppose that ϕ_{j} is a homomorphism from V_{j} into W_{j} for each j = 1, …, n, as modules over k. Consider the mapping from ∏_{j=1}^{n} V_{j} into ⨂_{j=1}^{n} W_{j} that sends (v_{1}, …, v_{n}) in ∏_{j=1}^{n} V_{j} to ϕ_{1}(v_{1}) ⊗ ⋯ ⊗ ϕ_{n}(v_{n}). (7.12.3)

It is easy to see that this mapping is multilinear over k. This leads to a unique module homomorphism ϕ from ⨂_{j=1}^{n} V_{j} into ⨂_{j=1}^{n} W_{j} such that

ϕ(v_{1} ⊗ ⋯ ⊗ v_{n}) (7.12.4)

is equal to (7.12.3) for every (v_{1}, …, v_{n}) ∈ ∏_{j=1}^{n} V_{j}.

Let Z_{1}, …, Z_{n} be another collection of n modules over k, and let ψ_{j} be a homomorphism from W_{j} into Z_{j} for each j = 1, …, n, as modules over k. This leads to a module homomorphism ψ from ⨂_{j=1}^{n} W_{j} into ⨂_{j=1}^{n} Z_{j}, as in the preceding paragraph. Note that ψ_{j} ∘ ϕ_{j} is a module homomorphism from V_{j} into Z_{j} for each j = 1, …, n. One can check that ψ∘ϕ is the same as the module homomorphism from ⨂_{j=1}^{n} V_{j} into ⨂_{j=1}^{n} Z_{j} obtained from ψ_{j} ∘ ϕ_{j}, 1 ≤ j ≤ n, as in the previous paragraph.

Let A be an associative algebra over k, and suppose that V_{l} is also a left or right module over A for some l ∈ {1, …, n}. One can define an action of A on ⨂_{j=1}^{n} V_{j} on the left or the right, as appropriate, so that ⨂_{j=1}^{n} V_{j} becomes a left or right module over A too. More precisely, if a ∈ A, then the corresponding module homomorphism from V_{l} into itself leads to a module homomorphism from ⨂_{j=1}^{n} V_{j} into itself as before, using the identity mapping on V_{j} when j ≠ l.

Similarly, let A be a Lie algebra over k, and suppose that V_{l} is a module over A for some l ∈ {1, …, n}. One can define an action of A on ⨂_{j=1}^{n} V_{j} in the same way as in the preceding paragraph, so that ⨂_{j=1}^{n} V_{j} becomes a module over A.

Suppose now that V_{l} is a module over A as a Lie algebra for each l = 1, …, n, which leads to an action of A on ⨂_{j=1}^{n} V_{j} for each l = 1, …, n, as in the previous paragraph. It is easy to see that these actions commute with each other on ⨂_{j=1}^{n} V_{j}. It follows that ⨂_{j=1}^{n} V_{j} is a module over A with respect to the sum of these actions, as in Section 7.1.

7.13 Functions on sets

Let k be a commutative ring with a multiplicative identity element, and let W be a module over k. Also let X be a nonempty set, and let c(X, W) be the space of W -valued functions on X. It is easy to see that c(X, W) is a module over k with respect to pointwise addition and scalar multiplication of functions. This is the same as the direct product of the family of copies of W indexed by X.

If f is a W -valued function on X, then the support of f on X is defined to be the set of x ∈ X such that f(x) ≠ 0. Let c_{00}(X, W) be the subset of c(X, W) consisting of functions with finite support. This is a submodule of c(X, W), as a module over k, which corresponds to the direct sum of the family of copies of W indexed by X. Of course, c_{00}(X, W) is the same as c(X, W) when X has only finitely many elements.

In particular, we can take W = k, considered as a module over itself. If x ∈ X, then let δ_{x} be the k-valued function on X equal to 1 at x and to 0 elsewhere. Every element of c_{00}(X, k) can be expressed in a unique way as a linear combination of finitely many δ_{x}’s with coefficients in k.

Let Z be another module over k. If ϕ is a homomorphism from c_{00}(X, k) into Z, as modules over k, then f(x) = ϕ(δ_{x}) defines a mapping from X into Z. It is easy to see that ϕ is uniquely determined by f, and that every Z-valued function f on X corresponds to a module homomorphism ϕ from c_{00}(X, k) into Z in this way. The mapping ϕ ↦ f defines an isomorphism from the space Hom_{k}(c_{00}(X, k), Z) of module homomorphisms from c_{00}(X, k) into Z onto c(X, Z), as modules over k.

Similarly, let ϕ be a homomorphism from c_{00}(X, W) into Z, as modules over k. If x ∈ X and w ∈ W, then δ_{x} w ∈ c_{00}(X, W), so that

ϕ_{x}(w) = ϕ(δ_{x} w) (7.13.1) defines an element of Z. This defines ϕ_{x} as a module homomorphism from W into Z, so that x ↦ ϕ_{x} is an element of c(X, Hom_{k}(W, Z)). One can check that ϕ is uniquely determined by x ↦ ϕ_{x}, and that every mapping from X into Hom_{k}(W, Z) corresponds to a module homomorphism ϕ from c_{00}(X, W) into Z in this way. This defines an isomorphism between the space Hom_{k}(c_{00}(X, W), Z) of module homomorhisms from c_{00}(X, W) into Z and c(X, Hom_{k}(W, Z)), as modules over k.

If f ∈ c(X, k) and w ∈ W, then f(x) w defines a W -valued function on X. This defines a mapping from c(X, k) × W into c(X, W) that is bilinear over k. The restriction of this mapping to c_{00}(X, k) × W maps into c_{00}(X, W).

Let μ be a mapping from c_{00}(X, k) × W into Z that is bilinear over k. If x ∈ X, then put μ_{x}(w) = μ(δ_{x}, w) (7.13.2)

for every w ∈ W, which defines a module homomorphism from W into Z. If f ∈ c_{00}(X, W), then

∑_{x∈X} μ_{x}(f(x)) (7.13.3)

defines an element of Z, where all but finitely many terms in the sum are equal to 0. This defines a homomorphism from c_{00}(X, W) into Z, as modules over k. One can use this to check that c_{00}(X, W) satisfies the requirements of the tensor product of c_{00}(X, k) and W, as modules over k.

Let V be another module over k, and suppose that ϕ is a homomorphism from V into c(X, Z), as modules over k. If x ∈ X and v ∈ V, then let ϕ_{x}(v) be the value of ϕ(v) at x, as a Z-valued function on X. This defines ϕ_{x} as a module homomorphism from V into Z for each x ∈ X, so that x ↦ ϕ_{x} is an element of c(X, Hom_{k}(V, Z)). Clearly ϕ is uniquely determined by x ↦ ϕ_{x}, and every mapping from X into Hom_{k}(V, Z) corresponds to a module homomorphism ϕ from V into c(X, Z) in this way. This defines an isomorphism between the space Hom_{k}(V, c(X, Z)) of module homomorphisms from V into c(X, Z) and c(X, Hom_{k}(V, Z)), as modules over k.

7.14 Some remarks about sums, products

Let k be a commutative ring with a multiplicative identity element, and let W be a module over k. Of course, k may be considered as a module over itself, and

(t, w) ↦ t w (7.14.1)

defines a mapping from k×W into W that is bilinear over k. Using this mapping, one can check that W satisfies the requirements of k ⨂ W.

Let I be a nonempty set, and let V_{j} be a module over k for every j ∈ I. Thus ⨁_{j∈I} V_{j} is a module over k, and one can verify that

( ⨁_{j∈I} V_{j}) ⨂ W = ⨁_{j∈I}(V_{j} ⨂ W) (7.14.2)

in a natural way. Of course, there is an analogous statement for tensor products with a direct sum in any factor.

If Z is a module over k, then

Hom_{k}(k, Z) = Z (7.14.3)

in a natural way, because a mapping from k into Z that is linear over k corresponds to multiplication by an element of Z. If I and V_{j}, j ∈ I, are as in the previous paragraph, then

Hom_{k}( ⨁_{j∈I} V_{j}, Z) = ∏_{j∈I} Hom_{k}(V_{j}, Z) (7.14.4)

in a natural way. More precisely, any module homomorphism from ⨁_{j∈I} V_{j} into Z leads to a module homomorphism from V_{l} into Z for every l ∈ I, using the natural inclusion of V_{l} into ⨁_{j∈I} V_{j}. Conversely, if one has a module homomorphism from V_{l} into Z for every l ∈ I, then one can get a module homomorphism from ⨁_{j∈I} V_{j} into Z using the given homomorphisms on each coordinate of an element of ⨁_{j∈I} V_{j}, and adding the results in Z.

Similarly, let V be a module over k, and let Z_{j} be a module over k for every j ∈ I. It is easy to see that

Hom_{k}(V, ∏_{j∈I} Z_{j}) = ∏_{j∈I} Hom_{k}(V, Z_{j}) (7.14.5)

in a natural way.

7.15 Some natural isomorphisms

Let k be a commutative ring with a multiplicative identity element, and let V_{1}, …, V_{n} be modules over k for some integer n ≥ 2. Also let σ be a permutation on {1, …, n}, which is to say a one-to-one mapping from {1, …, n} onto itself. Thus the tensor products ⨂_{j=1}^{n} V_{j} and ⨂_{j=1}^{n} V_{σ(j)} can be defined as modules over k as in Section 7.12. There is a unique module homomorphism from ⨂_{j=1}^{n} V_{j} into ⨂_{j=1}^{n} V_{σ(j)} with

v_{1} ⊗ ⋯ ⊗ v_{n} ↦ v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(n)} (7.15.1)

for every (v_{1}, …, v_{n}) ∈ ∏_{j=1}^{n} V_{j}. More precisely, one can start with the mapping from ∏_{j=1}^{n} V_{j} into ⨂_{j=1}^{n} V_{σ(j)} defined by

(v_{1}, …, v_{n}) ↦ v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(n)}, (7.15.2)

which is multilinear over k.

Let τ be another permutation on {1, …, n}, which leads to a unique module homomorphism from ⨂_{j=1}^{n} V_{σ(j)} into ⨂_{j=1}^{n} V_{τ(σ(j))} with

v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(n)} ↦ v_{τ(σ(1))} ⊗ ⋯ ⊗ v_{τ(σ(n))} (7.15.3) for every (v_{1}, …, v_{n}) ∈ ∏_{j=1}^{n} V_{j}, as before. The composition of this module homomorphism with the previous one that satisfies (7.15.1) is a module homomorphism from ⨂_{j=1}^{n} V_{j} into ⨂_{j=1}^{n} V_{τ(σ(j))} with

v_{1} ⊗ ⋯ ⊗ v_{n} ↦ v_{τ(σ(1))} ⊗ ⋯ ⊗ v_{τ(σ(n))} (7.15.4)

for every (v_{1}, …, v_{n}) ∈ ∏_{j=1}^{n} V_{j}. This is the same as the homomorphism associated to τ ∘ σ, by uniqueness. In particular, if τ ∘ σ is the identity mapping on {1, …, n}, then we get the identity mapping on ⨂_{j=1}^{n} V_{j}. It follows that the module homomorphism from ⨂_{j=1}^{n} V_{j} into ⨂_{j=1}^{n} V_{σ(j)} that satisfies (7.15.1) is an isomorphism, whose inverse is the analogous homomorphism from ⨂_{j=1}^{n} V_{σ(j)} into ⨂_{j=1}^{n} V_{j} associated to the inverse of σ.

Now let n_{1} and n_{2} be positive integers with n_{1} + n_{2} = n. The tensor products ⨂_{j=1}^{n_{1}} V_{j} and ⨂_{l=1}^{n_{2}} V_{n_{1}+l} can be defined as before when n_{1}, n_{2} ≥ 2, and otherwise they may be interpreted as being the given module over k when n_{1} or n_{2} is equal to 1. Thus the tensor product

( ⨂_{j=1}^{n_{1}} V_{j}) ⨂ ( ⨂_{l=1}^{n_{2}} V_{n_{1}+l}) (7.15.5)

is defined as a module over k as well. It is well known that there is a natural isomorphism between ⨂_{j=1}^{n} V_{j} and (7.15.5), as modules over k. More precisely, if (v_{1}, …, v_{n}) ∈ ∏_{j=1}^{n} V_{j}, then v_{1} ⊗ ⋯ ⊗ v_{n} corresponds to

(v_{1} ⊗ ⋯ ⊗ v_{n_{1}}) ⊗ (v_{n_{1}+1} ⊗ ⋯ ⊗ v_{n}) (7.15.6)

under this isomorphism.

Let V be a module over k, and suppose that V_{j} = V for each j = 1, …, n. In this case,

T^{n}V = ⨂_{j=1}^{n} V_{j} (7.15.7)

is called the nth tensor power of V. We can interpret T^{1}V as being equal to V, as in the preceding paragraph. If σ is a permutation on {1, …, n}, then we get a module automorphism on T^{n}V, as before. The elements of T^{n}V that are invariant under the module automorphisms associated to all permutations on {1, …, n} are said to be symmetric, and form a submodule of T^{n}V, as a module over k.

If V_{1}, V_{2}, V_{3} are modules over k, then (V_{1} ⨂ V_{2}) ⨂ V_{3} and V_{1} ⨂(V_{2} ⨂ V_{3}) are both isomorphic to V_{1} ⨂ V_{2} ⨂ V_{3}, as modules over k, as before. This leads to a natural isomorphism between (V_{1} ⨂ V_{2}) ⨂ V_{3} and V_{1} ⨂(V_{2} ⨂ V_{3}), as modules over k. More precisely, if v_{1} ∈ V_{1}, v_{2} ∈ V_{2}, and v_{3} ∈ V_{3}, then (v_{1} ⊗ v_{2}) ⊗ v_{3} corresponds to v_{1} ⊗ (v_{2} ⊗ v_{3}) under this isomorphism.

Formal series and ordered rings

8.1 Poles of finite order

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, and let T be an indeterminate. Consider the space A((T)) of formal series of the form

f(T) = ∑_{j=j_{0}}^{∞} f_{j} T^{j}, (8.1.1)

where j_{0} ∈ Z and f_{j} ∈ A for each j ≥ j_{0}. More precisely, A((T)) may be defined as the space of A-valued functions on Z that are equal to 0 for all but finitely many negative integers. Thus (8.1.1) corresponds to j ↦ f_{j} as an A-valued function on Z, where f_{j} = 0 when j < j_{0}. As in [4], elements of A((T)) may also be expressed as

f(T) = ∑_{j>>−∞} f_{j} T^{j}, (8.1.2)

to indicate that f_{j} = 0 for all but finitely many j < 0.

Note that A((T)) is a module over k with respect to termwise addition and scalar multiplication of these formal series, which corresponds to pointwise addition and scalar multiplication of the associated A-valued functions on Z. The space A[[T]] of formal power series in T with coefficients in A can be identified with the submodule of A((T)) consisting of formal series f(T) such that f_{j} = 0 for all j < 0. In particular, A can be identified with the submodule of A((T)) consisting of formal series f(T) with f_{j} = 0 when j ≠ 0.

If f(T) ∈ A((T)) and l ∈ Z, then

f(T) T^{l} = ∑_{j>>−∞} f_{j} T^{j+l} = ∑_{j>>−∞} f_{j−l} T^{j} (8.1.3)

defines an element of A((T)) as well. This is the same as f(T) when l = 0, and agrees with the analogous definition on A[[T]] in Section 4.3 when l ≥ 0. In this situation, f(T) ↦ f(T) T^{l} is a module automorphism on A((T)) for every l ∈ Z.

Let n ∈ Z be given, and let (A[[T]]) T^{n} be the subset of A((T)) of formal series of the form g(T) T^{n}, where g(T) ∈ A[[T]] is identified with an element of A[[T]] as before. Equivalently, this is the set of f(T) ∈ A((T)) that can be expressed as in (8.1.1), with j_{0} ≥ n. If n ≥ 1, then the elements of (A[[T]]) T^{n} correspond to formal power series in T with coefficients in A that vanish to order n − 1, as in Section 4.3. Clearly (A[[T]]) T^{n} is a submodule of A((T)) for every n ∈ Z, with (A[[T]]) T^{n} ⊆ (A[[T]]) T^{n+1} (8.1.4)

and

A((T)) = ⋃_{n=−∞}^{∞}(A[[T]]) T^{n}. (8.1.5)

Let us say that the elements of a subset E of A((T)) have poles of bounded order if E ⊆ (A[[T]]) T^{n} for some n ∈ Z. Of course, if E has only finitely many elements, then the elements of E have poles of bounded order. Suppose that E is a submodule of A((T)), as a module over k. If E is finitely generated, as a module over k, then it is easy to see that the elements of E have poles of bounded order.

8.2 Sequences and series

Let us continue with the same notation and hypotheses as in the previous section. Let l_{0} ∈ Z be given, and let

f_{l}(T) = ∑_{j>>−∞} f_{l,j} T^{j} (8.2.1)

be an element of A((T)) for each integer l ≥ l_{0}. As in Section 4.4, let us say that the sequence {f_{l}(T)}_{l=l_{0}}^{∞} is termwise eventually constant if for each j ∈ Z there is an integer L_{j} ≥ l_{0} such that f_{l,j} does not depend on l when l ≥ L_{j}. Similarly, let us say that {f_{l}(T)}_{l=l_{0}}^{∞} eventually agrees with f(T) ∈ A((T)) termwise if for every j ∈ Z there is an integer L ≥ l_{0} such that

f_{l,j} = f_{j} (8.2.2)

for every l ≥ L_{j}. Of course, this implies that {f_{l}(T)}_{l=l_{0}}^{∞} is termwise eventually constant. If {f_{l}(T)}_{l=l_{0}}^{∞} is termwise eventually constant, and if the f_{l}(T)’s have poles of bounded order, then {f_{l}(T)}_{l=l_{0}}^{∞} eventually agrees with an element of A((T)) termwise. However, a sequence of elements of A((T)) may eventually agree termwise with an element of A((T)) without having poles of bounded order.

Let α ∈ k and r ∈ Z be given, as well as another sequence {g_{l}(T)}_{l=l_{0}}^{∞} of elements of A((T)). If {f_{l}(T)}_{l=l_{0}}^{∞} and {g_{l}(T)}_{l=l_{0}}^{∞} are termwise eventually constant, then so are {α f_{l}(T)}_{l=l_{0}}^{∞}, {f_{l}(T) T^{r}}_{l=l_{0}}^{∞}, and {f_{l}(T) + g_{l}(T)}_{l=l_{0}}^{∞}. If the f_{l}(T)’s and g_{l}(T)’s have poles of bounded order, then the α f_{l}(T)’s and f_{l}(T) + g_{l}(T)’s have the same property. In this case, if r_{0} ∈ Z, then the f_{l}(T) T^{r}’s have poles of bounded order for l ≥ l_{0} and r ≥ r_{0}. If {f_{l}(T)}_{l=l_{0}}^{∞} and {g_{l}(T)}_{l=l_{0}}^{∞} eventually agree with f(T), g(T) ∈ A((T)) termwise, respectively, then {α f_{l}(T)}_{l=l_{0}}^{∞}, {f_{l}(T) T^{r}}_{l=l_{0}}^{∞}, and {f_{l}(T) + g_{l}(T)}_{l=l_{0}}^{∞} eventually agree with α f(T), f(T) T^{r}, and f(T) + g(T) termwise, respectively.

Let

a_{l}(T) = ∑_{j>>−∞} a_{l,j} T^{j} (8.2.3)

be an element of A((T)) for every integer l ≥ l_{0}. Suppose that {a_{l}(T)}_{l=l_{0}}^{∞} eventually agrees with 0 termwise, and that the poles of the a_{l}(T)’s have bounded order. Under these conditions, the partial sums

∑_{l=l_{0}}^{n} a_{l}(T) (8.2.4)

are termwise eventually constant and have poles of bounded order. This implies that the partial sums (8.2.4) eventually agree termwise with an element of A((T)), as before. Let us denote this element of A((T)) by

∑_{l=l_{0}}^{∞} a_{l}(T). (8.2.5)

If α ∈ k, then {α a_{l}(T)}_{l=l_{0}}^{∞} eventually agrees with 0 termwise, the α a_{l}(T)’s have poles of bounded order, and

∑_{l=l_{0}}^{∞} α a_{l}(T) = α ∑_{l=l_{0}}^{∞} a_{l}(T). (8.2.6)

Similarly, if r ∈ Z, then {a_{l}(T) T^{r}}_{l=l_{0}}^{∞} eventually agrees with 0 termwise, the a_{l}(T) T^{r}’s have poles of bounded order (in l), and

∑_{l=l_{0}}^{∞} a_{l}(T) T^{r} = ( ∑_{l=l_{0}}^{∞} a_{l}(T)) T^{r}. (8.2.7)

Let {b_{l}(T)}_{l=l_{0}}^{∞} be another sequence of elements of A((T)) that eventually agrees termwise with 0, and whose terms have poles of bounded order. This implies that {a_{l}(T) + b_{l}(T)}_{l=l_{0}}^{∞} eventually agrees termwise with 0 too, and that the a_{l}(T) + b_{l}(T)’s have poles of bounded order. It is easy to see that

∑_{l=l_{0}}^{∞}(a_{l}(T) + b_{l}(T)) = ∑_{l=l_{0}}^{∞} a_{l}(T) + ∑_{l=l_{0}}^{∞} b_{l}(T) (8.2.8)

8.3 Formal series and module homomorphisms

Let k be a commutative ring with a multiplicative identity element, let A, B be modules over k, and let T be an indeterminate. Thus B((T)) can be defined as a module over k as in Section 8.1. Let ϕ be a homomorphism from A into B((T)), as modules over k. If a ∈ A, then ϕ(a) can be expressed as

ϕ(a) = ∑_{j=−∞}^{∞} ϕ_{j}(a) T^{j}, (8.3.1)

where ϕ_{j}(a) ∈ B for every j ∈ Z, and ϕ_{j}(a) = 0 for all but finitely many j < 0. More precisely, for each j ∈ Z, ϕ_{j} is a homomorphism from A into B, as modules over k. Conversely, let ϕ_{j} be a module homomorphism from A into B for every j ∈ Z, and suppose that for each a ∈ A, ϕ_{j}(a) = 0 for all but finitely many j < 0. Under these conditions, (8.3.1) defines an element of B((T)) for every a ∈ A, and this defines ϕ as a homomorphism from A into B((T)), as modules over k.

Let ϕ be a module homomorphism from A into B((T)) again. Let us say that ϕ has poles of bounded order if the set of ϕ(a) with a ∈ A has poles of bounded order, as a subset of B((T)). Equivalently, this means that there is an integer n(ϕ) such that ϕ_{j}(a) = 0 for every a ∈ A and j < n(ϕ). This is the same as saying that ϕ_{j} = 0 for all but finitely many j < 0, as homomorphisms from A into B. If A is finitely generated as a module over k, then this follows automatically from the fact that ϕ(a) ∈ B((T)) for every a ∈ A.

Remember that the space Hom_{k}(A, B) of module homomorphisms from A into B is a module over k too, with respect to pointwise addition and scalar multiplication of mappings. Thus (Hom_{k}(A, B))((T)) can be defined as a module over k as before. Let

ϕ(T) = ∑_{l=l_{0}}^{∞} ϕ_{l} T^{l} (8.3.2)

be an element of (Hom_{k}(A, B))((T)), so that l_{0} ∈ Z and ϕ_{l} ∈ Hom_{k}(A, B) for every l ≥ l_{0}. If a ∈ A, then

(ϕ(T))(a) = ∑_{l=l_{0}}^{∞} ϕ_{l}(a) T^{l} (8.3.3)

defines an element of B((T)), and the mapping from a ∈ A to (8.3.3) is a homomorphism from A into B((T)), as modules over k. This homomorphism has poles of finite order, and every module homomorphism from A into B((T)) with poles of finite order corresponds to an element of (Hom_{k}(A, B))((T)) in this way.

The space Hom_{k}(A, B((T))) of module homomorphisms from A into B((T)) is a module over k as well. It is easy to see that the collection of module homomorphisms from A into B((T)) with poles of finite order is a submodule of Hom_{k}(A, B((T))). The mapping from (8.3.2) to (8.3.3) defines an injective module homomorphism from (Hom_{k}(A, B))((T)) onto this submodule of Hom_{k}(A, B((T))). If ϕ(T) ∈ (Hom_{k}(A, B))((T)) and r ∈ Z, then ϕ(T) T^{r} defines an element of (Hom_{k}(A, B))((T)) too, as in Section 8.1. Note that for every a ∈ A, (ϕ(T) T^{r})(a) = (ϕ(T))(a) T^{r}, (8.3.4)

as elements of B((T)).

8.4 Extending module homomorphisms

Let us continue with the notation and hypotheses in the previous section. Let ϕ(T) be an element of (Hom_{k}(A, B))((T)) as in (8.3.2) again, and let

a(T) = ∑_{m=m_{0}}^{∞} a_{m} T^{m} (8.4.1)

be an element of A((T)), where m_{0} ∈ Z. Thus

(ϕ(T))(a_{m}) = ∑_{l=l_{0}}^{∞} ϕ_{l}(a_{m}) T^{l} (8.4.2)

defines an element of B((T)) for every m ∈ Z, as in (8.3.3). Put

(ϕ(T))(a(T)) = ∑_{m=m_{0}}^{∞}(ϕ(T))(a_{m}) T^{m}, (8.4.3)

where the series on the right can be defined as an element of B((T)) as in Section 8.2. This defines a homomorphism from A((T)) into B((T)), as modules over k, associated to ϕ(T).

It is easy to see that

(ϕ(T))(a(T) T^{r}) = (ϕ(T))(a(T)) T^{r} (8.4.4)

for every a(T) ∈ A((T)) and r ∈ Z. We also have that

(ϕ(T))((A[[T]]) T^{n}) ⊆ (B[[T]]) T^{n+l_{0}} (8.4.5)

for every n ∈ Z. Note that (8.4.3) is linear over k in ϕ(T). If r ∈ Z, then ϕ(T) T^{r} defines an element of (Hom_{k}(A, B))((T)), as before. One can verify that (ϕ(T) T^{r})(a(T)) = (ϕ(T))(a(T)) T^{r} (8.4.6)

as elements of B((T)) for every a(T) ∈ A((T)).

Let a(T) ∈ A((T)) be given as in (8.4.1) again. As before, we take ϕ_{l} = 0 when l < l_{0}, and a_{m} = 0 when m < m_{0}. Let n ∈ Z be given, and observe that

ϕ_{l}(a_{n−l}) = 0 (8.4.7) when l < l_{0} and when n−l < m_{0}, which is to say that n−m_{0} < l. In particular, (8.4.7) holds for all but finitely many l ∈ Z, so that

((ϕ(T))(a(T)))_{n} = ∑_{l=−∞}^{∞} ϕ_{l}(a_{n−l}) (8.4.8)

defines an element of B. If n < l_{0} + m_{0}, then (8.4.7) holds for every l ∈ Z, so that (8.4.8) is equal to 0. Consider

(ϕ(T))(a(T)) = ∑_{n=l_{0}+m_{0}}^{∞}((ϕ(T))(a(T)))_{n} T^{n} (8.4.9)

as an element of B((T)). One can check that this is equivalent to (8.4.3). Let C be a third module over k, and let

ψ(T) = ∑_{r=r_{0}}^{∞} ψ_{r} T^{r} (8.4.10)

be an element of (Hom_{k}(B, C))((T)), where r_{0} ∈ Z. As usual, we take ψ_{r} = 0 when r < r_{0}. The composition ψ_{r} ∘ ϕ_{l} is defined as a module homomorphism from A into C for every l, r ∈ Z. Let n ∈ Z be given, and note that

ψ_{r} ∘ ϕ_{n−r} = 0 (8.4.11)

when r < r_{0} and when n − r < l_{0}, which means that n − l_{0} < r. It follows that (8.4.11) holds for all but finitely many r ∈ Z, so that

(ψ(T) ∘ ϕ(T))_{n} = ∑_{r=−∞}^{∞} ψ_{r} ∘ ϕ_{n−r} (8.4.12)

defines a module homomorphism from A into C. If n < l_{0} + r_{0}, then (8.4.11) holds for every r ∈ Z, and (8.4.12) is equal to 0. Put

ψ(T) ∘ ϕ(T) = ∑_{n=l_{0}+r_{0}}^{∞}(ψ(T) ∘ ϕ(T))_{n} T^{n}, (8.4.13)

which defines an element of (Hom_{k}(A, C))((T)). One can verify that the module homomorphism from A((T)) into C((T)) corresponding to (8.4.13) as before is the same as the composition of the homomorphisms from A((T)) into B((T)) and from B((T)) into C((T)) corresponding to ϕ(T) and ψ(T), respectively.

8.5 Homomorphisms from A((T)) into B((T))

Let k be a commutative ring with a multiplicative identity element again, let A, B be modules over k, and let T be an indeterminate. Also let ϕ be a homomorphism from A((T)) into B((T)), as modules over k, and suppose that

ϕ(f(T) T) = ϕ(f(T)) T (8.5.1)

8.5. HOMOMORPHISMS FROM A((T)) INTO B((T)) 173

for every f(T) ∈ A((T)). This implies that

ϕ(f(T) T^{r}) = ϕ(f(T)) T^{r} (8.5.2)

for every f(T) ∈ A((T)) and r ∈ Z. Remember that A[[T]] can be identified with the set of f(T) ∈ A((T)) such that f_{j} = 0 for every j < 0. Using (8.5.2), we get that ϕ is uniquely determined by its restriction to this subset of A((T)) corresponding to A[[T]].

Suppose in addition that there is an integer l_{0}(ϕ) such that

ϕ((A[[T]]) T^{n}) ⊆ (B[[T]]) T^{n+l_{0}(ϕ)} (8.5.3)

for every n ∈ Z. Note that this condition holds automatically when ϕ is obtained from an element of (Hom_{k}(A, B))((T)) as in the previous section. In order to verify this condition for any ϕ as in the preceding paragraph, it suffices to consider the case where n = 0, because of (8.5.2). The case where n = 0 can be reformulated as saying that if f(T) corresponds to an element of A[[T]], then ϕ(f(T)) T^{−l_{0}(ϕ)} corresponds to an element of B[[T]]. This means that

ϕ̃(f(T)) = ϕ(f(T)) T^{−l_{0}(ϕ)} (8.5.4)

maps the subset of A((T)) corresponding to A[[T]] into the subset of B((T)) corresponding to B[[T]]. Equivalently, the n = 0 case says that the collection of ϕ(f(T)) with f(T) ∈ A[[T]] has poles of bounded order in B((T)). Thus we may simply say that ϕ has poles of bounded order on A[[T]] in this situation.

Under these conditions, one can check that ϕ is uniquely determined by its restriction to the subset of A((T)) corresponding to A. This can also be obtained from the analogous statement in Section 4.8, applied to the mapping from the subset of A((T)) corresponding to A[[T]] into the subset of B((T)) corresponding to B[[T]] given by (8.5.4). In this situation, the restriction of ϕ to the subset of A((T)) corresponding to A has poles of bounded order, as in Section 8.3, and hence corresponds to an element of (Hom_{k}(A, B))((T)), as before. This element of (Hom_{k}(A, B))((T)) determines a homomorphism from A((T)) into B((T)), as in Section 8.4. In fact, ϕ is equal to this homomorphism on all of A((T)).

Let {a_{l}(T)}_{l=l_{0}}^{∞} be a sequence of elements of A((T)) starting at some l_{0} ∈ Z, and suppose that the set of a_{l}(T)’s, l ≥ l_{0}, has poles of bounded order in A((T)). This implies that the set of ϕ(a_{l}(T))’s, l ≥ l_{0}, has poles of finite order in B((T)), by (8.5.3). Suppose that {a_{l}(T)}_{l=l_{0}}^{∞} also eventually agrees with some a(T) in A((T)) termwise, as in Section 8.2. Under these conditions, one can check that {ϕ(a_{l}(T))}_{l=l_{0}}^{∞} eventually agrees with ϕ(a(T)) termwise. In particular, if {a_{l}(T)}_{l=l_{0}}^{∞} eventually agrees termwise with 0, then {ϕ(a_{l}(T))}_{l=l_{0}}^{∞} eventually agrees termwise with 0. In this situation, ∑_{l=l_{0}}^{∞} a_{l}(T) and ∑_{l=l_{0}}^{∞} ϕ(a_{l}(T)) can be defined as elements of A((T)) and B((T)), respectively, as in Section 8.2. It is easy to see that

ϕ( ∑_{l=l_{0}}^{∞} a_{l}(T)) = ∑_{l=l_{0}}^{∞} ϕ(a_{l}(T)), (8.5.5)

using the previous statement for the partial sums of these series.

8.6 Formal series and bilinear mappings

Let k be a commutative ring with a multiplicative identity element, let A, B, C be modules over k, and let T be an indeterminate. Thus C((T)) can be defined as a module over k as in Section 8.1, and we let β be a mapping from A × B into C((T)) that is bilinear over k. If a ∈ A and b ∈ B, then β(a, b) can be expressed as

β(a, b) = ∑_{r=−∞}^{∞} β_{r}(a, b) T^{r}, (8.6.1)

where β_{r}(a, b) ∈ C for every r ∈ Z, and

β_{r}(a, b) = 0 (8.6.2)

for all but finitely many r < 0. This defines β_{r} as a mapping from A × B into C that is bilinear over k for every r ∈ Z. Conversely, if β_{r} is a mapping from A × B into C that is bilinear over k for every r ∈ Z, and if for every a ∈ A and b ∈ B we have that (8.6.2) holds for all but finitely many r < 0, then (8.6.1) defines an element of C((T)) for every a ∈ A and b ∈ B, and this defines a mapping from A × B into C((T)) that is bilinear over k.

Let β be a mapping from A × B into C((T)) that is bilinear over k again. Let us say that β has poles of bounded order if the set of β(a, b) with a ∈ A and b ∈ B has poles of bounded order in C((T)). This means that there is an integer r(β) such that (8.6.2) holds for every a ∈ A, b ∈ B, and r < r(β). One can check that this holds automatically when A and B are finitely generated as modules over k. If β has poles of bounded order, then β corresponds to a formal series in T with poles of finite order whose coefficients are bilinear mappings from A × B into C.

Let β be a mapping from A × B into C((T)) that is bilinear over k and has poles of bounded order, so that there is an r(β) ∈ Z such that (8.6.2) holds for every a ∈ A, b ∈ B, and r < r(β). Also let f(T) = ∑_{j=j_{0}}^{∞} f_{j} T^{j} ∈ A((T)) and g(T) = ∑_{l=l_{0}}^{∞} g_{l} T^{l} ∈ B((T)) be given, where j_{0}, l_{0} ∈ Z. As before, we take f_{j} = 0 when j < j_{0}, and g_{l} = 0 when l < l_{0}. Let n ∈ Z be given, and observe that β(f_{j}, g_{n−j}) = 0 (8.6.3)

when j < j_{0}, and when n − j < l_{0}, which means that n − l_{0} < j. In particular, (8.6.3) holds for all but finitely many j ∈ Z, so that

h_{n} = ∑_{j=−∞}^{∞} β(f_{j}, g_{n−j}) (8.6.4)

defines an element of C((T)). Equivalently,

β(f_{n−l}, g_{l}) = 0 (8.6.5) when l < l_{0} and when n − l < j_{0}, which means that n − j_{0} < l. Thus (8.6.5) holds for all but finitely many l ∈ Z, and (8.6.4) is the same as

h_{n} = ∑_{l=−∞}^{∞} β(f_{n−l}, g_{l}). (8.6.6)

If n < j_{0} + l_{0}, then (8.6.3) holds for every j ∈ Z, which is the same as saying that (8.6.5) holds for every l ∈ Z, so that h_{n} = 0. Note that the coefficient of T^{r} in h_{n} is equal to 0 when r < r(β), because of the corresponding hypothesis on β. Put

h(T) = ∑_{n=j_{0}+l_{0}}^{∞} h_{n} T^{n}, (8.6.7)

where the series on the right defines an element of C((T)) as in Section 8.2.

Put β(f(T), g(T)) = h(T), (8.6.8)

which defines a mapping from A((T)) × B((T)) into C((T)) that is bilinear over k. This mapping agrees with the initial mapping from A × B into C((T)), when A and B are identified with submodules of A((T)) and B((T)), respectively, as in Section 8.1. One can verify that

β(f(T) T^{m_{1}}, g(T) T^{m_{2}}) = β(f(T), g(T)) T^{m_{1}+m_{2}} (8.6.9)

for every f(T) ∈ A((T)), g(T) ∈ B((T)), and m_{1}, m_{2} ∈ Z. The coefficient of T^{r} in (8.6.7) is equal to 0 when

r < j_{0} + l_{0} + r(β), (8.6.10)

because of the analogous statement for h_{n}. Equivalently, if f(T) ∈ (A[[T]]) T^{m_{1}} and g(T) ∈ (B[[T]]) T^{m_{2}} for some m_{1}, m_{2} ∈ Z, then

β(f(T), g(T)) ∈ (C[[T]]) T^{m_{1}+m_{2}+r(β)}. (8.6.11)

Suppose that A = B. If the initial mapping β from A × A into C((T)) is symmetric or antisymmetric, then the extension of β to A((T)) × A((T)) just defined has the same property, because (8.6.4) and (8.6.6) are the same. Similarly, if β(a, a) = 0 for every a ∈ A, then

β(f(T), f(T)) = 0 (8.6.12)

for every f(T) ∈ A((T)). To see this, it suffices to verify that

∑_{j=−∞}^{∞} β(f_{j}, f_{n−j}) = 0 (8.6.13)

for every n ∈ Z. Remember that β is antisymmetric on A × A in this situation, as in Section 2.1. If n is odd, then (8.6.13) follows from the antisymmetry of β on A × A. If n is even, then (8.6.13) follows from the antisymmetry of β and the fact that β(f_{n/2}, f_{n/2}) = 0, by hypothesis.

8.7 Bilinear mappings on A((T)) × B((T))

Let k be a commutative ring with a multiplicative identity element again, let A, B, and C be modules over k, and let T be an indeterminate. Suppose that β is a mapping from A((T)) × B((T)) into C((T)) that is bilinear over k and satisfies β(f(T) T, g(T)) = β(f(T), g(T) T) = β(f(T), g(T)) T (8.7.1)

for every f(T) ∈ A((T)) and g(T) ∈ B((T)). This implies that

β(f(T) T^{m_{1}}, g(T) T^{m_{2}}) = β(f(T), g(T)) T^{m_{1}+m_{2}} (8.7.2)

for every f(T) ∈ A((T)), g(T) ∈ B((T)), and m_{1}, m_{2} ∈ Z. It is easy to see that β is uniquely determined by its restriction to the subset of A((T)) × B((T)) corresponding to A[[T]] × B[[T]], using (8.7.2).

As before, let us ask in addition that there be an integer r(β) such that if f(T) ∈ (A[[T]]) T^{m_{1}} and g(T) ∈ (B[[T]]) T^{m_{2}} for some m_{1}, m_{2} ∈ Z, then

β(f(T), g(T)) ∈ (C[[T]]) T^{m_{1}+m_{2}+r(β)}. (8.7.3)

Remember that this condition holds when β is obtained from a bilinear mapping from A × B into C((T)) with poles of bounded order as in the previous section. To verify this condition for any β as in the preceding paragraph, it is enough to consider the case where m_{1} = m_{2} = 0, because of (8.7.2). This case can be reformulated as saying that if f(T) and g(T) correspond to elements of A[[T]] and B[[T]], respectively, then β(f(T), g(T)) T^{−r(β)} corresponds to an element of C[[T]]. Equivalently, this means that

β̃(f(T), g(T)) = β(f(T), g(T)) T^{−r(β)} (8.7.4)

maps the subset of A((T)) × B((T)) corresponding to A[[T]] × B[[T]] into the subset of C((T)) corresponding to C[[T]]. The m_{1} = m_{2} = 0 case of (8.7.3) is the same as saying that the collection of β(f(T), g(T)) with f(T) ∈ A[[T]] and g(T) ∈ B[[T]] has poles of bounded order in C((T)). In this situation, we may simply say that β has poles of bounded order on A[[T]] × B[[T]].

One can check that β is uniquely determined by its restriction to the subset of A((T)) × B((T)) that corresponds to A × B under these conditions. This can also be seen using the analogous statement in Section 4.8 for mappings from A[[T]] × B[[T]] into C[[T]], applied to the mapping that corresponds to (8.7.4). Of course, the restriction of β to A × B has poles of bounded order on A × B. Thus the restriction of β to A × B can be extended to A((T)) × B((T)) as in the previous section. This extension agrees with β on all of A((T)) × B((T)) in this situation.

Let {a_{m}(T)}_{m=m_{0}}^{∞} and {b_{r}(T)}_{r=r_{0}}^{∞} be sequences of elements of A((T)) and B((T)), respectively. Suppose that the sets of a_{m}(T)’s, m ≥ m_{0}, and b_{r}(T)’s, r ≥ r_{0}, have poles of bounded order in A((T)) and B((T)), respectively. This implies that the set of β(a_{m}(T), b_{r}(T))’s, m ≥ m_{0}, r ≥ r_{0}, has poles of bounded 8.8. ALGEBRAS AND MODULES OVER K((T)) 177

order in C((T)), by (8.7.3). Suppose that {a_{m}(T)}_{m=m_{0}}^{∞} and {b_{r}(T)}_{r=r_{0}}^{∞} eventually agree with some a(T) ∈ A((T)) and b(T) ∈ B((T)) termwise, respectively, as in Section 8.2. One can verify that {β(a_{r}(T), b_{r}(T))}_{r=max(m_{0},r_{0})}^{∞} eventually agrees termwise with β(a(T), b(T)).

Suppose now that {a_{m}(T)}_{m=m_{0}}^{∞} and {b_{r}(T)}_{r=r_{0}}^{∞} eventually agree termwise with 0 in A((T)) and B((T)), respectively, in addition to having poles of bounded order. This implies that ∑_{m=m_{0}}^{∞} a_{m}(T) and ∑_{r=r_{0}}^{∞} b_{r}(T) define elements of A((T)) and B((T)), respectively, as in Section 8.2. If N is an integer with N ≥ m_{0} + r_{0}, then put

c_{N}(T) = ∑_{m=m_{0}}^{N−r_{0}} β(a_{m}(T), b_{N−m}(T)), (8.7.5)

which is an element of C((T)). Equivalently, this is the sum of

β(a_{m}(T), b_{r}(T)) (8.7.6)

over m ≥ m_{0} and r ≥ r_{0} with m + r = N. Note that the set of c_{N}(T)’s, N ≥ m_{0} + r_{0}, has poles of bounded order in C((T)), because of the analogous statement for (8.7.6) in the preceding paragraph. One can check that (8.7.6) vanishes to arbitrarily large order in T when m or r is sufficiently large, because of (8.7.3). In particular, {c_{N}(T)}_{N=m_{0}+r_{0}}^{∞} eventually agrees termwise with 0. Thus ∑_{N=m_{0}+r_{0}}^{∞} c_{N}(T) defines an element of C((T)), as in Section 8.2. One can verify that

∑_{N=m_{0}+r_{0}}^{∞} c_{N}(T) = β( ∑_{m=m_{0}}^{∞} a_{m}(T), ∑_{r=r_{0}}^{∞} b_{r}(T)), (8.7.7)

as in Section 4.2.

8.8 Algebras and modules over k((T))

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Also let T be an indeterminate, and let f(T) = ∑_{j=j_{0}}^{∞} f_{j} T^{j} and g(T) = ∑_{l=l_{0}}^{∞} g_{l} T^{l} be elements of A((T)), where j_{0}, l_{0} ∈ Z. As usual, we take f_{j} = 0 when j < j_{0}, and g_{l} = 0 when l < l_{0}. Let n ∈ Z be given, and observe that f_{j} g_{n−j} = 0 (8.8.1)

when j < j_{0} and when n − j < l_{0}, which means that n − l_{0} < j. In particular, (8.8.1) holds for all but finitely many j ∈ Z, so that

h_{n} = ∑_{j=−∞}^{∞} f_{j} g_{n−j} (8.8.2)

reduces to a finite sum in A. Equivalently,

f_{n−l} g_{l} = 0 (8.8.3)

when l < l_{0} and when n − l < j_{0}, which means that n − j_{0} < l. Thus (8.8.3) holds for all but finitely many l ∈ Z, and (8.8.2) is the same as

h_{n} = ∑_{l=−∞}^{∞} f_{n−l} g_{l}. (8.8.4)

If n < j_{0} + l_{0}, then (8.8.1) holds for every j ∈ Z, which is the same as saying that (8.8.3) holds for every l ∈ Z, so that h_{n} = 0. Put

f(T) g(T) = h(T) = ∑_{n=j_{0}+l_{0}}^{∞} h_{n} T^{n}, (8.8.5)

which defines another element of A((T)). Of course, this is the same as in (8.6.7), with the bilinear mapping β given by multiplication on A. As before, one can check that

(f(T) T^{m_{1}}) (g(T) T^{m_{2}}) = (f(T) g(T)) T^{m_{1}+m_{2}} (8.8.6)

for every m_{1}, m_{2} ∈ Z. If f(T) ∈ (A[[T]]) T^{m_{1}} and g(T) ∈ (A[[T]]) T^{m_{2}} for some m_{1}, m_{2} ∈ Z, then f(T) g(T) ∈ (A[[T]]) T^{m_{1}+m_{2}}. (8.8.7)

This extends multiplication on A to a mapping from A((T)) × A((T)) into A((T)) that is bilinear over k, which makes A((T)) into an algebra over k in the strict sense. In particular, this agrees with the extension of multiplication on A to A[[T]] discussed in Section 4.6, so that A[[T]] may be considered as a subalgebra of A((T)). If A is a commutative algebra over k, then A((T)) is commutative as well, by the remark about symmetry of β in the previous section. Similarly, if A is an associative algebra over k, then one can verify that A((T)) is an associative algebra. If A has a multiplicative identity element e, then e corresponds to the multiplicative identity element in A((T)) too.

Suppose that A is an associative algebra over k with a multiplicative identity element e, and let f(T) = ∑_{j=j_{0}}^{∞} f_{j} T^{j} be an element of A((T)). Thus f(T) T^{−j_{0}} corresponds to an element of A[[T]]. If f_{j_{0}} is invertible in A, then f(T) T^{−j_{0}} is invertible in A[[T]], as in Section 4.7. This implies that f(T) is invertible in A((T)).

Applying the earlier remarks to k as a commutative associative algebra over itself, we get that k((T)) is a commutative associative algebra over k with a multiplicative identity element. Let A be a module over k, and let f(T) ∈ k((T)) and g(T) ∈ A((T)) be given. Under these conditions, f(T) g(T) can be defined as an element of A((T)) as in (8.8.5), where the terms on the right side of (8.8.2) are defined using scalar multiplication on A. This is the same as extending scalar multiplication on A, as a mapping from k × A into A that is bilinear over k, to 8.9. ABSOLUTE VALUES ON K((T)) 179

a mapping from k((T)) × A((T)) into A((T)), as in the previous section. One can check that this makes A((T)) into a module over k((T)). Note that (8.8.6) holds for every m_{1}, m_{2} ∈ Z in this situation. Similarly, if f(T) ∈ (k[[T]]) T^{m_{1}} and g(T) ∈ (A[[T]]) T^{m_{2}} for some m_{1}, m_{2} ∈ Z, then (8.8.7) holds.

Let B be another module over k, and let ϕ(T) ∈ (Hom_{k}(A, B))((T)) be given. This leads to a homomorphism from A((T)) into B((T)) as modules over k, as in Section 8.4. More precisely, one can verify that this is a homomorphism from A((T)) into B((T)) as modules over k((T)). One can also look at this in terms of k-linear mappings satisfying (8.5.1) and (8.5.3).

Similarly, let C be a third module over k, and let β be a mapping from A×B into C((T)) that is bilinear over k and has poles of bounded order. One can check that the extension of β to a mapping from A((T)) × B((T)) into C((T)) defined in Section 8.6 is bilinear over k((T)). One can also look at this in terms of mappings from A((T)) × B((T)) into C((T)) that are bilinear over k and satisfy (8.7.1) and (8.7.3). In particular, if A is an algebra over k in the strict sense, then A((T)) may be considered as an algebra over k((T)) in the strict sense.

Let (A, [·, ·]_{A}) be a Lie algebra over k. If f(T), g(T) ∈ A((T)), then (8.8.2) should be expressed as

h_{n} = ∑_{j=−∞}^{∞}[f_{j}, g_{n−j}]_{A} (8.8.8)

for each n ∈ Z, so that (8.8.5) corresponds to

[f(T), g(T)]_{A((T))} = h(T) = ∑_{n=−∞}^{∞} h_{n} T^{n}. (8.8.9)

As in Section 8.6, [f(T), f(T)]_{A((T))} = 0 for every f(T) ∈ A((T)), because [a, a]_{A} = 0 for every a ∈ A. One can also check that [·, ·]_{A((T))} satisfies the Jacobi identity on A((T)), using the Jacobi identity for [·, ·]_{A} on A. Thus A((T)) is a Lie algebra with respect to (8.8.9) over k, and in fact over k((T)).

8.9 Absolute values on k((T))

Let k be a field, and let T be an indeterminate. If f(T) is a nonzero element of k((T)), then f(T) is invertible in k((T)), because nonzero elements of k are invertible in k, and using the remark about invertibility in the previous section. Thus k((T)) is a field. Let f(T) = ∑_{j>>−∞} f_{j} T^{j} ∈ k((T)) be given. If f(T) ≠ 0, then there is a unique minimal integer j_{0}(f(T)) such that f_{j_{0}(f(T))} ≠ 0, which is to say that f_{j} = 0 when j < j_{0}(f(T)). If f(T) = 0, then it is convenient to put j_{0}(f) = +∞. Observe that

j_{0}(f(T) + g(T)) ≥ min(j_{0}(f(T)), j_{0}(g(T))) (8.9.1)

and j_{0}(f(T) g(T)) = j_{0}(f(T)) + j_{0}(g(T)) (8.9.2) for every f(T), g(T) ∈ k((T)), with suitable interpretations when any of these terms is +∞.

Let r be a positive real number with r ≤ 1. If f(T) ∈ k((T)), then put

|f(T)|_{r} = r^{j_{0}(f(T))} (8.9.3)

when f(T) ≠ 0, and |f(T)|_{r} = 0 when f(T) = 0. Using (8.9.1) and (8.9.2), we get that |f(T) + g(T)|_{r} ≤ max(|f(T)|_{r}, |g(T)|_{r}) (8.9.4)

and |f(T) g(T)|_{r} = |f(T)|_{r} |g(T)|_{r} (8.9.5)

for every f(T), g(T) ∈ k((T)). Thus |f(T)|_{r} defines an ultrametric absolute value function on k((T)), which is the same as the trivial absolute value function on k((T)) when r = 1. If a is a positive real number, then 0 < r^{a} ≤ 1, and

|f(T)|_{r}^{a} = |f(T)|_{r^{a}} (8.9.6)

for every f(T) ∈ k((T)).

It follows that d_{r}(f(T), g(T)) = |f(T) − g(T)|_{r} (8.9.7)

is an ultrametric on k((T)), which is the discrete metric when r = 1. Let us suppose from now on in this section that r < 1. If l ∈ Z, then the closed ball in k((T)) centered at 0 with radius r^{l} with respect to (8.9.7) is given by

(8.9.8) \overline{B}(0, r^{l}) = \{f(T) ∈ k((T)) : |f(T)|_{r} ≤ r^{l}\}
= {f(T) ∈ k((T)) : j_{0}(f(T)) ≥ l} = (k[[T]]) T^{l}.

Let l ∈ Z be given, so that (8.9.8) can be identified with the space of kvalued functions on the set of integers j ≥ l. This may be considered as the Cartesian product of copies of k indexed by integers j ≥ l. One can check that the topology determined on (8.9.8) by the restriction of the ultrametric (8.9.7) to (8.9.8) corresponds to the product topology on the Cartesian product just mentioned, using the discrete topology on k.

One can verify that k((T)) is complete with respect to the ultrametric (8.9.7), as follows. Any Cauchy sequence in k((T)) with respect to (8.9.7) is contained in (8.9.8) for some l ∈ Z, because a Cauchy sequence in any metric space is bounded. It is easy to see that for each j ∈ Z, the corresponding sequence of coefficients of T^{j} in the terms of the Cauchy sequence is eventually constant, as a sequence of elements of k. This leads to an element of (8.9.8), for which the coefficient of T^{j} is the eventual constant value of the sequence in k just mentioned, for each j ∈ Z. The given Cauchy sequence in k((T)) converges to this element of (8.9.8) with respect to the ultrametric (8.9.7), by the description of the topology determined on (8.9.8) by the restriction of the ultrametric (8.9.7) in the preceding paragraph.

8.10 Formal series and algebra homomorphisms

Let k be a commutative ring with a multiplicative identity element, and let T be an indeterminate. Also let A, B be algebras over k in the strict sense, where multiplication of x, y is expressed as x y. Remember that multiplication on A and B can be extended to A((T)) and B((T)), respectively, so that A((T)) and B((T)) become algebras in the strict sense over k((T)), as in Section 8.8. In particular, they may be considered as algebras over k.

Let ϕ be a homomorphism from A into B((T)), as modules over k for the moment. As in Section 8.3, ϕ can be expressed as

ϕ(a) = ∑_{j=−∞}^{∞} ϕ_{j}(a) T^{j} (8.10.1)

for each a ∈ A, where ϕ_{j} is a module homomorphism from A into B for every j ∈ Z, and for every a ∈ A we have that ϕ_{j}(a) = 0 for all but finitely many j < 0. Of course, ϕ is a homomorphism from A into B((T)) as algebras over k when ϕ(a a′) = ϕ(a) ϕ(a′) (8.10.2)

for every a, a′ ∈ A. Let a, a′ ∈ A be given, so that

ϕ(a a′) = ∑_{n=−∞}^{∞} ϕ_{n}(a a′) T^{n}, (8.10.3)

as in (8.10.1), where ϕ_{n}(a a′) = 0 for all but finitely many n < 0. We also have that

(8.10.4)  ϕ(a) ϕ(a′) = ( ∑_{j=−∞}^{∞} ϕ_{j}(a) T^{j}) ( ∑_{l=−∞}^{∞} ϕ_{l}(a′) T^{l})

= ∑_{n=−∞}^{∞} ( ∑_{j=−∞}^{∞} ϕ_{j}(a) ϕ_{n−j}(a′)) T^{n},

as in Section 8.8. Remember that for each n ∈ Z, ϕ_{j}(a) ϕ_{n−j}(a′) = 0 for all but finitely many j ∈ Z, and that for all but finitely many n < 0, this condition holds for every j ∈ Z. Comparing (8.10.3) and (8.10.4), we get that (8.10.2) holds if and only if

ϕ_{n}(a a′) = ∑_{j=−∞}^{∞} ϕ_{j}(a) ϕ_{n−j}(a′) (8.10.5)

for every n ∈ Z.

Suppose that ϕ has poles of bounded order on A, so that there is an integer r(ϕ) such that ϕ_{j} = 0 on A when j < r(ϕ). Let f(T) = ∑_{j=j_{0}}^{∞} f_{j} T^{j} be an element of A((T)). As in Section 8.4, ϕ can be extended to a module homomorphism from A((T)) into B((T)), by putting

ϕ(f(T)) = ∑_{j=j_{0}}^{∞} ϕ(f_{j}) T^{j}, (8.10.6)

where the sum on the right is defined as an element of B((T)) as in Section 8.2. Let g(T) = ∑_{l=l_{0}}^{∞} g_{l} T^{l} be another element of A((T)), so that

ϕ(g(T)) = ∑_{l=l_{0}}^{∞} ϕ(g_{l}) T^{l}. (8.10.7)

Remember that f(T) g(T) = h(T) = ∑_{n=j_{0}+l_{0}}^{∞} h_{n} T^{n} is defined in A((T)) by putting

h_{n} = ∑_{j=−∞}^{∞} f_{j} g_{n−j} (8.10.8)

for each n, which reduces to a finite sum in A. Thus

ϕ(f(T) g(T)) = ϕ(h(T)) = ∑_{n=j_{0}+l_{0}}^{∞} ϕ(h_{n}) T^{n}, (8.10.9)

where the sum on the right is defined as an element of B((T)) as in Section 8.2 again. If ϕ is an algebra homomorphism from A into B((T)), then

ϕ(h_{n}) = ∑_{j=−∞}^{∞} ϕ(f_{j} g_{n−j}) = ∑_{j=−∞}^{∞} ϕ(f_{j}) ϕ(g_{n−j}) (8.10.10)

for each n, where the sums reduce to finite sums in B((T)). We would like to say that

∑_{n=j_{0}+l_{0}}^{∞} ( ∑_{j=−∞}^{∞} ϕ(f_{j}) ϕ(g_{n−j})) T^{n} = ( ∑_{j=j_{0}}^{∞} ϕ(f_{j}) T^{j}) ( ∑_{l=l_{0}}^{∞} ϕ(g_{l}) T^{l}), (8.10.11)

as elements of B((T)). If ϕ maps A into B, then this follows from the definition of the extension of multiplication on B to B((T)), as in Section 8.8. Otherwise, the ϕ(f_{j})’s, j ≥ j_{0}, and ϕ(g_{l})’s, l ≥ l_{0}, are elements of B((T)) with poles of bounded order, and (8.10.11) can be obtained from (8.7.7). It follows that

ϕ(f(T) g(T)) = ϕ(f(T)) ϕ(g(T)), (8.10.12)

so that the extension of ϕ to A((T)) is an algebra homomorphism as well.

There are analogous statements for opposite algebra homomorphisms, as usual.

8.11 Involutions and formal series

Let k be a commutative ring with a multiplicative identity element, and let T be an indeterminate. Also let A, B be modules over k, and let ϕ be a module homomorphism from A into B. If f(T) = ∑_{j>>−∞} f_{j} T^{j} ∈ A((T)), then

ϕ(f(T)) = ∑_{j>>−∞} ϕ(f_{j}) T^{j} (8.11.1)

defines an element of B((T)). This defines a homomorphism from A((T)) into B((T)), as modules over k((T)). This may be seen as a simple case of the situation discussed in Section 8.4, by identifying ϕ with an element of (Hom_{k}(A, B))((T)). In particular, we have that

ϕ(f(T) T^{r}) = ϕ(f(T)) T^{r} (8.11.2)

for every f(T) ∈ A((T)) and r ∈ Z. Of course, if f(T) corresponds to an element of A[T] or A[[T]], then ϕ(f(T)) corresponds to an element of B[T] or B[[T]], as appropriate. If ϕ is a one-to-one mapping from A onto B, then (8.11.1) defines a one-to-one mapping from A((T)) onto B((T)).

Now let A, B be algebras over k in the strict sense, where multiplication of x, y is expressed as x y. If ϕ is an algebra homomorphism from A into B, then (8.11.1) defines an algebra homomorphism from A((T)) into B((T)), as in the previous section. Similarly, if ϕ is an opposite algebra homomorphism from A into B, then (8.11.1) defines an opposite algebra homomorphism from A((T)) into B((T)).

Let x ↦ x^{∗} be an algebra involution on A, and put

f(T)^{∗} = ∑_{j>>−∞} f_{j}^{∗} T^{j} (8.11.3)

for every f(T) ∈ A((T)). This defines an algebra involution on A((T)), as before. Clearly f(T) ∈ A((T)) is self-adjoint with respect to this involution if and only if f_{j} is self-adjoint in A for every j. Similarly, f(T) is anti-self-adjoint with respect to this involution if and only if f_{j} is anti-self-adjoint in A for every j.

If k is the field C of complex numbers, then there are analogous statements for conjugate-linear mappings. More precisely, if A and B are vector spaces over C, then A((T)) and B((T)) may be considered as vector spaces over C too. If ϕ is a conjugate-linear mapping from A into B, then (8.11.1) defines a conjugatelinear mapping from A((T)) into B((T)), as vector spaces over C. Remember that complex vector spaces may be considered as real vector spaces, and that conjugate-linear mappings between complex vector spaces may be considered as real-linear mappings between the corresponding real vector spaces. This can be used to reduce statements about conjugate-linear mappings to the analogous statements for real-linear mappings, as before.

In particular, if a(T) = ∑_{j>>−∞} a_{j} T^{j} ∈ C((T)), then

\overline{a(T)} = ∑_{j>>−∞} \overline{a_{j}} T^{j} (8.11.4)

defines an element of C((T)) too, and this defines a conjugate-linear automorphism of C((T)) as an algebra over C. If a(T) corresponds to an element of C[T] or C[[T]], then \overline{a(T)} corresponds to an element of C[T] or C[[T]], as appropriate. One can use this to define conjugate-linearity over C[T], C[[T]], and C((T)). These conjugate-linearity conditions amount to ordinary conjugatelinearity over C, together with the appropriate “real” linearity condition over R[T], R[[T]], or R((T)), respectively.

8.12 Ordered rings

Let R be a ring with a nonzero multiplicative identity element e. Of course, R may be considered as an associative algebra over Z, and in particular as a module over Z. Suppose that certain nonzero elements x of R have been designated as positive, which may be expressed by

x > 0. (8.12.1)

We say that R is an ordered ring if the following conditions are satisfied. First, if x and y are positive elements of R, then

x + y > 0, x y > 0. (8.12.2)

Second, if x is any nonzero element of R, then either x or −x is positive. This corresponds to the definition on p261 of [19].

Let R be an ordered ring. If x, y are nonzero elements of R, then x y ≠ 0. More precisely, x y = (−x) (−y) > 0 (8.12.3)

when x, y > 0, and when −x, −y > 0. Similarly,

−x y = (−x) y = x (−y) > 0 (8.12.4)

when −x, y > 0, and when x, −y > 0.

If R is commutative, then it follows that R is an integral domain. This corresponds to the definition of an ordered integral domain on p9 of [3]. Similarly, an ordered field is an ordered ring that is also a field, as in [3, 19].

Let R be an ordered ring again. Note that x ∈ R and −x cannot both be positive, because that would imply that x + (−x) = 0 is positive. If x ∈ R and x ≠ 0, then x^{2} > 0, (8.12.5)

because x or −x is positive, and x^{2} = (−x)^{2}. Of course, this corresponds to (8.12.3), with y = x. In particular,

e = e^{2} > 0 (8.12.6)

in R. If x, y ∈ R, then put x < y (8.12.7)

when y − x > 0. This defines a linear ordering on R, which is invariant under translations on R.

Alternatively, one might start with a translation-invariant linear ordering on R, and define x ∈ R to be positive when x > 0 with respect to this ordering. One can check that the sum of positive elements of R is positive in this situation. If products of positive elements of R are positive too, then R is an ordered ring. This is how ordered fields are defined on p7 in [20].

Clearly Z is an ordered ring with respect to the standard ordering. In fact, this is the only ordering on Z for which Z is an ordered ring. More precisely, if Z is an ordered ring with respect to some ordering, then 1 has to be positive in Z with respect to that ordering, by (8.12.6). This implies that all sums of 1 have to be positive with respect to this ordering on Z. One can check that these are the only elements of Z that can be positive, so that this ordering on Z is the same as the usual one.

Let R be an ordered ring, and let R_{0} be a subring of R that contains e. It is easy to see that R_{0} is an ordered ring too, with respect to the restriction of the restriction of the ordering on R to R_{0}.

8.13 Some additional features

Let R be a ring, with multiplicative identity element e. If n is a positive integer, then n · e is the sum of n e’s in R, as usual. We can extend this to integers n ≤ 0 in the obvious way, by putting 0 · e = 0 in R, and n · e = −((−n) · e) when n < 0. This defines a ring homomorphism from Z into R.

Suppose that R is an ordered ring. If n ∈ Z_{+}, then

n · e > 0 (8.13.1)

in R, by (8.12.6). Of course, this implies that

−(n · e) = (−n) · e > 0 (8.13.2)

when −n ∈ Z_{+}. Thus n ↦ n · e is an injective order-preserving mapping from Z into R, with respect to the standard ordering on Z.

If x ∈ R, x > 0, and x has a multiplicative inverse x^{−1} in R, then

x^{−1} > 0. (8.13.3)

Otherwise, if −x^{−1} > 0, then −e = x (−x^{−1}) > 0, contradicting (8.12.6).

If x ∈ R, then the absolute value |x| of x may be defined as an element of R by

(8.13.4)  |x| = x when x ≥ 0
= −x when − x ≥ 0,

as on p10 of [3], and p264 of [19]. Note that |x| ≥ 0 and

−|x| ≤ x ≤ |x| (8.13.5)

for every x ∈ R. One can check that

|x y| = |x| |y| (8.13.6)

and |x + y| ≤ |x| + |y| (8.13.7)

for every x, y ∈ R. More precisely, (8.13.6) is basically the same as (8.12.3) and (8.12.4). To get (8.13.7), and one can use (8.13.5) and its analogue for y.

Let T be an indeterminate, and let R((T)) be as before. More precisely, R may be considered as an associative algebra over Z, so that R((T)) is an associative algebra over Z too. If f(T) ∈ R((T)) can be expressed as ∑_{j=j_{0}}^{∞} f_{j} T^{j}, where f_{j_{0}} > 0 in R, then let us say that f(T) is positive as an element of R((T)). Let us check that this makes R((T)) into an ordered ring, as in the discussion on p284-5 in [19]. Note that the elements of R((T)) are called extended formal power series in [19].

Let f(T) be as in the preceding paragraph, and let g(T) = ∑_{l=l_{0}}^{∞} g_{l} T^{l} be another positive element of R((T)), with g_{l_{0}} > 0. Put h(T) = f(T) g(T), so that h(T) = ∑_{n=j_{0}+l_{0}}^{∞} h_{n} T^{n}, with

h_{j_{0}+l_{0}} = f_{j_{0}} g_{l_{0}}. (8.13.8)

This implies that h_{j_{0}+l_{0}} > 0 in R, so that h(T) > 0 in R((T)).

One can verify that f(T) + g(T) > 0 (8.13.9)

in R((T)), directly from the definitions. More precisely, if l_{0} > j_{0}, then (8.13.9) holds when f_{j_{0}} > 0 in R, without additional conditions on g(T). Similarly, if j_{0} > l_{0}, then (8.13.9) holds when g_{l_{0}} > 0 in R, without additional conditions on f(T).

If a(T) is any nonzero element of R((T)), then a(T) can be expressed as ∑_{r=r_{0}}^{∞} a_{r} T^{r}, where a_{r_{0}} ≠ 0. Because R is an ordered ring, either a_{r_{0}} > 0 or −a_{r_{0}} > 0 in R. This implies that a(T) > 0 or −a(T) > 0 in R((T)), as appropriate.

8.14 Ordered fields

Note that Q is an ordered field with respect to the standard ordering. One can check that this is the only ordering on Q for which Q is an ordered field. Indeed, if Q is an ordered field with respect to some ordering, then every n ∈ Z_{+} is positive with respect to this ordering on Q, as before. This implies that 1/n is positive with respect to this ordering on Q, as in the previous section, and hence that quotients of elements of Z_{+} are positive with respect to this ordering on Q. One can verify that these are the only elements of Q that can be positive, so that this ordering on Q is the standard ordering.

Let k be an ordered field. Note that k has characteristic 0, because sums of 1 are positive in k. The usual homomorphism from Z into k extends to a field isomorphism from Q onto a subfield of k. This isomorphism is also compatible with the standard ordering on Q, as in the preceding paragraph. This corresponds to the corollary on p266 of [19].

The classical version of the archimedean property can be stated for k as follows: if x, y are positive elements of k, then there is a positive integer n such that n · x > y. (8.14.1)

Of course, R has the archimedean property with respect to the standard ordering. If k has the archimedean property, then every subfield of k has the archimedean property.

Let k_{1} and k_{2} be ordered fields. To say that k_{1} and k_{2} are isomorphic as ordered fields means that there is a field isomorphism from k_{1} onto k_{2} that preserves order as well. Clearly the archimedean property is invariant under order-preserving field isomorphisms.

Suppose that k_{1} is an ordered field with the archimedean property. It is well known that k_{1} is isomorphic as an ordered field to a subfield of R, with the ordering induced by the standard ordering on R. This corresponds to Exercise 10 on p286 of [19].

Let k be an ordered field again, and let T be an indeterminate. Thus k((T)) is an ordered field with respect to the ordering obtained from the one on k described in the previous section. It is easy to see that k((T)) does not have the archimedean property with respect to this ordering, even if k has the archimedean property, as on p285 of [19]. More precisely, T is a positive element of k((T)), because 1 > 0 in k. However,

n · T < 1 (8.14.2)

for every positive integer n.

Some topics related to inner products on vector spaces over ordered fields will be discussed in Section 11.9.

Part II

Solvability, nilpotence, and semisimplicity Solvability and nilpotence

9.1 Some basic isomorphism theorems

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense. If A_{0} is a two-sided ideal in A, then the quotient A/A_{0} can be defined as an algebra over k in the strict sense too, as in Section 2.11. Let q_{0} be the canonical quotient mapping from A onto A/A_{0}, which is an algebra homomorphism.

Suppose that B is another algebra over k in the strict sense, and that ϕ is an algebra homomorphism from A into B. The kernel ker ϕ of ϕ is a two-sided ideal in A, as mentioned in Section 2.11. If A_{0} ⊆ ker ϕ, then there is a unique algebra homomorphism ψ from A/A_{0} onto ϕ(A) ⊆ B such that

ψ ∘ q_{0} = ϕ. (9.1.1)

If A_{0} = ker ϕ, then ψ is injective.

Let A_{1}, A_{2} be two-sided ideals in A such that A_{1} ⊆ A_{2}, and let q_{1}, q_{2} be the canonical quotient mappings from A onto A/A_{1}, A/A_{2}, respectively. There is a natural algebra homomorphism Ψ from A/A_{1} onto A/A_{2} such that

Ψ ∘ q_{1} = q_{2}, (9.1.2)

as in the preceding paragraph.

Note that A_{1} may be considered as a two-sided ideal in A_{2}, so that A_{2}/A_{1} can be defined as an algebra over k in the strict sense. Of course, A_{2}/A_{1} is essentially the same as q_{1}(A_{2}) ⊆ A/A_{1}. It is easy to see that q_{1}(A_{2}) is a twosided ideal in A/A_{1}, because A_{2} is a two-sided ideal in A.

The kernel of Ψ is equal to q_{1}(A_{2}), by construction. Thus Ψ can be identified with the quotient mapping from A/A_{1} onto (A/A_{1})/q_{1}(A_{2}), which is the same as (A/A_{1})/(A_{2}/A_{1}). This leads to a natural algebra isomorphism between this quotient and A/A_{2}.

Let A_{3}, A_{4} be two-sided ideals in A, and observe that A_{3} ∩ A_{4} is a two-sided ideal in A as well. Remember that A_{3} + A_{4} is a two-sided ideal in A too, as in Section 7.1. Let q_{4} be the canonical quotient mapping from A onto A/A_{4}. The restriction of q_{4} to A_{3} + A_{4} is essentially the same as the canonical quotient mapping from A_{3} + A_{4} onto (A_{3} + A_{4})/A_{4}, as before.

Observe that q_{4}(A_{3}) = q_{4}(A_{3} +A_{4}). The kernel of the restriction of q_{4} to A_{3} is equal to A_{3} ∩ A_{4}. The restriction of q_{4} to A_{3}, as an algebra homomorphism from A_{3} onto q_{4}(A_{3}), can be identified with the quotient mapping from A_{3} onto A_{3}/(A_{3} ∩ A_{4}). This leads to a natural algebra isomorphism between this quotient and q_{4}(A_{3} + A_{4}), which is essentially the same as (A_{3} + A_{4})/A_{4}.

These isomorphism theorems are stated for Lie algebras (over fields) on p7-8 of [14].

9.2 Products of ideals

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Also let A_{1} and A_{2} be submodules of A, as a module over k. The product A_{1} · A_{2} of A_{1} and A_{2} is defined to be the subset of A consisting of all finite sums of elements of A of the form a_{1} a_{2}, where a_{1} ∈ A_{1} and a_{2} ∈ A_{2}. It is easy to see that A_{1} · A_{2} is a submodule of A as well, as a module over k. If multiplication on A is commutative or anti-commutative, then

A_{1} · A_{2} = A_{2} · A_{1}. (9.2.1)

If A_{1} is a right ideal in A, then

A_{1} · A_{2} ⊆ A_{1}. (9.2.2)

Similarly, if A_{2} is a left ideal in A, then

A_{1} · A_{2} ⊆ A_{2}. (9.2.3)

Suppose for the moment that A is an associative algebra over k. If A_{1} is a left ideal in A, then A_{1} · A_{2} is a left ideal in A too. If A_{2} is a right ideal in A, then A_{1} · A_{2} is a right ideal in A.

Let B be a subalgebra of A, as an algebra over k in the strict sense. If B_{1} and B_{2} are submodules of B, as a module over k, then B_{1} and B_{2} are submodules of A too, so that [B_{1}, B_{2}] can be defined as a submodule of A as before. This is the same as defining [B_{1}, B_{2}] as a submodule of B in the analogous way.

Let C be another algebra over k in the strict sense, and let ϕ be an algebra homomorphism from A into C. If A_{1} and A_{2} are submodules of A, as a module over k, then ϕ(A_{1}) and ϕ(A_{2}) are submodules of C, so that ϕ(A_{1}) · ϕ(A_{2}) can be defined as a submodule of C as before. Observe that

ϕ(A_{1} · A_{2}) = ϕ(A_{1}) · ϕ(A_{2}). (9.2.4)

Suppose now that (A, [·, ·]_{A}) is a Lie algebra over k. In this situation, A_{1} ·A_{2} may be denoted [A_{1}, A_{2}], and consists of finite sums of elements of A of the form [a_{1}, a_{2}]_{A}, where a_{1} ∈ A_{1} and a_{2} ∈ A_{2}, as before. Note that

[A_{1}, A_{2}] = [A_{2}, A_{1}], (9.2.5)

as in (9.2.1), because of anticommutativity of the Lie bracket on A. If A_{1} and A_{2} are ideals in A, then it is easy to see that [A_{1}, A_{2}] is an ideal in A, using the Jacobi identity.

In particular, we can apply this to A_{1} = A_{2} = A, to get that [A, A] is an ideal in A. This is known as the derived algebra associated to A. By construction, A/[A, A] is commutative as a Lie algebra over k.

Let A_{0} be an ideal in A, and suppose that A/A_{0} is commutative as a Lie algebra over k. If a_{1}, a_{2} ∈ A, then the image of [a_{1}, a_{2}]_{A} in A/A_{0} is the same as the Lie bracket of the images of a_{1} and a_{2} in A/A_{0}, which is equal to 0. This means that [a_{1}, a_{2}] ∈ A_{0}, so that

[A, A] ⊆ A_{0}. (9.2.6)

Equivalently, if ϕ is a homomorphism from A into a commutative Lie algebra over k, then one can take A_{0} to be the kernel of ϕ.

Let B be a Lie subalgebra of A, so that B may be considered as a Lie algebra over k too. If B_{1}, B_{2} are ideals in B, then [B_{1}, B_{2}] is an ideal in B, as before. In particular, [B, B] is an ideal in B, and

[B, B] ⊆ [A, A]. (9.2.7)

9.3 Solvable Lie algebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. If j is a nonnegative integer, then A^{(j)} is defined inductively by putting A^{(0)} = A, A^{(1)} = [A, A], and

A^{(j+1)} = [A^{(j)}, A^{(j)}] (9.3.1)

for each j ≥ 0, where the right side is as defined in the preceding section. It is easy to see that A^{(j)} is an ideal in A for every j ≥ 0, using induction and a remark in the previous section. This sequence of ideals is called the derived series of A. Here we use the notation on p10 of [14], while on p35 of [25] the notation D^{j} A is used for A^{(j−1)} when j ≥ 1.

Observe that A^{(j+1)} ⊆ A^{(j)} (9.3.2)

for every j ≥ 0. Of course, A^{(1)} = {0} exactly when A is commutative as a Lie algebra. If A^{(j)} = {0} for some j ≥ 0, then A is said to be solvable as a Lie algebra.

If B is a Lie subalgebra of A, then B^{(j)} can be defined as an ideal in B for each j ≥ 0 as before. One can check that

B^{(j)} ⊆ A^{(j)} (9.3.3) for every j ≥ 0, using (9.2.7) and induction. In particular, if A is solvable, then B is solvable.

By construction, A^{(j+l)} = (A^{(j)})^{(l)} (9.3.4)

for every j, l ≥ 0. If A^{(j)} is solvable for some j ≥ 0, then it follows that A is solvable.

Let C be another Lie algebra over k, and let ϕ be a Lie algebra homomorphism from A onto C. One can verify that

ϕ(A^{(j)}) = C^{(j)} (9.3.5)

for every j ≥ 0, using (9.2.4) and induction. If A is solvable, then it follows that C is solvable.

Suppose that C is solvable, so that C^{(n)} = {0} for some nonnegative integer n. This implies that ϕ(A^{(n)}) = C^{(n)} = {0}, (9.3.6)

as in (9.3.5), which means that

A^{(n)} ⊆ ker ϕ. (9.3.7)

Suppose that the kernel of ϕ is solvable as a Lie algebra over k as well, so that (ker ϕ)^{(l)} = {0} for some nonnegative integer l. Under these conditions, we get that A^{(n+l)} = (A^{(n)})^{(l)} ⊆ (ker ϕ)^{(l)} = {0}, (9.3.8)

and hence that A is solvable.

These properties correspond to parts (a) and (b) of the proposition on p11 of [14], and to part of Exercise 1 on p43 of [25]. Alternatively, let A_{0}, A_{1}, …, A_{n} be finitely many Lie subalgebras of A, with A_{0} = A, and A_{j+1} an ideal in A_{j} for j = 0, …, n − 1. Suppose that A_{j}/A_{j+1} is commutative as a Lie algebra for each j = 0, …, n − 1, which is the same as saying that

[A_{j}, A_{j}] ⊆ A_{j+1} (9.3.9)

for every j = 0, …, n − 1. This implies that

A^{(j)} ⊆ A_{j} (9.3.10)

for each j = 0, 1, …, n, by induction. In particular, if A_{n} = {0}, then it follows that A is solvable. Conversely, if A is solvable, then A^{(n)} = {0} for some n ≥ 0, and one can simply take A_{j} = A^{(j)} for j = 0, 1, …, n. This characterization of solvability is mentioned on p35-6 of [25], and in Exercise 2 on p14 of [14].

9.4 The solvable radical

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let A_{1} and A_{2} be ideals in A, so that A_{1} ∩ A_{2} and A_{1} + A_{2} are ideals in A as well. If A_{1} and A_{2} are solvable as Lie algebras over k, then it is well known that A_{1} + A_{2} is solvable as a Lie algebra too. To see this, it suffices to check that (A_{1} + A_{2})/A_{2} is solvable as a Lie algebra over k, because A_{2} is solvable, as in the previous section. Remember that (A_{1} + A_{2})/A_{2} is isomorphic to A_{1}/(A_{1} ∩ A_{2}), as in Section 9.1. Of course, A_{1}/(A_{1} ∩ A_{2}) is solvable as a Lie algebra, because A_{1} is solvable, as in the previous section. Thus (A_{1} + A_{2})/A_{2} is solvable, as desired. This is part (c) of the proposition on p11 of [14], which is mentioned on p44 of [25].

One often considers Lie algebras A over a field, with finite dimension as a vector space over the field. In this case, it is easy to see that there is a maximal solvable ideal in A, by taking a solvable ideal in A of maximal dimension, as a linear subspace of A. The remarks in the previous paragraph imply that a maximal solvable ideal in A is unique, and in fact contains every other solvable ideal in A. This maximal solvable ideal is called the (solvable) radical of A, and may be denoted Rad A.

Let A be a Lie algebra over a commutative ring k with a multiplicative identity element again. It may still happen that A has a maximal solvable ideal, which can still be called the radical of A, and denoted Rad A. In particular, A has a maximal solvable ideal when solvable ideals in A satisfy an ascending chain condition. If a maximal solvable ideal exists, then it is unique, and contains all other solvable ideals in A, as before. If

Rad A = {0}, (9.4.1)

then A may be called semisimple as a Lie algebra. More precisely, let us say that A is semisimple as a Lie algebra if {0} is the only solvable ideal in A, in which case it is automatically maximal. Equivalently, A is not semisimple when A contains a nonzero solvable ideal, without asking for a maximal solvable ideal.

Suppose that A is not semisimple, and let B be a nonzero solvable ideal in A. Note that the derived subalgebra B^{(j)} is an ideal in A for every nonnegative integer j. This follows from a remark in Section 9.2 when j = 1, and can be verified using induction otherwise. Because B ≠ {0} is solvable, there is a nonnegative integer j_{0} such that B^{(j_{0})} ≠ {0} and B^{(j_{0}+1)} = {0}. This means that B^{(j_{0})} is commutative as a Lie algebra, since [B^{(j_{0})}, B^{(j_{0})}] = B^{(j_{0}+1)} = {0}. Of course, if A has a nonzero ideal that is commutative as a Lie algebra, then A is not semisimple, because commutative Lie algebras are solvable. Thus A is not semisimple exactly when A has a nonzero ideal that is commutative as a Lie algebra, as on p22 of [14] and p44 of [25].

Suppose that the radical of A exists, so that the quotient A/ Rad A can be defined as a Lie algebra over k, and let q be the canonical quotient mapping from A onto A/ Rad A. In this situation, A/ Rad A is automatically semisimple, as on p11 of [14]. Indeed, if C is any ideal in A/ Rad A, then q^{−1}(C) is an ideal in A. If C is solvable as a Lie algebra over k, then q^{−1}(C) is solvable as a Lie algebra too, because Rad A is solvable, as in the previous section. This implies that q^{−1}(C) = Rad A, so that C = {0}, as desired.

9.5 Nilpotent Lie algebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. If j is a nonnegative integer, then A^{j} is defined inductively by putting A^{0} = A, A^{1} = [A, A], and

A^{j+1} = [A, A^{j}] (9.5.1)

for every j ≥ 0, where the right side is as defined in Section 9.2. One can check that A^{j} is an ideal in A for every j ≥ 0, using induction and a remark in Section 9.2. This sequence of ideals is called the descending central series or lower central series in A. This uses the notation on p11 in [14], and the notation C^{j}A is used on p32 of [25] for A^{j−1} when j ≥ 1.

The fact that A^{j} is an ideal in A says exactly that

A^{j+1} ⊆ A^{j} (9.5.2)

for every j ≥ 0. If A^{j} = {0} for some j ≥ 0, then A is said to be nilpotent as a Lie algebra.

Observe that A^{(j)} ⊆ A^{j} (9.5.3)

for every j ≥ 0, with equality when j = 0, 1. It follows that nilpotent Lie algebras are solvable. If A is a commutative Lie algebra, then A^{1} = A^{(1)} = {0}, and hence A is nilpotent.

If B is a Lie subalgebra of A, then B^{j} can be defined as an ideal in B in the same way as before, so that B^{0} = B and

B^{j+1} = [B, B^{j}] (9.5.4)

for every j ≥ 0. It is easy to see that

B^{j} ⊆ A^{j} (9.5.5)

for every j ≥ 0, by induction. If A is nilpotent, then it follows that B is nilpotent.

Let ϕ be a Lie algebra homomorphism from A onto another Lie algebra C over k. One can check that ϕ(A^{j}) = C^{j} (9.5.6)

for every j ≥ 0, using (9.2.4) and induction. If A is nilpotent, then it follows that C is nilpotent as well.

Remember that Z(A) is the center of A as a Lie algebra, as in Section 7.6. If B is a submodule of A, as a module over k, then

B ⊆ Z(A) (9.5.7)

if and only if [A, B] = {0}. (9.5.8)

If A^{j} ⊆ Z(A) (9.5.9)

for some j ≥ 0, then it follows that

A^{j+1} = [A, A^{j}] = {0}, (9.5.10)

so that A is nilpotent as a Lie algebra. If A/Z(A) is nilpotent as a Lie algebra, then (9.5.9) holds for some j ≥ 0, because of (9.5.6). This implies that A is nilpotent as a Lie algebra, as before.

Let B be a submodule of A again, as a module over k, and let B_{0} be an ideal in A. It is easy to see that [A, B] ⊆ B_{0} (9.5.11)

if and only if the image of B in A/B_{0} is contained in the center of A/B_{0}, as a Lie algebra.

Let A_{0}, A_{1}, …, A_{n} be finitely many ideals in A, with A_{0} = A, and

A_{j+1} ⊆ A_{j} (9.5.12)

for each j = 0, …, n − 1. Suppose that A_{j}/A_{j+1} is contained in the center of A/A_{j+1} for each j = 0, …, n − 1, which is the same as saying that

[A, A_{j}] ⊆ A_{j+1} (9.5.13)

for every j = 0, …, n−1, as in the preceding paragraph. Under these conditions, we get that A^{j} ⊆ A_{j} (9.5.14)

for each j = 0, …, n, by induction. If A_{n} = {0}, then it follows that A is nilpotent as a Lie algebra. Conversely, if A is nilpotent as a Lie algebra, then A^{n} = {0} for some nonnegative integer n, and one can take A_{j} = A^{j} for j = 0, …, n.

These basic properties of nilpotent Lie algebras correspond to parts (a) and (b) of the proposition on p12 of [14], part of Theorem 2.1 on p32 in [25], and part of Exercise 1 on p43 of [25].

Suppose that A is nilpotent as a Lie algebra, and that A ≠ {0}. Let j be the largest nonnegative integer such that A^{j} ≠ {0}, which means that A^{j+1} = {0}. This implies that A^{j} ⊆ Z(A), and in particular that Z(A) ≠ {0}. This is part (c) of the proposition on p12 of [14].

9.6 Two-dimensional Lie algebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. If A is generated, as a module over k, by a single element, then it is easy to see that A is commutative as a Lie algebra. Suppose now that A is generated by a_{0}, b_{0} ∈ A, as a module over k, so that every element of A can be expressed as

α a_{0} + β b_{0} (9.6.1) for some α, β ∈ k. This implies that A^{(1)} = [A, A] consists of elements of A of the form γ [a_{0}, b_{0}]_{A}, (9.6.2)

where γ ∈ k. It follows that A^{(2)} = [A^{(1)}, A^{(1)}] = {0}, and in particular that A is solvable.

By hypothesis, [a_{0}, b_{0}]_{A} = α_{0} a_{0} + β_{0} b_{0} (9.6.3)

for some α_{0}, β_{0} ∈ k. Of course, if [a_{0}, b_{0}]_{A} = 0, then A is commutative as a Lie algebra. Suppose that [a_{0}, b_{0}]_{A} ≠ 0, and that (9.6.1) is not equal to 0 in A when α, β ∈ k satisfy α ≠ 0 or β ≠ 0. If k has no nonzero nilpotent elements, then one can check that A is not nilpotent as a Lie algebra. This corresponds to the first part of Exercise 5 on p14 of [14], and part of Exercise 2 on p43 of [25]. Suppose for the moment that k is a field, so that A is a two-dimensional vector space over k. One can choose a basis a, b for A such that

[a, b]_{A} = a, (9.6.4)

as on p5 of [14], and the other part of Exercise 2 on p43 of [25].

Let A be a module over k, and let [a, b]_{A} be a mapping from A × A into A that is bilinear over k and satisfies

[a, a]_{A} = 0 (9.6.5)

for every a ∈ A. In order for [·, ·]_{A} to define a Lie bracket on A, one should verify that the Jacobi identity holds for any triple of elements x, y, and z of A. If x = y = z, then each of the three terms in the Jacobi identity is equal to 0, because of (9.6.5). If any two of x, y, and z are the same element of A, then one of the terms in the Jacobi identity is automatically equal to 0, by (9.6.5) again. In this case, the Jacobi identity can be obtained using the antisymmetry of [·, ·]_{A}, which follows from (9.6.5), as usual.

Suppose that A is generated as a module over k by a_{0}, b_{0} ∈ A, so that every element of A can be expressed as in (9.6.1). In order to show that [·, ·]_{A} satisfies the Jacobi identity on A, it suffices to consider triples of elements x, y, and z of A where each of x, y, and z is equal to either a_{0} or b_{0}, because of bilinearity. This means that at least two of the elements x, y, and z are the same. In this situation, the Jacobi identity can be obtained from (9.6.5), as in the preceding paragraph.

9.7 Nilpotency conditions

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Remember that if x ∈ A, then ad x = ad_{x} is the homomorphism from A into itself, as a module over k, defined by

ad_{x}(y) = [x, y]_{A} (9.7.1) for every y ∈ A, as in Section 2.4. Let n be a positive integer, and let x_{1}, …, x_{n} be n elements of A. Thus ad_{x_{j}} is a module homomorphism from A into itself for each j = 1, …, n, so that compositions of the ad_{x_{j}}’s are defined as module homomorphisms from A into itself. If y ∈ A, then

(ad_{x_{1}} ∘ ad_{x_{2}} ∘ ⋯ ∘ ad_{x_{n}})(y) = [x_{1}, [x_{2}, […, [x_{n}, y]_{A} …]_{A}]_{A}]_{A}. (9.7.2)

This is an element of the ideal A^{n} defined in Section 9.5. If A^{n} = {0}, then (9.7.2) is equal to 0 for every y ∈ A, so that

ad_{x_{1}} ∘ ad_{x_{2}} ∘ ⋯ ∘ ad_{x_{n}} = 0 (9.7.3)

as a mapping from A into itself. Conversely, if (9.7.3) holds for every x_{1}, …, x_{n} in A, then (9.7.2) is equal to 0 for all x_{1}, …, x_{n}, y ∈ A, and which implies that A^{n} = {0}. This corresponds to part of Theorem 2.1 on p32 of [25], and is also mentioned on p12 of [14].

Let B be an associative algebra over k, where multiplication of b, b′ ∈ B is expressed as b b′. An element b of B is said to be nilpotent if b^{l} = 0 for some positive integer l. If b_{1} and b_{2} are commuting nilpotent elements of B, then it is easy to see that b_{1} + b_{2} is nilpotent as well. More precisely, if b_{1}^{l_{1}} = 0 and b_{2}^{l_{2}} = 0 for some positive integers l_{1} and l_{2}, then

(b_{1} + b_{2})^{l_{1}+l_{2}−1} = 0. (9.7.4)

Indeed, the left side of (9.7.4) can be expressed as a sum of terms of the form b_{1}^{j_{1}} b_{2}^{j_{2}}, where j_{1} and j_{2} are nonnegative integers with j_{1} + j_{2} = l_{1} + l_{2} − 1, which implies that either j_{1} ≥ l_{1} or j_{2} ≥ l_{2}.

Let B_{Lie} be B as a Lie algebra over k, with respect to the commutator bracket [b, b′]_{B} = b b′ − b′ b corresponding to multiplication on B. Let n be a positive integer, and suppose that

b_{1} b_{2} ⋯ b_{n} b_{n+1} = 0 (9.7.5)

for every b_{1}, b_{2}, …, b_{n}, b_{n+1} ∈ B. This implies that

[b_{1}, [b_{2}, […, [b_{n}, b_{n+1}]_{B} …]_{B}]_{B}]_{B} = 0 (9.7.6)

for every b_{1}, b_{2}, …, b_{n}, b_{n+1} ∈ B, because the left side of (9.7.6) can be expanded into a sum of products of n+1 elements of B. It follows that B_{Lie}^{n} = {0}, where B_{Lie}^{n} is defined as in Section 9.5. In particular, this means that B_{Lie} is nilpotent as a Lie algebra.

Let A be a Lie algebra over k again. An element x of A is said to be ad-nilpotent if ad_{x} is nilpotent as an element of the algebra of module homomorphisms from A into itself, as on p12 of [14]. If A^{n} = {0} for some positive integer n, then (ad_{x})^{n} = 0 as a module homomorphism from A into itself for every x ∈ A, as in (9.7.3). In particular, if A is nilpotent as a Lie algebra, then every element of A is ad-nilpotent.

Let B be an associative algebra over k again, with corresponding Lie algebra B_{Lie}. If b ∈ B, then ad_{b} is defined as a module homomorphism from B into itself, by ad_{b}(c) = [b, c]_{B} = b c − c b (9.7.7)

for every c ∈ B. Equivalently,

ad_{b} = M_{b} − M̃_{b}, (9.7.8)

where M_{b} and M̃_{b} are the operators of left and right multiplication by b on B, respectively, as in Sections 2.2 and 2.7. Remember that M_{b} and M̃_{b} commute as module homomorphisms from B into itself. If b^{l} = 0 for some positive integer l, then (M_{b})^{l} = M_{b^{l}} = 0 and (M̃_{b})^{l} = M̃_{b^{l}} = 0, as module homomorphisms from B into itself. This implies that (ad_{b})^{2 l−1} = 0, as a module homomorphism from B into itself, as in (9.7.4). Thus b is ad-nilpotent as an element of B_{Lie} when b is nilpotent in B. This corresponds to the lemma on p12 of [14], and Step 2 on p34 of [25].

9.8 Maximal Lie subalgebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. If x ∈ A, then

{α x : α ∈ k} (9.8.1)

is a Lie subalgebra of A that contains x as an element. If x ≠ 0 and k is a field, then (9.8.1) is one-dimensional as a linear subspace of A.

Let B be a Lie subalgebra of A that is proper, so that B ≠ A. As usual, B is said to be maximal as a proper Lie subalgebra of A with respect to inclusion if any proper Lie subalgebra of A that contains B is equal to B. If B_{0} is any proper Lie subalgebra of A, then in some situations one can get the existence of a maximal proper Lie subalgebra B of A that contains B_{0}. In particular, if k is a field and A is finite-dimensional as a vector space over k, then one can take B to be a Lie subalgebra of A that contains B_{0} and whose dimension is strictly less than the dimension of A and maximal. One can also get such maximal proper Lie subalgebras of A when Lie subalgebras of A satisfy an ascending chain condition.

Let B_{0} be a submodule of A, as a module over k. The normalizer N_{A}(B_{0}) of B_{0} in A is defined to be the set of x ∈ A such that

[x, y]_{A} ∈ B_{0} (9.8.2)

for every y ∈ B_{0}. It is easy to see that N_{A}(B_{0}) is a submodule of A, as a module over k, because B_{0} is a submodule of A. One can check that N_{A}(B_{0}) is a Lie subalgebra of A, using the Jacobi identity. If B_{0} is a Lie subalgebra of A, then

B_{0} ⊆ N_{A}(B_{0}), (9.8.3) and in fact B_{0} is an ideal in N_{A}(B_{0}), as a Lie algebra over k. In this case, N_{A}(B_{0}) may be described as the largest Lie subalgebra of A that contains A as an ideal, as on p7 of [14]. If B_{0} = N_{A}(B_{0}), then B_{0} is said to be self-normalizing in A. If B_{0} is a maximal Lie subalgebra of A, then it follows that B_{0} is either self-normalizing in A, or B_{0} is an ideal in A.

Let B be an ideal in A. If x ∈ A, then it is easy to see that

{α x + y : α ∈ k, y ∈ B} (9.8.4)

is a Lie subalgebra of A. Of course, (9.8.4) contains B and x. If B is a maximal proper Lie subalgebra of A, and x ∈ A \ B, then (9.8.4) is equal to A.

Suppose that A ≠ {0}, and that {0} is maximal as a proper Lie subalgebra of A. Let x be a nonzero element of A, so that (9.8.1) is equal to A. Observe that {α ∈ k : α x = 0} (9.8.5)

is a proper ideal in k, because x ≠ 0. One can check that (9.8.5) is a maximal ideal in k in this situation. This uses the fact that ideals in k correspond to submodules of (9.8.1), which are Lie subalgebras of A.

9.9 Nilpotent linear mappings

Let k be a field, and let V be a vector space over k of positive finite dimension. Remember that the space L(V) of linear mappings from V into itself is an associative algebra over k with respect to composition of mappings. In particular, a linear mapping T from V into itself is said to be nilpotent if T^{n} = 0 for some positive integer n, which is to say that T is nilpotent as an element of L(V). As in Section 2.10, we may use gl(V) to denote the space of linear mappings from V into itself as a Lie algebra over k, with respect to the corresponding commutator bracket. Let A be a Lie subalgebra of gl(V), and suppose that every element of A is nilpotent as an element of L(V). Under these conditions, it is well known that there is a v ∈ V such that v ≠ 0 and

a(v) = 0 (9.9.1)

for every a ∈ A. This is the theorem stated at the bottom of p12 in [14], which corresponds to Theorem 3.2’ on p33 in [25], as in Step 1 on p34 in [25].

Note that A is a finite-dimensional vector space over k. The theorem is proved using induction on the dimension of A. Of course, the theorem is trivial when A = {0}. If the dimension of A is equal to 1, then the theorem reduces to the fact that a nilpotent linear mapping from V into itself has a nontrivial kernel.

Suppose now that A has positive dimension, and that the theorem holds for Lie algebras over k of smaller dimension. Let B be a proper Lie subalgebra of A, which has dimension less than the dimension of A.

If x ∈ gl(V), then let ad_{x} be the linear mapping from gl(V) into itself defined by ad_{x}(y) = [x, y] (9.9.2) for every y ∈ gl(V), as in Section 2.4. If x is nilpotent as an element of L(V), then ad_{x} is nilpotent as a linear mapping from gl(V) into itself, as in Section 9.7. If x ∈ A, then ad_{x} maps A into itself, because A is a Lie subalgebra of gl(V), by hypothesis. More precisely, ad_{x} is nilpotent as a linear mapping from A into itself, because x is nilpotent as an element of L(V), by hypothesis.

If x ∈ B, then ad_{x} maps B into itself, because B is a Lie subalgebra of A. In particular, B is a linear subspace of A, so that the quotient A/B can be defined as a vector space over k. Let q be the canonical quotient mapping from A onto A/B. If T is a linear mapping from A into itself that maps B into itself, then q ∘T is a linear mapping from A into A/B whose kernel contains B. This means that for a ∈ A, q(T (a)) only depends on q(a), so that there is a unique linear mapping T^{A/B} from A/B into itself such that

T^{A/B} ∘ q = q ∘ T. (9.9.3)

Note that T^{A/B} is nilpotent on A/B when T is nilpotent on A. Of course, A/B ≠ {0}, because B ≠ A, by hypothesis.

If x ∈ B, then we can apply this to T = ad_{x}, to get a linear mapping ad_{x}^{A/B} from A/B into itself. More precisely, ad_{x}^{A/B} is nilpotent on A/B, because ad_{x} is nilpotent on A, as before. Let C be the collection of ad_{x}^{A/B}, with x ∈ B. This is a Lie subalgebra of gl(A/B). The dimension of C, as a vector space over k, is less than or equal to the dimension of B.

Hence the dimension of C is strictly less than the dimension of A. The induction hypothesis permits us to apply the theorem to C, to get that there is a nonzero element q(a_{0}) of A/B, a_{0} ∈ A, such that

ad_{x}^{A/B}(q(a_{0})) = 0 (9.9.4)

for every x ∈ B. Equivalently, this means that q(ad_{x}(a_{0})) = 0 for every x ∈ B, which is the same as saying that

[x, a_{0}] = ad_{x}(a_{0}) ∈ B (9.9.5)

for every x ∈ B. This shows that a_{0} is an element of the normalizer N_{A}(B) of B in A. Note that a_{0} ∉ B, because q(a_{0}) ≠ 0 in A/B.

Let us now take B to be a maximal proper Lie subalgebra of A. In this case, we get that B is an ideal in A, as in the previous section. Put

(9.9.6) W = {v ∈ V : x(v) = 0 for every x ∈ B},

which is a linear subspace of V. Using the induction hypothesis again, we get that W ≠ 0. Let us check that

a(W) ⊆ W (9.9.7)

for every a ∈ A. If x ∈ B and v ∈ W, then

x(a(v)) = a(x(v)) − ([a, x])(v) = 0, (9.9.8) because [a, x] ∈ B, as before. This implies that a(v) ∈ W, as desired.

Let a_{1} be any element of A \ B. The restriction of a_{1} to W is a nilpotent linear mapping from W into itself, and hence there is a v_{1} ∈ W such that v_{1} ≠ 0 and a_{1}(v_{1}) = 0. If a is any element of A, then a can be expressed as the sum of an element of B and a scalar multiple of a_{1}, as in the previous section. It follows that a(v_{1}) = 0, because v_{1} ∈ W. Thus (9.9.1) holds, as desired.

9.10 Engel’s theorem

Let k be a field, and let V be a vector space over k of positive finite dimension n. A finite sequence F = {V_{j}}_{j=0}^{n} of n linear subspaces in V is said to be a flag in V if V_{0} = {0}, V_{n} = V, V_{j} ⊆ V_{j+1} for j = 0, …, n − 1, and the dimension of V_{j} is equal to j for each j = 0, …, n. If l is a nonnegative integer with l ≤ n, then let U_{l}(F) be the collection of linear mappings T from V into itself such that T (V_{j}) ⊆ V_{j−l} (9.10.1)

for each j = l, …, n. This is a subalgebra of the algebra L(V) of all linear mappings from V into itself, as an associative algebra over k with respect to composition of mappings. Note that U_{0}(F) contains the identity mapping I = I_{V} on V, U_{n}(F) = 0, and U_{l_{2}}(F) ⊆ U_{l_{1}}(F) (9.10.2)

when l_{1} ≤ l_{2}.

More precisely, if l_{1}, l_{2} are nonnegative integers with l_{1}+l_{2} ≤ n, T_{1} ∈ U_{l_{1}}(F), and T_{2} ∈ U_{l_{2}}(F), then T_{1} ∘ T_{2} ∈ U_{l_{1}+l_{2}}(F). (9.10.3)

In particular, U_{l}(F) is an ideal in U_{0}(F) for each l. If T_{1}, T_{2}, …, T_{n} ∈ U_{1}(F), then T_{1} ∘ T_{2} ∘ ⋯ ∘ T_{n} = 0, (9.10.4)

because the n-fold composition on the left is an element of U_{n}(F). We may consider U_{l}(F) as a Lie subalgebra of the Lie algebra gl(V) of linear mappings from V into itself with respect to the commutator bracket for each l = 0, 1, …, n. It follows from (9.10.4) that U_{1}(F) is nilpotent as a Lie algebra over k, with respect to the commutator bracket, as in Section 9.7.

Let A be a Lie subalgebra of gl(V), and suppose that every element of A is nilpotent as a linear mapping on V, as in the previous section. Under these conditions, it is well known that there is a flag F in V such that

A ⊆ U_{1}(F). (9.10.5)

This is the corollary stated on p13 of [14], which corresponds to Theorem 3.2 on p33 of [25]. If V ≠ {0}, then one can first get a one-dimensional linear subspace V_{1} of V on which the elements of A vanish, as in the previous section. In order to repeat the process, one can look at the induced linear mappings on the quotient V/V_{1}.

Now let A be any finite-dimensional Lie algebra over k. If every element of A is ad-nilpotent, as in Section 9.7, then it is well known that A is nilpotent as a Lie algebra. This is the theorem stated on the middle of p12 in [14], which corresponds to Theorem 3.1 on p33 of [25]. In the argument on the bottom of p34 of [25], one applies the theorem mentioned in the preceding paragraph to the image of A under the adjoint representation, as a Lie algebra of linear mappings from A into itself. This leads to a flag of linear subspaces of A, which are in fact ideals in A, and which can be used to show that A is nilpotent as a Lie algebra, as in Section 9.5. Alternatively, one can use the previous theorem to get that the image of A under the adjoint representation is nilpotent as a Lie algebra. This implies that A is nilpotent as a Lie algebra, because the kernel of the adjoint representation is the center Z(A) of A, as a Lie algebra. The proof on the middle of p13 in [14] applies the theorem mentioned in the previous section to the image of the adjoint representation of A when A ≠ {0} to get that Z(A) ≠ {0}. One can repeat the process on A/Z(A) to get that A is nilpotent.

Let A be a finite-dimensional nipotent Lie algebra over k, and let B be an ideal in A with B ≠ {0}. If x ∈ A, then ad_{x} is a linear mapping from A into itself that maps B into itself, because B is an ideal in A. Consider the collection A_{B} of linear mappings from B into itself obtained by restricting ad_{x} to B for each x ∈ A. This is a Lie subalgebra of gl(B), because of the usual properties of the adjoint representation on A, as in Section 2.4. Remember that ad_{x} is nilpotent as a Lie mapping from A into itself for each x ∈ A, because A is nilpotent as a Lie algebra, as in Section 9.7. This implies that the elements of A_{B} are nilpotent as linear mappings from B into itself. It follows that there is a y ∈ B such that y ≠ 0 and ad_{x}(y) = 0 for every x ∈ A, as in the previous section. Equivalently, this means that B ∩ Z(A) ≠ {0}, as in the lemma on p13 of [14].

9.11 Flags and matrices

Let k be a field, let V be a vector space of positive finite dimension n, and let F = {V_{j}}_{j=0}^{n} be a flag in V. If T is an element of the algebra U_{0}(F) defined in the previous section, then T induces a linear mapping from V_{j}/V_{j−1} into itself for each j = 1, …, n. This linear mapping corresponds to multiplication by an element ϕ_{j}(T) of k, because V_{j}/V_{j−1} is a one-dimensional vector space over k. This defines an algebra homomorphism ϕ_{j} from U_{0}(F) into k for each j = 1, …, n. Let us consider k^{n} as a commutative associative algebra over k, with respect to coordinatewise addition and multiplication. Thus we get an algebra homomorphism ϕ from U_{0}(F) into k^{n}, whose jth coordinate is the algebra homomorphism ϕ_{j} from U_{0}(F) into k just mentioned. The kernel of ϕ is the ideal U_{1}(F) of U_{0}(F) defined in the previous section. In particular, if T_{1}, T_{2} ∈ U_{0}(F), then

T_{1} ∘ T_{2} − T_{2} ∘ T_{1} ∈ U_{1}(F). (9.11.1)

Remember that U_{0}(F) is a Lie subalgebra of gl(V), so that U_{0}(F) may be considered as a Lie algebra over k with respect to the commutator bracket associated to composition of linear mappings on V. Using (9.11.1), we get that

[U_{0}(F), U_{0}(F)] ⊆ U_{1}(F). (9.11.2)

This implies that U_{0}(F) is solvable as a Lie algebra, because U_{1}(F) is nilpotent as a Lie algebra.

Now let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let A be an associative algebra over k, where multiplication of x, y ∈ A is expressed as x y. Remember that the space M_{n}(A) of n × n matrices with entries in A is an associative algebra over k with respect to matrix multiplication, as in Section 2.8. If r is a nonnegative integer with r ≤ n, then let T_{n,r}(A) be the collection of a = (a_{j,l}) ∈ M_{n}(A) such that

a_{j,l} = 0 (9.11.3)

when l ≤ j +r −1. Equivalently, this means that a_{j,l} may be nonzero only when l ≥ j + r. Thus T_{n,0}(A) consists of upper-triangular matrices, T_{n,1}(A) consists of strictly upper-triangular matrices, and T_{n,n}(A) = {0}. Clearly

T_{n,r_{2}}(A) ⊆ T_{n,r_{1}}(A) (9.11.4)

when r_{1} ≤ r_{2}. If a ∈ T_{n,r_{1}}(A) and b ∈ T_{n,r_{2}}(A) for some nonnegative integers r_{1}, r_{2} with r_{1} + r_{2} ≤ n, then

a b ∈ T_{n,r_{1}+r_{2}}(A). (9.11.5)

In particular, T_{n,0}(A) is a subalgebra of M_{n}(A), T_{n,r}(A) is an ideal in T_{n,0}(A) for each 0 ≤ r ≤ n, and the product of n elements of T_{n,1}(A) is equal to 0.

If j is a positive integer with j ≤ n and a ∈ T_{n,0}(A), then put ψ_{j}(a) = a_{j,j}, which defines an algebra homomorphism from T_{n,0}(A) onto A. Let ψ be the mapping from T_{n,0}(A) into A^{n} such that the jth coordinate of ψ(a) is equal to ψ_{j}(a) for every j = 1, …, n and a ∈ T_{n,0}(A). This defines an algebra homomorphism from T_{n,0}(A) onto A^{n}, where A^{n} is considered as an associative algebra over k with respect to coordinatewise addition and multiplication. The kernel of ψ is equal to T_{n,1}(A). If A is a commutative algebra over k, then

a b − b a ∈ T_{n,1}(A) (9.11.6)

for every a, b ∈ T_{n,0}(A).

Remember that gl_{n}(A) is the same as M_{n}(A), but considered as a Lie algebra over k with respect to the commutator bracket. Similarly, we may use t_{n,r}(A) for T_{n,r}(A), considered as a Lie subalgebra of gl_{n}(A). As in Section 9.7, t_{n,1}(A) is nilpotent as a Lie algebra, because the product of n elements of T_{n,1}(A) is equal to 0. If A is commutative, then

[t_{n,0}(A), t_{n,0}(A)] ⊆ t_{n,1}(A), (9.11.7) by (9.11.6). This implies that t_{n,0}(A) is solvable as a Lie algebra, because t_{n,1}(A) is nilpotent.

Let k be a field again, and let V be a vector space over k of dimension n ∈ Z_{+}. If v_{1}, …, v_{n} is a basis for V, then we can get a flag F = {V_{j}}_{j=0}^{n} in V by taking V_{j} to be the linear span of v_{1}, …, v_{j} for each j = 1, …, n. Of course, every flag in V corresponds to a basis for V in this way. Using this basis for V, elements of M_{n}(k) correspond to linear mappings from V into itself, as in Section 2.10. Similarly, T_{n,r}(k) corresponds to U_{r}(F) for each r = 0, 1, …, n.

9.12 A useful lemma

Let k be a field of characteristic 0, let (A, [·, ·]) be a Lie algebra over k, and let B be an ideal in A. Also let V be a finite-dimensional vector space over k, and suppose that V is a module over A, as a Lie algebra over k. Let v be a nonzero element of V, and suppose that χ is a mapping from B into k such that

b · v = χ(b) v (9.12.1)

for every b ∈ B. If a ∈ A and b ∈ B, then [a, b] ∈ B, and in fact

χ([a, b]) = 0. (9.12.2)

This is the Main Lemma stated on p36 of [25], which corresponds to part of the proof of Step (3) on p16 of [14].

Equivalently, let ρ be the representation of A on V, which makes V into a module over A as a Lie algebra over k. The hypothesis (9.12.1) says that for each b ∈ B, v is an eigenvector of ρ_{b}, with eigenvalue χ(b). The conclusion (9.12.2) says that ([a, b]) · v = ρ_{[a,b]}(v) = 0 (9.12.3)

for every a ∈ A and b ∈ B. One could also reduce to the case where A is a Lie subalgebra of gl(V), by considering the Lie algebra of linear mappings from V into itself of the form ρ_{a} for some a ∈ A. In particular, the discussion in [14] is given in this setting.

Let a ∈ A be given, and put V_{0} = {0}. If j is a positive integer, then let V_{j} be the linear span of v, ρ_{a}(v), …, (ρ_{a})^{j−1}(v) (9.12.4)

in V. Thus V_{j} is a linear subspace of V for every j ≥ 0, with V_{j} ⊆ V_{j+1}. Let n be the smallest positive integer such that

V_{n} = V_{n+1}. (9.12.5)

This uses the finite-dimensionality of V, to get that this condition holds for some positive integer. Note that V_{n} has dimension equal to n as a vector space over k, and that ρ_{a}(V_{n}) ⊆ V_{n+1} = V_{n}. (9.12.6)

In particular, n is less than or equal to the dimension of V.

If b ∈ B, then we would like to show that

ρ_{b}((ρ_{a})^{j}(v)) = χ(b) (ρ_{a})^{j}(v) modulo V_{j} (9.12.7)

for each j ≥ 0, using induction on j. This is the same as (9.12.1) when j = 0. If j ≥ 1, then

(9.12.8) ρ_{b}((ρ_{a})^{j}(v)) = ρ_{b}(ρ_{a}((ρ_{a})^{j−1}(v)))
= ρ_{a}(ρ_{b}((ρ_{a})^{j−1}(v))) − ([ρ_{a}, ρ_{b}])((ρ_{a})^{j−1}(v))
= ρ_{a}(ρ_{b}((ρ_{a})^{j−1}(v))) − ρ_{[a,b]}((ρ_{a})^{j−1}(v)).

Of course,

ρ_{b}((ρ_{a})^{j−1}(v)) = χ(b) (ρ_{a})^{j−1}(v) modulo V_{j−1}, (9.12.9)

by the induction hypothesis. This implies that

ρ_{a}(ρ_{b}((ρ_{a})^{j−1}(v))) = χ(b) (ρ_{a})^{j}(v) modulo V_{j}, (9.12.10)

because ρ_{a}(V_{j−1}) ⊆ V_{j} by construction. Similarly,

ρ_{[a,b]}((ρ_{a})^{j−1}(v)) = χ([a, b]) (ρ_{a})^{j−1}(v) modulo V_{j−1}, (9.12.11)

by the induction hypothesis, because [a, b] ∈ B. It follows that the left side of (9.12.11) is an element of V_{j}. Combining this with (9.12.8) and (9.12.10), we get that (9.12.7) holds, as desired.

In particular, (9.12.7) implies that ρ_{b} maps V_{n} into itself when b ∈ B. Observe that

tr_{V_{n}}ρ_{b} = n · χ(b) (9.12.12)

for every b ∈ B, by (9.12.7), where more precisely the left side is the trace of the restriction of ρ_{b} to V_{n}. We also have that

tr_{V_{n}}ρ_{[a,b]} = tr_{V_{n}}([ρ_{a}, ρ_{b}]) = 0 (9.12.13)

for every b ∈ B, using the fact that ρ_{a} and ρ_{b} both map V_{n} into itself in the second step. Thus

n · χ([a, b]) = 0 (9.12.14)

for every b ∈ B, by (9.12.12) applied to [a, b]. This implies (9.12.2), because k is supposed to have characteristic 0. Note that this also works when k has positive characteristic and the dimension of V is strictly less than the characteristic of k, because n is less than or equal to the dimension of V. This is related to Exercise 2 on p20 of [14].

9.13 Lie’s theorem

Let k be an algebraically closed field of characteristic 0, let (A, [·, ·]) be a solvable Lie algebra over k, and let ρ be a representation of A as a Lie algebra on a finitedimensional vector space V over k. If V ≠ {0}, then there exists a v ∈ V such that v ≠ 0 and v is an eigenvector for ρ_{a} for every a ∈ A. This corresponds to Theorem 5.1’ on p36 of [25], and the theorem on p15 of [14]. As before, one can reduce to the case where A is a Lie subalgebra of gl(V), by considering the Lie algebra of linear mappings from V into itself of the form ρ_{a} for some a ∈ A. The theorem on p15 of [14] is stated in this way, so that A is finite-dimensional as a vector space over k in particular. The finite-dimensionality of A is implicit in Theorem 5.1’ in [25], as mentioned at the beginning of Chapter 5 in [25]. Let us suppose now that A is finite-dimensional as a vector space over k too.

The proof uses induction on the dimension of A, as a vector space over k. Of course, if A = {0}, then the statement is trivial. Suppose now that A ≠ {0}, and note that [A, A] ≠ A, because A is solvable. Let B be a linear subspace of V of codimension 1 that contains [A, A], which implies that B is an ideal in A. The induction hypothesis implies that there is a v ∈ V with v ≠ 0 and a mapping χ from B into k such that

ρ_{b}(v) = χ(b) v (9.13.1)

for every b ∈ B. Put

(9.13.2) W = {w ∈ V : ρ_{b}(w) = χ(b) w for every b ∈ B},

which is a linear subspace of V with v ∈ W, so that W ≠ {0}. We would like to verify that ρ_{a}(W) ⊆ W (9.13.3)

for every a ∈ A. If a ∈ A, b ∈ B, and w ∈ W, then

ρ_{b}(ρ_{a}(w)) = ρ_{a}(ρ_{b}(w)) − ρ_{[a,b]}(w) = χ(b) ρ_{a}(w) − χ([a, b]) w, (9.13.4)

using the fact that [a, b] ∈ B in the second step. Combining this with (9.12.2), we obtain that ρ_{b}(ρ_{a}(w)) = χ(b) ρ_{a}(w), (9.13.5)

as desired.

Let a_{0} be any element of A not in B. Because k is algebraically closed, there is a w_{0} ∈ W such that w_{0} ≠ 0 and w_{0} is an eigenvector for ρ_{a_{0}}. If a is any element of A, then a can be expressed as the sum of a multiple of a_{0} and an element b of B, because B has codimension 1 in A. It follows that w_{0} is an eigenvector for ρ_{a}, as desired, because w_{0} is an eigenvector for ρ_{b}, by definition of W. Note that this also works when k has positive characteristic strictly larger than the dimension of V, as in Exercise 2 on p20 of [14].

Under these conditions, Lie’s theorem states that there is a flag F = {V_{j}}_{j=0}^{n} in V such that ρ_{a}(V_{j}) ⊆ V_{j} for every a ∈ A and j = 0, 1, …, n. This is Theorem 5.1 on p36 of [25], which corresponds to Corollary A on p16 of [14]. More precisely, one can get V_{1} as in the previous paragraphs. One can repeat the process, by considering the induced linear mappings on V/V_{1}.

In particular, if A is a solvable Lie algebra over k that is finite-dimensional as a vector space over k, then there is a flag in A consisting of ideals in A. This is Corollary 5.2 on p37 of [25], and Corollary B on p16 of [14]. This follows from the statement in the preceding paragraph, applied to the adjoint representation of A.

9.14 Structure constants

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Remember that the space k^{n} of n-tuples of elements of k is a (free) module over k with respect to coordinatewise addition and scalar multiplication. Let u_{1}, …, u_{n} be the “standard basis” elements of k^{n}, so that the jth coordinate of u_{l} is equal to 1 when j = l and to 0 otherwise. If x = (x_{1}, …, x_{n}) ∈ k^{n}, then

x = ∑_{l=1}^{n} x_{l} u_{l}. (9.14.1)

Of course, if k is a field and V is an n-dimensional vector space over k, then V can be identified with k^{n} by choosing a basis for V.

Let [·, ·]_{k^{n}} be a mapping from k^{n} × k^{n} into k^{n} that is bilinear over k. We can express [u_{j}, u_{l}]_{k^{n}} as

[u_{j}, u_{l}]_{k^{n}} = ∑_{r=1}^{n} c_{j,l}^{r} u_{r} (9.14.2)

for each j, l = 1, …, n, where c_{j,l}^{r} are elements of k for every j, l, r = 1, …, n. This implies that

([x, y]_{k^{n}})_{r} = ∑_{j=1}^{n} ∑_{l=1}^{n} c_{j,l}^{r} x_{j} y_{l} (9.14.3)

for every x, y ∈ k^{n} and r = 1, …, n, where the left side is the rth coordinate of [x, y]_{k^{n}}. More precisely, this uses (9.14.1) and the bilinearity of [·, ·]_{k^{n}} over k. Conversely, if c_{j,l}^{r} ∈ k for every j, l, r = 1, …, n, then (9.14.3) defines a mapping from k^{n} × k^{n} into k^{n} that is bilinear over k.

Clearly [x, y]_{k^{n}} = −[y, x]_{k^{n}} (9.14.4)

for every x, y ∈ k^{n} if and only if

c_{j,l}^{r} = −c_{l,j}^{r} (9.14.5)

for every j, l, r = 1, …, n. One can check that

[x, x]_{k^{n}} = 0 (9.14.6) for every x ∈ k^{n} if and only if (9.14.5) holds and

c_{j,j}^{r} = 0 (9.14.7)

for every j, r = 1, …, n. If 1 + 1 has a multiplicative inverse in k, then (9.14.4) implies (9.14.6), and (9.14.5) implies (9.14.7), as usual. The Jacobi identity for [·, ·]_{k^{n}} holds if and only if

∑_{h=1}^{n}(c_{j,l}^{h} c_{h,m}^{r} + c_{l,m}^{h} c_{h,j}^{r} + c_{m,j}^{h} c_{h,l}^{r}) = 0 (9.14.8)

for every j, l, m, r = 1, …, n, as on p5 of [14]. Let us suppose from now on in this section that the c_{j,l}^{r}’s satisfy these conditions, so that [·, ·]_{k^{n}} defines a Lie bracket on k^{n}.

Let A be a commutative associative algebra over k. Note that the space A^{n} of n-tuples of elements of A is a module over k with respect to coordinatewise addition and scalar multiplication. If a, b ∈ A^{n}, then define [a, b]_{A^{n}} as an element of A^{n} by

([a, b]_{A^{n}})_{r} = ∑_{j=1}^{n} ∑_{l=1}^{n} c_{j,l}^{r} a_{j} b_{l} (9.14.9)

for each r = 1, …, n, where the left side is the rth coordinate of [a, b]_{A^{n}}. The conditions on the c_{j,l}^{r}’s in the preceding paragraph imply that A^{n} is a Lie algebra over k with respect to (9.14.9). If k^{n} is solvable or nilpotent as a Lie algebra with respect to [·, ·]_{k^{n}}, then one can check that A^{n} has the same property with respect to [·, ·]_{A^{n}}.

Suppose that A has a multiplicative identity element e. In this case, A^{n} may be considered as a module over A with respect to coordinatewise addition and scalar multiplication, and as a Lie algebra over A with respect to (9.14.9). Of course, t ↦ t e (9.14.10)

defines a ring homomorphism from k into A, which leads to a Lie algebra homomorphism from k^{n} into A^{n}, as Lie algebras over k. Suppose that (9.14.10) is injective, which implies that the corresponding Lie algebra homomorphism from k^{n} into A^{n} is injective. If A^{n} is solvable or nilpotent as a Lie algebra, then it follows that k^{n} has the same property.

9.15 Another corollary

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]) be a Lie algebra over k. If x ∈ A, then we may use ad_{A} x = ad_{A,x} to denote the usual mapping

(ad_{A}x)(y) = ad_{A,x}(y) = [x, y] (9.15.1) from A into itself. Similarly, if B is a Lie subalgebra of A and x ∈ B, then ad_{B} x = ad_{B,x} is a module homomorphism from B into itself. In this situation, ad_{B,x} is the same as the restriction of (9.15.1) to y ∈ B.

Suppose that k is an algebraically closed field of characteristic 0, and that (A, [·, ·]) be a solvable Lie algebra over k that is finite-dimensional as a vector space over k. Under these conditions, [A, A] is nilpotent as a Lie algebra over k. This is Corollary C on p16 of [14], and part of Corollary 5.3 on p37 of [25]. Remember that there is a flag of ideals in A, as a consequence of Lie’s theorem in Section 9.13. If x ∈ A, then ad_{A,x} maps these ideals into themselves. If x ∈ [A, A], then ad_{A,x} maps the nonzero ideals in the flag into the next smaller one, as in Section 9.11. This implies that ad_{A,x} is nilpotent as a mapping from A into itself. It follows that ad_{[A,A],x} is nilpotent as a mapping from [A, A] into itself, because this mapping is the same as the restriction of ad_{A,x} to [A, A], as in the preceding paragraph. This implies that [A, A] is nilpotent as a Lie algebra, as in Section 9.10.

Alternatively, one can use the same type of argument to get that the image of [A, A] under the adjoint representation of A is nilpotent as a Lie algebra over k. One can use this to get that [A, A] is nipotent as a Lie algebra, because the kernel of the adjoint representation of A is the center of A.

Suppose now that k is a field of characteristic 0, and that A is a finitedimensional solvable Lie algebra over k. Corollary 5.3 on p37 of [25] states that [A, A] is still nilpotent as a Lie algebra over k, without asking k to be algebraically closed. To see this, let k_{1} be an algebraically closed field that contains k. The statement is trivial when A = {0}, and so we may suppose that the dimension n of A as a vector space over k is positive. Thus A is isomorphic to k^{n} as a vector space over k, and we may as well suppose that A = k^{n} with some Lie bracket. This leads to a Lie bracket on k_{1}^{n}, as in the previous section. If k^{n} is solvable as a Lie algebra, then k_{1}^{n} is solvable as a Lie algebra too, as before. This implies that [k_{1}^{n}, k_{1}^{n}] is nilpotent as a Lie algebra, by the earlier arguments for algebraically closed fields. Note that [k^{n}, k^{n}] may be considered as a Lie subalgebra of [k_{1}^{n}, k_{1}^{n}], as a Lie algebra over k. It follows that [k^{n}, k^{n}] is solvable as a Lie algebra over k, as desired.

Matrices and traces

10.1 Some remarks about gl_{n}(k)

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. The space k^{n} of n-tuples of elements of k is a (free) module over k with respect to coordinatewise addition and scalar multiplication, as usual. If a = (a_{j,l}) is an n × n matrix with entries in k and x ∈ k^{n}, then T_{a}(x) is defined as the element of k^{n} whose jth coordinate is given by

(T_{a}(x))_{j} = ∑_{l=1}^{n} a_{j,l} x_{l} (10.1.1)

for each j = 1, …, n. This defines a module homomorphism from k^{n} into itself, and a ↦ T_{a} is an algebra isomorphism from the algebra M_{n}(k) of n × n matrices with entries in k with respect to matrix multiplication onto the algebra Hom_{k}(k^{n}, k^{n}) of module homomorphisms from k^{n} into itself with respect to composition of mappings.

Let u_{1}, …, u_{n} be the n “standard basis” elements of k^{n}, so that the lth coordinate of u_{r} is equal to 1 when l = r, and to 0 otherwise. Thus

x = ∑_{r=1}^{n} x_{r} u_{r} (10.1.2)

for every x = (x_{1}, …, x_{n}) ∈ k^{n}. If a ∈ M_{n}(k), then

(T_{a}(u_{r}))_{j} = a_{j,r} (10.1.3)

for every j, r = 1, …, n.

Similarly, if h, m ∈ {1, …, n}, then let e_{h,m} be the element of M_{n}(k) whose (h, m) entry is equal to 1, and all of whose other entries are equal to 0. If a = (a_{j,l}) is any element of M_{n}(k), then a can be expressed as

a = ∑_{h=1}^{n} ∑_{m=1}^{n} a_{h,m} e_{h,m}. (10.1.4)

10.1. SOME REMARKS ABOUT GL_{N}(K) 211

It is sometimes convenient to let T_{h,m} be the module homomorphism from k^{n} into itself associated to e_{h,m} as in (10.1.1) for each h, m = 1, …, n, so that T_{h,m} = T_{e_{h,m}}. (10.1.5) If a = (a_{j,l}) ∈ M_{n}(k), then T_{a} can be expressed as

T_{a} = ∑_{h=1}^{n} ∑_{m=1}^{n} a_{h,m} T_{h,m}. (10.1.6)

If q, r ∈ {1, …, n}, then let δ_{q,r} ∈ k be equal to 1 when q = r, and to 0 otherwise, as usual. Observe that T_{h,m}(u_{r}) = δ_{m,r} u_{h} (10.1.7) for every h, m, r = 1, …, n. We also have that e_{h,m} e_{q,r} = δ_{m,q} e_{h,r} (10.1.8) for every h, m, q, r = 1, …, n. Equivalently, T_{h,m} ∘ T_{q,r} = δ_{m,q} T_{h,r} (10.1.9) for every h, m, q, r = 1, …, n. It follows from (10.1.8) that [e_{h,m}, e_{q,r}] = e_{h,m} e_{q,r} − e_{q,r} e_{h,m} = δ_{m,q} e_{h,r} − δ_{r,h} e_{q,m} (10.1.10) for every h, m, q, r = 1, …, n. In particular, if h ≠ r and m ≠ q, then [e_{h,m}, e_{q,r}] = 0, (10.1.11) because each of the two terms on the right side of (10.1.10) is equal to 0. Otherwise, [e_{h,m}, e_{m,r}] = e_{h,r} (10.1.12) when h ≠ r, and [e_{h,m}, e_{q,h}] = −e_{q,m} (10.1.13) when m ≠ q. Of course, these two cases are equivalent, because of the antisymmetry of the commutator bracket. Similarly, [e_{h,m}, e_{m,h}] = e_{h,h} − e_{m,m} (10.1.14) for every h, m = 1, …, n.

Remember that gl_{n}(k) is the same as M_{n}(k) as a module over k, but considered as a Lie algebra over k with respect to the corresponding commutator bracket, as in Section 2.9. Similarly, sl_{n}(k) is the ideal in gl_{n}(k) consisting of matrices with trace 0, as before. In fact, [gl_{n}(k), gl_{n}(k)] = sl_{n}(k), (10.1.15) where the left side is as defined in Section 9.2. More precisely, the inclusion of the left side of (10.1.15) in the right side follows from basic properties of the trace, as in Section 2.9. The opposite inclusion can be obtained from (10.1.12) and (10.1.14). This corresponds to Exercise 2 on p9 of [14]. One can also verify that [gl_{n}(k), sl_{n}(k)] = sl_{n}(k), (10.1.16)

using the same argument.

10.2 Some basic properties of sl_{2}(k)

Let k be a commutative ring with a multiplicative identity element, and remember that sl_{2}(k) is the space of 2 × 2 matrices with entries in k and trace 0. This is a Lie algebra over k with respect to the usual commutator bracket. Consider the elements of sl_{2}(k) given by

x = \binom{0 1}{0 0}, y = \binom{0 0}{1 0}, and h = \binom{1 0}{0 − 1}. (10.2.1)

It is easy to see that every element of sl_{2}(k) can be expressed in a unique way as a linear combination of x, y, and h with coefficients in k. Thus sl_{2}(k) is isomorphic to the free module k^{3} of rank 3 over k, as a module over k.

Equivalently, using the notation in the previous section, with n = 2, we have that x = e_{1,2}, y = e_{2,1}, and h = e_{1,1} − e_{2,2}. (10.2.2)

One can check that

[x, y] = h, [h, x] = 2 · x, [h, y] = −2 · y, (10.2.3)

as on p6 of [14]. Here 2 · a = a + a for each a ∈ sl_{2}(k), as usual. If 1 + 1 = 0 in k, then h is the same as the identity matrix, so that

[h, x] = [h, y] = 0, (10.2.4)

as in (10.2.3). In this case, it follows that sl_{2}(k) is nilpotent as a Lie algebra over k, as in Exercise 3 on p14 of [14].

Suppose for the moment that k is a field with characteristic 2. Elements of sl_{2}(k) correspond to linear mappings from k^{2} into itself, as a two-dimensional vector space over k, as before. It is well known and not difficult to check that the linear mappings on k^{2} corresponding to elements of sl_{2}(k) do not have a (nonzero) simultaneous eigenvector. This shows that the results discussed in Section 9.13 can fail in positive characteristic, as mentioned on p37 of [25].

If 1 + 1 has a multiplicative inverse in k, then we get that

[sl_{2}(k), sl_{2}(k)] = sl_{2}(k), (10.2.5)

where the left side is as defined in Section 9.2. This corresponds to part of Exercise 9 on p5 of [14]. Similarly, if 2^{j} · 1 = 0 in k for some j ∈ Z_{+}, then one can check that sl_{2}(k) is nilpotent as a Lie algebra over k. However, if for each j ∈ Z_{+}, 2^{j} · 1 ≠ 0 in k, then sl_{2}(k) is not solvable as a Lie algebra over k.

If n is any positive integer with n ≥ 3, then

[sl_{n}(k), sl_{n}(k)] = sl_{n}(k). (10.2.6)

This corresponds to part of Exercise 9 on p5 of [14] again. Of course, the left side of (10.2.6) is contained in the right side, as in (10.1.15). To get the opposite inclusion, one can verify that

e_{h,r} ∈ [sl_{n}(k), sl_{n}(k)] (10.2.7)

for every h, r ∈ {1, …, n} with h ≠ r, using (10.1.12) with m ≠ h, r. We also have that e_{h,h} − e_{m,m} ∈ [sl_{n}(k), sl_{n}(k)] (10.2.8)

for every h, m = 1, …, n with h ≠ m, by (10.1.14).

10.3 Scalar and diagonal matrices

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let A be an associative algebra over k, where multiplication of x, y ∈ A is expressed as x y. As usual, an n × n matrix a = (a_{j,l}) with entries in A is said to be a diagonal matrix if a_{j,l} = 0 when j ≠ l. Let D_{n}(A) be the space of these diagonal matrices, which is a subalgebra of the algebra M_{n}(A) of n × n matrices with entries in A, as an associative algebra over k with respect to matrix multiplication. If multiplication on A is commutative, then matrix multiplication is commutative on D_{n}(A).

An element a of D_{n}(A) is said to be a scalar matrix if the diagonal entries a_{j,j} of a are all equal to each other. Let S_{n}(A) be the space of these scalar matrices, which is a subalgebra of D_{n}(A). If multiplication on A is commutative, then the elements of S_{n}(A) commute with all other elements of M_{n}(A), with respect to matrix multiplication.

As before, gl_{n}(A) is the same as M_{n}(A), but considered as a Lie algebra over k with respect to the corresponding commutator bracket. Similarly, let d_{n}(A) and s_{n}(A) be the same as D_{n}(A) and S_{n}(A), respectively, considered as Lie subalgebras of gl_{n}(A).

Let us now simply take A = k. Remember that the identity matrix I in M_{n}(k) is the diagonal matrix with diagonal entries equal to the multiplicative identity element 1 in k, which is the multiplicative identity element in M_{n}(k). The scalar matrices in M_{n}(k) are the same as scalar multiples of I by elements of k.

Using the notation in Section 10.1, we have that

e_{h,m} ∈ sl_{n}(k) (10.3.1)

for every h, m = 1, …, n with h ≠ m, and

e_{h,h} − e_{m,m} ∈ sl_{n}(k) (10.3.2) for every h, m = 1, …, n. As on p2 of [14], we may consider the e_{h,m}’s with h ≠ m, together with the matrices e_{h,h} − e_{h+1,h+1} for h = 1, …, n − 1 when n ≥ 2, as the “standard basis elements” of sl_{n}(k). One can check that every element of sl_{n}(k) can be expressed in a unique way as a linear combination of these standard basis elements with coefficients in k. Note that there are (n^{2} − n) + (n − 1) = n^{2} − 1 of these standard basis elements in sl_{n}(k). Thus sl_{n}(k) is isomorphic to the free module k^{n^{2}−1} of rank n^{2} −1 over k, as a module over k.

Of course, every element of gl_{n}(k) can be expressed in a unique way as a linear combination of the e_{h,m}’s, h, m = 1, …, n, with coefficients in k, so that gl_{n}(k) is isomorphic to the free module k^{n^{2}} of rank n^{2} over k, as a module over k. Alternatively, one can verify that every element of gl_{n}(k) can be expressed in a unique way as a linear combination of the standard basis elements for sl_{n}(k) mentioned in the preceding paragraph together with e_{1,1}, with coefficients in k. In particular, every element of gl_{n}(k) can be expressed in a unique way as the sum of an element of sl_{n}(k) and a multiple of e_{1,1} by an element of k, so that gl_{n}(k) is isomorphic to the direct sum of sl_{n}(k) and k, as modules over k. Indeed, if a ∈ gl_{n}(k), then

a − (tr a) e_{1,1} ∈ sl_{n}(k), (10.3.3)

because tr e_{1,1} = 1. Thus a can be expressed as the sum of (10.3.3) and (tr a) e_{1,1}, and one can use the trace again to see that this is unique.

Note that tr I = n · 1, as an element of k. If n · 1 = 0 in k, then n × n scalar matrices with entries in k have trace equal to 0. Suppose for the moment that n · 1 has a multiplicative inverse in k. If a ∈ gl_{n}(k), then

a − \frac{tr a}{n · 1} I ∈ sl_{n}(k), (10.3.4)

and tr a/(n · 1) is the unique element of k with this property. This implies that every element of gl_{n}(k) can be expressed in a unique way as the sum of elements of sl_{n}(k) and s_{n}(k), as in Exercise 7 on p5 in [14].

10.4 Centrality in gl_{n}(k), sl_{n}(k)

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Remember that a = (a_{j,l}) ∈ gl_{n}(k) can be expressed as

a = ∑_{h=1}^{n} ∑_{m=1}^{n} a_{h,m} e_{h,m}, (10.4.1)

as in (10.1.4). If q, r ∈ {1, …, n}, then

[a, e_{q,r}] = ∑_{h=1}^{n} ∑_{m=1}^{n} a_{h,m} [e_{h,m}, e_{q,r}]
(10.4.2)  = ∑_{h≠r} a_{h,q} e_{h,r} − ∑_{m≠q} a_{r,m} e_{q,m} + a_{r,q} (e_{r,r} − e_{q,q}),

by (10.1.11), (10.1.12) with m = q, (10.1.13) with h = r, and (10.1.14) with h = r and m = q. More precisely, the two sums on the right side of (10.4.2) are taken over h, m = 1, …, n with h ≠ r and m ≠ q, respectively. If q ≠ r, then we get that

[a, e_{q,r}] = ∑_{h≠q,r} a_{h,q} e_{h,r} − ∑_{m≠q,r} a_{r,m} e_{q,m} + (a_{q,q} − a_{r,r}) e_{q,r}

+ a_{r,q} (e_{r,r} − e_{q,q}), (10.4.3)

where the sums are taken over h, m = 1, …, n with h, m ≠ q, r, respectively.

Suppose that [a, e_{q,r}] = 0 for every q, r ∈ {1, …, n} with q ≠ r. In this case, one can use (10.4.3) to get that a is a diagonal matrix whose diagonal entries are equal to each other, so that a is a scalar matrix. In particular, the center of gl_{n}(k) as a Lie algebra over k is the Lie subalgebra s_{n}(k) of scalar matrices. This corresponds to the first part of Exercise 3 on p10 of [14].

Similarly, the center Z(sl_{n}(k)) of sl_{n}(k) as a Lie algebra over k is the intersection of s_{n}(k) with sl_{n}(k). This consists of matrices of the form t I, where t ∈ k satisfies n · t = 0. If n · 1 has a multiplicative inverse in k, then Z(sl_{n}(k)) = {0}. If n · 1 = 0 in k, then Z(sl_{n}(k)) = s_{n}(k). This corresponds to the second part of Exercise 3 on p10 of [14].

If r ∈ {1, …, n}, then

[a, e_{r,r}] = ∑_{h≠r} a_{h,r} e_{h,r} − ∑_{m≠r} a_{r,m} e_{r,m}, (10.4.4)

by (10.4.2) with q = r. Let d_{n}(k) be the Lie subalgebra of gl_{n}(k) consisting of diagonal matrices, as in the previous section. Remember that a ∈ gl_{n}(k) is in the normalizer of d_{n}(k) in gl_{n}(k) if [a, b] ∈ d_{n}(k) for every b ∈ d_{n}(k), as in Section 9.8. In this case, it is easy to see that a ∈ d_{n}(k), using (10.4.4). This corresponds to part of Exercise 7 on p10 of [14].

If a ∈ d_{n}(k), then [a, e_{q,r}] = (a_{q,q} − a_{r,r}) e_{q,r} (10.4.5)

for every q, r ∈ {1, …, n} with q ≠ r, by (10.4.3). This also works when q = r, in which the right side is equal to 0, by (10.4.4). This corresponds to Exercise 6 on p5 of [14].

10.5 Solvability and traces

Let k be a field of characteristic 0, and let V be a vector space over k of positive finite dimension n. Remember that the space gl(V) of linear mappings from V into itself is a Lie algebra over k, with respect to the usual commutator bracket. Let A be a Lie subalgebra of gl(V), and suppose that A is solvable as a Lie algebra over k. If T ∈ A and R ∈ [A, A], then

tr(T ∘ R) = 0, (10.5.1) as in Theorem 7.1 on p42 of [25]. Here [A, A] is the derived algebra of A, as in Section 9.2, as usual.

Suppose for the moment that k is algebraically closed. Lie’s theorem implies that there is a flag F = {V_{j}}_{j=0}^{n} in V such that T (V_{j}) ⊆ V_{j} for every T ∈ A and j = 0, 1, …, n, as in Section 9.13. If R ∈ [A, A], then R(V_{j}) ⊆ V_{j−1} for j = 1, …, n, as in Section 9.11. This implies (10.5.1), using a basis for V that is compatible with F. This corresponds to Exercise 7 on p21 of [14].

Now let k be any field of characteristic 0. Of course, V is isomorphic to k^{n} as a vector space over k, and we may as well take V = k^{n}. We can reformulate (10.5.1) in terms of matrices, as follows. If A_{0} is a Lie subalgebra of gl_{n}(k) that is solvable as a Lie algebra over k, then

tr(T_{0} R_{0}) = 0 (10.5.2)

for every T_{0} ∈ A_{0} and R_{0} ∈ [A_{0}, A_{0}]. More precisely, this uses matrix multiplication and the trace on the space M_{n}(k) of n × n matrices with entries in k.

Let k_{1} be an algebraically closed field that contains k as a subfield, so that gl_{n}(k) is contained in gl_{n}(k_{1}). Let A_{1} be the linear span of A_{0} in gl_{n}(k_{1}), as a vector space over k_{1}. It is easy to see that A_{1} is a Lie subalgebra of gl_{n}(k_{1}), as a Lie algebra over k_{1}, because A_{0} is a Lie subalgebra of gl_{n}(k). Similarly, one can check that A_{1} is solvable as a Lie algebra over k_{1}, because A_{0} is solvable as a Lie algebra over k. It follows that

tr(T_{1} R_{1}) = 0 (10.5.3)

for every T_{1} ∈ A_{1} and R_{1} ∈ [A_{1}, A_{1}], by the earlier argument for algebraically closed fields. This uses matrix multiplication and the trace on M_{n}(k_{1}), which contains M_{n}(k). This implies (10.5.2), because A_{0} ⊆ A_{1}, and hence [A_{0}, A_{0}] is contained in [A_{1}, A_{1}].

Let k be a commutative ring with a multiplicative identity element, and remember that [gl_{2}(k), gl_{2}(k)] = sl_{2}(k), as in Section 10.1. Suppose that 1+1 = 0 in k, so that the identity matrix in gl_{2}(k) is in sl_{2}(k). Remember that sl_{2}(k) is nilpotent as a Lie algebra over k in this case, as in Section 10.2, so that gl_{2}(k) is solvable as a Lie algebra over k. In this situation, (10.5.2) would say that every element of gl_{2}(k) has trace 0, which is of course not the case.

10.6 Diagonalizable linear mappings

Let k be a field, let V be a vector space over k, and let T be a linear mapping from V into itself. If λ ∈ k, then put

E_{λ}(T) = {v ∈ V : T (v) = λ(v)}, (10.6.1)

which is a linear subspace of V. As usual, λ is said to be an eigenvalue of T when E_{λ}(T) ≠ {0}, in which case the elements of E_{λ}(T) are said to be eigenvectors of T corresponding to λ. If R is another linear mapping from V into itself that commutes with T and v ∈ E_{λ}(T), then

T (R(v)) = R(T (v)) = λ R(v). (10.6.2)

This means that R(v) ∈ E_{λ}(T), so that

R(E_{λ}(T)) ⊆ E_{λ}(T). (10.6.3)

Suppose that λ_{1}, …, λ_{l} are finitely many distinct elements of k, and let r ∈ {1, …, n} be given. Consider the linear mapping

∏_{j≠r}(T − λ_{j} I) (10.6.4)

on V, where more precisely I is the identity mapping on V, and the product is the composition of T_{j} − λ_{j} I for j = 1, …, n and j ≠ r. This linear mapping is equal to 0 on E_{λ_{j}}(T) when j ≠ r, and it is equal to

∏_{j≠r}(λ_{r} − λ_{j}) (10.6.5)

times the identity mapping on E_{λ_{r}}(T). As before, (10.6.5) is the product of λ_{r}−λ_{j} for j = 1, …, n and j ≠ r, which is a nonzero element of k. If v_{j} ∈ E_{λ_{j}}(T) for each j = 1, …, l and

∑_{j=1}^{l} v_{j} = 0, (10.6.6)

then it is easy to see that v_{r} = 0 for every r = 1, …, l, by applying (10.6.4) to the sum on the left.

Suppose from now on in this section that V has positive finite dimension, as a vector space over k. Note that T can have only finitely many distinct eigenvalues, and that the sum of the dimensions of the nontrivial eigenspaces of T is less than or equal to the dimension of V, by the remark at the end of the preceding paragraph. If every element of V can be expressed as a sum of eigenvectors of T, then T is said to be diagonalizable on V. This means that V is the direct sum of the nontrivial eigenspaces of T, and hence that there is a basis for V consisting of eigenvectors for T.

Suppose that T is diagonalizable on V, with distinct eigenvalues λ_{1}, …, λ_{l}. If r ∈ {1, …, n}, then (10.6.4) maps V onto E_{λ_{r}}(T). Let W be a linear subspace of V such that T (W) ⊆ W. (10.6.7)

Observe that (10.6.4) maps W into itself for each r = 1, …, n. Of course, every w ∈ W can be expressed as a sum of eigenvectors of T, by hypothesis. In this situation, these eigenvectors of T are also elements of W, because (10.6.4) maps W into itself for each r. This implies that the restriction of T to W is diagonalizable.

Let R be another linear mapping from V into itself that commutes with T again. Thus R maps E_{λ_{j}}(T) into itself for each j = 1, …, n, as in (10.6.3). If R is diagonalizable on V, then the restriction of R to E_{λ_{j}}(T) is diagonalizable for every r = 1, …, n, as in the previous paragraph. This implies that R and T are simultaneously diagonalizable on V, which is to say that there is a basis of V consisting of vectors that are eigenvectors for both R and T. In particular, it follows that R + T and R ∘ T are diagonalized by the same basis for V.

10.7 Nilpotent vectors

Let k be a field, and let A be an associative algebra over k with a multiplicative identity element e, where multiplication of a, b ∈ A is expressed as a b. If a ∈ A is nilpotent, λ ∈ k, and λ ≠ 0, then λ e + a has a multiplicative inverse in A. This follows from a remark in Section 3.1 when λ = 1, and otherwise one can reduce to that case.

Let V be a vector space over k, and let T be a linear mapping from V into itself. Remember that

(I − T) ∑_{j=0}^{n} T^{j} = ( ∑_{j=0}^{n} T^{j}) (I − T) = I − T^{n+1} (10.7.1)

for every nonnegative integer n, as in Section 3.1, where I is the identity mapping on V. Suppose for the moment that for each v ∈ V we have that T^{l}(v) = 0 for some nonnegative integer l, which implies that T^{j}(v) = 0 when j ≥ l. This permits us to define

∑_{j=0}^{∞} T^{j}(v) (10.7.2)

as an element of V for every v ∈ V, so that ∑_{j=0}^{∞} T^{j} is defined as a linear mapping from V into itself. One can check that

(I − T) ∑_{j=0}^{∞} T^{j} = ( ∑_{j=0}^{∞} T^{j}) (I − T) = I, (10.7.3)

using (10.7.1). This implies that I − T is invertible on V, with

(I − T)^{−1} = ∑_{j=0}^{∞} T^{j}. (10.7.4)

If λ ∈ k and λ ≠ 0, then λ I −T = λ (I −(1/λ) T) is invertible on V too, because (1/λ) T satisfies the same condition on V.

Let T be any linear mapping from V into itself again, and put

(10.7.5) E_{0}(T) = {v ∈ V : T^{l}(v) = 0 for some l ∈ Z_{+}}.

This is a linear subspace of V that contains the kernel of T. If v ∈ E_{0}(T) and v ≠ 0, and if j is the smallest positive integer such that T^{j}(v) = 0, then T^{j−1}(v) is a nonzero element of the kernel of T. Thus E_{0}(T) ≠ {0} if and only if the kernel of T is nontrivial. Note that T maps E_{0}(T) into itself. If λ ∈ k and λ ≠ 0, then λ I − T is invertible as a linear mapping on E_{0}(T), by the remarks in the previous paragraph. If R is a linear mapping from V into itself that commutes with T and v ∈ E_{0}(T), then

T^{l}(R(v)) = R(T^{l}(v)) = 0 (10.7.6)

for some l ∈ Z_{+}. This means that R(v) ∈ E_{0}(T), and hence

R(E_{0}(T)) ⊆ E_{0}(T). (10.7.7)

Let λ ∈ k be given, and put

(10.7.8) E_{λ}(T) = E_{0}(T − λ I) = {v ∈ V : (T − λ I)^{l}(v) = 0 for some l ∈ Z_{+}}.

This is a linear subspace of V, which reduces to (10.7.5) when λ = 0. Of course,

E_{λ}(T) ⊆ E_{λ}(T), (10.7.9)

where E_{λ}(T) is as in (10.6.1). As before, E_{λ}(T) ≠ {0} if and only if the kernel of T − λ I is nontrivial, which means that λ is an eigenvalue of T. If R is a linear mapping from V into itself that commutes with T, then

R(E_{λ}(T)) ⊆ E_{λ}(T), (10.7.10)

as in (10.7.7). In particular, T maps E_{λ}(T) into itself. If μ ∈ k and μ ≠ λ, then

μ I − T = (μ − λ) I − (T − λ I) (10.7.11)

is invertible as a linear mapping from E_{λ}(T) into itself, by the analogous statement in the preceding paragraph.

Let λ_{1}, …, λ_{m} be finitely many distinct eigenvalues of T, and let v_{j} ∈ E_{λ_{j}}(T) be given for j = 1, …, m. If

∑_{j=1}^{m} v_{j} = 0, (10.7.12)

then v_{r} = 0 for each r = 1, …, m. To see this, one can apply suitable powers of T − λ_{j} I to the sum on the left side of (10.7.12) for j ≠ r, to get a product of powers of T − λ_{j} I with j ≠ r applied to v_{r}. This can only be 0 when v_{r} = 0, because of the invertibility of T − λ_{j} I on E_{λ_{r}}(T) when j ≠ r.

10.8 Jordan–Chevalley decompositions

Let k be a field, and let V be a vector space over k of positive finite dimension. If T is a linear mapping from V into itself, then one may wish to be able to express T as T = T_{1} + T_{2}, (10.8.1) where T_{1} is a diagonalizable linear mapping from V into itself, T_{2} is a nilpotent linear mapping from V into itself, and T_{1}, T_{2} commute with each other, and hence with T. Suppose that this is possible, so that V corresponds to the direct sum of the eigenspaces of T_{1}, as in Section 10.6, and T_{2} maps each of these eigenspaces into itself, because T_{1} and T_{2} commute. Let λ be an eigenvalue of T_{1}, and let E_{λ}(T_{1}) be the corresponding eigenspace of T_{1}, as before. On E_{λ}(T_{1}), T − λ I = T_{2}, so that T − λ I is nilpotent on E_{λ}(T). This implies that

E_{λ}(T_{1}) ⊆ E_{λ}(T), (10.8.2)

where the right side is as in (10.7.8). In particular, λ is an eigenvalue of T as well.

Let λ_{1}, …, λ_{m} be a list of all of the distinct eigenvalues of T_{1}, so that every element of V can be expressed as a sum of elements of the corresponding eigenspaces E_{λ_{j}}(T_{1}), j = 1, …, m. An element of V can be expressed in at most one way as a sum of elements of the E_{λ_{j}}(T)’s, j = 1, …, m, as in the previous section. Using this and (10.8.2), we get that

E_{λ_{j}}(T_{1}) = E_{λ_{j}}(T) (10.8.3)

for each j = 1, …, m. One can also verify that λ_{1}, …, λ_{m} are all of the eigenvalues of T. If one already knows that every element of V can be expressed as a sum of elements of the E_{λ}(T)’s, where λ is an eigenvalue of T, then one can get T_{1} and T_{2} using the remarks in the previous section. It is well known that this holds when the characteristic polynomial of T can be factored into a product of linear factors. In particular, this happens when k is algebraically closed.

Note that T_{1} and T_{2} can be expressed as polynomials in T with coefficients in k and no constant term, as in part (b) of the proposition on p17 of [14], and Lemma 6.1 on p40 of [25]. Equivalently, this means that T_{1} and T_{2} can be expressed as linear combinations of positive powers of T. In particular, if R is any linear mapping from V into itself that commutes with T, then R commutes with T_{1} and T_{2}. Alternatively, if R commutes with T, then one can check that R commutes with T_{1}, using (10.7.10) and (10.8.3). This implies that R commutes with T_{2} as well, by (10.8.1).

Suppose that T_{3} and T_{4} are commuting linear mappings from V into itself such that T = T_{3} + T_{4}, (10.8.4)

T_{3} is diagonalizable on V, and T_{4} is nilpotent on V. Note that T_{3} and T_{4} commute with T, and hence with T_{1} and T_{2}, as in the preceding paragraph. Of course, T_{1} − T_{3} = T_{4} − T_{2}, (10.8.5)

by (10.8.1) and (10.8.4). In this situation, T_{1} − T_{3} is diagonalizable on V, as in Section 10.6. We also have that T_{4} − T_{2} is nilpotent on V, as in Section 9.7. The only linear mapping from V into itself that is both diagonalizable and nilpotent is equal to 0 on V, so that T_{1} = T_{3} and T_{2} = T_{4}. This is the uniqueness statement in [14, 25]. One could obtain T_{1} = T_{3} from (10.8.3) too.

Suppose that W_{0} and W are linear subspaces of V with

T (W) ⊆ W_{0} ⊆ W. (10.8.6)

Under these conditions, T_{1}(W), T_{2}(W) ⊆ W_{0}, (10.8.7)

as in part (c) of the proposition on p17 of [14], and Consequence 6.2 on p40 of [25]. This follows from the expressions for T_{1} and T_{2} in terms of polynomials in T mentioned earlier.

10.9 Some related situations

Let k be a field, and let V be a vector space over k of positive finite dimension. If A and B are linear mappings from V into itself, then put

ad_{A}(B) = [A, B] = A ∘ B − B ∘ A, (10.9.1)

as before. This defines ad_{A} as a linear mapping from L(V) into itself, where L(V) is the space of linear mappings from V into itself. If A is diagonalizable on V, then ad_{A} is diagonalizable on L(V). This can be seen using a basis for V consisting of eigenvectors for A, to reduce to the analogous statement for matrices, as in Section 10.4. Similarly, if A is nilpotent on V, then ad_{A} is nilpotent on L(V), as in Section 9.7. Let T, T_{1}, and T_{2} be as in (10.8.1), so that

ad_{T} = ad_{T_{1}} + ad_{T_{2}}. (10.9.2)

Note that ad_{T_{1}} is diagonalizable on L(V), and that ad_{T_{2}} is nilpotent on L(V), by the corresponding properties of T_{1} and T_{2} on V, and the previous remarks. We also have that [ad_{T_{1}}, ad_{T_{2}}] = ad_{[T_{1},T_{2}]} = 0, (10.9.3)

so that ad_{T_{1}} and ad_{T_{2}} commute on L(V), because T_{1} and T_{2} commute on V. Thus (10.9.2) is the analogue of (10.8.1) for ad_{T}, as in Lemma A on p18 of [14]. This also corresponds to Lemma 6.3 on p41 of [25], with p = q = 1.

Let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Also let δ be a derivation on A. If a, b ∈ A are eigenvectors of δ with eigenvalues λ, μ ∈ k, respectively, then

δ(a b) = δ(a) b + a δ(b) = (λ a) b + a (μ b) = (λ + μ) (a b), (10.9.4)

so that a b is an eigenvector of A with eigenvalue λ + μ when a b ≠ 0. It follows that the linear span in A of the eigenvectors of δ forms a subalgebra of A. This is stated in Exercise 12 on p6 of [14] for derivations on Lie algebras that come from the adjoint representation.

If λ ∈ k, then put

(10.9.5) E_{λ}(δ) = {a ∈ A : (δ − λ I)^{l}(a) = 0 for some l ∈ Z_{+}}, as in Section 10.7, where I is the identity mapping on A. This is a linear subspace of A which is nontrivial exactly when λ is an eigenvalue of δ on A, as before. Let a, b ∈ A and λ, μ ∈ k be given, and observe that

(10.9.6) (δ − (λ + μ) I)(a b) = δ(a) b + a δ(b) − λ a b − μ a b
= ((δ − λ I)(a)) b + a ((δ − μ I)(b)).

If a ∈ E_{λ}(δ) and b ∈ E_{μ}(δ), then one can use this repeatedly to get that

(δ − (λ + μ) I)^{j}(a b) = 0 (10.9.7)

when j is sufficiently large, so that a b ∈ E_{λ+μ}(δ). If a b ≠ 0, then E_{λ+μ}(δ) ≠ {0}, which implies that λ + μ is an eigenvalue of δ, as before.

Suppose that k is algebraically closed, and that A has positive finite dimension as a vector space over k. As in the previous section, there are linear mappings δ_{1}, δ_{2} from A into itself such that

δ = δ_{1} + δ_{2}, (10.9.8)

δ_{1} is diagonalizable on A, δ_{2} is nilpotent on A, and δ_{1}, δ_{2} commute on A. More precisely, E_{λ}(δ_{1}) = E_{λ}(δ) (10.9.9)

for every λ ∈ k, and in particular the eigenvalues of δ and δ_{1} are the same. If a ∈ E_{λ}(δ_{1}) = E_{λ}(δ) and b ∈ E_{μ}(δ_{1}) = E_{μ}(δ) for some λ, μ ∈ k, then a b is an element of E_{λ+μ}(δ) = E_{λ+μ}(δ_{1}), as in the preceding paragraph. Note that a b may be equal to 0, in which case λ + μ need not be an eigenvalue of δ, δ_{1}. In this situation,

δ_{1}(a b) = (λ + μ) a b = (λ a) b + a (μ b) = δ_{1}(a) b + a δ_{1}(b). (10.9.10)

One can use this to get that δ_{1} is a derivation on A, because A is spanned by the eigenspaces of δ_{1}. It follows that δ_{2} is a derivation on A as well, by (10.9.8). This corresponds to Lemma B on p18-19 of [14].

10.10 Replicas

Let k be a field, let V be a vector space over k of positive finite dimension n, and let A be a diagonalizable linear mapping from V into itself. Thus V corresponds to the direct sum of the nontrivial eigenspaces of A. If ϕ is a k-valued function on the set of eigenvalues of A, then ϕ(A) can be defined as a linear mapping from V into itself by putting ϕ(A) = λ I on E_{λ}(A) for each eigenvalue λ ∈ k of A. This corresponds to Definition 6.4 on p41 of [25], but with fewer conditions on ϕ for the moment, as in a remark just after Definition 6.4 in [25]. Note that ϕ(A) can be expressed as a polynomial in A with coefficients in k, using a polynomial whose values at the eigenvalues of A are the same as the values of ϕ.

Equivalently, let v_{1}, …, v_{n} be a basis of V consisting of eigenvectors of A with eigenvalues a_{1}, …, a_{n} ∈ k, so that

A(v_{j}) = a_{j} v_{j} (10.10.1)

for every j = 1, …, n. Using this basis, ϕ(A) can be characterized by

(ϕ(A))(v_{j}) = ϕ(a_{j}) v_{j} (10.10.2)

for each j = 1, …, n. If h, m ∈ {1, …, n}, then let E_{h,m} be the linear mapping from V into itself such that

E_{h,m}(v_{j}) = δ_{m,j} v_{h} (10.10.3)

for every j = 1, …, n, where δ_{m,j} is as in Section 10.1. The E_{h,m}’s form a basis for the space L(V) of linear mappings from V into itself, as a vector space over k. Let ad_{A} be the linear mapping from L(V) into itself corresponding to A as in (10.9.1). One can check that

ad_{A}(E_{h,m}) = (a_{h} − a_{m}) E_{h,m} (10.10.4)

for every h, m = 1, …, n, as in (10.4.5). Similarly,

ad_{ϕ(A)}(E_{h,m}) = (ϕ(a_{h}) − ϕ(a_{m})) E_{h,m} (10.10.5)

for every h, m = 1, …, n.

Suppose now that ϕ is defined on a subgroup of k, as a commutative group with respect to addition, that contains the eigenvalues of A. Suppose also that ϕ is a group homomorphism from this subgroup of k into k, with respect to addition. Under these conditions, (10.10.5) implies that

ad_{ϕ(A)}(E_{h,m}) = ϕ(a_{h} − a_{m}) E_{h,m} (10.10.6)

for every h, m = 1, …, n. This means that

ad_{ϕ(A)} = ϕ(ad_{A}), (10.10.7)

where the right side is obtained from ad_{A} as a diagonalizable linear mapping from L(V) into itself in the same way as before. This corresponds to Lemma 6.5 on p41 of [25], with p = q = 1, using a remark near the top of p41 in [25].

If 0 is not an eigenvalue of A, then ϕ(A) can be expressed as a polynomial in A with coefficients in k and constant term equal to 0, using a polynomial whose values at the eigenvalues of A are the same as the values of ϕ, and whose value at 0 is equal to 0. We can also do this when 0 is an eigenvalue of A, as long as ϕ(0) = 0. Of course, ϕ(0) = 0 automatically when ϕ is a homomorphism from an additive subgroup of k into k, as in the preceding paragraph. In this case, we can apply the same argument to (10.10.7) on L(V), to get that (10.10.7) can be expressed as a polynomial in ad_{A} with coefficients in k and constant term equal to 0.

If k has characteristic 0, then k may be considered as a vector space over the field Q rational numbers. In this case, we may consider mappings from the linear span of the set of eigenvalues of A in k, as a vector space over Q, into k that are linear over Q, as in the proof of the lemma on p19 of [14]. In [25], one simply considers mappings ϕ from k into itself that are linear over Q. The corresponding linear mappings ϕ(A) are called replicas of A by Chevalley, as mentioned on p42 of [25].

10.11 Two lemmas about traces

Let k be an algebraically closed field of characteristic 0, and let V be a vector space over k of positive finite dimension n. Also let T be a linear mapping from V into itself, and let T_{1}, T_{2} be the corresponding linear mappings from V into itself discussed in Section 10.8, so that

T = T_{1} + T_{2}, (10.11.1)

T_{1} is diagonalizable on V, T_{2} is nilpotent on V, and T_{1}, T_{2} commute. If ϕ is a mapping from the linear span of the eigenvalues of T_{1} in k, as a vector space over Q, into k that is linear over Q, then ϕ(T_{1}) can be defined as a linear mapping from V into itself, as in the previous section. If

tr(T ∘ ϕ(T_{1})) = 0 (10.11.2)

for all such ϕ, then T_{1} = 0, so that T = T_{2} is nilpotent on V. This corresponds to Lemma 6.7 on p42 of [25], rephrased a bit in the way that it is used in the proof of the lemma on p19 of [14].

Remember that T_{2} maps the eigenspaces of T_{1} into themselves, because T_{1} and T_{2} commute. If ϕ is as in the preceding paragraph, then it follows that T_{2} commutes with ϕ(T_{1}), by definition of ϕ(T_{1}). In particular, T_{2} ∘ ϕ(T_{1}) is nilpotent, and tr(T_{2} ∘ ϕ(T_{1})) = 0. (10.11.3)

Thus (10.11.2) is the same as saying that

tr(T_{1} ∘ ϕ(T_{1})) = 0. (10.11.4)

Let λ_{1}, …, λ_{l} ∈ k be a list of the eigenvalues of T_{1}, and let m_{1}, …, m_{l} ∈ Z_{+} be the dimensions of the corresponding eigenspaces of T_{1} in V. Using (10.11.4) we get that

∑_{j=1}^{l} m_{j} · λ_{j} ϕ(λ_{j}) = 0, (10.11.5)

because ϕ(T_{1}) = ϕ(λ_{j}) I on E_{λ_{j}}(T_{1}) for each j = 1, …, l, by construction. If we also ask that ϕ(λ_{j}) be in the subfield of k corresponding to Q for each j = 1, …, l, then we can apply ϕ to (10.11.5), to obtain that

∑_{j=1}^{l} m_{j} · ϕ(λ_{j})^{2} = 0. (10.11.6)

This means that ϕ(λ_{j}) = 0 for each j = 1, …, n, because ϕ(λ_{j}) is in the subfield of k corresponding to Q. This implies that 0 is the only eigenvalue of T_{1}, because the previous statement holds for all mappings ϕ from the linear span of the eigenvalues of T_{1} in k as a vector space over Q into the subfield of k corresponding to Q that are linear over Q. Thus T_{1} = 0 on V, because T_{1} is diagonalizable on V, by hypothesis, as desired.

If k is the field C of complex numbers, then a variant of this argument was remarked by Bergman, as mentioned on p42 of [25]. Namely, if (10.11.5) holds with ϕ equal to complex conjugation on C, then

∑_{j=1}^{l} m_{j} |λ_{j}|^{2} = 0. (10.11.7)

This implies directly that 0 is the only eigenvalue of T_{1}, as desired.

Let A and B be linear subspaces of the space L(V) of all linear mappings from V into itself, with A ⊆ B. (10.11.8)

Put (10.11.9) M = {T ∈ L(V) : [T, B] ∈ A for every B ∈ B}.

If T ∈ M satisfies tr(T ∘ R) = 0 (10.11.10)

for every R ∈ M, then T is nilpotent on V. This is the lemma stated on p19 of [14].

Let T ∈ M be given, and let T_{1}, T_{2} ∈ L(V) be as in (10.11.1) again. Also let ϕ be a mapping from the linear span of the eigenvalues of T_{1} in k, as a vector space over Q, into k that is linear over Q, as before. Thus ϕ(T_{1}) can be defined as a linear mapping from V into itself as in the previous section, and we would like to check that ϕ(T_{1}) ∈ M. (10.11.11)

This means that (10.11.2) follows from (10.11.10), so that we can reduce to the previous statement.

If A is a linear mapping from V into itself, then ad_{A} denotes the corresponding linear mapping from L(V) into itself, as in (10.9.1). The condition that T ∈ M means exactly that

ad_{T} (B) ⊆ A. (10.11.12)

Remember that ad_{T} = ad_{T_{1}} + ad_{T_{2}} (10.11.13)

is the analogue of (10.11.1) for ad_{T} as a linear mapping from V into itself, as in Section 10.9. This implies that ad_{T_{1}} can be expressed as a polynomial in ad_{T} with coefficients in k and constant term equal to 0, as in Section 10.8. It follows from this and (10.11.12) that

ad_{T_{1}}(B) ⊆ A, (10.11.14) so that T_{1} ∈ M.

We also have that ad_{ϕ(T_{1})} = ϕ(ad_{T_{1}}), (10.11.15)

as in (10.10.7). Remember that (10.11.15) can be expressed as a polynomial in ad_{T_{1}} with coefficients in k and constant term equal to 0, as in the previous section. This implies that ad_{ϕ(T_{1})}(B) ⊆ A, (10.11.16)

because of (10.11.14). Thus (10.11.11) holds, as desired.

10.12 Cartan’s criterion

Let k be a field of characteristic 0, and let V be a vector space over k of positive finite dimension n. Also let A be a Lie subalgebra of gl(V), and suppose that

tr(T ∘ R) = 0 (10.12.1)

for every T ∈ A and R ∈ [A, A]. Under these conditions, A is solvable as a Lie algebra over k. This is part of Theorem 7.1 on p42 of [25]. Note that the converse was discussed in Section 10.5.

Suppose first that k is algebraically closed, which corresponds to the theorem on p20 in [14]. In order to show that A is solvable as a Lie algebra, it suffices to show that [A, A] is nilpotent as a Lie algebra over k. To do this, it is enough to show that every element of [A, A] is nilpotent as a linear mapping on V, as in Section 9.10.

Put (10.12.2) M = {Z ∈ gl(V) : [Z, T] ∈ [A, A] for every T ∈ A},

and observe that A ⊆ M. We would like to show that

tr(R ∘ Z) = 0 (10.12.3)

for every R ∈ [A, A] and Z ∈ M. This will imply that every R ∈ [A, A] is nilpotent as a linear mapping on V, as in the previous section.

In order to get (10.12.3), it is enough to check that

tr(([T_{1}, T_{2}]) ∘ Z) = 0 (10.12.4)

for every T_{1}, T_{2} ∈ A and Z ∈ M. Remember that

tr(([T_{1}, T_{2}]) ∘ Z) = − tr(T_{2} ∘ ([T_{1}, Z])), (10.12.5)

as in Section 7.8. If T_{1} ∈ A and Z ∈ M, then [T_{1}, Z] ∈ [A, A], by the definition (10.12.2) of M. If we also have that T_{2} ∈ A, then it follows that the right side of (10.12.5) is equal to 0, by (10.12.1). This implies (10.12.4), as desired.

Now let k be any field of characteristic 0. We may as well take V = k^{n}, because V is isomorphic to k^{n} as a vector space over k. The earlier statement can be reformulated in terms of matrices as saying that if A_{0} is a Lie subalgebra of gl_{n}(k) such that tr(T_{0} R_{0}) = 0 (10.12.6)

for every T_{0} ∈ A_{0} and R_{0} ∈ [A_{0}, A_{0}], then A_{0} is solvable as a Lie algebra over k. This uses matrix multiplication and the trace on the space M_{n}(k) of n × n matrices with entries in k.

Let k_{1} be an algebraically closed field that contains k as a subfield, so that gl_{n}(k) may be considered as a subset of gl_{n}(k_{1}). If A_{1} is the linear span of A_{0} in gl_{n}(k_{1}), as a vector space over k_{1}, then A_{1} is a Lie subalgebra of gl_{n}(k_{1}), as a Lie algebra over k_{1}, as mentioned in Section 10.5. Note that [A_{0}, A_{0}] is an ideal in A_{0} as a Lie algebra over k, and that [A_{1}, A_{1}] is defined analogously as an ideal in A_{1}, as a Lie algebra over k_{1}, as in Section 9.2. One can check that [A_{1}, A_{1}] is the same as the linear span of [A_{0}, A_{0}] in A_{1}, as a vector space over k_{1}.

It follows that tr(T_{1} R_{1}) = 0 (10.12.7)

for every T_{1} ∈ A_{1} and R_{1} ∈ [A_{1}, A_{1}], because of (10.12.6). This uses matrix multiplication and the trace on M_{n}(k_{1}). Thus A_{1} is solvable as a Lie algebra over k_{1}, by the earlier argument for algebraically closed fields of characteristic 0. This implies that A_{0} is solvable as a Lie algebra over k, as desired.

10.13 Comparing radicals

Let k be a field, and let (A, [·, ·]) be a Lie algebra over k. If x ∈ A, then ad_{x} = ad_{A,x} is the linear mapping from A into itself defined by

ad_{x}(z) = ad_{A,x}(z) = [x, z] (10.13.1)

for every z ∈ A, as in Section 2.4. Suppose that A has positive finite dimension as a vector space over k, so that the Killing form

β(x, y) = tr_{A}(ad_{x} ∘ ad_{y}) (10.13.2)

is defined as an element of k for every x, y ∈ A. Remember that the radical

(10.13.3) A^{β} = {x ∈ A : β(x, y) = 0 for every y ∈ A}

of (10.13.2) in A is an ideal in A as a Lie algebra over k, as in Section 7.11.

Of course, the space of linear mappings from A into itself, as a vector space over k, is an associative algebra over k with respect to composition of mappings, and hence a Lie algebra over k with respect to the corresponding commutator bracket. Let A_{0} be a Lie subalgebra of A. The image

{ad_{x} : x ∈ A_{0}} (10.13.4)

of A_{0} under the adjoint representation of A is a Lie subalgebra of the Lie algebra of all linear mappings from A into itself. Suppose that β(x, y) = 0 for every x ∈ [A_{0}, A_{0}] and y ∈ A_{0}. If k has characteristic 0, then Cartan’s criterion implies that (10.13.4) is solvable as a Lie algebra over k. It follows that A_{0} is solvable as a Lie algebra over k, because the kernel of the adjoint representation on A is the center of A. This corresponds to the corollary on p20 of [14] when A_{0} = A.

The radical (10.13.3) of β in A automatically satisfies the conditions on A_{0} mentioned in the preceding paragraph. If k has characteristic 0, then we get that A^{β} is solvable as a Lie algebra over k. Let Rad A be the solvable radical of A, as in Section 9.4. It follows that

A^{β} ⊆ Rad A (10.13.5)

when k has characteristic 0, because A^{β} is an ideal in A as a Lie algebra.

Remember that A is said to be semisimple as a Lie algebra when Rad A = {0}. If A is semisimple and k has characteristic 0, then A^{β} = {0}, by (10.13.5), which means that the Killing form (10.13.2) on A is nondegenerate. This corresponds to parts of the theorem on p22 of [14] and Theorem 2.1 on p44 of [25].

If B is an ideal in A and B is commutative as a Lie algebra over k, then B is contained in the radical (10.13.3) of β in A, as in Section 7.11. If the Killing form on A is nondegenerate, then it follows that B = {0}. Remember that A is semisimple as a Lie algebra when {0} is the only ideal in A that is commutative as a Lie algebra, as in Section 9.4. This means that A is semisimple as a Lie algebra when the Killing form on A is nondegenerate. This corresponds to the other parts of the theorem on p22 of [14] and Theorem 2.1 on p44 of [25].

10.14 Complementary ideals

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]) be a Lie algebra over k. Suppose that A is isomorphic to k^{r} as a module over k for some positive integer r, so that the Killing form β can be defined on A as in (10.13.2). Remember that β has the associativity or invariance property

β([x, w], y) = β(x, [w, y]) (10.14.1)

for every w, x, y ∈ A, as in Section 7.11. If B is a submodule of A, as a module over k, then put

(10.14.2) B^{⊥} = B^{⊥,β} = {x ∈ A : β(x, y) = 0 for every y ∈ B}.

This is a submodule of A, as a module over k, which is the same as the radical of β in A when B = A. If B is an ideal in A as a Lie algebra over k, then it is easy to see that (10.14.2) is an ideal in A, using (10.14.1). This can also be obtained from statements in Sections 6.10 and 7.7.

Let us suppose from now on in this section that k is a field, and that A has positive finite dimension as a vector space over k. Let B be a linear subspace of A, so that (10.14.2) is a linear subspace of A too, as before. If the Killing form β is nondegenerate as a bilinear form on A, then the sum of the dimensions of B and B^{⊥} is equal to the dimension of A, by standard arguments.

Now let B be an ideal in A as a Lie algebra over k, so that (10.14.2) is an ideal in A too, as before. Thus

A_{0} = B ∩ B^{⊥} (10.14.3)

is an ideal in A as well, and β(x, y) = 0 for every x, y ∈ A_{0}, by construction. If k has characteristic 0, then it follows that A_{0} is solvable as a Lie algebra over k, as in the previous section. If A is semisimple as a Lie algebra over k, then we get that A_{0} = {0}. This implies that

[x, y] = 0 (10.14.4)

for every x ∈ B and y ∈ B^{⊥}, because [x, y] ∈ B ∩ B^{⊥}.

If k has characteristic 0 and A is semisimple as a Lie algebra over k, then the Killing form β is nondegenerate on A, as in the previous section. Under these conditions, we have that

B + B^{⊥} = {x + y : x ∈ B, y ∈ B^{⊥}} = A. (10.14.5)

More precisely, the dimension of B + B^{⊥} is equal to the sum of the dimensions of B and B^{⊥}, because B ∩ B^{⊥} = {0}, as in the preceding paragraph. We have also seen that the sum of the dimensions of B and B^{⊥} is equal to the dimension of A, because β is nondegenerate on A. This implies (10.14.5), and hence that A is isomorphic to the direct sum of B and B^{⊥} as a Lie algebra over k, because of (10.14.4).

This corresponds to the first step in the proof of the theorem on p23 in [14], and to Theorem 2.2 on p44 of [25]. Note that ideals in B and B^{⊥} are ideals in A in this situation, because of (10.14.4) and (10.14.5). This implies that B and B^{⊥} are semisimple as Lie algebras over k, because A is semisimple. It follows that A/B is semisimple as a Lie algebra over k, because it is isomorphic to B^{⊥}.

10.15 Simple Lie algebras

Let k be a field. A Lie algebra (A, [·, ·]) over k is said to be simple if A is not commutative as a Lie algebra, and if the only ideals in A are A itself and {0}. See p6 of [14], and Definition 2.3 on p44 of [25]. Note that the trivial Lie algebra {0} is not considered to be simple, nor is the one-dimensional Lie algebra k with respect to the trivial Lie bracket. If A is a simple Lie algebra over k, then the center Z(A) of A as a Lie algebra is trivial, because Z(A) is an ideal in A, and A is not commutative as a Lie algebra. Similarly, if A is simple and [A, A] is as in Section 9.2, then [A, A] = A, because [A, A] is an ideal in A, and A is not commutative. In particular, this means that A is not solvable as a Lie algebra, and in fact that A is semisimple as a Lie algebra.

Let (A, [·, ·]) be a Lie algebra over A again, and let A_{1}, …, A_{n} be finitely many ideals in A. Suppose that every element of A can be expressed in a unique way as the sum of elements of A_{1}, …, A_{n}, so that A corresponds to the direct sum of A_{1}, …, A_{n}, as a vector space over k. Let j, l ∈ {1, …, n} be given, with j ≠ l, so that A_{j} ∩ A_{l} = {0}. If a_{j} ∈ A_{j} and a_{l} ∈ A_{l}, then [a_{j}, a_{l}] ∈ A_{j} ∩ A_{l}, because A_{j} and A_{l} are ideals in A. This implies that

[a_{j}, a_{l}] = 0, (10.15.1)

because A_{j} ∩ A_{l} = {0}. This means that A corresponds to the direct sum of A_{1}, …, A_{n} as a Lie algebra over k, as remarked on p22 of [14]. Note that ideals in any A_{j} as a Lie algebra over k are ideals in A, because of (10.15.1).

Suppose from now on in this section that k has characteristic 0, and that A is a semisimple Lie algebra over k with positive finite dimension as a vector space over k. Under these conditions, there are finitely many ideals A_{1}, …, A_{n} in A such that A corresponds to the direct sum of the A_{j}’s, as in the preceding paragraph, and A_{j} is simple as a Lie algebra over k for each j = 1, …, n. Of course, if A is already simple as a Lie algebra over k, then this holds with n = 1. Otherwise, A corresponds to the direct sum of two proper ideals, each of which is semisimple as a Lie algebra over k, as in the previous section. One can repeat the process as needed until all of the ideals are simple as Lie algebras over k. At each step, any ideal in any of the ideals already obtained in A is an ideal in A too, as in the previous paragraph. This corresponds to the first part of the first theorem on p23 of [14], and to Corollary 1 on p45 of [25].

Remember that [A_{j}, A_{j}] = A_{j} for each j = 1, …, n, because the A_{j}’s are simple Lie algebras. This implies that [A, A] = A, as in the corollary on p32 in [14], and Corollary 2 on p45 of [25].

Let C be a simple Lie algebra over k, and suppose that ϕ is a Lie algebra homomorphism from A onto C. Note that ϕ(A_{j}) is an ideal in C for each j = 1, …, n, because A_{j} is an ideal in A and ϕ(A) = C. It follows that for each j = 1, …, n, ϕ(A_{j}) is either trivial or equal to C, because C is simple. Clearly ϕ(A_{j_{1}}) ≠ {0} for some j_{1} ∈ {1, …, n}, because ϕ(A) = C. This can happen for at most one element of {1, …, n}, by (10.15.1) and the noncommutativity of C. The restriction of ϕ to A_{j_{1}} is injective, because A_{j_{1}} is simple as a Lie algebra over k. This is the uniqueness property discussed on p45 of [25].

Let B_{0} be a simple ideal in A. Note that the center Z(A) of A is trivial, because A is semisimple. This implies that [A, B_{0}] ≠ {0}, because B_{0} ≠ {0}. It follows that [A_{j_{0}}, B_{0}] ≠ {0} for some j_{0} ∈ {1, …, n}. Remember that [A_{j_{0}}, B_{0}] is an ideal in A that is contained in A_{j_{0}} and B_{0}, because A_{j_{0}} and B_{0} are ideals in A, as in Section 9.2. Thus A_{j_{0}} = [A_{j_{0}}, B_{0}] = B_{0}, because A_{j_{0}} and B_{0} are simple. This is the uniqueness part of the first theorem on p23 of [14]. See also Theorem 4’ on p6 of [24].

If B is a nontrivial proper ideal in A, then one can start the process for expressing A as a direct sum of simple ideals using B, as before. Thus B is also expressed as a direct sum of some of these ideals, as in the corollary on p23 of [14], and Corollary 3 on p45 of [25].

Some examples and related properties

11.1 Simplicity and solvability

Let k be a field, and remember that x = \binom{0 1}{0 0}, y = \binom{0 0}{1 0}, and h = \binom{1 0}{0 −1} form a basis for sl_{2}(k), as a vector space over k. We also have that [x, y] = h, [h, x] = 2 · x, and [h, y] = −2 · y, as in Section 10.2. It is not difficult to check directly that sl_{2}(k) is simple as a Lie algebra over k when the characteristic of k is different from 2, as on p6-7 of [14]. More precisely, suppose that A is a nonzero ideal in sl_{2}(k). Let a x + b y + c h (11.1.1)

be a nonzero element of A, where a, b, c ∈ k. One can take the commutator of (11.1.1) with x twice to get that −2 · b x ∈ A, and similarly one can take the commutator of (11.1.1) with y twice to get that −2 · a y ∈ A. This implies that x ∈ A when b ≠ 0, and that y ∈ A when a ≠ 0, because the characteristic of A is not 2. Otherwise, if a = b = 0, then c ≠ 0, and it follows that h ∈ A. Thus A contains at least one of x, y, or h. One can use this to get that A contains each of x, y, and h, because the characteristic of k is not 2. This means that A = sl_{2}(k), as desired.

Let k be any field, and let (A, [·, ·]_{A}) be a Lie algebra over k. Suppose that the dimension of A is 3, as a vector space over k, and that [A, A] = A, where [A, A] is as in Section 9.2. Under these conditions, A is simple as a Lie algebra over k, as in Exercise 5 on p10 of [14]. To see this, let A_{0} be an ideal in A, so that the quotient A/A_{0} is a Lie algebra over k too. It is easy to see that

[A/A_{0}, A/A_{0}] = A/A_{0} (11.1.2)

in this situation, as in Section 9.3. However, if A/A_{0} has dimension 1 as a vector space over k, then the left side of (11.1.2) is {0}. Similarly, if A/A_{0} has dimension 2 as a vector space over k, then the left side of (11.1.2) has dimension less than or equal to 1. It follows that A/A_{0} has dimension 0 or 3, which is to say that A_{0} = {0} or A, as desired.

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k again. Suppose that the derived subalgebra A^{(1)} = [A, A] is generated by two elements a, b as a module over k, so that every element of A^{(1)} can be expressed as a linear combination of a and b, with coefficients in k. This implies that A^{(2)} = [A^{(1)}, A^{(1)}] consists of multiples of [a, b]_{A}, with coefficients in k. It follows that A^{(3)} = {0}, so that A is solvable as a Lie algebra. Of course, if A^{(1)} consists of the multiples of a single element of A with coefficients in k, then A^{(2)} = {0}.

Suppose that k is a field, and that A has dimension 3 as a vector space over k. If [A, A] is a proper subset of A, then [A, A] has dimension less than or equal to 2, as a vector space over k. This implies that A is solvable as a Lie algebra, as in the preceding paragraph.

11.2 Traces on M_{n}(k)

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Remember that the space M_{n}(k) of n × n matrices with entries in k is an associative algebra over k with respect to matrix multiplication, as in Section 2.8. If a, x ∈ M_{n}(k), then

L_{a}(x) = a x (11.2.1)

is defined as an element of M_{n}(k) using matrix multiplication. This defines L_{a} as a homomorphism from M_{n}(k) into itself, as a module over k. If b ∈ M_{n}(k) too, then L_{a} ∘ L_{b} = L_{a b}, (11.2.2)

as in Section 2.8. It is not difficult to see that

tr_{M_{n}(k)}L_{a} = n · tr a (11.2.3)

for every a ∈ M_{n}(k), where tr a is the ordinary trace of a, as an n × n matrix with entries in k. The left side of (11.2.3) is the trace of L_{a} as a module homomorphism from M_{n}(k) into itself, as in Section 7.8. More precisely, the definition of tr_{M_{n}(k)} uses the fact that M_{n}(k) is isomorphic to k^{n^{2}}, as a module over k. It follows that

tr_{M_{n}(k)}(L_{a} ∘ L_{b}) = tr_{M_{n}(k)}L_{a b} = n · tr(a b) (11.2.4)

for every a, b ∈ M_{n}(k), using (11.2.2) in the first step.

Similarly, if a ∈ M_{n}(k), then

R_{a}(x) = x a (11.2.5)

defines R_{a} as a module homomorphism from M_{n}(k) into itself. If b ∈ M_{n}(k) too, then R_{a} ∘ R_{b} = R_{b a}, (11.2.6)

11.2. TRACES ON M_{N}(K) 233

as in Section 2.7. One can check that tr_{M_{n}(k)}R_{a} = n · tr a (11.2.7) for every a ∈ M_{n}(k). This implies that (11.2.8) tr_{M_{n}(k)}(R_{a} ∘ R_{b}) = tr_{M_{n}(k)}(R_{b a}) = n · tr(b a) = n · tr(a b) for every a, b ∈ M_{n}(k), using (11.2.6) in the first step.

Remember that L_{a} and R_{b} commute with each other on M_{n}(k) for every a, b ∈ M_{n}(k), as in Section 2.7. One can verify that tr_{M_{n}(k)}(L_{a} ∘ R_{b}) = (tr a) (tr b) (11.2.9) for every a, b ∈ M_{n}(k).

If a ∈ M_{n}(k), then ad_{a}(x) = [a, x] = a x − x a = L_{a}(x) − R_{a}(x) (11.2.10) defines a module homomorphism from M_{n}(k) into itself, as usual. Equivalently, ad_{a} = L_{a} − R_{a}. (11.2.11) If b ∈ M_{n}(k) too, then

(11.2.12) ad_{a} ∘ ad_{b} = (L_{a} − R_{a}) ∘ (L_{b} − R_{b})
= L_{a} ∘ L_{b} − L_{a} ∘ R_{b} − L_{b} ∘ R_{a} + R_{a} ∘ R_{b}
= L_{a b} − L_{a} ∘ R_{b} − L_{b} ∘ R_{a} + R_{b a}.

It follows that

tr_{M_{n}(k)}(ad_{a} ∘ ad_{b}) = tr_{M_{n}(k)}L_{a b} − tr_{M_{n}(k)}(L_{a} ∘ R_{b})

−tr_{M_{n}(k)}(L_{b} ∘ R_{a}) + tr_{M_{n}(k)}R_{b a} (11.2.13)

= n · tr(a b) − 2 · (tr a) (tr b) + n · tr(b a)
= 2 n · tr(a b) − 2 · (tr a) (tr b).

Note that the right side is automatically equal to 0 when either a or b is a multiple of the identity matrix.

Of course, gl_{n}(k) is the same as M_{n}(k) as a module over k, so that traces over gl_{n}(k) are the same as traces over M_{n}(k). If a ∈ sl_{n}(k), then let ad_{sl_{n}(k),a} be the mapping from sl_{n}(k) into itself defined in the usual way, which is the same as the restriction of ad_{a} to sl_{n}(k). Remember that sl_{n}(k) is isomorphic to k^{n^{2}−1} as a module over k, and that gl_{n}(k) is isomorphic to the direct sum of sl_{n}(k) and k, as modules over k, as in Section 10.3. If a, b ∈ sl_{n}(k), then we have that (11.2.14) tr_{sl_{n}(k)}(ad_{sl_{n}(k),a} ∘ ad_{sl_{n}(k),b}) = tr_{M_{n}(k)}(ad_{a} ∘ ad_{b}), as in Section 7.10. This implies that (11.2.15) tr_{sl_{n}(k)}(ad_{sl_{n}(k),a} ∘ ad_{sl_{n}(k),b}) = 2 n · tr(a b), by (11.2.13).

11.3 A nice criterion

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Of course, the center Z(A) of A as a Lie algebra is a solvable ideal in A. If every solvable ideal in A is contained in Z(A), then A is said to be reductive as a Lie algebra over k, as in Exercise 5 on p30 and p102 of [14]. This means that the solvable radical Rad A of A is equal to Z(A), as in Section 9.4. In particular, this holds when A is commutative or semisimple as a Lie algebra.

Now let k be an algebraically closed field of characteristic 0, and let V be a vector space over k of positive finite dimension. Also let A be a Lie subsalgebra of the Lie algebra gl(V) of linear mappings from V into itself. Thus V may be considered as a module over A, as a Lie algebra over k. Suppose that V is irreducible as a module over A. Under these conditions, A is reductive as a Lie algebra over k, and every element of the center Z(A) of A is a scalar multiple of the identity mapping I = I_{V} on V. In particular, the dimension of Z(A) as a vector space over k is less than or equal to 1. If A ⊆ sl(V), then A is semisimple as a Lie algebra over k. This corresponds to part (b) of the proposition on p102 of [14], and is related to Theorem 5.1 on p50 of [25].

Remember that the solvable radical Rad A of A is the maximal solvable ideal in A, as in Section 9.4. Lie’s theorem implies that there is a v_{0} ∈ V such that v_{0} ≠ 0 and v_{0} is an eigenvector of every element of Rad A, as in Section 9.13. This means that there is a linear functional λ on Rad A such that

T (v_{0}) = λ(T) v_{0} (11.3.1)

for every T ∈ Rad A.

If R ∈ A and T ∈ Rad A, then [R, T] = R ∘ T − T ∘ R ∈ Rad A, because Rad A is an ideal in A. Under these conditions,

λ([R, T]) = 0, (11.3.2)

as in Section 9.12.

Put (11.3.3) V_{λ} = {v ∈ V : T (v) = λ(T) v for every T ∈ Rad A}.

This is a linear subspace of V that contains v_{0}. Let R ∈ A, T ∈ Rad A, and v ∈ V_{λ} be given, and observe that

(11.3.4) T (R(v)) = R(T (v)) − ([R, T])(v) = R(λ(T) v) − λ([R, T]) v
= λ(T) R(v).

This uses the fact that [R, T] ∈ Rad A in the second step, and (11.3.2) in the third step. It follows that R(v) ∈ V_{λ}, so that R(V_{λ}) ⊆ V_{λ}.

Because V is irreducible as a module over A, we get that V_{λ} = V. This means that every T ∈ Rad A is equal to λ(T) I. In particular, Rad A ⊆ Z(A), which implies that Rad A = Z(A), as desired.

A basic property of reductive Lie algebras will be discussed in Section 13.9.

11.4 Ideals and structure constants

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Thus the space k^{n} of n-tuples of elements of k is a (free) module over k with respect to coordinatewise addition and scalar multiplication. Let c_{j,l}^{r} be an element of k for every j, l, r = 1, …, n. If x, y ∈ k^{n}, then let [x, y]_{k^{n}} be the element of k^{n} whose rth coordinate is given by

([x, y]_{k^{n}})_{r} = ∑_{j=1}^{n} ∑_{l=1}^{n} c_{j,l}^{r}x_{j} y_{l} (11.4.1)

for every r = 1, …, n. This defines a mapping from k^{n} × k^{n} into k^{n} that is bilinear over k. Suppose that

c_{j,l}^{r} = −c_{l,j}^{r} and c_{j,j}^{r} = 0 (11.4.2)

for every j, l, r = 1, …, n, so that [x, x]_{k^{n}} = 0 for every x ∈ k^{n}. Suppose also that the c_{j,l}^{r}’s satisfy (9.14.8), so that (11.4.1) satisfies the Jacobi identity, as before. This means that k^{n} is a Lie algebra over k with respect to (11.4.1).

Let m < n be a positive integer, and let us use k^{m} × {0} to denote the space of x ∈ k^{n} such that x_{j} = 0 when j ≥ m+1. Suppose that for every r = 1, …, n,

c_{j,l}^{r} = 0 when j ≥ m + 1 or l ≥ m + 1. (11.4.3)

This implies that k^{m} ×{0} is an ideal in k^{n}, as a Lie algebra over k with respect to (11.4.1). Note that this condition is necessary for k^{m} × {0} to be an ideal in k^{n} with respect to (11.4.1).

Let K be a commutative associative algebra over k, and note that the space K^{n} of n-tuples of elements of K may be considered as a module over k with respect to coordinatewise addition and scalar multiplication. If a, a′ ∈ K^{n}, then let [a, a′]_{K^{n}} be the element of K^{n} whose rth coordinate is equal to

([a, a′]_{K^{n}})_{r} = ∑_{j=1}^{n} ∑_{l=1}^{n} c_{j,l}^{r} a_{j} a′_{l} (11.4.4)

for each r = 1, …, n. As in Section 9.14, K^{n} is a Lie algebra over k with respect to (11.4.4). We also have that K^{m} × {0} is an ideal in K^{n} with respect to (11.4.4), because of (11.4.3). If k^{m} × {0} is solvable as a Lie algebra over k, then K^{m} × {0} is solvable as a Lie algebra over k too, as before.

Suppose that K has a multiplicative identity element e, so that K^{n} may be considered as a module over K, and as a Lie algebra over K with respect to (11.4.4). Remember that t ↦ t e defines a ring homomorphism from k into K, which leads to a homomorphism from k^{n} into K^{n}, as Lie algebras over k. If t ↦ t e is injective as a mapping from k into K, then the corresponding mapping from k^{n} into K^{n} is injective.

Now let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k of positive finite dimension n, as a vector space over k. Any choice of basis for A, as a vector space over k, leads to an isomorphism between A and k^{n}, as vector spaces over k. As in Section 9.14, there are structure constants c_{j,l}^{r} ∈ k such that (11.4.1) corresponds to [·, ·]_{A} with respect to this isomorphism.

Let K be a field that contains k as a subfield. Using the structure constants c_{j,l}^{r} just mentioned, K^{n} becomes a Lie algebra over K with respect to (11.4.4). Any other choice of basis for A will lead to isomorphic Lie algebra structures on k^{n} and K^{n}.

Suppose that A_{0} is a proper nonzero ideal in A, of dimension m. We can choose a basis for A that contains a basis for A_{0}, so that A_{0} corresponds to k^{m} × {0} in k^{n}. If A_{0} is solvable as a Lie algebra over k, then k^{m} × {0} is solvable as a Lie algebra over k, and K^{m} × {0} is solvable as a Lie algebra over K.

If K^{n} is semisimple as a Lie algebra over K, then it follows that A is semisimple as a Lie algebra over k. Similarly, if K^{n} is simple as a Lie algebra over K, then A is simple as a Lie algebra over k. This corresponds to part of Theorem 9 on p9 of [24] and the remark following it when k = R and K = C.

11.5 Nondegeneracy and bilinear forms

Let k be a field, and let V be a vector space over k of finite positive dimension n. If v_{1}, …, v_{n} is a basis for V and l ∈ {1, …, n}, then there is a unique linear functional λ_{l} on V such that λ_{l}(v_{j}) is equal to 1 when j = l and to 0 when j ≠ l. If μ is any linear functional on V, then it is easy to see that

μ = ∑_{l=1}^{n} μ(v_{l}) λ_{l}. (11.5.1)

In fact, λ_{1}, …, λ_{n} forms a basis for the dual space V′ of all linear functionals on V, as a vector space over k.

Let β be a bilinear form on V, so that

β_{w}(v) = β(v, w) (11.5.2)

is a linear functional on V for each w ∈ W. Observe that

β_{v_{l}} = ∑_{j=1}^{n} β_{v_{l}}(v_{j}) λ_{j} = ∑_{j=1}^{n} β(v_{j}, v_{l}) λ_{j} (11.5.3)

for each l = 1, …, n, using (11.5.1) in the first step. Of course, w ↦ β_{w} defines a linear mapping from V into V′, and (11.5.3) expresses this linear mapping in terms of a matrix. In particular, β is nondegenerate as a bilinear form on V if and only if (β(v_{j}, v_{l})) is invertible as an n × n matrix with entries in k.

Now let (A, [·, ·]_{A}) be a Lie algebra over k with finite positive dimension n as a vector space over k. Using a basis for A, we get an isomorphism between A and k^{n}, as vector spaces over k. As before, there are structure constants c_{j,l}^{r} ∈ k for j, l, r = 1, …, n that satisfy (11.4.2) and (9.14.8) such that (11.4.1) corresponds to [·, ·]_{A} with respect to the isomorphism just mentioned. If x ∈ k^{n}, then ad_{k^{n},x}(z) = [x, z]_{k^{n}} (11.5.4)

defines a linear mapping from k^{n} into itself, as in Section 2.4. The Killing form on k^{n} is defined by

b_{k^{n}}(x, y) = tr_{k^{n}}(ad_{k^{n},x} ∘ ad_{k^{n},y}) (11.5.5)

for every x, y ∈ k^{n}, as in Section 7.9.

Let K be a field that contains k as a subfield, so that K^{n} becomes a Lie algebra over K with respect to (11.4.4). As before,

ad_{K^{n},x} = [x, z]_{K^{n}} (11.5.6)

defines a linear mapping from K^{n} into itself for each x ∈ K^{n}, and the Killing form on K^{n} is defined by

b_{K^{n}}(x, y) = tr_{K^{n}}(ad_{K^{n},x} ∘ ad_{K^{n},y}) (11.5.7)

for every x, y ∈ K^{n}. If x ∈ k^{n}, then (11.5.4) and (11.5.6) are the same on k^{n}, because (11.4.1) and (11.4.4) are the same on k^{n}. This implies that (11.5.5) and (11.5.7) are the same when x, y ∈ k^{n}.

Let u_{1}, …, u_{n} be the standard basis elements of k^{n}, so that the jth coordinate of u_{l} is equal to 1 when j = l and to 0 when j ≠ l. Note that u_{1}, …, u_{n} form a basis for K^{n} as well, as a vector space over K. As in the preceding paragraph, b_{k^{n}}(u_{j}, u_{l}) = b_{K^{n}}(u_{j}, u_{l}) (11.5.8)

for every j, l = 1, …, n. It follows that (11.5.5) is nondegenerate as a bilinear form on k^{n} if and only if (11.5.7) is nondegenerate as a bilinear form on K^{n}. More precisely, this happens exactly when the determinant of (11.5.8), as an n × n matrix with entries in k, is not 0. Remember that nondegeneracy of the Killing form is equivalent to semisimplicity of a finite-dimensional Lie algebra over a field of characteristic 0, as in Section 10.13. If k has characteristic 0, then K has characteristic 0, and we get that k^{n} is semisimple as a Lie algebra over k if and only if K^{n} is semisimple as a Lie algebra over K. This corresponds to part of Theorem 9 on p9 of [24] when k = R and K = C.

11.6 Bilinear forms and adjoints

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. The space k^{n} of n-tuples of elements of k is a module over k with respect to coordinatewise addition and scalar multiplication, as usual. Let β(·, ·) be a bilinear form on k^{n}, which is to say a mapping from k^{n} × k^{n} into k that is bilinear over k. Remember that there is a unique n × n matrix (β_{j,l}) with entries in k such that

β(x, y) = ∑_{j=1}^{n} ∑_{l=1}^{n} β_{j,l} x_{j} y_{l} (11.6.1)

for every x, y ∈ k^{n}, as in Section 3.12. It is easy to see that β(·, ·) is symmetric or antisymmetric as a bilinear form on k^{n} if and only if (β_{j,l}) is symmetric or antisymmetric as a matrix, respectively, as before. Similarly, β(x, x) = 0 for every x ∈ k^{n} if and only if (β_{j,l}) is antisymmetric and β_{j,j} = 0 for every j = 1, …, n. If 1 + 1 has a multiplicative inverse in k, then this is equivalent to the antisymmetry of β(·, ·) or (β_{j,l}).

Remember that M_{n}(k) is the space of n × n matrices with entries in k, which is an associative algebra over k with respect to matrix multiplication. If a ∈ M_{n}(k), then a^{t} denotes the transpose of a, as usual. Let us also use β to denote (β_{j,l}) as an element of M_{n}(k), and let us suppose for the rest of the section that β is invertible in M_{n}(k). Put

a^{∗} = (β^{−1} a β)^{t} = β^{t} a^{t} (β^{t})^{−1} (11.6.2)

for each a ∈ M_{n}(k). It is easy to see that a ↦ a^{∗} defines an opposite algebra automorphism on M_{n}(k), because of the same property of a ↦ a^{t}.

If a = (a_{j,l}) ∈ M_{n}(k), then

(T_{a}(x))_{j} = ∑_{l=1}^{n} a_{j,l} x_{l} (11.6.3)

defines a module homomorphism T_{a} from k^{n} into itself, as before. Remember that a ↦ T_{a} defines an isomorphism from the space M_{n}(k) of n × n matrices with entries in k onto the space of module homomorphisms from k^{n} into itself, as algebras over k. By construction,

β(T_{a}(x), y) = β(x, T_{a^{∗}}(y)) (11.6.4)

for every a ∈ M_{n}(k) and x, y ∈ k^{n}, as in Section 3.12. More precisely, a^{∗} is uniquely determined by this property, and we may call (T_{a})^{∗} = T_{a^{∗}} the adjoint of T_{a} with respect to β. This defines an opposite algebra automorphism on the space of module homomorphisms from k^{n} into itself.

Observe that

(11.6.5) (a^{∗})^{∗} = β^{t} (a^{∗})^{t} (β^{t})^{−1} = β^{t} (β^{t} a^{t} (β^{t})^{−1})^{t} (β^{t})^{−1}
= β^{t} ((β)^{t})^{−1})^{t} (a^{t})^{t} (β^{t})^{t} (β^{t})^{−1}
= β^{t} β^{−1} a β (β^{t})^{−1}

for every a ∈ M_{n}(k). If β is either symmetric or antisymmetric, then we get that (a^{∗})^{∗} = a (11.6.6)

for every a ∈ M_{n}(k), and (11.6.2) is the same as

a^{∗} = β a^{t} β^{−1}. (11.6.7)

Thus a ↦ a^{∗} defines an algebra involution on M_{n}(k) in each of these two cases. Equivalently, T ↦ T^{∗} is an algebra involution on the space of module homomorphisms from k^{n} into itself in both cases. This may be considered as an instance of a remark in Section 3.14 as well.

11.7 Bilinear forms and symmetry conditions

Let us continue with the same notation and hypotheses as in the previous section, so that β = (β_{j,l}) is an invertible element of M_{n}(k), and β(·, ·) is the corresponding bilinear form on k^{n}, as in (11.6.1). Remember that a ∈ M_{n}(k) is invertible exactly when the determinant det a of a is invertible in k, and that the determinant of the transpose of a is the same as det a. If a is antisymmetric, then (11.7.1) det a = det a^{t} = det(−a) = (−1)^{n} det a.

If a is invertible too, then it follows that (−1)^{n} = 1 in k. This implies that −1 = 1 in k when n is odd.

Let a ∈ M_{n}(k) be given, and let T_{a} be the corresponding module homomorphism from k^{n} into itself, as in (11.6.3). Observe that

β(T_{a}(x), y) (11.7.2)

defines another bilinear form on k^{n}. One can check that every bilinear form on k^{n} corresponds to a unique a ∈ M_{n}(k) in this way, because β is supposed to be invertible. This defines an isomorphism between M_{n}(k) and the space of bilinear forms on k^{n}, as modules over k. Equivalently, this defines an isomorphism between the space of module homomorphisms from k^{n} into itself and the space of bilinear forms on k^{n}, as modules over k.

Remember that T_{a} is said to be symmetric with respect to β when

β(T_{a}(x), y) = β(x, T_{a}(y)) (11.7.3)

for every x, y ∈ k^{n}. This is equivalent to asking that a be self-adjoint with respect to (11.6.2), which is to say that a^{∗} = a. If β(·, ·) is symmetric as a bilinear form on k^{n}, then (11.7.3) holds if and only if

β(T_{a}(x), y) = β(T_{a}(y), x) (11.7.4)

for every x, y ∈ k^{n}, which means that (11.7.2) is symmetric as a bilinear form on k^{n}. If β(·, ·) is antisymmetric as a bilinear form on k^{n}, then (11.7.3) holds if and only if β(T_{a}(x), y) = −β(T_{a}(y), x) (11.7.5)

for every x, y ∈ k^{n}, which means that (11.7.2) is antisymmetric as a bilinear form on k^{n}.

Similarly, T_{a} is antisymmetric with respect to β when

β(T_{a}(x), y) = −β(x, T_{a}(y)) (11.7.6)

for every x, y ∈ k^{n}. This is equivalent to asking that a be anti-self-adjoint with respect to (11.6.2), so that a^{∗} = −a. If β(·, ·) is symmetric as a bilinear form on k^{n}, then (11.7.6) holds if and only if (11.7.5) holds, so that (11.7.2) is antisymmetric as a bilinear form on k^{n}. If β(·, ·) is antisymmetric as a bilinear form on k^{n}, then (11.7.6) holds if and only if (11.7.4) holds, so that (11.7.2) is symmetric as a bilinear form on k^{n}.

Of course, if γ = (γ_{j,l}) ∈ M_{n}(k), then

γ(x, y) = ∑_{j=1}^{n} ∑_{l=1}^{n} γ_{j,l} x_{j} y_{l} (11.7.7)

defines a bilinear form on k^{n}. As usual, γ(·, ·) is symmetric or antisymmetric as a bilinear form on k^{n} exactly when γ is symmetric or antisymmetric as a matrix, respectively. Note that (11.7.2) corresponds to (11.7.7) with

γ = β a. (11.7.8)

Thus (11.7.2) is symmetric or antisymmetric as a bilinear form on k^{n} exactly when (11.7.8) is symmetric or antisymmetric as a matrix, respectively.

11.8 Traces and involutions

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Also let a ↦ a^{∗} be an algebra involution on the algebra M_{n}(k) of n × n matrices with entries in k. Suppose that

tr a^{∗} = tr a (11.8.1)

for every a ∈ M_{n}(k) too, where tr a ∈ k is the usual trace of a. Note that this condition holds automatically when a^{∗} is as in (11.6.2). Remember that a ↦ a^{∗} is an involution on M_{n}(k) in that situation when β is symmetric or antisymmetric.

Put (a, b)_{M_{n}(k)} = tr(a b^{∗}) (11.8.2)

for every a, b ∈ M_{n}(k), which defines a bilinear form on M_{n}(k). Observe that

(a, b)_{M_{n}(k)} = tr(a b^{∗})^{∗} = tr((b^{∗})^{∗} a^{∗}) = tr(b a^{∗}) = (b, a)_{M_{n}(k)} (11.8.3)

for every a, b ∈ M_{n}(k), using (11.8.1) in the first step. If c ∈ M_{n}(k), then let

L_{c}(a) = c a and R_{c}(a) = a c (11.8.4)

be the corresponding operators of left and right multiplication on M_{n}(k) by c, respectively. Thus

(L_{c}(a), b)_{M_{n}(k)} = tr(c a b^{∗}) = tr(a b^{∗} c)
(11.8.5)  = tr(a (c^{∗} b)^{∗}) = (a, L_{c^{∗}}(b))_{M_{n}(k)}

for every a, b ∈ M_{n}(k). Similarly,

(R_{c}(a), b)_{M_{n}(k)} = tr(a c b^{∗}) = tr(a (b c^{∗})^{∗}) = (a, R_{c^{∗}}(b))_{M_{n}(k)} (11.8.6)

11.9. INNER PRODUCTS OVER ORDERED FIELDS 241

for every a, b ∈ M_{n}(k).

Let a and b be elements of M_{n}(k) again. If b is self-adjoint with respect to the given involution on M_{n}(k), so that b^{∗} = b, then (a, b)_{M_{n}(k)} = tr(a b). (11.8.7) If a is self-adjoint and b is arbitrary, then (a, b)_{M_{n}(k)} = (b, a)_{M_{n}(k)} = tr(b a) = tr(a b). (11.8.8) If a is arbitrary and b is anti-self-adjoint, so that b^{∗} = −b, then (a, b)_{M_{n}(k)} = − tr(a b). (11.8.9) If a is anti-self-adjoint and b is arbitrary, then (a, b)_{M_{n}(k)} = (b, a)_{M_{n}(k)} = − tr(b a) = − tr(a b). (11.8.10)

If c ∈ M_{n}(k), then put C_{c}(a) = [c, a] = c a − a c = L_{c}(a) − R_{c}(a) (11.8.11) for every a ∈ M_{n}(k), so that C_{c} is a homomorphism from M_{n}(k) into itself, as a module over k. Note that (C_{c}(a), b)_{M_{n}(k)} = (a, C_{c^{∗}}(b))_{M_{n}(k)} (11.8.12) for every a, b ∈ M_{n}(k), by (11.8.5) and (11.8.6). Of course, (C_{c}(a))^{∗} = ([c, a])^{∗} = −[c^{∗}, a^{∗}] = −C_{c^{∗}}(a^{∗}) (11.8.13) for every a ∈ M_{n}(k).

Suppose that 1 + 1 has a multiplicative inverse in k. If a ∈ M_{n}(k) is selfadjoint and b ∈ M_{n}(k) is anti-self-adjoint, then (a, b)_{M_{n}(k)} = 0, (11.8.14) by (11.8.8) and (11.8.9). This also holds when a is anti-self-adjoint and b is self-adjoint, by (11.8.3).

11.9 Inner products over ordered fields

Let k be an ordered field, as in Section 8.12, and let V be a vector space over k. Also let ⟨v, w⟩_{V} be an inner product on V, which is to say a symmetric bilinear form on V that is positive definite, in the sense that ⟨v, v⟩_{V} > 0 (11.9.1) for every v ∈ V with v ≠ 0. As usual, v, w ∈ V are said to be orthogonal with respect to ⟨·, ·⟩_{V} when ⟨v, w⟩_{V} = 0. (11.9.2)

Suppose that e_{1}, …, e_{n} are finitely many nonzero vectors in V that are pairwise orthogonal, so that ⟨e_{j}, e_{l}⟩_{V} = 0 (11.9.3)

when j ≠ l. If v ∈ V, then put

w = ∑_{j=1}^{n} \frac{⟨v, e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} e_{j}, (11.9.4)

which is an element of the linear span of e_{1}, …, e_{n}. Observe that

⟨w, e_{l}⟩_{V} = ⟨v, e_{l}⟩_{V} (11.9.5)

for every l = 1, …, n. Equivalently, ⟨w − v, e_{l}⟩_{V} = 0 for each l = 1, …, n, so that ⟨w − v, u⟩_{V} = 0 (11.9.6)

for every u in the linear span of e_{1}, …, e_{n} in V. If v is in the linear span of e_{1}, …, e_{n} in V, then we get that

v = w, (11.9.7)

because w is in the linear span of e_{1}, …, e_{n} in V, by construction.

Suppose from now on in this section that V has positive finite dimension n, as a vector space over k. In this case, there are nonzero pairwise-orthogonal vectors e_{1}, …, e_{n} in V whose linear span is equal to V, so that they form a basis for V as a vector space over k. This can be obtained from the Gram–Schmidt process, using the remarks in the preceding paragraph. If v ∈ V, then we get that

v = ∑_{j=1}^{n} \frac{⟨v, e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} e_{j}, (11.9.8)

as before. Of course, if k = R, then we can take the e_{j}’s to be orthonormal in V, so that ⟨e_{j}, e_{j}⟩_{V} = 1 for every j = 1, …, n.

Remember that L(V) denotes the algebra of linear mappings from V into itself. If T ∈ L(V), then there is a unique adjoint mapping T^{∗} ∈ L(V) such that ⟨T (v), w⟩_{V} = ⟨v, T^{∗}(w)⟩_{V} (11.9.9)

for every v, w ∈ V, as usual. We have also seen that T ↦ T^{∗} is an algebra involution on L(V). Observe that

T (e_{l}) = ∑_{j=1}^{n} \frac{⟨T (e_{l}), e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} e_{j} (11.9.10)

for each l = 1, …, n, so that

tr_{V} T = ∑_{j=1}^{n} \frac{⟨T (e_{j}), e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} . (11.9.11)

11.9. INNER PRODUCTS OVER ORDERED FIELDS 243

This implies that

tr_{V} T^{∗} = ∑_{j=1}^{n} \frac{⟨T^{∗}(e_{j}), e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} = ∑_{j=1}^{n} \frac{⟨e_{j}, T (e_{j})⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} = tr_{V} T. (11.9.12)

If T_{1}, T_{2} ∈ L(V), then put ⟨T_{1}, T_{2}⟩_{L(V)} = tr_{V} (T_{1} ∘ T_{2}^{∗}) = tr_{V} (T_{2}^{∗} ∘ T_{1}), (11.9.13) which defines a bilinear form on L(V). Using (11.9.12), we get that ⟨T_{1}, T_{2}⟩_{L(V)} = tr_{V} (T_{1} ∘ T_{2}^{∗})^{∗} = tr_{V} (T_{2} ∘ T_{1}^{∗}) = ⟨T_{2}, T_{1}⟩_{L(V)}, (11.9.14) as in the previous section. Observe that

(11.9.15) ⟨T, T ⟩_{L(V)} = tr_{V} (T^{∗} ∘ T) = ∑_{j=1}^{n} \frac{⟨(T^{∗}(T (e_{j})), e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}}
= ∑_{j=1}^{n} \frac{⟨T (e_{j}), T (e_{j})⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}}

for every T ∈ L(V). If T ≠ 0, then it follows that ⟨T, T ⟩_{L(V)} > 0, (11.9.16) because each of the terms on the right side of (11.9.15) is greater than or equal to 0 in k, and at least one term is strictly positive. Thus (11.9.13) defines an inner product on L(V), as a vector space over k.

Suppose that T_{1} ∈ L(V) is self-adjoint and T_{2} ∈ L(V) is anti-self-adjoint, so that T_{1}^{∗} = T_{1} and T_{2}^{∗} = −T_{2}. Under these conditions, we have that ⟨T_{1}, T_{2}⟩_{L(V)} = −tr_{V} (T_{2} ∘ T_{1}) = −⟨T_{2}, T_{1}⟩_{L(V)}. (11.9.17) This means that ⟨T_{1}, T_{2}⟩_{L(V)} = 0. (11.9.18)

If A ∈ L(V), then put L_{A}(T) = A ∘ T and R_{A}(T) = T ∘ A (11.9.19) for every T ∈ L(V), as before. Observe that

⟨L_{A}(T_{1}), T_{2}⟩_{L(V)} = tr_{V} (A ∘ T_{1} ∘ T_{2}^{∗}) = tr_{V} (T_{1} ∘ T_{2}^{∗} ∘ A)
(11.9.20)  = tr_{V} (T_{1} ∘ (A^{∗} ∘ T_{2})^{∗}) = ⟨T_{1}, L_{A^{∗}}(T_{2})⟩_{L(V)}

and

⟨R_{A}(T_{1}), T_{2}⟩_{L(V)} = tr_{V} (T_{1} ∘ A ∘ T_{2}^{∗}) = tr_{V} (T_{1} ∘ (T_{2} ∘ A^{∗})^{∗})
(11.9.21)  = ⟨T_{1}, R_{A^{∗}}(T_{2})⟩_{L(V)}

for every T_{1}, T_{2} ∈ L(V). Put

C_{A}(T) = [A, T] = A ∘ T − T ∘ A = L_{A}(T) − R_{A}(T) (11.9.22)

for every T ∈ L(V). Thus

⟨C_{A}(T_{1}), T_{2}⟩_{L(V)} = ⟨T_{1}, C_{A^{∗}}(T_{2})⟩_{L(V)} (11.9.23)

for every T_{1}, T_{2} ∈ L(V), by (11.9.21) and (11.9.22).

11.10 Self-adjoint Lie subalgebras

Let us continue with the same notation and hypotheses as in the previous section. As before, we use gl(V) to denote L(V), considered as a Lie algebra over k with respect to the corresponding commutator bracket. If R ∈ gl(V), then put C_{R}(T) = [R, T] = R ∘ T − T ∘ R (11.10.1)

for every T ∈ gl(V), as in (11.9.22), which is the same as ad_{gl(V),R}(T). Remember that ⟨C_{R}(T_{1}), T_{2}⟩_{L(V)} = ⟨T_{1}, C_{R^{∗}}(T_{2})⟩_{L(V)} (11.10.2)

for every T_{1}, T_{2} ∈ gl(V), as in (11.9.23). This means that C_{R^{∗}} is the adjoint of C_{R} as a linear mapping from gl(V) = L(V) into itself, with respect to the inner product (11.9.13).

Let A be a Lie subalgebra of gl(V), as a Lie algebra over k.. If R ∈ A, then C_{R} maps A into itself, and the restriction of C_{R} to A is the same as ad_{A,R}. If R_{1}, R_{2} ∈ A, then

(11.10.3) tr_{A}(ad_{A,R_{1}} ∘ ad_{A,R_{2}}) = tr_{A}(C_{R_{1}} ∘ C_{R_{2}})

is the same as the Killing form on A evaluated at R_{1}, R_{2}. This is the trace of ad_{A,R_{1}} ∘ ad_{A,R_{2}} as a linear mapping from A into itself, where A is considered as a finite-dimensional vector space over k. In the right side of (11.10.3), one should use the restrictions of C_{R_{1}} and C_{R_{2}} to A, as indicated by taking the trace is taken over A.

If A is any subset of L(V), then put

A^{∗} = {T^{∗} : T ∈ A}, (11.10.4)

which is a subset of L(V) as well. Let us say that A is self-adjoint as a subset of L(V) when A^{∗} = A. Let A be a Lie subalgebra of gl(V) again, and suppose that A is self-adjoint as a subset of L(V). If R ∈ A, then R^{∗} ∈ A, so that C_{R} and C_{R^{∗}} both map A into itself. The restriction of C_{R^{∗}} to A is the same as the adjoint of the restriction of C_{R} to A, with respect to the restriction of the inner product (11.9.13) on L(V) to A.

Let L(A) be the space of all linear mappings from A into itself, as usual. The restriction of the inner product (11.9.13) on L(V) to A defines an inner product on A, as a vector space over k. If Z ∈ L(A), then let Z^{A,∗} ∈ L(A) be the adjoint of Z with respect to the inner product on A just mentioned. Put

⟨Z_{1}, Z_{2}⟩_{L(A)} = tr_{A}(Z_{1} ∘ Z_{2}^{A,∗}) (11.10.5)

for every Z_{1}, Z_{1} ∈ L(A). This defines an inner product on L(A), as a vector space over k, as in the previous section.

Let R_{1}, R_{2} ∈ A be given, and remember that ad_{A,R_{2}^{∗}} is the same as the adjoint of ad_{A,R_{2}} with respect to the inner product on A mentioned in the preceding paragraph. It follows that

(11.10.6) tr_{A}(ad_{A,R_{1}} ∘ ad_{A,R_{2}}) = ⟨ad_{A,R_{1}}, ad_{A,R_{2}^{∗}}⟩_{L(A)},

where the right side is as in (11.10.5). In particular,

(11.10.7) tr_{A}(ad_{A,R} ∘ ad_{A,R^{∗}}) = ⟨ad_{A,R}, ad_{A,R}⟩_{L(A)}

for every R ∈ A. The right side is automatically greater than or equal to 0 in k, because (11.10.5) is an inner product on L(A). More precisely, the right side of (11.10.7) is equal to 0 exactly when ad_{A,R} = 0 as a linear mapping from A into itself.

11.11 Comparing involutions

Let A be a commutative group, where the group operations are expressed additively, and let x ↦ x^{∗,1} (11.11.1)

and x ↦ x^{∗,2} (11.11.2)

be group homomorphisms from A into itself. Suppose for the moment that these two group homomorphisms commute on A, so that

(x^{∗,1})^{∗,2} = (x^{∗,2})^{∗,1} (11.11.3)

for every x ∈ A. Of course, one can define self-adjointness and anti-self-adjointness of elements of A with respect to (11.11.1) and (11.11.2) in the usual way. If x ∈ A is self-adjoint with respect to (11.11.2), then it follows that x^{∗,1} is self-adjoint with respect to (11.11.2) as well. Similarly, if x is anti-self-adjoint with respect to (11.11.2), then x^{∗,1} is anti-self-adjoint with respect to (11.11.2).

Now let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e, where multiplication of a, b ∈ A is expressed as a b. Suppose that (11.11.1) is an algebra involution on A, and let c be an invertible element of A. Put

x^{∗,2} = c^{−1} x^{∗,1} c (11.11.4) for every x ∈ A. This defines (11.11.2) as an opposite algebra automorphism on A. Let us suppose from now on in this section that

c^{∗,1} = c (11.11.5)

or c^{∗,1} = −c. (11.11.6)

In either case, (11.11.2) is an algebra involution on A as well, as in Section 3.14. Note that c^{∗,2} = c (11.11.7)

when (11.11.5) holds, and that

c^{∗,2} = −c (11.11.8)

when (11.11.6) holds.

If x ∈ A, then (x^{∗,1})^{∗,2} = c^{−1} (x^{∗,1})^{∗,1} c = c^{−1} x c (11.11.9)

and

(x^{∗,2})^{∗,1} = (c^{−1} x^{∗,1} c)^{∗,1} = c^{∗,1} (x^{∗,1})^{∗,1} (c^{∗,1})^{−1} = c^{∗,1} x (c^{∗,1})^{−1}. (11.11.10)

This reduces to (x^{∗,2})^{∗,1} = c x c^{−1} (11.11.11)

when c satisfies (11.11.5) or (11.11.6). If

c^{2} = t e (11.11.12)

for some t ∈ k with a multiplicative inverse in k, then we get that (11.11.3) holds. If k = C, then (11.11.1) may be conjugate-linear, as usual.

Let A_{1} be a subalgebra of A, and suppose that (11.11.1) maps A_{1} into itself. There are some situations where c is not necessarily in A_{1}, but c^{−1} x c ∈ A_{1} for every x ∈ A_{1}. If x ∈ A_{1}, then it follows that (11.11.4) is an element of A_{1} as well.

11.12 Some projections

Let k be a field, let V be a vector space over k, and let b(v, w) be a bilinear form on V. Also let V_{0} be a finite-dimensional linear subspace of V, and suppose that the restriction of b(v, w) to v, w ∈ V_{0} is nondegenerate on V_{0}. If z ∈ V, then put λ_{z}(v) = b(v, z) (11.12.1)

for every v ∈ V_{0}, which defines a linear functional on V_{0}. Because b(·, ·) is nondegenerate on V_{0}, there is a unique element P_{0}(z) of V_{0} such that

λ_{z}(v) = b(v, P_{0}(z)) (11.12.2)

11.12. SOME PROJECTIONS 247

for every v ∈ V_{0}. Equivalently, this means that b(v, z − P_{0}(z)) = 0 (11.12.3) for every v ∈ V. It is easy to see that P_{0} defines a linear mapping from V into V_{0}. More precisely, P_{0}(z) = z (11.12.4) when z ∈ V_{0}, so that P_{0} maps V onto V_{0}. It follows that P_{0} ∘ P_{0} = P_{0} (11.12.5) on V, so that P_{0} is a projection on V.

Let Z_{0} be the kernel of P_{0}, which is a linear subspace of V. Note that V_{0} ∩ Z_{0} = {0}, (11.12.6) by (11.12.4). If z ∈ V, then z − P_{0}(z) ∈ Z_{0}, (11.12.7) by (11.12.5). Thus z ∈ V_{0} + Z_{0}, so that V = V_{0} + Z_{0}. This shows that V corresponds to the direct sum of V_{0} and Z_{0}, as a vector space over k. If z ∈ Z_{0}, then b(v, z) = 0 (11.12.8) for every v ∈ V_{0}, by (11.12.3). This is equivalent to (11.12.3), because of (11.12.7).

Suppose that b(·, ·) is either symmetric or antisymmetric on V. This implies that b(z, v) = 0 (11.12.9) for every v ∈ V_{0} and z ∈ Z_{0}, by (11.12.8). If u, w ∈ V, then P_{0}(u), P_{0}(w) ∈ V_{0}, u − P_{0}(u), w − P_{0}(w) ∈ Z_{0}, and hence b(P_{0}(u), w − P_{0}(w)) = b(u − P_{0}(u), P_{0}(w)) = 0, (11.12.10) by (11.12.8) and (11.12.9). It follows that b(u, w) = b(P_{0}(u), P_{0}(w)) + b(u − P_{0}(u), w − P_{0}(w)). (11.12.11) Thus b(·, ·) corresponds, as a bilinear form on V, to the bilinear form on the direct sum of V_{0} and Z_{0} obtained from the restrictions of b(·, ·) to V_{0} and Z_{0}.

Suppose for the moment that b(w, w) = 0 (11.12.12) for every v ∈ V. Remember that this implies that b(·, ·) is antisymmetric on V, and that the converse holds when the characteristic of k is different from 2, as in Section 2.1. Let x, y be elements of V such that b(x, y) ≠ 0. (11.12.13)

This implies that x and y are linearly independent in V, because of (11.12.12). Thus the linear span V_{0} of x and y in V is a two-dimensional linear subspace of V. It is easy to see that the restriction of b(·, ·) to V_{0} is nondegenerate on V_{0}. It follows that there is a complementary linear subspace Z_{0} of V_{0} with the properties discussed in the previous paragraphs.

Suppose that V has finite dimension as a vector space over k. If b(·, ·) ≢ 0 on V, then there are x, y ∈ V as in the preceding paragraph. Repeating the process, we get that V corresponds to the direct sum of finitely many two-dimensional linear subspaces on which b(·, ·) is nondegenerate, and possibly an additional linear subspace on which b(·, ·) ≡ 0. This additional linear subspace is not needed when b(·, ·) is nondegenerate on V. By construction, b(·, ·) corresponds to the bilinear form on this direct sum obtained from the restrictions of b(·, ·) to these linear subspaces.

Let V be any vector space over k again, and suppose that b(·, ·) is symmetric on V. Let x be an element of V such that

b(x, x) ≠ 0, (11.12.14)

and let V_{0} be the linear span of x in V. Clearly V_{0} is a one-dimensional linear subspace of V, and the restriction of b(·, ·) to V_{0} is nondegenerate on V_{0}. This leads to a complementary subspace Z_{0} of V_{0} in V as before, so that b(·, ·) corresponds to the bilinear form on the direct sum of V_{0} and Z_{0} obtained from the restrictions of b(·, ·) to V_{0} and Z_{0}.

Suppose that V has finite dimension as a vector space over k again. Repeating the argument from the preceding pagraph, we get that V can be expressed as the direct sum of finitely many one-dimensional linear subspaces on which b(·, ·) ≢ 0, and possibly an additional linear subspace W such that (11.12.12) holds for every w ∈ W. As before, b(·, ·) corresponds to the bilinear form on the direct sum obtained from the restrictions of b(·, ·) to these linear subspaces. If the characteristic of k is different from 2, then b(·, ·) ≡ 0 on W, because b(·, ·) is both symmetric and antisymmetric on W. In this case, the additional subspace W is not needed when b(·, ·) is nondegenerate on V.

11.13 Antisymmetric bilinear forms

Let k be a field with characteristic different from 2, and let V be a vector space over k of positive finite dimension. Also let β(·, ·) be a nondegenerate antisymmetric bilinear form on V. Under these conditions, there is a basis for V consisting of vectors x_{1}, …, x_{n}, y_{1}, …, y_{n} for some positive integer n with the following properties. First,

β(x_{j}, x_{l}) = β(y_{j}, y_{l}) = 0 (11.13.1)

for every j, l = 1, …, n, and β(x_{j}, y_{l}) = 0 (11.13.2)

11.13. ANTISYMMETRIC BILINEAR FORMS 249

when j ≠ l. Second, for each j = 1, …, n, β(x_{j}, y_{j}) ≠ 0. (11.13.3) This well-known representation for β(·, ·) follows from some of the remarks in the previous section. More precisely, we may ask that β(x_{j}, y_{j}) = 1 (11.13.4) for every j = 1, …, n, by multiplying the x_{j}’s or y_{j}’s by suitable nonzero elements of k, if necessary.

If v ∈ V, then v can be expressed in a unique way as

v = ∑_{j=1}^{n} v_{x_{j}} x_{j} + ∑_{j=1}^{n} v_{y_{j}} y_{j}, (11.13.5)

where v_{x_{j}}, v_{y_{j}} ∈ k for every j = 1, …, n. Put

⟨v, w⟩_{V} = ∑_{j=1}^{n} v_{x_{j}} w_{x_{j}} + ∑_{j=1}^{n} v_{y_{j}} w_{y_{j}} (11.13.6)

for every v, w ∈ V, where w_{x_{j}}, w_{y_{j}} ∈ k correspond to w as in (11.13.5). This defines a symmetric bilinear form on V, with

⟨v, v⟩_{V} = ∑_{j=1}^{n} v_{x_{j}}^{2} + ∑_{j=1}^{n} v_{y_{j}}^{2} (11.13.7)

for every v ∈ V. If k is an ordered field, then (11.13.7) is strictly positive when v ≠ 0, and hence (11.13.6) defines an inner product on V. Of course, x_{1}, …, x_{n}, y_{1}, …, y_{n} is an orthonormal basis for V with respect to (11.13.6).

Let B be the unique linear mapping from V into itself that satisfies B(x_{j}) = y_{j}, B(y_{j}) = −x_{j} (11.13.8) for every j = 1, …, n. It is easy to see that B^{2} = −I_{V} , (11.13.9) where I_{V} is the identity mapping on V. One can check that β(v, w) = ⟨B(v), w⟩_{V} (11.13.10) for every v, w ∈ V, by reducing to the cases where v = x_{j} or y_{j} and w = x_{l} or y_{l}, 1 ≤ j, l ≤ n. We also have that B^{∗} = −B, (11.13.11) where B^{∗} is the adjoint of B with respect to (11.13.6). This corresponds to the antisymmetry of β(·, ·) on V, because of (11.13.10).

If T is a linear mapping from V into itself, then the adjoint T^{∗,β} of T with respect to β(·, ·) is the unique linear mapping from V into itself such that

β(T (v), w) = β(v, T^{∗,β}(w)) (11.13.12)

for every v, w ∈ V. This is the same as saying that

⟨B(T (v)), w⟩_{V} = ⟨B(v), T^{∗,β}(w)⟩_{V} (11.13.13)

for every v, w ∈ V, by (11.13.10). This is equivalent to

⟨B(T (v)), w⟩_{V} = −⟨v, B(T^{∗,β}(w))⟩_{V} (11.13.14)

for every v, w ∈ V, because of (11.13.11). This means that

(B ∘ T)^{∗} = −B ∘ T^{∗,β}, (11.13.15)

where the left side is the adjoint of B ∘ T with respect to the inner product (11.13.6). Thus

T^{∗,β} = −B^{−1} ∘ (B ∘ T)^{∗} = −B^{−1} ∘ T^{∗} ∘ B^{∗} = B^{−1} ∘ T^{∗} ∘ B. (11.13.16)

11.14 Symmetric bilinear forms

Let k be an ordered field, and let V be a vector space over k of positive finite dimension n. In this section, we consider a nondegenerate symmetric bilinear form β(·, ·) on V. It is well known that there is a basis e_{1}, …, e_{n} for V such that β(e_{j}, e_{l}) = 0 (11.14.1)

when j ≠ l, and for each j = 1, …, n,

β(e_{j}, e_{j}) ≠ 0, (11.14.2)

as in Section 11.12. Remember that the absolute value |t| of t ∈ k can be defined as an element of k, as in Section 8.13. If k = R, then one can choose the e_{j}’s so that β(e_{j}, e_{j}) = ±1 for each j = 1, …, n.

If v ∈ V, then let

v = ∑_{j=1}^{n} v_{j} e_{j} (11.14.3)

be the unique representation of v as a linear combination of the e_{j}’s with coefficients v_{j} ∈ k. Put

⟨v, w⟩_{V} = ∑_{j=1}^{n} |β(e_{j}, e_{j})| v_{j} w_{j} (11.14.4)

for every v, w ∈ V, where w_{j} ∈ k corresponds to w as in (11.14.3). This defines a symmetric bilinear form on V, and

⟨v, v⟩_{V} = ∑_{j=1}^{n} |β(e_{j}, e_{j})| v_{j}^{2} (11.14.5)

11.14. SYMMETRIC BILINEAR FORMS 251 for every v ∈ V. If v ≠ 0, then (11.14.5) is strictly positive in k, so that (11.14.4) is an inner product on V. Note that the e_{j}’s are pairwise orthogonal with respect to (11.14.4).

Let B be the unique linear mapping from V into itself that satisfies

(11.14.6)  B(e_{j}) = e_{j} when β(e_{j}, e_{j}) > 0
= −e_{j} when − β(e_{j}, e_{j}) > 0.

Clearly B^{2} = I_{V} , (11.14.7) where I_{V} is the identity mapping on V. It is easy to see that B^{∗} = B, (11.14.8) where B^{∗} is the adjoint of B with respect to the inner product (11.14.4). Of course,

⟨B(v), w⟩_{V} = ∑_{j=1}^{n} β(e_{j}, e_{j}) v_{j} w_{j} (11.14.9)

for every v, w ∈ V, by construction. One can check that β(v, w) = ⟨B(v), w⟩_{V} (11.14.10) for every v, w ∈ V, by reducing to the case where v = e_{j} and w = e_{l}, 1 ≤ j, l ≤ n.

If T is a linear mapping from V into itself, then the adjoint T^{∗,β} of T with respect to β is the unique linear mapping from V into itself such that β(T (v), w) = β(v, T^{∗,β}(w)) (11.14.11) for every v, w ∈ V, as before. This means that ⟨B(T (v)), w⟩_{V} = ⟨B(v), T^{∗,β}(w)⟩_{V} (11.14.12) for every v, w ∈ V, because of (11.14.10). This is the same as saying that ⟨B(T (v)), w⟩_{V} = ⟨v, B(T^{∗,β}(w))⟩_{V} (11.14.13) for every v, w ∈ V, by (11.14.8). This is equivalent to asking that (B ∘ T)^{∗} = B ∘ T^{∗,β}, (11.14.14) where the left side is the adjoint of B ∘ T with respect to the inner product (11.14.4). This shows that T^{∗,β} = B^{−1} ∘ (B ∘ T)^{∗} = B^{−1} ∘ T^{∗} ∘ B^{∗} = B^{−1} ∘ T^{∗} ∘ B. (11.14.15)

Some complex versions

12.1 Complexifying ordered fields

Let k be a ordered ring, as in Section 8.12, and suppose that k is commutative and has a multiplicative identity element 1 = 1_{k}. Also let a be an element of k such that a > 0. Thus, for each x ∈ k, we have that x^{2} ≠ −a, because x^{2} ≥ 0. As usual, we can get a commutative ring k[\sqrt{−a}] by adjoining a square root \sqrt{−a} of −a to k. More precisely, we can define k[\sqrt{−a}] to be the space k^{2} of ordered pairs of elements of k. An element of k[\sqrt{−a}] may be expressed in a unique way as z = x + y \sqrt{−a}, (12.1.1)

with x, y ∈ k, which corresponds to (x, y) ∈ k^{2}. Let us identify x ∈ k with x + 0 \sqrt{−a} in k[\sqrt{−a}], so that k corresponds to a subset of k[\sqrt{−a}]. Let

w = u + v \sqrt{−a} (12.1.2)

be another element of k[\sqrt{−a}], with u, v ∈ k. Addition and multiplication on k can be extended to k[\sqrt{−a}] by putting

z + w = (x + u) + (y + v) \sqrt{−a} (12.1.3)

and z w = (x u − y v a) + (x v + y u) \sqrt{−a}, (12.1.4)

as usual. One can verify that k[\sqrt{−a}] is a commutative ring, and that k corresponds to a subring of k[\sqrt{−a}]. The multiplicative identity element 1 in k corresponds to the multiplicative identity element in k[\sqrt{−a}]. Of course, (\sqrt{−a})^{2} = −a in k[\sqrt{−a}], by construction.

If z ∈ k[\sqrt{−a}] is as in (12.1.1), then the conjugate \overline{z} of z in k[\sqrt{−a}] is defined as usual by \overline{z} = x − y \sqrt{−a}. (12.1.5)

One can verify that \overline{z + w} = \overline{z} + \overline{w} (12.1.6)

12.2. SESQUILINEARITY OVER K[\sqrt{−A}] 253

and \overline{z w} = \overline{z} \overline{w} (12.1.7)

for every z, w ∈ k[\sqrt{−a}]. Clearly

\overline{(\overline{z})} = z (12.1.8)

for every z ∈ k[\sqrt{−a}], and \overline{z} = z exactly when z corresponds to an element of k. If z ∈ k[\sqrt{−a}] is as in (12.1.1) again, then

z + \overline{z} = 2 · x (12.1.9)

and z \overline{z} = x^{2} + a y^{2} (12.1.10)

correspond to elements of k. Note that

z \overline{z} > 0 (12.1.11)

in k when z ≠ 0.

Suppose now that k is an ordered field. If z ∈ k[\sqrt{−a}] and z ≠ 0, then z \overline{z} corresponds to a nonzero element of k, which has a multiplicative inverse in k. This implies that z has a multiplicative inverse in k[\sqrt{−a}], which is \overline{z} (1/(z \overline{z})). Thus k[\sqrt{−a}] is a field in this case as well. If k is a subfield of R, then we can take \sqrt{−a} to be i \sqrt{a} ∈ C, so that k[\sqrt{−a}] corresponds to a subfield of C.

Let k be an ordered field again, let a be a positive element of k, and let k[\sqrt{−a}] be as before. Also let V and W be vector spaces over k[\sqrt{−a}], which may be considered as vector spaces over k too. A linear mapping T from V into W, as vector spaces over k[\sqrt{−a}], may be called k[\sqrt{−a}]-linear, or linear over k[\sqrt{−a}], as usual. Similarly, if T is a linear mapping from V into W as vector spaces over k, the we may say that T is k-linear, or linear over k. Thus a mapping T from V into W is k[\sqrt{−a}]-linear if and only if T is k-linear and

T (\sqrt{−a} v) = \sqrt{−a} T (v) (12.1.12)

for every v ∈ V. A k-linear mapping T from V into W is said to be conjugatelinear if T (\sqrt{−a} v) = −\sqrt{−a} T (v) (12.1.13)

for every v ∈ V. This implies that

T (t v) = \overline{t} T (v) (12.1.14)

for every t ∈ k[\sqrt{−a}] and v ∈ V.

12.2 Sesquilinearity over k[\sqrt{−a}]

Let k be an ordered field, let a be a positive element of k, and let k[\sqrt{−a}] be as in the previous section. Also let V be a vector space over k, and let b(v, w) be a function defined for v, w ∈ V with values in k[\sqrt{−a}]. We say that b(·, ·) is sesquilinear if b(v, w) is k[\sqrt{−a}]-linear in v for every w ∈ V, and conjugatelinear in w for every v ∈ V. In particular, this means that b(·, ·) is bilinear over k, where V and k[\sqrt{−a}] are considered as vector spaces over k. As before, b(·, ·) is said to be Hermitian-symmetric on V if

b(w, v) = \overline{b(v, w)} (12.2.1)

for every v, w ∈ V.

Let b(·, ·) be a sesquilinear form on V, and let T be a k[\sqrt{−a}]-linear mapping from V into itself. As before, we say that T is self-adjoint with respect to b(·, ·) if b(T (v), w) = b(v, T (w)) (12.2.2)

for every v, w ∈ V, and we say that T is anti-self-adjoint with respect to b(·, ·) if b(T (v), w) = −b(v, T (w)) (12.2.3)

for every v, w ∈ V. It is easy to see that T is anti-self-adjoint with respect to b(·, ·) if and only if \sqrt{−a} T is self-adjoint with respect to b(·, ·). Remember that the space L(V) of k[\sqrt{−a}]-linear mappings from V into itself is a vector space over k[\sqrt{−a}], and thus may be considered as a vector space over k too. The spaces of self-adjoint and anti-self-adjoint linear mappings from V into itself with respect to b(·, ·) are k-linear subspaces of L(V), which is to say that they are linear subspaces of L(V), as a vector space over k. If T_{1}, T_{2} ∈ L(V) are anti-self-adjoint with respect to b(·, ·), then their commutator [T_{1}, T_{2}] = T_{1} ∘ T_{2} − T_{2} ∘ T_{1} with respect to composition is anti-self-adjoint with respect to b(·, ·) as well. This means that the space of anti-self-adjoint linear mappings from V into itself with respect to b(·, ·) is a Lie subalgebra of gl(V) as a Lie algebra over k with respect to the commutator bracket.

Suppose from now on in this section that V has finite dimension as a vector space over k[\sqrt{−a}]. If w ∈ V, then

b_{w}(v) = b(v, w) (12.2.4)

defines a linear functional on V, and w ↦ b_{w} is a conjugate-linear mapping from V into its dual space V′. The image

{b_{w} : w ∈ V } (12.2.5)

of this mapping is a linear subspace of V′, as a vector space over k[\sqrt{−a}]. As before, one can verify that (12.2.5) is equal to V′ exactly when w ↦ b_{w} is injective, because V and V′ have the same dimension as vector spaces over k[\sqrt{−a}].

We say that b(·, ·) is nondegenerate as a sesquilinear form on V if for every v ∈ V with v ≠ 0 there is a w ∈ V such that b(v, w) ≠ 0. Equivalently, this means that the intersections of the kernels of the b_{w}’s, w ∈ V, is the trivial subspace of V. This holds exactly when (12.2.5) is equal to V′, as in the complex case.

12.3. INNER PRODUCTS OVER K[\sqrt{−A}] 255

Let b(·, ·) be a nondegenerate sesquilinear form on V, let T be a k[\sqrt{−a}]- linear mapping from V into itself, and let w ∈ V be given. Thus b(T (v), w) defines a linear functional on V, as a function of v, so that there is a unique element T^{∗}(w) of V such that

b(T (v), w) = b(v, T^{∗}(w)) (12.2.6)

for every v ∈ V. One can check that T^{∗} is a k[\sqrt{−a}]-linear mapping from V into itself, which is called the adjoint of T with respect to b(·, ·). However, T ↦ T^{∗} is conjugate-linear as a mapping from L(V) into itself, as a vector space over k[\sqrt{−a}]. Note that T is self-adjoint with respect to b(·, ·) when T^{∗} = T, and that T is anti-self-adjoint with respect to b(·, ·) when T^{∗} = −T. If T_{1}, T_{2} ∈ L(V), then one can check that (T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗}, (12.2.7)

as usual. If b(·, ·) is Hermitian-symmetric on V, then one can verify that

(T^{∗})^{∗} = T (12.2.8)

for every T ∈ L(V), as in Section 2.15.

12.3 Inner products over k[\sqrt{−a}]

Let k be an ordered field again, let a be a positive element of k, and let k[\sqrt{−a}] be as in Section 12.1. Also let V be a vector space over k[\sqrt{−a}], and let ⟨·, ·⟩_{V} be a Hermitian form on V, which is to say a Hermitian-symmetric sesquilinear form on V. Note that ⟨v, v⟩_{V} ∈ k (12.3.1)

for every v ∈ V, because \overline{⟨v, v⟩_{V}} = ⟨v, v⟩_{V}, by Hermitian symmetry. Suppose that ⟨v, v⟩_{V} > 0 (12.3.2)

for every v ∈ V with v ≠ 0, in which case ⟨·, ·⟩_{V} is said to be an inner product on V. If v, w ∈ V satisfy ⟨v, w⟩_{V} = 0, (12.3.3)

then v and w are said to be orthogonal with respect to ⟨·, ·⟩_{V}, as usual.

Let e_{1}, …, e_{n} be finitely many pairwise-orthogonal nonzero vectors in V, and let v ∈ V be given. Thus

w = ∑_{j=1}^{n} \frac{⟨v, e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} e_{j} (12.3.4)

is an element of the linear span of e_{1}, …, e_{n} in V, and

⟨w, e_{l}⟩_{V} = ⟨v, e_{l}⟩_{V} (12.3.5) for every l = 1, …, n. This means that ⟨w − v, e_{l}⟩_{V} = 0 for each l = 1, …, n, so that ⟨w − v, u⟩_{V} = 0 (12.3.6)

for every element u of the linear span of e_{1}, …, e_{n} in V. If v is an element of the linear span of e_{1}, …, e_{n} in V, then it follows that v = w.

Let us suppose from now on in this section that V has positive finite dimension n, as a vector space over k[\sqrt{−a}]. Using the Gram–Schmidt process, one can get nonzero pairwise-orthogonal vectors e_{1}, …, e_{n} in V that form a basis for V. If v ∈ V, then

v = ∑_{j=1}^{n} \frac{⟨v, e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} e_{j}, (12.3.7)

as in the preceding paragraph. Let T be a linear mapping from V into itself, so that there is a unique T^{∗} ∈ L(V) such that

⟨T (v), w⟩_{V} = ⟨v, T^{∗}(w)⟩_{V} (12.3.8)

for every v, w ∈ V, as before. Of course,

T (e_{l}) = ∑_{j=1}^{n} \frac{⟨T (e_{l}), e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} e_{j} (12.3.9)

for every l = 1, …, n, as in (12.3.7). This implies that

tr_{V} T = ∑_{j=1}^{n} \frac{⟨T (e_{j}), e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} . (12.3.10)

Thus

(12.3.11) tr_{V} T^{∗} = ∑_{j=1}^{n} \frac{⟨T^{∗}(e_{j}), e_{j}⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} = ∑_{j=1}^{n} \frac{⟨e_{j}, T (e_{j})⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}}

= ∑_{j=1}^{n} \frac{\overline{⟨T (e_{j}), e_{j}⟩_{V}}}{⟨e_{j}, e_{j}⟩_{V}} = \overline{tr_{V} T} .

Put ⟨T_{1}, T_{2}⟩_{L(V)} = tr_{V} (T_{1} ∘ T_{2}^{∗}) = tr_{V} (T_{2}^{∗} ∘ T_{1}) (12.3.12)

for every T_{1}, T_{2} ∈ L(V). This defines a sesquilinear form on L(V), because T ↦ T^{∗} is conjugate-linear on L(V). Observe that

\overline{⟨T_{1}, T_{2}⟩_{L(V)}} = \overline{tr_{V} (T_{1} ∘ T_{2}^{∗})} = tr_{V} (T_{1} ∘ T_{2}^{∗})^{∗}
(12.3.13)  = tr_{V} (T_{2} ∘ T_{1}^{∗}) = ⟨T_{2}, T_{1}⟩_{L(V)}

12.4. SELF-ADJOINTNESS IN GL(V) 257

for every T_{1}, T_{2} ∈ L(V), using (12.3.11) in the second step. Thus (12.3.12) is Hermitian-symmetric on L(V). If T ∈ L(V), then

(12.3.14)  ⟨T, T ⟩_{L(V)} = tr_{V} (T^{∗} ∘ T) = ∑_{j=1}^{n} \frac{⟨T^{∗}(T (e_{j}))⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}}

= ∑_{j=1}^{n} \frac{⟨T (e_{j}), T (e_{j})⟩_{V}}{⟨e_{j}, e_{j}⟩_{V}} .

This implies that ⟨T, T ⟩_{L(V)} > 0 when T ≠ 0, because each of the terms on the right side of (12.3.14) is greater than or equal to 0 in k, and at least one term is strictly positive. It follows that (12.3.12) defines an inner product on L(V), as a vector space over k[\sqrt{−a}].

Let A ∈ L(V) be given, and put

L_{A}(T) = A ∘ T, R_{A}(T) = T ∘ A (12.3.15)

for every T ∈ L(V), as before. As in Sections 11.8 and 11.9, we have that

⟨L_{A}(T_{1}), T_{2}⟩_{L(V)} = tr_{V} (A ∘ T_{1} ∘ T_{2}^{∗}) = tr_{V} (T_{1} ∘ T_{2}^{∗} ∘ A)
(12.3.16)  = tr_{V} (T_{1} ∘ (A^{∗} ∘ T_{2})^{∗}) = ⟨T_{1}, L_{A^{∗}}(T_{2})⟩_{L(V)}

and

⟨R_{A}(T_{1}), T_{2}⟩_{L(V)} = tr_{V} (T_{1} ∘ A ∘ T_{2}^{∗}) = tr_{V} (T_{1} ∘ (T_{2} ∘ A^{∗})^{∗})
(12.3.17)  = ⟨T_{1}, R_{A^{∗}}(T_{2})⟩_{L(V)}

for every T_{1}, T_{2} ∈ L(V). If we put

C_{A}(T) = [A, T] = L_{A}(T) − R_{A}(T) (12.3.18)

for each T ∈ L(V), then we get that

⟨C_{A}(T_{1}), T_{2}⟩_{L(V)} = ⟨T_{1}, C_{A^{∗}}(T_{2})⟩_{L(V)} (12.3.19)

for every T_{1}, T_{2} ∈ L(V), as before.

12.4 Self-adjointness in gl(V)

Let us continue with the same notation and hypotheses as in the previous section. We would like to consider the analogues of the remarks in Section 11.10 in this situation. Remember that gl(V) is the same as L(V), considered as a Lie algebra over k[\sqrt{−a}] with respect to the corresponding commutator bracket. If R, T ∈ gl(V), then C_{R}(T) = [R, T] = R ∘ T − T ∘ R (12.4.1)

is the same as ad_{gl(V),R}(T). As before,

⟨C_{R}(T_{1}), T_{2}⟩_{L(V)} = ⟨T_{1}, C_{R^{∗}}(T_{2})⟩_{L(V)} (12.4.2) for every R, T_{1}, T_{2} ∈ gl(V).

Let A be a Lie subalgebra of gl(V), as a Lie algebra over k[\sqrt{−a}]. Thus, for each R ∈ A, C_{R} maps A into itself, and the restriction of C_{R} to A is the same as ad_{A,R}. Let R_{1}, R_{2} ∈ A be given, so that

(12.4.3) tr_{A}(ad_{A,R_{1}} ∘ ad_{A,R_{2}}) = tr_{A}(C_{R_{1}} ∘ C_{R_{2}})

is the same as the Killing form on A evaluated at R_{1}, R_{2}, as before. This uses the trace on A, as a finite-dimensional vector space over k[\sqrt{−a}]. More precisely, one should use the restrictions of C_{R_{1}} and C_{R_{2}} to A on the right side of (12.4.3).

If A is any subset of L(V), then let A^{∗} be the subset of L(V) defined by

A^{∗} = {T^{∗} : T ∈ A}, (12.4.4)

as before. Let us say that A is self-adjoint in L(V) when A^{∗} = A. Let A be a Lie subalgebra of gl(V) again, as a Lie algebra over k[\sqrt{−a}], and suppose that A is self-adjoint in L(V). Let R ∈ A be given, so that R^{∗} ∈ A too, and C_{R} and C_{R^{∗}} both map A into itself. The restriction of C_{R^{∗}} to A is the adjoint of the restriction of C_{R} to A, as an inner product space over k[\sqrt{−a}] with respect to the restriction of the inner product (12.3.12) on L(V) to A.

Let L(A) be the space of all linear mappings from A into itself, which is in particular a vector space over k[\sqrt{−a}]. The restriction of the inner product (12.3.12) on L(V) to A defines an inner product on A as a vector space over k[\sqrt{−a}], as before. If Z ∈ L(A), then let Z^{A,∗} ∈ L(A) be the adjoint of Z with respect to this inner product. If Z_{1}, Z_{2} ∈ L(A), then put

⟨Z_{1}, Z_{2}⟩_{L(A)} = tr_{A}(Z_{1} ∘ Z_{2}^{A,∗}). (12.4.5)

This defines an inner product on L(A), as a vector space over k[\sqrt{−a}], as in the previous section.

Let R_{1}, R_{2} ∈ A be given, so that R_{2}^{∗} ∈ A as well, and ad_{A,R_{2}^{∗}} is the same as the adjoint of ad_{A,R_{2}} with respect to the inner product on A mentioned in the preceding paragraph. This implies that

(12.4.6) tr_{A}(ad_{A,R_{1}} ∘ ad_{A,R_{2}}) = ⟨ad_{A,R_{1}}, ad_{A,R_{2}^{∗}}⟩_{L(A)},

where the right side is as in (12.4.5). If R ∈ A, then we can take R_{1} = R and R_{2} = R^{∗}, to get that

(12.4.7) tr_{A}(ad_{A,R} ∘ ad_{A,R^{∗}}) = ⟨ad_{A,R}, ad_{A,R}⟩_{L(A)}.

The right side is automatically greater than or equal to 0 in k, because (12.4.5) is an inner product on L(A). Similarly, the right side is strictly positive when ad_{A,R} ≠ 0 on A.

12.5. PROJECTIONS AND SESQUILINEAR FORMS 259

12.5 Projections and sesquilinear forms

Let k be an ordered field, let a be a positive element of k again, and let k[\sqrt{−a}] be as in Section 12.1. Also let V be a vector space over the complex numbers, and let b(v, w) be a sesquilinear form on V, as in Section 12.2. Suppose that V_{0} is a finite-dimensional linear subspace of V, and that the restriction of b(v, w) to v, w ∈ V_{0} is nondegenerate on V_{0}. Let z ∈ V be given, and put λ_{z}(v) = b(v, z) (12.5.1) for every v ∈ V_{0}, so that λ_{z} defines a linear functional on V_{0}. As in Section 11.12, there is a unique element P_{0}(z) of V_{0} such that λ_{z}(v) = b(v, P_{0}(z)) (12.5.2) for every v ∈ V_{0}, because b(·, ·) is nondegenerate on V_{0}. Equivalently, b(v, z − P_{0}(z)) = 0 (12.5.3) for every v ∈ V. Of course, P_{0}(z) = z (12.5.4) when z ∈ V_{0}. One can check that P_{0} defines a linear mapping from V onto V_{0}, as before. Note that P_{0} ∘ P_{0} = P_{0}, so that P_{0} is a projection on V.

Let Z_{0} be the kernel of P_{0}, and observe that V_{0} ∩ Z_{0} = {0}, (12.5.5) by (12.5.4). We also have that z − P_{0}(z) ∈ Z_{0} for every z ∈ V, so that V = V_{0} + Z_{0}. (12.5.6) Thus V corresponds to the direct sum of V_{0} and Z_{0}, as a vector space over k[\sqrt{−a}]. Note that Z_{0} is the same as the set of z ∈ V such that b(v, z) = 0 (12.5.7) for every v ∈ V_{0}.

Suppose from now on in this section that b(·, ·) is Hermitian-symmetric on V. Using this and (12.5.7), we get that b(z, v) = 0 (12.5.8) for every v ∈ V_{0} and z ∈ Z_{0}. Let u, w ∈ V be given, so that P_{0}(u), P_{0}(w) ∈ V_{0}, u − P_{0}(u), w − P_{0}(w) ∈ Z_{0}, and b(P_{0}(u), w − P_{0}(w)) = b(u − P_{0}(u), P_{0}(w)) = 0. (12.5.9) Thus b(u, w) = b(P_{0}(u), P_{0}(w)) + b(u − P_{0}(u), w − P_{0}(w)). (12.5.10)

This means that b(·, ·) corresponds to the sesquilinear form on the direct sum of V_{0} and Z_{0} obtained from the restrictions of b(·, ·) to V_{0} and Z_{0}.

Suppose that x ∈ V satisfies

b(x, x) ≠ 0, (12.5.11)

and let V_{0} be the linear span of x in V. Thus V_{0} is a one-dimensional linear subspace of V, and the restriction of b(·, ·) to V_{0} is nondegenerate on V_{0}. It follows that V corresponds to the direct sum of V_{0} and another linear subspace Z_{0}, in such a way that b(·, ·) corresponds to the sesquilinear form on the direct sum obtained from the restrictions of b(·, ·) to V_{0} and Z_{0}.

Suppose now that V has finite dimension as a vector space over k[\sqrt{−a}]. Repeating the previous argument, we can express V as the direct sum of finitely many one-dimensional subspaces on which b(·, ·) ≢ 0, and perhaps an additional linear subspace W such that b(w, w) = 0 (12.5.12)

for every w ∈ W. We also have that b(·, ·) corresponds to the sesquilinear form on the direct sum obtained from the restrictions of b(·, ·) to these linear subspaces. One can check that b(·, ·) ≡ 0 on W, using polarization arguments. If b(·, ·) is nondegenerate on V, then this additional subspace W is not needed.

12.6 Nondegenerate Hermitian forms

Let k be an ordered field, let a be a positive element of k, and let k[\sqrt{−a}] be as in Section 12.1 again. Also let V be a vector space over k[\sqrt{−a}] of positive finite dimension n, and let β(·, ·) be a nondegenerate Hermitian form on V. Under these conditions, there is a basis e_{1}, …, e_{n} for V such that

β(e_{j}, e_{l}) = 0 (12.6.1)

when j ≠ l, and for each j = 1, …, n,

β(e_{j}, e_{j}) ≠ 0, (12.6.2)

as in the previous section. Note that

β(e_{j}, e_{j}) ∈ k (12.6.3)

for each j = 1, …, n, by Hermitian symmetry. If k = R, then one choose the e_{j}’s so that β(e_{j}, e_{j}) = ±1 for each j = 1, …, n.

Each v ∈ V can be expressed in a unique way as

v = ∑_{j=1}^{n} v_{j} e_{j}, (12.6.4)

where v_{j} ∈ k[\sqrt{−a}] for every j = 1, …, n. Remember that the absolute value |t| of t ∈ k is defined as an element of k as in Section 8.13. If v, w ∈ V, then put

⟨v, w⟩_{V} = ∑_{j=1}^{n} |β(e_{j}, e_{j})| v_{j} \overline{w_{j}}, (12.6.5)

where w_{j} ∈ k[\sqrt{−a}] corresponds to w as in (12.6.4). This defines a Hermitian form on V, as a vector space over k[\sqrt{−a}]. In particular,

⟨v, v⟩_{V} = ∑_{j=1}^{n} |β(e_{j}, e_{j})| v_{j} \overline{v_{j}} (12.6.6)

is an element of k for each v ∈ V. If v ≠ 0, then it is easy to see that (12.6.6) is positive in k. Thus (12.6.5) defines an inner product on V, as a vector space over k[\sqrt{−a}]. Of course, the e_{j}’s are orthogonal with respect to (12.6.5), by construction.

Let B be the unique linear mapping from V into itself such that

(12.6.7)  B(e_{j}) = e_{j} when β(e_{j}, e_{j}) > 0
= −e_{j} when − β(e_{j}, e_{j}) > 0.

Note that B^{2} is the identity mapping I = I_{V} on V. One can check that B is self-adjoint with respect to (12.6.5). It is easy to see that

β(v, w) = ⟨B(v), w⟩_{V} (12.6.8)

when v = e_{j} and w = e_{l}, 1 ≤ j, l ≤ n. This implies that (12.6.8) holds for every v, w ∈ V.

Let T be a linear mapping from V into itself, and remember that the adjoint T^{∗,β} ∈ L(V) of T with respect to β(·, ·) is characterized by the condition that

β(T (v), w) = β(v, T^{∗,β}(w)) (12.6.9)

for every v, w ∈ V. This is the same as saying that

⟨B(T (v)), w⟩_{V} = ⟨B(v), T^{∗,β}(w)⟩_{V} (12.6.10)

for every v, w ∈ V, by (12.6.8). Equivalently, this means that

⟨B(T (v)), w⟩_{V} = ⟨v, B(T^{∗,β}(w))⟩_{V} (12.6.11)

for every v, w ∈ V, because B is self-adjoint with respect to (12.6.5). This is the same as asking that (B ∘ T)^{∗} = B ∘ T^{∗,β}, (12.6.12)

where the left side is the adjoint of B ∘ T with respect to the inner product (12.6.5). Thus

T^{∗,β} = B^{−1} ∘ (B ∘ T)^{∗} = B^{−1} ∘ T^{∗} ∘ B^{∗} = B^{−1} ∘ T^{∗} ∘ B. (12.6.13)

12.7 Real parts

Let k be an ordered field, let a be a positive element of k, and let k[\sqrt{−a}] be as in Section 12.1, as before. If z = x + y \sqrt{−a} ∈ k[\sqrt{−a}], with x, y ∈ k, then let us call Re_{k[\sqrt{−a}]} z = x = (1/2) (z + \overline{z}) (12.7.1)

the real part of z. Note that

Re_{k[\sqrt{−a}]}(\sqrt{−a} z) = −a y = (\sqrt{−a}/2) (z − \overline{z}). (12.7.2)

Let V = V_{k[\sqrt{−a}]} be a finite-dimensional vector space over k[\sqrt{−a}], and let ⟨·, ·⟩_{V_{k[\sqrt{−a}]}} be an inner product on V. We shall use V_{k} to denote V considered as a vector space over k. It is easy to see that

⟨v, w⟩_{V_{k}} = Re_{k[\sqrt{−a}]} (⟨v, w⟩_{V_{k[\sqrt{−a}]}}) (12.7.3)

defines an inner product on V_{k}, as a vector space over k.

Let L(V) = L_{k[\sqrt{−a}]}(V_{k[\sqrt{−a}]}) be the algebra of k[\sqrt{−a}]-linear mappings from V into itself, as usual, and let L_{k}(V_{k}) be the algebra of k-linear mappings from V into itself. If T ∈ L_{k[\sqrt{−a}]}(V_{k[\sqrt{−a}]}), then its adjoint with respect to ⟨·, ·⟩_{V_{k[\sqrt{−a}]}}

is the unique k[\sqrt{−a}]-linear mapping T^{∗,V_{k[\sqrt{−a}]}} from V into itself such that

⟨T (v), w⟩_{V_{k[\sqrt{−a}]}} = ⟨v, T^{∗,V_{k[\sqrt{−a}]}}(w)⟩_{V_{k[\sqrt{−a}]}} (12.7.4)

for every v, w ∈ V. Similarly, if T ∈ L_{k}(V_{k}), then its adjoint with respect to (12.7.3) is the unique k-linear mapping T^{∗,V_{k}} from V into itself such that

⟨T (v), w⟩_{V_{k}} = ⟨v, T^{∗,V_{k}}(w)⟩_{V_{k}} (12.7.5)

for every v, w ∈ V. If T ∈ L_{k[\sqrt{−a}]}(V_{k[\sqrt{−a}]}), then T may be considered as an element of L_{k}(V_{k}), and T^{∗,V_{k}} = T^{∗,V_{k[\sqrt{−a}]}}. (12.7.6)

Put J_{a}(v) = \sqrt{−a} v (12.7.7)

for every v ∈ V, which defines a k[\sqrt{−a}]-linear mapping from V into itself. By construction, J_{a}^{2} = −a I, (12.7.8)

where I is the identity mapping on V, and

⟨J_{a}(v), w⟩_{V_{k}} = Re_{k[\sqrt{−a}]} (\sqrt{−a} ⟨v, w⟩_{V_{k[\sqrt{−a}]}}) (12.7.9)

for every v, w ∈ V. One can check that

J_{a}^{∗,V_{k}} = J_{a}^{∗,V_{k[\sqrt{−a}]}} = −J_{a}. (12.7.10)

This means that (12.7.9) is an antisymmetric bilinear form on V_{k}. Remember that a k-linear mapping from V into itself is linear over k[\sqrt{−a}] exactly when it commutes with J_{a}.

Let β_{V_{k[\sqrt{−a}]}}(·, ·) be a sesquilinear form on V_{k[\sqrt{−a}]}, and put

β_{V_{k}}(v, w) = Re_{k[\sqrt{−a}]} (β_{V_{k[\sqrt{−a}]}}(v, w)) (12.7.11)

for every v, w ∈ V. This defines a bilinear form on V_{k}, which is to say a k-valued function on V × V that is bilinear over k. Observe that

β_{V_{k}}(J_{a}(v), w) = −β_{V_{k}}(v, J_{a}(w)) = Re_{k[\sqrt{−a}]} (\sqrt{−a} β_{V_{C}}(v, w)) (12.7.12)

for every v, w ∈ V. If β_{V_{k[\sqrt{−a}]}}(·, ·) is Hermitian-symmetric on V_{C}, then (12.7.11) is symmetric on V_{R}, and (12.7.12) is antisymmetric on V_{R}. Of course, (12.7.11) and (12.7.12) correspond to (12.7.3) and (12.7.9) when β_{V_{C}}(·, ·) = ⟨·, ·⟩_{V_{C}}.

Suppose now that β_{V_{k[\sqrt{−a}]}}(·, ·) is nondegenerate as a sesquilinear form on V_{k[\sqrt{−a}]}, which implies that (12.7.11) and (12.7.12) are nondegenerate as bilinear

forms on V_{k}. If T ∈ L_{k[\sqrt{−a}]}(V_{k[\sqrt{−a}]}), then there is a unique T^{∗,β_{V_{k[\sqrt{−a}]}}} in L_{k[\sqrt{−a}]}(V_{k[\sqrt{−a}]}) such that

β_{V_{k[\sqrt{−a}]}}(T (v), w) = β_{V_{k[\sqrt{−a}]}}(v, T^{∗,β_{V_{k[\sqrt{−a}]}}} (w)) (12.7.13)

for every v, w ∈ V, as usual. Similarly, if T ∈ L_{k}(V_{k}), then there is a unique T^{∗,β_{V_{k}}} ∈ L_{k}(V_{k}) such that

β_{V_{k}}(T (v), w) = β_{V_{k}}(v, T^{∗,β_{V_{k}}} (w)) (12.7.14)

for every v, w ∈ V. If T ∈ L_{k[\sqrt{−a}]}(V_{k[\sqrt{−a}]}), then T ∈ L_{k}(V_{k}), and

T^{∗,β_{V_{R}}} = T^{∗,β_{V_{C}}} , (12.7.15)

as before. In particular, J_{a}^{∗,β_{V_{k}}} = J_{a}^{∗,β_{V_{k[\sqrt{−a}]}}} = −J_{a}.

Let α_{V_{k}}(·, ·) be a bilinear form on V_{k} such that

α_{V_{k}}(J_{a}(v), w) = −α_{V_{k}}(v, J_{a}(w)) (12.7.16)

for every v, w ∈ V. Put

α_{V_{k[\sqrt{−a}]}}(v, w) = α_{V_{k}}(v, w) + (1/\sqrt{−a}) α_{V_{k}}(J_{a}(v), w) (12.7.17)

for every v, w ∈ V. One can check that this defines a sesquilinear form on V_{k[\sqrt{−a}]}. If α_{V_{k}}(·, ·) is symmetric on V_{k}, then (12.7.16) is antisymmetric on V_{k}, and (12.7.17) is Hermitian-symmetric on V_{k[\sqrt{−a}]}. If α_{V_{k}}(·, ·) is nondegenerate on V_{k}, then (12.7.16) is nondegenerate on V_{k}, and (12.7.17) is nondegenerate on V_{k[\sqrt{−a}]}. If α_{V_{k}}(·, ·) is an inner product on V_{k}, then (12.7.17) is an inner product on V_{k[\sqrt{−a}]}. If α_{V_{k}}(·, ·) is equal to (12.7.11), then (12.7.17) is equal to β_{V_{C}}(·, ·), by (12.7.12).

12.8 Bilinear forms over k[\sqrt{−a}]

Let k be an ordered field, let a be a positive element of k, and let k[\sqrt{−a}] be as in Section 12.1, as usual. Also let V = V_{k[\sqrt{−a}]} be a finite-dimensional vector space over k[\sqrt{−a}] again, and let γ_{V_{k[\sqrt{−a}]}}(·, ·) be a bilinear form on V_{k[\sqrt{−a}]},

which is to say a k[\sqrt{−a}]-valued function on V ×V that is bilinear over k[\sqrt{−a}]. It follows that γ_{V_{k}}(v, w) = Re_{k[\sqrt{−a}]} (γ_{V_{k[\sqrt{−a}]}}(v, w)) (12.8.1)

defines a bilinear form on V_{k}, which is to say a k-valued function on V × V that is bilinear over k. Put J_{a}(v) = \sqrt{−a} v for every v ∈ V, as in the previous section. Observe that

γ_{V_{k}}(J_{a}(v), w) = γ_{V_{k}}(v, J_{a}(w)) = Re_{k[\sqrt{−a}]} (\sqrt{−a} γ_{V_{k[\sqrt{−a}]}}(v, w)) (12.8.2)

for every v, w ∈ V. If γ_{V_{k[\sqrt{−a}]}}(·, ·) is symmetric or antisymmetric on V_{k[\sqrt{−a}]}, then (12.8.1) and (12.8.2) have the same property on V_{k}. If γ_{V_{k[\sqrt{−a}]}}(·, ·) is nondegenerate as a bilinear form on V_{k[\sqrt{−a}]}, then (12.8.1) and (12.8.2) are nondegenerate as bilinear forms on V_{k}.

Let α_{V_{k}}(·, ·) be a bilinear form on V_{k} such that

α_{V_{k}}(J_{a}(v), w) = α_{V_{k}}(v, J_{a}(w)) (12.8.3)

for every v, w ∈ V. Put

α_{V_{k[\sqrt{−a}]}}(v, w) = α_{V_{k}}(v, w) + (1/\sqrt{−a}) α_{V_{k}}(J_{a}(v), w) (12.8.4)

for every v, w ∈ V. One can check that this defines a bilinear form on V_{k[\sqrt{−a}]}. If α_{V_{k}}(·, ·) is symmetric or antisymmetric, then (12.8.3) and (12.8.4) have the same property. Similarly, if α_{V_{k}}(·, ·) is nondegenerate, then (12.8.3) and (12.8.4) are nondegenerate.

Suppose for the moment that γ_{V_{k[\sqrt{−a}]}}(·, ·) is nondegenerate on V_{k[\sqrt{−a}]}, so that γ_{V_{k}}(·, ·) is nondegenerate on V_{k}. If T ∈ L_{k[\sqrt{−a}]}(V_{k[\sqrt{−a}]}), then there is a

unique T^{∗,γ_{V_{k[\sqrt{−a}]}}} ∈ L_{k[\sqrt{−a}]}(V_{k[\sqrt{−a}]}) such that

γ_{V_{k[\sqrt{−a}]}}(T (v), w) = γ_{V_{k[\sqrt{−a}]}}(v, T^{∗,γ_{V_{k[\sqrt{−a}]}}} (w)) (12.8.5)

for every v, w ∈ V. Similarly, if T ∈ L_{k}(V_{k}), then there is a unique T^{∗,γ_{V_{k}}} in L_{k}(V_{k}) such that γ_{V_{k}}(T (v), w) = γ_{V_{k}}(v, T^{∗,γ_{V_{k}}} (w)) (12.8.6)

for every v, w ∈ V. If T ∈ L_{k[\sqrt{−a}]}(V_{k[\sqrt{−a}]}), then T ∈ L_{k}(V_{k}), and

T^{∗,γ_{V_{k}}} = T^{∗,γ_{V_{k[\sqrt{−a}]}}} . (12.8.7)

Note that J_{a}^{∗,γ_{V_{k}}} = J_{a}^{∗,γ_{V_{k[\sqrt{−a}]}}} = J_{a}. (12.8.8)

Let ⟨·, ·⟩_{V_{k[\sqrt{−a}]}} be an inner product on V_{k[\sqrt{−a}]}. If w ∈ V, then there is a unique element C(w) of V such that

γ_{V_{k[\sqrt{−a}]}}(v, w) = ⟨v, C(w)⟩_{V_{k[\sqrt{−a}]}} (12.8.9)

for every v ∈ V, by standard arguments. More precisely, the left side may be considered as a linear functional on V_{k[\sqrt{−a}]} as a function of v, which can be represented in terms of the inner product as on the right side. It is easy to see that C is conjugate-linear as a mapping from V into itself, because the left side of (12.8.9) is linear over k[\sqrt{−a}] in w, and the inner product ⟨·, ·⟩_{V_{k[\sqrt{−a}]}} is conjugate-linear in the second variable.

Suppose that γ_{V_{k[\sqrt{−a}]}}(·, ·) is nondegenerate on V_{k[\sqrt{−a}]} again, which implies

that C is a one-to-one mapping from V onto itself. If T is a k[\sqrt{−a}]-linear mapping from V into itself, then

⟨T (v), C(w)⟩_{V_{k[\sqrt{−a}]}} = ⟨v, C(T^{∗,γ_{V_{k[\sqrt{−a}]}}} (w))⟩_{V_{k[\sqrt{−a}]}} (12.8.10)

for every v, w ∈ V, by (12.8.5) and (12.8.9). We also have that

⟨T (v), C(w)⟩_{V_{k[\sqrt{−a}]}} = ⟨v, T^{∗,V_{k[\sqrt{−a}]}}(C(w))⟩_{V_{k[\sqrt{−a}]}} (12.8.11)

for every v, w ∈ V, where T^{∗,V_{k[\sqrt{−a}]}} is the adjoint of T with respect to the inner product ⟨·, ·⟩_{V_{k[\sqrt{−a}]}} on V_{k[\sqrt{−a}]}. It follows that

C ∘ T^{∗,γ_{V_{k[\sqrt{−a}]}}} = T^{∗,V_{k[\sqrt{−a}]}} ∘ C. (12.8.12)

12.9 Conjugate-linearity and adjoints

Let k be an ordered field, let a be a positive element of k, and let k[\sqrt{−a}] be as in Section 12.1. Also let V = V_{k[\sqrt{−a}]} be a finite-dimensional vector space over k[\sqrt{−a}], and let V_{k} be V considered as a vector space over k, as before. Note that the composition of two conjugate-linear mappings from V into itself is linear over k[\sqrt{−a}]. Similarly, the composition of a conjugate-linear mapping from V into itself with a k[\sqrt{−a}]-linear mapping from V into itself, in either order, is conjugate-linear.

Let ⟨·, ·⟩_{V_{k[\sqrt{−a}]}} be an inner product on V_{k[\sqrt{−a}]}, and let ⟨·, ·⟩_{V_{k}} be its real part, as in (12.7.3). Thus ⟨·, ·⟩_{V_{k}} is an inner product on V_{k} as a vector space over k, as before. Let C be a conjugate-linear mapping from V into itself, which is linear over k in particular. The adjoint C^{∗,V_{k}} of C with respect to ⟨·, ·⟩_{V_{k}} is defined as a k-linear mapping from V into itself in the usual way. Remember that J_{a} is the mapping from V into itself defined by multiplication by \sqrt{−a}, as in (12.7.7). Conjugate-linearity of C means that

C ∘ J_{a} = −J_{a} ∘ C, (12.9.1) as in Section 12.1. This implies that

C^{∗,V_{k}} ∘ J_{a} = −J_{a} ∘ C^{∗,V_{k}}, (12.9.2)

because J_{a}^{∗,V_{k}} = −J_{a}, as in (12.7.10). This shows that C^{∗,V_{k}} is conjugate-linear on V as well.

Observe that

⟨v, C(w)⟩_{V_{k}} = ⟨C^{∗,V_{k}}(v), w⟩_{V_{k}} = ⟨w, C^{∗,V_{k}}(v)⟩_{V_{k}} (12.9.3)

for every v, w ∈ V, by definition of the adjoint. In fact, we have that

⟨v, C(w)⟩_{V_{k[\sqrt{−a}]}} = ⟨w, C^{∗,V_{k}}(v)⟩_{V_{k[\sqrt{−a}]}} (12.9.4)

for every v, w ∈ V. More precisely, the real parts of both sides are the same, by (12.9.3). One can get (12.9.4) using this and the fact that both sides are k[\sqrt{−a}]-linear in w. This gives another way to see the conjugate-linearity of C^{∗,V_{k}} on V too.

Put γ_{V_{k[\sqrt{−a}]}}(v, w) = ⟨v, C(w)⟩_{V_{k[\sqrt{−a}]}} (12.9.5)

for every v, w ∈ V. This defines a bilinear form on V_{k[\sqrt{−a}]}, because C is conjugate-linear, and ⟨·, ·⟩_{V_{k[\sqrt{−a}]}} is sesquilinear. Similarly, put

γ_{V_{k}}(v, w) = ⟨v, C(w)⟩_{V_{k}} (12.9.6)

for every v, w ∈ V, which is the same as the real part of (12.9.5). Thus

γ_{V_{k[\sqrt{−a}]}}(v, w) = ⟨w, C^{∗,V} (v)⟩_{V_{k[\sqrt{−a}]}} (12.9.7)

and γ_{V_{k}}(v, w) = ⟨w, C^{∗,V_{k}}(v)⟩_{V_{k}} (12.9.8)

for every v, w ∈ V. It follows that the self-adjointness or anti-self-adjointness of C with respect to ⟨·, ·⟩_{V_{k}} is equivalent to the symmetry or anti-symmetry of (12.9.5), (12.9.6), as appropriate.

12.10 Antisymmetric forms over k[\sqrt{−a}]

Let k be an ordered field, let a be a positive element of k, and let k[\sqrt{−a}] be as in Section 12.1. Also let V be a vector space over k[\sqrt{−a}] of positive finite dimension, and let γ(·, ·) be a nondegenerate antisymmetric bilinear form on V. As in Section 11.13, there is a basis for V consisting of vectors x_{1}, …, x_{n}, y_{1}, …, y_{n} for some positive integer n such that

γ(x_{j}, x_{l}) = γ(y_{j}, y_{l}) = 0 (12.10.1)

for every j, l = 1, …, n, γ(x_{j}, y_{l}) = 0 (12.10.2)

12.10. ANTISYMMETRIC FORMS OVER K[\sqrt{−A}] 267

when j ≠ l, and γ(x_{j}, y_{j}) = 1 (12.10.3)

for every j = 1, …, n. It is easy to define an inner product on V, as a vector space over k[\sqrt{−a}], for which x_{1}, …, x_{n}, y_{1}, …, y_{n} are orthonormal, as in the next paragraph. Using this, we can express γ(·, ·) in terms of a conjugate-linear mapping C from V into itself, as in the previous sections.

Each v ∈ V can be expressed in a unique way as

v = ∑_{j=1}^{n} v_{x_{j}} x_{j} + ∑_{j=1}^{n} v_{y_{j}} y_{j}, (12.10.4)

where v_{x_{j}}, v_{y_{j}} ∈ k[\sqrt{−a}] for every j = 1, …, n. Similarly, if w ∈ V, then let w_{x_{j}}, w_{y_{j}} ∈ k[\sqrt{−a}] be as in (12.10.4). Put

⟨v, w⟩_{V} = ∑_{j=1}^{n} v_{x_{j}} \overline{w_{x_{j}}} + ∑_{j=1}^{n} v_{y_{j}} \overline{w_{y_{j}}} (12.10.5)

for every v, w ∈ V. It is easy to see that this defines an inner product on V, as a vector space over k[\sqrt{−a}]. Note that x_{1}, …, x_{n}, y_{1}, …, y_{n} are orthonormal in V with respect to (12.10.5), by construction.

Let C be the unique conjugate-linear mapping from V into itself such that

C(x_{j}) = −y_{j}, C(y_{j}) = x_{j} (12.10.6)

for each j = 1, …, n. Thus, if v ∈ V is as in (12.10.4), then

C(v) = − ∑_{j=1}^{n} \overline{v_{j}} y_{j} + ∑_{j=1}^{n} \overline{w_{j}} x_{j}. (12.10.7)

One can verify that γ(v, w) = ⟨v, C(w)⟩_{V} (12.10.8)

for every v, w ∈ V. More precisely, one can first check that this holds when v, w are among the basis vectors x_{1}, …, x_{n}, y_{1}, …, y_{n}. This implies that (12.10.8) holds for all v, w ∈ V, because both sides of (12.10.8) are k[\sqrt{−a}]-linear in v and w.

Observe that C^{2} = −I_{V} , (12.10.9)

where I_{V} is the identity mapping on V. Let V_{k} be V considered as a vector space over k, as before. Also let ⟨v, w⟩_{V_{k}} be the real part of (12.10.5), which defines an inner product on V_{k}. One can check that C is anti-self-adjoint as a k-linear mapping from V into itself with respect to ⟨·, ·⟩_{V_{k}}. This corresponds to the antisymmetry of (12.10.8) as a bilinear form on V, as in the previous section.

12.11 Symmetric forms over k[\sqrt{−a}]

Let k be an ordered field, let a be a positive element of k, and let k[\sqrt{−a}] be as in Section 12.1 again. Also let V be a vector space over k[\sqrt{−a}] of positive finite dimension n, and let e_{1}, …, e_{n} be a basis for V. Suppose that γ(·, ·) is a bilinear form on V such that γ(e_{j}, e_{l}) = 0 (12.11.1)

when j ≠ l, and for each j = 1, …, n,

γ(e_{j}, e_{j}) ≠ 0 (12.11.2)

and γ(e_{j}, e_{j}) ∈ k. (12.11.3)

Note that γ(·, ·) is symmetric and nondegenerate on V, because of (12.11.1) and (12.11.2). Conversely, it is well known that for any nondegenerate symmetric bilinear form on V, there is a basis for V for which (12.11.1) and (12.11.2) hold, as in Sections 11.12 and 11.14. Because of (12.11.3), we can reduce further to the case where γ(e_{j}, e_{j}) > 0 (12.11.4)

for every j = 1, …, n, using scalar multiplication by \sqrt{−a}, when needed. If k = R, so that k[\sqrt{−a}] = C, then we can reduce directly to the case where

γ(e_{j}, e_{j}) = 1 (12.11.5)

for every j = 1, …, n, without asking that (12.11.3) hold.

Every v ∈ V can be expressed in a unique way as

v = ∑_{j=1}^{n} v_{j} e_{j}, (12.11.6)

where v_{j} ∈ k[\sqrt{−a}] for each j = 1, …, n. If v, w ∈ V, then put

⟨v, w⟩_{V} = ∑_{j=1}^{n} |γ(e_{j}, e_{j})| v_{j} \overline{w_{j}}, (12.11.7)

where w_{j} ∈ k[\sqrt{−a}] corresponds to w as in (12.11.6). Remember that the absolute value |γ(e_{j}, e_{j})| of γ(e_{j}, e_{j}) is defined as an element of k as in Section 8.13. Of course, this is the same as γ(e_{j}, e_{j}) when (12.11.4) holds. It is easy to see that (12.11.7) defines an inner product on V, for which the e_{j}’s are pairwise orthogonal.

Let C be the unique conjugate-linear mapping from V into itself such that

(12.11.8)  C(e_{j}) = e_{j} when γ(e_{j}, e_{j}) > 0
= −e_{j} when − γ(e_{j}, e_{j}) > 0.

If (12.11.4) holds, then C(e_{j}) = e_{j} for every j = 1, …, n, so that

C(v) = ∑_{j=1}^{n} \overline{v_{j}} e_{j} (12.11.9)

for every v ∈ V as in (12.11.6). Observe that

⟨v, C(w)⟩_{V} = ∑_{j=1}^{n} γ(e_{j}, e_{j}) v_{j} w_{j} (12.11.10)

for every v, w ∈ V. Using this, one can check that

γ(v, w) = ⟨v, C(w)⟩_{V} (12.11.11)

for every v, w ∈ V. One can also verify that

C^{2} = I_{V} , (12.11.12)

where I_{V} is the identity mapping on V.

Let V_{k} be V considered as a vector space over k, as usual. The real part ⟨v, w⟩_{V_{k}} of (12.11.7) defines an inner product on V_{k}, as before. One can check that C is self-adjoint with respect to ⟨·, ·⟩_{V_{k}}, as a k-linear mapping from V into itself. This corresponds to the fact that (12.11.11) is symmetric on V, as in Section 12.9.

12.12 Nonnegative self-adjoint operators

Let (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) be finite-dimensional inner product spaces, both defined over the real numbers or both defined over the complex numbers, and let ∥ · ∥_{V}, ∥ · ∥_{W} be the corresponding norms on V, W, respectively. Also let T be a linear mapping from V into W, and let T^{∗} be the corresponding adjoint mapping from W into V. Thus T^{∗} ∘ T maps V into itself, and it is easy to see that T^{∗} ∘ T is self-adjoint. Observe that

⟨(T^{∗} ∘ T)(v), v⟩_{V} = ⟨T (v), T (v)⟩_{W} = ∥T (v)∥_{W}^{2} (12.12.1)

for every v ∈ V. Of course, one could consider infinite-dimensional Hilbert spaces as well.

Let A be a linear mapping from V into itself that is self-adjoint with respect to the inner product ⟨·, ·⟩, so that

⟨A(v), w⟩ = ⟨v, A(w)⟩ (12.12.2)

for every v, w ∈ V. In the complex case, (12.12.2) implies that

⟨A(v), v⟩ = ⟨v, A(v)⟩ = \overline{⟨A(v), v⟩} (12.12.3) for every v ∈ V, so that ⟨A(v), v⟩ ∈ R for every v ∈ V. In both the real and complex cases, A is said to be nonnegative on V if

⟨A(v), v⟩ ≥ 0 (12.12.4)

for every v ∈ V. Similarly, A is said to be strictly positive on V if

⟨A(v), v⟩_{V} > 0 (12.12.5)

for every v ∈ V with v ≠ 0. In particular, this implies that the kernel of A is trivial. If T is a linear mapping from V into W, as before, then T^{∗} ∘ T is automatically nonnegative, by (12.12.1). More precisely, T^{∗} ∘ T is strictly positive exactly when the kernel of T is trivial.

If A is any self-adjoint linear mapping from V into itself, then there is an orthonormal basis for V consisting of eigenvectors for A, because V has finite dimension. Note that the eigenvalues of A are real, even in the complex case. It is easy to see that A is nonnegative on V if and only if the eigenvalues of A are nonnegative real numbers. Similarly, A is strictly positive on V if and only if the eigenvalues of A are positive real numbers. If A is nonnegative on V and the kernel of A is trivial, then it follows that A is strictly positive on V.

If A is nonnegative on V, then there is a nonnegative self-adjoint linear mapping B from V into itself such that B^{2} = A. This can be obtained using a diagonalization for A, as in the preceding paragraph. If A is strictly positive on V, then B is strictly positive on V as well. Note that B automatically commutes with A in this situation. More precisely, B commutes with any linear mapping C from V into itself that commutes with A. Indeed, if C commutes with A, then C maps the eigenspaces of A into themselves. On each eigenspace of A, B is equal to a nonnegative multiple of the identity, by construction.

Let B_{1} be any nonnegative self-adjoint linear mapping from V into itself such that B_{1}^{2} = A. Thus B_{1} commutes with A, so that B_{1} maps the eigenspaces of A into themselves. One can check that the restriction of B_{1} to each eigenspace of A is a nonnegative multiple of the identity. This can be obtained using a diagonalization of B_{1} with respect to an orthonormal basis for each eigenspace of A. It follows that B_{1} is uniquely determined by A under these conditions.

12.13 Polar decompositions

Let us continue with the same notation and hypotheses as in the previous section. Let T be a linear mapping from V into W again, so that T^{∗} ∘ T is a nonnegative self-adjoint linear mapping from V into itself. It follows that there is a unique nonnegative self-adjoint linear mapping R from V into itself such that R^{2} = T^{∗} ∘ T, (12.13.1)

as in the previous section. If u, v ∈ V, then

⟨(T^{∗} ∘ T)(u), v⟩_{V} = ⟨T (u), T (v)⟩_{W} (12.13.2) and ⟨R^{2}(u), v⟩_{V} = ⟨R(u), R(v)⟩_{V} . (12.13.3)

This implies that ⟨R(u), R(v)⟩_{V} = ⟨T (u), T (v)⟩_{W} , (12.13.4)

by (12.13.1). In particular,

∥R(v)∥_{V} = ∥T (v)∥_{W} (12.13.5)

for every v ∈ V, by taking u = v in (12.13.4). Let us suppose from now on in this section that the kernel of T is trivial. This implies that the kernel of R is trivial too, by (12.13.5). Thus R is invertible as a mapping from V into itself, because V has finite dimension.

Put U = T ∘ R^{−1}, (12.13.6)

which defines a linear mapping from V into W. If v, v′ ∈ V, then

⟨U(R(v)), U(R(v′))⟩_{V} = ⟨T (v), T (v′)⟩_{W} = ⟨R(v), R(v′)⟩_{V} , (12.13.7)

using the definition of U in the first step, and (12.13.4) in the second step. This means that ⟨U(v), U(v′)⟩_{W} = ⟨v, v′⟩_{V} (12.13.8)

for every v, v′ ∈ V, because R is invertible on V. It follows that

∥U(v)∥_{W} = ∥v∥_{V} (12.13.9)

for every v ∈ V, by taking v′ = v in (12.13.8). Let us suppose from now on in this section that V = W. This implies that T maps V onto itself, because V has finite dimension. Thus U is a one-to-one mapping from V onto itself. More precisely, U^{−1} = U^{∗}, (12.13.10)

because of (12.13.8), as in Section 3.8.

Suppose that T is normal on V, in the sense that T commutes with T^{∗}. Of course, this implies that T commutes with T^{∗} ∘ T. It follows that R commutes with T, as in the previous section. This means that U commutes with R and T, by the definition of U.

Suppose for the moment that T is self-adjoint on V, which implies that T is normal in particular. In this case, we get that

U^{2} = T^{2} ∘ R^{−2} = I, (12.13.11)

the identity operator on V. Equivalently, one can check that U is self-adjoint in this situation.

Similarly, if T is anti-self-adjoint on V, then T is normal, and T^{∗} ∘T = −T^{2}. This implies that U^{2} = T^{2} ∘ R^{−2} = −I. (12.13.12)

Alternatively, one can verify directly that U is anti-self-adjoint in this case.

12.14 Bilinear forms and inner products

Let (V, ⟨·, ·⟩) be a finite-dimensional inner product space over the real numbers. If B is a linear mapping from V into itself, then

β(v, w) = ⟨B(v), w⟩ (12.14.1)

defines a bilinear form on V. Every bilinear form on V corresponds to a unique linear mapping on V in this way. Note that β is symmetric or antisymmetric on V exactly when B is self-adjoint or anti-self-adjoint on V with respect to ⟨·, ·⟩, as appropriate. Let us suppose from now on that B is invertible on V, so that β is nondegenerate on V.

Of course, the adjoint B^{∗} of B with respect to ⟨·, ·⟩ is invertible on V too, because B is invertible. Note that B ∘ B^{∗} is a strictly positive self-adjoint linear mapping from V into itself, which corresponds to taking T = B^{∗} in the previous section. As before, there is a unique nonnegative self-adjoint linear mapping A from V into itself such that A^{2} = B ∘ B^{∗}. (12.14.2)

More precisely, A is invertible on V, so that A is strictly positive on V. Put

B_{0} = A^{−1} ∘ B, (12.14.3)

which is invertible on V as well. By construction,

B_{0} ∘ B_{0}^{∗} = A^{−1} ∘ B ∘ B^{∗} ∘ A^{−1} = I, (12.14.4)

so that B_{0}^{−1} = B_{0}^{∗}. (12.14.5)

If we take T = B^{∗} in the previous section, then A corresponds to R, and B_{0} corresponds to U^{∗}.

If B is normal on V, then A commutes with B, and hence B_{0} commutes with A and B, as before. If B is self-adjoint on V, and normal in particular, then B_{0}^{2} = A^{−2} ∘ B^{2} = I. (12.14.6)

Equivalently, one can check that B_{0} is self-adjoint on V. Similarly, if B is anti-self-adjoint on V, then B is normal, and

B_{0}^{2} = A^{−2} ∘ B^{2} = −I. (12.14.7)

Alternatively, one can verify that B_{0} is anti-self-adjoint in this case.

Clearly B = A ∘ B_{0}, (12.14.8)

by (12.14.3). Put ⟨v, w⟩_{A} = ⟨A(v), w⟩ (12.14.9) for every v, w ∈ V. This defines an inner product on V, because A is strictly positive and self-adjoint on V. Observe that

β(v, w) = ⟨B_{0}(v), w⟩_{A} (12.14.10)

for every v, w ∈ V. If B is normal with respect to ⟨·, ·⟩ on V, then

⟨B_{0}(v), w⟩_{A} = ⟨A(B_{0}(v)), w⟩ = ⟨B_{0}(A(v)), w⟩
(12.14.11)  = ⟨A(v), B_{0}^{∗}(w)⟩ = ⟨v, B_{0}^{∗}(w)⟩_{A}

for every v, w ∈ V, because B_{0} commutes with A.

Now let (V, ⟨·, ·⟩) be a finite-dimensional inner product space over the complex numbers. If B is a linear mapping from V into itself, then (12.14.1) defines a sesquilinear form β on V, and every sesquilinear form on V corresponds to a unique linear mapping from V into itself in this way. One can check that β is Hermitian-symmetric on V exactly when B is self-adjoint on V with respect to ⟨·, ·⟩. Suppose from now on in this section that B is invertible on V, so that β is nondegenerate on V. This implies that the adjoint B^{∗} of B with respect to ⟨·, ·⟩ is invertible on V as well.

As before, B∘B^{∗} is a strictly positive self-adjoint linear mapping from V into itself. Hence there is a unique nonnegative self-adjoint linear mapping A from V into itself that satisfies (12.14.2). In fact, A is invertible on V, and strictly positive. Let B_{0} be as in (12.14.3), which is invertible on V, with inverse equal to B_{0}^{∗}, as in (12.14.5). If B is normal on V, then A commutes with B, and B_{0} commutes with A and B. If B is self-adjoint on V, then B is normal, and

B_{0}^{2} = I. (12.14.12)

Alternatively, one can check directly that B_{0} is self-adjoint in this case.

Let ⟨v, w⟩_{A} be defined for v, w ∈ V as in (12.14.9), which defines an inner product on V, because A is strictly positive and self-adjoint on V. Thus (12.14.10) holds for every v, w ∈ V again, by construction. If B is normal on V with respect to ⟨·, ·⟩, then the adjoint of B_{0} with respect to ⟨·, ·⟩_{A} is the same as the adjoint with respect to ⟨·, ·⟩, as in (12.14.11).

12.15 Complex bilinear forms

Let (V, ⟨·, ·⟩) be a finite-dimensional inner product space over the complex numbers again. If C is a conjugate-linear mapping from V into itself, then

γ(v, w) = ⟨v, C(w)⟩ (12.15.1)

is a bilinear form on V, as before. Every bilinear form on V corresponds to a conjugate-linear mapping C on V in this way, as in Section 12.8.

Let V_{R} be V considered as a vector space over the real numbers, and let ⟨·, ·⟩_{V_{R}} be the real part of ⟨·, ·⟩, which is an inner product on V_{R}. Also let C be a conjugate-linear mapping from V into itself, and let C^{∗,V_{R}} be the adjoint of C, as a real-linear mapping from V into itself, with respect to ⟨·, ·⟩_{V_{R}}. Thus C^{∗,V_{R}} is conjugate-linear on V too, as in Section 12.9. It follows that C ∘ C^{∗,V_{R}} is a complex-linear mapping from V into itself, as before.

Note that C ∘ C^{∗,V_{R}} is nonnegative and self-adjoint with respect to ⟨·, ·⟩_{V_{R}}. This implies that there is a unique real-linear mapping A from V into itself that is nonnegative and self-adjoint with respect ⟨·, ·⟩_{V_{R}} and satisfies

A^{2} = C ∘ C^{∗,V_{R}}. (12.15.2)

Remember that A commutes with any real-linear mapping from V into itself that commutes with C ∘ C^{∗,V_{R}}. In particular, A commutes with the mapping J from V into itself that corresponds to multiplication by i, because C ∘ C^{∗,V_{R}} is complex linear. This means that A is complex linear on V as well, and one can check that A is self-adjoint and nonnegative with respect to ⟨·, ·⟩.

Alternatively, C ∘ C^{∗,V_{R}} is self-adjoint as a complex-linear mapping from V into itself with respect to ⟨·, ·⟩, because C ∘ C^{∗,V_{R}} is complex linear and selfadjoint as a real-linear mapping from V into itself with respect to ⟨·, ·⟩_{V_{R}}, as in Section 12.7. Similarly, it is easy to see that C ∘ C^{∗,V_{R}} is nonnegative as a selfadjoint complex-linear mapping from V into itself with respect to ⟨·, ·⟩, because C ∘ C^{∗,V_{R}} is nonnegative as a self-adjoint real-linear mapping from V into itself with respect to ⟨·, ·⟩_{V_{R}}. Thus one can take A to be the unique complex-linear mapping from V into itself that is nonnegative and self-adjoint with respect to ⟨·, ·⟩ and satisfies (12.15.2).

Suppose from now on in this section that C is invertible on V, so that (12.15.1) is nondegenerate as a bilinear form on V. This implies that C^{∗,V_{R}} is invertible on V too, and that C ∘ C^{∗,V_{R}} is strictly positive on V with respect to ⟨·, ·⟩_{V_{R}}, and hence with respect to ⟨·, ·⟩. It follows that A is invertible on V, and strictly positive with respect to ⟨·, ·⟩_{V_{R}}, ⟨·, ·⟩. Thus

⟨v, w⟩_{V_{R},A} = ⟨A(v), w⟩_{V_{R}} (12.15.3)

defines an inner product on V_{R}, and

⟨v, w⟩_{A} = ⟨A(v), w⟩ (12.15.4)

defines an inner product on V. Of course,

⟨v, w⟩_{V_{R},A} = Re ⟨v, w⟩_{A} (12.15.5)

for every v, w ∈ V, by construction.

Put C_{0} = A^{−1} ∘ C, (12.15.6)

which is invertible as a real-linear mapping on V. In fact, C_{0} is conjugatelinear on V, because A is complex-linear and C is conjugate-linear. As before, C_{0} ∘ C_{0}^{∗,V_{R}} = I, so that C_{0}^{−1} = C_{0}^{∗,V_{R}}. (12.15.7)

Of course, C = A ∘ C_{0}, so that

(12.15.8) γ(v, w) = ⟨v, C(w)⟩ = ⟨v, A(C_{0}(w))⟩
= ⟨A(v), C_{0}(w)⟩ = ⟨v, C_{0}(w)⟩_{A}

for every v, w ∈ V.

Suppose for the moment that C is normal with respect to ⟨·, ·⟩_{V_{R}}, so that C commutes with C^{∗,V_{R}}. This implies that C commutes with C ∘ C^{∗,V_{R}}, and hence that A commutes with C. It follows that C_{0} commutes with A and C in this situation. Thus

⟨C_{0}(v), w⟩_{V_{R},A} = ⟨A(C_{0}(v)), w⟩_{V_{R}} = ⟨C_{0}(A(v)), w⟩_{V_{R}}
(12.15.9)  = ⟨A(v), C_{0}^{∗,V_{R}}(w)⟩_{V_{R}} = ⟨v, C_{0}^{∗,V_{R}}(w)⟩_{V_{R},A}

for every v, w ∈ V. This means that the adjoint of C_{0} with respect to ⟨·, ·⟩_{V_{R},A} is the same as C_{0}^{∗,V_{R}} in this case.

If γ(·, ·) is symmetric or antisymmetric on V, then

γ_{R}(v, w) = Re γ(v, w) = ⟨v, C(w)⟩_{V_{R}} (12.15.10)

has the analogous property as a bilinear form on V_{R}. This means that C is selfadjoint or antiself-adjoint with respect to ⟨·, ·⟩_{V_{R}}. In both cases, C is normal with respect to ⟨·, ·⟩_{V_{R}}. If C is self-adjoint with respect to ⟨·, ·⟩_{V_{R}}, then C_{0} is self-adjoint with respect to ⟨·, ·⟩_{V_{R}}, and

C_{0}^{2} = I, (12.15.11)

as in the previous section. Similarly, if C is anti-self-adjoint with respect to ⟨·, ·⟩_{V_{R}}, then C_{0} is anti-self-adjoint with respect to ⟨·, ·⟩_{V_{R}}, and

C_{0}^{2} = −I. (12.15.12)

Semisimplicity

13.1 Semisimple modules

Let k be a commutative ring with a multiplicative identity element, let A, V be modules over k, and let ρ be a bilinear action of A on V, as in Section 6.1. Suppose that V corresponds to the direct sum of a family of submodules, as a module over k, where each of these submodules is mapped into itself by the action of ρ. Suppose also that the action of ρ on each of these submodules is irreducible, or equivalently that each of these submodules is simple with respect to the action of ρ, as in Section 6.14. Under these conditions, ρ is said to be completely reducible on V, or equivalently V is said to be semisimple with respect to the action of ρ.

In particular, these notions may be applied to representations of associative algebras over k, or to Lie algebras over k. Equivalently, one may consider modules over associative algebras over k, or Lie algebras over k. One can always reduce to the case of a representation of an associative algebra, using the algebra of homomorphisms from V into itself, as a module over k, generated by the mappings on V defined by the bilinear action. Of course, one may include the identity mapping on V in this algebra, without affecting the invariant submodules.

Let V_{0} be a submodule of V, as a module over k, that is mapped into itself by the action of ρ on V. Consider the following condition on V_{0}:

(13.1.1) V corresponds to the direct sum of V_{0} and another submodule V_{1}, as a module over k, where V_{1} is also invariant under the action of ρ on V.

In fact, ρ is completely reducible on V exactly when this condition holds for all such V_{0}. This is Proposition 4.1 on p11 of [5] for modules over rings. This corresponds to a remark at the bottom of p45 of [25] for representations of Lie algebras as well.

Indeed, suppose that V corresponds to the direct sum of a family {V_{j}}_{j∈I} of submodules, as a module over k, where for each j ∈ I, V_{j} is mapped into itself by the action of ρ on V, and V_{j} is irreducible under the action of ρ. If I_{0} ⊆ I, then let V (I_{0}) be the submodule of V generated by V_{j}, j ∈ I_{0}. Let W be a submodule of V, as a module over k, that is mapped into itself by the action of ρ on V. Under these conditions, there is an I_{1} ⊆ I such that

V (I_{1}) ∩ W = {0}, (13.1.2)

and I_{1} is maximal with respect to inclusion. More precisely, one can find such an I_{1} in a fairly straightforward way when I has only finitely or countably many elements, and otherwise one can use Zorn’s lemma or Hausdorff’s maximality principle.

If j ∈ I \ I_{1}, then V (I_{1} ∪ {j}) ∩ W ≠ {0}, (13.1.3)

by the maximality of I_{1}. Thus there is a nonzero element of W that can be expressed as the sum of elements of V (I_{1}) and V_{j}, and this element of V_{j} has to be nonzero, by (13.1.2). This means that there is a nonzero element of V_{j} that can be expressed as a sum of elements of V (I_{1}) and W, so that

V_{j} ∩ (V (I_{1}) + W) ≠ {0}. (13.1.4)

It follows that V_{j} ⊆ V (I_{1}) + W, (13.1.5)

because V_{j} ∩ (V (I_{1}) + W) is a submodule of V, as a module over k, that is invariant under the action of ρ, and V_{j} is irreducible with respect to the action of ρ. This implies that V = V (I_{1}) + W, (13.1.6)

so that V corresponds to the direct sum of V (I_{1}) and W, as a module over k, by (13.1.2).

Before considering the converse, note that (13.1.1) holds for a submodule V_{0} of V, as a module over k, that is mapped into itself by the action of ρ on V, if and only if

(13.1.7) there is a homomorphism ϕ_{0} from V onto V_{0}, as modules over k, that intertwines the action of ρ on V, and is equal to the identity mapping on V_{0}.

Of course, if (13.1.1) holds, then one can take ϕ_{0} to be the corresponding projection from V onto V_{0}. If (13.1.7) holds, then one can take V_{1} to be the kernel of ϕ_{0} in (13.1.1).

Suppose now that (13.1.1) holds for every submodule V_{0} of V, as a module over k, that is mapped into itself by the action of ρ on V, so that (13.1.7) holds for every such V_{0}. Let W be a submodule of V, and let W_{0} be a submodule of W, as modules over k, such that W and W_{0} are mapped into themselves by the action of ρ on V. Using (13.1.7) with V_{0} = W_{0}, we get a homomorphism from V onto W_{0}, as modules over k, that intertwines the action of ρ on V, and is equal to the identity mapping on W_{0}. The restriction of this homomorphism to W is a homomorphism from W onto W_{0}, as modules over k, that intertwines the action of ρ on W, and is equal to the identity mapping on W_{0}. This implies that W corresponds to the direct sum of W_{0} and another submodule W_{1}, as a module over k, where W_{1} is also mapped into itself by the action of ρ, as before.

If k is a field, and V has finite dimension, as a vector space over k, then it is easy to get that ρ is completely reducible on V, using the remarks in the preceding paragraph. Otherwise, complete reducibility can be obtained as in Proposition 4.1 on p11 of [5].

13.2 Complete reducibility

Let k be a field of characteristic 0, and let A be a Lie algebra over k with finite dimension as a vector space over k. Suppose that A is semisimple as a Lie algebra over k.

Remember that [A, A] = A, as in Section 10.15. This implies that any representation of A on a one-dimensional vector space over k is trivial, as in the lemma on p28 of [14], and remarked on p47 of [25].

A famous theorem going back to Weyl states that every finite-dimensional representation of A is completely reducible, as on p28 of [14] and p46 of [25]. In order to prove this, we shall begin with the following splitting principle. Weyl’s theorem will be obtained from this in Section 13.7.

Let V be a finite-dimensional module over A, and suppose that W is a submodule of V, as a module over A, such that W has codimension one in V as a vector space over k. This means that the quotient space V/W has dimension one as a vector space over k, so that the induced action of A on V/W is trivial, as before. Equivalently, the action of A on V actually maps V into W. Under these conditions, we would like to show that V corresponds to the direct sum of W and a one-dimensional submodule of V, as a module over A.

The case where W is irreducible as a module over A will be discussed in Section 13.6. This will use a suitable Casimir element, which will be discussed in the next two sections.

To reduce to the case where W is irreducible, we use induction on the dimension of W. Suppose that W is not irreducible as a module over A, so that there is a proper nonzero submodule Z of W, as a module over A. Under these conditions, V/Z is an module over A too, W/Z is a submodule of V/Z, as a module over A, and W/Z has codimension one in V/Z, as a vector space over k. The dimension of W/Z is strictly less than the dimension of W, as vector spaces over k. Using induction, we get that V/Z corresponds to the direct sum of W/Z and a one-dimensional submodule of V/Z, as a module over A.

This one-dimensional submodule of V/Z can be expressed as U/Z, where U is a linear subspace of V that contains Z as a codimension-one subspace, and U is a submodule of V, as a module over A. Note that the dimension of Z is strictly less than the dimension of W, as vector spaces over k. Using the induction hypothesis again, we get that U corresponds to the direct sum of Z and a one-dimensional submodule of U, as a module over A. One can check that V corresponds to the direct sum of W and this one-dimensional submodule of U, as desired.

13.3 Casimir elements

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k. Suppose that A has positive finite dimension n as a vector space over k, and let β be a nondegenerate symmetric bilinear form on A. Suppose also that β is associative on A, or equivalently that β is invariant with respect to the adjoint representation on A, so that β([x, w]_{A}, y) = β(x, [w, y]_{A}) (13.3.1)

for every w, x, y ∈ A, as in Sections 6.10 and 7.11. Let u_{1}, …, u_{n} be a basis for A, as a vector space over k. Under these conditions, there is a basis w_{1}, …, w_{n} for A such that β(u_{j}, w_{l}) = δ_{j,l} (13.3.2)

for every j, l = 1, …, n. Here δ_{j,l} ∈ k is equal to 1 when j = l, and to 0 when j ≠ l, as usual. This is the dual basis for A with respect to β.

Let x ∈ A be given, and let (a_{j,l}) and (b_{j,l}) be the n×n matrices with entries in k such that

[x, u_{j}]_{A} = ∑_{l=1}^{n} a_{j,l} u_{l} (13.3.3)

and

[x, w_{j}]_{A} = ∑_{l=1}^{n} b_{j,l} w_{l} (13.3.4)

for every j = 1, …, n. Observe that

β([x, u_{j}]_{A}, w_{r}) = ∑_{l=1}^{n} a_{j,l} β(u_{l}, w_{r}) = a_{j,r} (13.3.5)

for every j, r = 1, …, n, and similarly

β(u_{j}, [x, w_{r}]_{A}) = ∑_{l=1}^{n} b_{r,l} β(u_{j}, w_{l}) = b_{r,j} (13.3.6)

for every j, r = 1, …, n. Using (13.3.1), we get that

a_{j,r} = −b_{r,j} (13.3.7)

for every j, r = 1, …, n.

Let V be a vector space over k, and let ρ be a representation of A on V. Put

c_{ρ}(β) = ∑_{j=1}^{n} ρ_{u_{j}} ∘ ρ_{w_{j}}, (13.3.8)

which defines a linear mapping from V into itself. This is the Casimir element of the space L(V) of linear mappings from V into itself associated to β and ρ. If x ∈ A is as in the preceding paragraph, then

[ρ_{x}, c_{ρ}(β)] = ∑_{j=1}^{n}[ρ_{x}, ρ_{u_{j}} ∘ ρ_{w_{j}}], (13.3.9)

using the commutator bracket on L(V) corresponding to composition of linear mappings on V. It follows that

[ρ_{x}, c_{ρ}(β)] = ∑_{j=1}^{n}[ρ_{x}, ρ_{u_{j}}] ∘ ρ_{w_{j}} + ∑_{j=1}^{n} ρ_{u_{j}} ∘ [ρ_{x}, ρ_{w_{j}}], (13.3.10)

as in Section 2.5. Because ρ is a Lie algebra representation, we get that

[ρ_{x}, c_{ρ}(β)] = ∑_{j=1}^{n} ρ_{[x,u_{j}]_{A}} ∘ ρ_{w_{j}} + ∑_{j=1}^{n} ρ_{u_{j}} ∘ ρ_{[x,w_{j}]_{A}}

(13.3.11)  = ∑_{j=1}^{n} ∑_{l=1}^{n} a_{j,l} ρ_{u_{l}} ∘ ρ_{w_{j}} + ∑_{j=1}^{n} ∑_{l=1}^{n} b_{j,l} ρ_{u_{j}} ∘ ρ_{w_{l}} = 0,

using (13.3.3) and (13.3.4) in the second step, and (13.3.7) in the third step. This corresponds to part of the discussion on p27 of [14].

Let u′_{1}, …, u′_{n} and w′_{1}, …, w′_{n} be bases for A as a vector space over k. We can express these bases in terms of the u_{j}’s and w_{l}’s, so that

u′_{h} = ∑_{j=1}^{n} μ_{h,j} u_{j} (13.3.12)

for every h = 1, …, n and

w′_{r} = ∑_{l=1}^{n} ν_{r,l} w_{l} (13.3.13)

for every r = 1, …, n, where μ = (μ_{h,j}) and ν = (ν_{r,l}) are invertible n × n matrices with entries in k. Thus

(13.3.14)  β(u′_{h}, w′_{r}) = ∑_{j=1}^{n} ∑_{l=1}^{n} μ_{h,j} ν_{r,l} β(u_{j}, w_{l})

= ∑_{j=1}^{n} ∑_{l=1}^{n} μ_{h,j} ν_{r,l} δ_{j,l} = ∑_{j=1}^{n} μ_{h,j} ν_{r,j}

for every h, r = 1, …, n, using (13.3.2) in the second step. It follows that

β(u′_{h}, w′_{r}) = δ_{h,r} (13.3.15)

for every h, r = 1, …, n if and only if

∑_{j=1}^{n} μ_{h,j} ν_{r,j} = δ_{h,r} (13.3.16)

for every h, r = 1, …, n. Of course, (13.3.16) is the same as saying that μ times the transpose of ν is the identity matrix. This is equivalent to the condition that the transpose of ν times μ be the identity matrix, which means that

∑_{h=1}^{n} ν_{h,l} μ_{h,j} = δ_{l,j} (13.3.17)

for every j, l = 1, …, n. If ρ is as in the previous paragraph, then

∑_{h=1}^{n} ρ_{u′_{h}} ∘ ρ_{w′_{h}} = ∑_{h=1}^{n} ∑_{j=1}^{n} ∑_{l=1}^{n} μ_{h,j} ν_{h,l} ρ_{u_{j}} ∘ ρ_{w_{l}}. (13.3.18)

If (13.3.17) holds, then we get that

∑_{h=1}^{n} ρ_{u′_{h}} ∘ ρ_{w′_{h}} = ∑_{j=1}^{n} ∑_{l=1}^{n} δ_{l,j} ρ_{u_{j}} ∘ ρ_{w_{l}} = ∑_{j=1}^{n} ρ_{u_{j}} ∘ ρ_{w_{j}}, (13.3.19)

which is the same as (13.3.8).

The hypothesis that β be symmetric on A does not seem to have been used so far, but it does give some additional properties. If β is symmetric on A, then the conditions on u_{1}, …, u_{n} and w_{1}, …, w_{n} are symmetric in these two bases for A.

13.4 Another perspective

Let us continue with the same notation and hypotheses as in the previous section. Let A′ be the space of linear functionals on A, as a vector space over k. Remember that the tensor products A ⨂ A and A ⨂ A′ can be defined as vector spaces over k, as in Section 7.12. The product of an element of A and a linear functional on A defines a linear mapping from A into itself. This defines a bilinear mapping from A × A′ into the space L(A) of linear mappings from A into itself, as a vector space over k. This leads to a linear mapping from A ⨂ A′ into L(A), which is an isomorphism in this case, because A has finite dimension as a vector space over k. If z ∈ A, then

β_{z}(x) = β(x, z) (13.4.1) defines a linear functional on A, and z ↦ β_{z} is a linear mapping from A into A′. More precisely, z ↦ β_{z} is a one-to-one linear mapping from A onto A′, because β is nondegenerate on A. This leads to a one-to-one linear mapping from A ⨂ A onto A ⨂ A′.

The condition (13.3.2) on the bases u_{1}, …, u_{n} and w_{1}, …, w_{n} for A is the same as saying that

∑_{j=1}^{n} u_{j} β(x, w_{j}) = x (13.4.2)

for every x ∈ A. Consider

∑_{j=1}^{n} u_{j} ⊗ w_{j}, (13.4.3)

as an element of A ⨂ A. This corresponds to the identity mapping on A under the isomorphisms mentioned in the preceding paragraph, because of (13.4.2). Remember that A may be considered as a module over itself, as a Lie algebra over k, using the adjoint representation. It follows that A′, A, and the related tensor products may be considered as modules over A too, in the usual way. The invariance condition on β implies that z ↦ β_{z} is a homomorphism from A into A′, as modules over A. One can use this to get that (13.4.3) is invariant under the corresponding representation of A on A ⨂ A, because the identity mapping on A is invariant under the respresentation of A on L(A). This can also be verified using (13.3.3), (13.3.4), and (13.3.7), as before.

The image of (13.4.3) in the universal eveloping algebra of A is called the Casimir element associated to β, as on p46 of [25]. One can get (13.3.8) from (13.4.3) using the action of the universal enveloping algebra on V associated to the representation ρ, as in [25]. This amounts to using ρ to get a bilinear mapping from A × A into L(V), and thus a linear mapping from A ⨂ A into L(V). Related matters are discussed on p118-9 of [14].

Let u′_{1}, …, u′_{n} and w′_{1}, …, w′_{n} be bases for A again, which can be expressed in terms of the u_{j}’s and w_{l}’s as in (13.3.12) and (13.3.13), respectively. Thus

∑_{h=1}^{n} u′_{h} ⊗ w′_{h} = ∑_{h=1}^{n} ∑_{j=1}^{n} ∑_{l=1}^{n} μ_{h,j} ν_{h,l} u_{j} ⊗ w_{l}. (13.4.4)

If (13.3.17) holds, then it follows that

∑_{h=1}^{n} u′_{h} ⊗ w′_{h} = ∑_{j=1}^{n} ∑_{l=1}^{n} δ_{l,j} u_{j} ⊗ w_{l} = ∑_{j=1}^{n} u_{j} ⊗ w_{j}. (13.4.5)

This can also be obtained from the fact that (13.4.3) corresponds to the identity mapping on A under the isomorphisms mentioned earlier.

If β is symmetric on A, then (13.4.3) is symmetric as an element of A ⨂ A, as in Section 7.15. This follows from (13.4.5), because (13.3.2) is symmetric in u_{1}, …, u_{n} and w_{1}, …, w_{n} in this case.

13.5 A more particular situation

Let k be a field of characteristic 0, and let (A, [·, ·]_{A}) be a Lie algebra over k with positive finite dimension n as a vector space over k. Also let V be a finite-dimensional vector space over k, and let ρ be a representation of A on V. Put β_{ρ}(x, y) = tr_{V} (ρ_{x} ∘ ρ_{y}) (13.5.1)

for every x, y ∈ A, which defines a symmetric bilinear form on A. Remember that (13.5.1) satisfies the associativity or invariance condition (13.3.1), as in Section 7.9. Suppose from now on in this section that A is semisimple as a Lie algebra over k, and that ρ is injective as a Lie algebra homomorphism from A into gl(V).

Remember that the radical of (13.5.1) in A is defined by

(13.5.2) A^{β_{ρ}} = {x ∈ A : β_{ρ}(x, y) = 0 for every y ∈ A},

as in Section 7.11. This is an ideal in A, because (13.5.1) is associative on A, as before. The image {ρ_{x} : x ∈ A^{β_{ρ}}} (13.5.3)

of A^{β_{ρ}} under ρ is a Lie subalgebra of gl(V). Using Cartan’s criterion, we get that (13.5.3) is solvable as a Lie algebra over k. This implies that (13.5.2) is solvable, because ρ is injective. It follows that A^{β_{ρ}} = {0}, because A is semisimple. This shows that (13.5.1) is nondegenerate on A under these conditions.

Let u_{1}, …, u_{n} be a basis for A, as a vector space over k. As in Section 13.3, there is a basis w_{1}, …, w_{n} for A that satisfies (13.3.2), with β = β_{ρ}. This leads to a Casimir element c_{ρ} = c_{ρ}(β_{ρ}) of L(V) associated to ρ as in (13.3.8). Observe that

tr_{V} c_{ρ} = ∑_{j=1}^{n} tr_{V} (ρ_{u_{j}} ∘ ρ_{w_{j}}) = ∑_{j=1}^{n} β_{ρ}(u_{j}, w_{j}) = n. (13.5.4)

In particular, this means that c_{ρ} ≠ 0, because we are assuming for convenience that A ≠ {0}.

Remember that c_{ρ} commutes with the action of ρ on V, as in (13.3.11). If ρ is irreducible on V, then it follows that c_{ρ} is a one-to-one mapping from V onto itself, by Schur’s lemma, as in Section 6.14. Otherwise, if W is a linear subspace of V that is invariant under the action of ρ, then c_{ρ} maps W into itself as well. This corresponds to parts of the discussions on p27 of [14] and p46 of [25].

13.6 A splitting theorem

Let k be a field of characteristic 0, and let A be a finite-dimensional semisimple Lie algebra over k. Also let V be a finite-dimensional vector space over k, and let ρ^{V} be a representation of A on V. Suppose that W is a codimension-one linear subspace of V, and that the action of ρ^{V} on V maps W into itself. We would like to show that V is the direct sum of W and a one-dimensional subspace that is mapped to itself by ρ^{V}, as in Section 13.2. Remember that the induced action of ρ^{V} on V/W is trivial, as before.

Let ρ^{W} be the representation of A on W obtained by restricting ρ^{V} from V to W. In this section, we consider the case where W is irreducible with respect to ρ^{W}. This corresponds to arguments on p28-9 of [14], and p47 of [25]. These arguments focus on V and W, respectively, as we shall see. Note that the roles of V and W are exchanged in the notation used on p47 of [25].

If x, y ∈ A, then put

β_{ρ^{V}} (x, y) = tr_{V} (ρ_{x}^{V} ∘ ρ_{y}^{V}) (13.6.1)

and β_{ρ^{W}} (x, y) = tr_{W} (ρ_{x}^{W} ∘ ρ_{y}^{W}), (13.6.2)

as in (13.5.1). In this situation, (13.6.1) and (13.6.2) are the same, because the action induced on V/W by ρ^{V} is trivial. This also uses the remarks in Section 7.10.

Let us begin with the argument in [14]. Without loss of generality, we may suppose that ρ^{V} is injective as a Lie algebra homomorphism from A into gl(V). Otherwise, we can replace A with its quotient by the kernel of ρ^{V}. This quotient of A is semisimple as a Lie algebra over k too, as in Section 10.14. We may suppose that A ≠ {0} as well, since otherwise the problem is very easy.

Let c_{ρ^{V}} be the Casimir element of L(V) associated to ρ^{V} as in the previous section. Thus c_{ρ^{V}} maps W into itself, because W is invariant under ρ^{V}, by hypothesis. The kernel of c_{ρ^{V}} in V is invariant under the action of ρ^{V}, because c_{ρ^{V}} commutes with the action of ρ^{V} on V, as in (13.3.11).

Because c_{ρ^{V}} (W) ⊆ W, c_{ρ^{V}} induces a linear mapping from V/W into itself. This induced mapping is equal to 0, because of the corresponding statement for ρ^{V}, and the definition of c_{ρ^{V}}. If c_{ρ^{V}} ≡ 0 on W, then it follows that tr_{V} c_{ρ^{V}} = 0, as in Section 7.10. This contradicts (13.5.4), because A ≠ {0}.

Thus c_{ρ^{V}} ≢ 0 on W, so that the restriction of c_{ρ^{V}} to W is a one-to-one mapping onto W, by Schur’s lemma, as in Section 6.14. However, c_{ρ^{V}} is not invertible on V, because the induced mapping on V/W is equal to 0. Hence the kernel of c_{ρ^{V}} is a one-dimensional linear subspace of V whose intersection with W is trivial. This gives an invariant complement of W in V, as desired.

Now let us consider the argument in [25]. Let A_{0} be the kernel of ρ^{W}, as a Lie algebra homomorphism from A into gl(W). Thus A_{0} is an ideal in A. If x ∈ A_{0}, then ρ_{x}^{V} = ρ_{x}^{W} = 0 on W, and ρ_{x}^{V} (V) ⊆ W, because the mapping on V/W induced by ρ_{x}^{V} is equal to 0. If y ∈ [A_{0}, A_{0}], then it follows that ρ_{y}^{V} = 0 on V. Remember that A_{0} is semisimple as a Lie algebra over k, because A is semisimple, as in Section 10.14. This implies that A_{0} = [A_{0}, A_{0}], as in Section 10.15. It follows that ρ_{y}^{V} = 0 on V for every y ∈ A_{0}. Note that A/A_{0} is semisimple as a Lie algebra over k, as in Section 10.14 again. This permits us to reduce to the case where ρ^{W} is injective as a Lie algebra homomorphism from A into gl(W), since otherwise we could replace A with A/A_{0}.

We may suppose that A ≠ {0} too, as before. Because ρ^{W} is injective as a Lie algebra homomorphism from A into gl(W), (13.6.2) is nondegenerate on A, as in the previous section. Let c_{ρ^{V}} (β_{ρ^{W}}) be the Casimir element of L(V) that corresponds to ρ^{V} and (13.6.2), as in Section 13.3. Note that c_{ρ^{V}} (β_{ρ^{W}}) maps V into W, because of the corresponding property of ρ^{V}. The restriction of c_{ρ^{V}} (β_{ρ^{W}}) to W is the same as the Casimir element c_{ρ^{W}} = c_{ρ^{W}} (β_{ρ^{W}}) of L(W) that corresponds to ρ^{W} and (13.6.2), by the definitions of c_{ρ^{V}} (β_{ρ^{W}}) and ρ^{W}. The trace of c_{ρ^{W}} on W is equal to the dimension of A, as in (13.5.4). Thus c_{ρ^{W}} ≠ 0, because A ≠ {0}. This implies that c_{ρ^{W}} is invertible on W, by Schur’s lemma, because ρ^{W} is irreducible on W. It follows that the kernel of c_{ρ^{V}} (β_{ρ^{W}}) is a one-dimensional subspace of V complementary to W, as before.

More precisely, the argument in [25] is formulated in terms of the Casimir element of the universal enveloping algebra of A associated to (13.6.2), as in Section 13.3. This is used again in the remarks following the proof, with the roles of V and W interchanged in the notation of [25], as before.

13.7 Weyl’s theorem

Let k be a field of characteristic 0 again, let A be a finite-dimensional semisimple Lie algebra over k, and let V be a module over A that is finite-dimensional as a vector space over k. Also let W be a nonzero proper submodule of V. We would like to show that V corresponds to a direct sum of W and another submodule of V. This will imply Weyl’s theorem, as in Section 13.2.

Remember that the space L(V, W) of all linear mappings from V into W may be considered as a module over A too, as in Section 7.5. More precisely, if a ∈ A and T ∈ L(V, W), then a · T is defined as a linear mapping from V into W by putting (a · T)(v) = a · (T (v)) − T (a · v) (13.7.1)

for every v ∈ V. This uses the action of A on V in both terms on the right, and the fact that this action sends W into itself, by hypothesis.

Let Z be the collection of linear mappings T from V into W for which there is an α(T) ∈ k such that T (w) = α(T) w (13.7.2)

for every w ∈ W. It is easy to see that Z is a linear subspace of L(V, W), and that α defines a linear mapping from Z onto k. If a ∈ A, T ∈ Z, and w ∈ W, then

(a · T)(w) = a · (T (w)) − T (a · w) = a · (α(T) w) − α(T) (a · w) = 0. (13.7.3)

This implies that a · T ∈ Z, with α(a · T) = 0. In particular, Z is a submodule of L(V, W), as a module over A.

Let Z_{0} be the collection of linear mappings T from V into W such that T ≡ 0 on W. Equivalently, this means that T ∈ Z, with α(T) = 0. Note that Z_{0} is a submodule of Z, as a module over A. More precisely, if a ∈ A and T ∈ Z, then a · T ∈ Z_{0}, as in the preceding paragraph. The codimension of Z_{0} in Z is equal to one, because Z_{0} is the kernel of α on Z.

The splitting theorem discussed in Section 13.2 and the previous section implies that there is a one-dimensional submodule of Z, as a module over A, that is complementary to Z_{0}. Let R be a nonzero element of this one-dimensional complementary submodule. Note that α(R) ≠ 0, since otherwise R ∈ Z_{0}, which would imply that R = 0. We may as well suppose that α(R) = 1, by multiplying R by 1/α(R). If a ∈ A, then a · R ∈ Z_{0}, as before, which implies that a · R = 0, because a · R is in the submodule of Z complementary to Z_{0}. This means that R commutes with the actions of A on V and W. Thus the kernel of R is a submodule of V, as a module over A. The kernel of R is also complementary to W in V, as desired.

13.8 Symmetric forms and tensors

Let k be a field, let A be a finite-dimensional vector space over k, and let β(·, ·) be a bilinear form on A. Thus

β_{z}(x) = β(x, z) (13.8.1)

defines a linear functional on A for each z ∈ A, and z ↦ β_{z} defines a linear mapping from A into the dual space A′ of all linear functionals on A. Let y, z ∈ A be given, and put

T_{y,z}(x) = β_{z}(x) y = β(x, z) y (13.8.2)

for every x ∈ A, which defines T_{y,z} as a linear mapping from A into itself. Observe that β(T_{y,z}(x), w) = β(x, z) β(y, w) (13.8.3)

and β(x, T_{z,y}(w)) = β(x, z) β(w, y) (13.8.4)

for every w, x, y, z ∈ A. If β(·, ·) is symmetric as a bilinear form on A, then we get that β(T_{y,z}(x), w) = β(x, T_{z,y}(w)) (13.8.5)

for every w, x, y, z ∈ A.

Let us suppose from now on in this section that β(·, ·) is nondegenerate on A. If T is any linear mapping from A into itself, then there is a unique adjoint linear mapping T^{∗} from A into itself such that

β(T (x), w) = β(x, T^{∗}(w)) (13.8.6)

for every w, x ∈ A, as in Section 2.14. If β(·, ·) is symmetric on A, then

(T_{y,z})^{∗} = T_{z,y} (13.8.7) for every y, z ∈ A, by (13.8.5). Remember that A ⨂ A and A ⨂ A′ are defined as vector spaces over k, as in Section 7.12. Clearly

(y, z) ↦ T_{y,z} (13.8.8)

defines a mapping from A × A into the space L(A) of linear mappings from A into itself that is bilinear over k. This leads to a linear mapping from A ⨂ A into L(A), with y ⊗ z ↦ T_{y,z} (13.8.9)

for every y, z ∈ A. More precisely, we have seen that there is a natural isomorphism from A ⨂ A′ onto L(A), as vector spaces over k, as in Section 13.4. Because β(·, ·) is nondegenerate on A, z ↦ β_{z} is an isomorphism from A onto A′, as vector spaces over k. This leads to an isomorphism from A ⨂ A onto A ⨂ A′, as vector spaces over k, as before. We can compose these mappings to get an isomorphism from A ⨂ A onto L(A), as vector spaces over k. This is the same as the mapping determined by (13.8.9), by construction.

There is a natural automorphism on A ⨂ A, as a vector space over k, with

y ⊗ z ↦ z ⊗ y (13.8.10)

for every y, z ∈ A, as in Section 7.15. Let us suppose from now on in this section that β(·, ·) is symmetric on A. Of course,

(y, z) ↦ T_{z,y} = (T_{y,z})^{∗} (13.8.11)

defines a mapping from A × A into L(A), which is bilinear over k. This leads to a linear mapping from A ⨂ A into L(A), with

y ⊗ z ↦ T_{z,y} = (T_{y,z})^{∗} (13.8.12)

for every y, z ∈ A. This is the same as the composition of the mapping from A ⨂ A into L(A) determined by (13.8.9) with T ↦ T^{∗}, as a linear mapping from L(A) onto itself. This is also the same as the composition of the automorphism on A ⨂ A determined by (13.8.10) with the linear mapping from A ⨂ A into L(A) determined by (13.8.9). More precisely, these linear mappings from A ⨂ A into L(A) are the same, because they correspond to the same bilinear mapping from A × A into L(A).

An element of T^{2}A = A ⨂ A is said to be symmetric if it is invariant under the automorphism determined by (13.8.10), as in Section 7.15. Symmetric elements of A ⨂ A correspond exactly to linear mappings from A into itself that are self-adjoint with respect to β(·, ·), under the vector space isomorphism from A ⨂ A onto L(A) determined by (13.8.9), as in the preceding paragraph. Note that the identity mapping on A is automatically self-adjoint with respect to β(·, ·). Thus the element of A ⨂ A that corresponds to the identity mapping on A under the isomorphism just mentioned is symmetric in A ⨂ A under these conditions. This gives another way to look at the symmetry condition mentioned at the end of Section 13.4.

13.9 Reductive Lie algebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Remember that A may be considered as a module over itself, using the adjoint representation. The kernel of the adjoint representation, as a Lie algebra homomorphism from A into gl(A), is the same as the center Z(A) of A as a Lie algebra. Of course, the quotient A/Z(A) may be considered as a Lie algebra over k too, because Z(A) is an ideal in A. Thus A may be considered as a module over A/Z(A), as a Lie algebra over k.

An ideal in A as a Lie algebra is the same as a submodule of A, as a module over itself with respect to the adjoint representation. Similarly, ideals in A as a Lie algebra are the same as submodules of A as a module over A/Z(A).

Now let k be a field of characteristic 0, and suppose that A is a finitedimensional Lie algebra over k. Suppose also that A is reductive as a Lie algebra, as in Section 11.3. Thus Z(A) is the same as the solvable radical of A. In this case, A/Z(A) is semisimple as a Lie algebra over k, as in Section 9.4.

Of course, Z(A) is a submodule of A, as a module over A/Z(A). Because A/Z(A) is semisimple as a Lie algebra over k, there is a submodule B of A, as a module over A/Z(A), such that A corresponds to the direct sum of Z(A) and B, as in Section 13.7. This means that

Z(A) ∩ B = {0}, Z(A) + B = A, (13.9.1)

and that B is an ideal in A, as before.

Observe that [A, A] = [B, B] ⊆ B. (13.9.2)

We also have that [A/Z(A), A/Z(A)] = A/Z(A), (13.9.3)

because A/Z(A) is semisimple as a Lie algebra over k, as in Section 10.15. The canonical quotient mapping q from A onto A/Z(A) maps [A, A] onto the left side of (13.9.3). In this situation, the restriction of q to B is a one-to-one mapping from B onto A/Z(A). It follows that

B = [A, A], (13.9.4)

and that [A, A] is semisimple as a Lie algebra over k. This shows that A corresponds to the direct sum of Z(A) and [A, A] as a Lie algebra over k, by definition of Z(A). This corresponds to the second part of part (a) of Exercise 5 on p30 of [14], and to part (a) of the proposition on p102 of [14]. This also seems to correspond to a comment at the bottom of p50 in [25].

13.10 Semisimple ideals

Let k be a field of characteristic 0, and let (B, [·, ·]_{B}) be a Lie algebra over k with finite dimension as a vector space over k. Suppose that A is an ideal in B that is semisimple as a Lie algebra over k. Under these conditions, there is a unique ideal B_{0} in B such that B corresponds to the direct sum of A and B_{0}, as a Lie algebra over k. This is Corollary 1 on p47 of [25].

Of course, B may be considered as a module over itself, using the adjoint representation. We can restrict the action of B on itself to an action of A on B, so that B becomes a module over A, as a Lie algebra over k. Note that A may be considered as a submodule of B, as a module over A. As in Section 13.7, there is a submodule B_{0} of B, as a module over A, such that B corresponds to the direct sum of A and B_{0}, as a module over A. In particular, B corresponds to the direct sum of A and B_{0}, as a vector space over k, so that

A ∩ B_{0} = {0} and A + B_{0} = B. (13.10.1)

Let us check that [A, B_{0}] = {0}, (13.10.2)

where the left side is defined as in Section 9.2, as usual. Clearly

[A, B_{0}] ⊆ [A, B] ⊆ A, (13.10.3)

because A is an ideal in B. We also have that [A, B_{0}] ⊆ B_{0}, because B_{0} is a submodule of B, as a module over A. Thus [A, B_{0}] ⊆ A ∩ B_{0} = {0}, as desired.

Suppose that y ∈ B satisfies [a, y]_{B} = 0 for every a ∈ A. Because A + B_{0} = B, there are x ∈ A and z ∈ B_{0} such that y = x + z. We already know that [a, z]_{B} = 0, by (13.10.2), and so we get that [a, x]_{B} = 0 for every a ∈ A. This implies that x = 0, because x ∈ A and A is semisimple, so that the center of A is trivial. It follows that y = z ∈ B_{0}. This shows that B_{0} is exactly the set of y ∈ B such that [a, y]_{B} = 0 for every a ∈ A. Thus B_{0} is uniquely determined by the properties of being a submodule of B, as a module over A, that is complementary to A.

Equivalently, B_{0} is the centralizer of A in B, as in Section 7.6. It is easy to see that B_{0} is an ideal in B, as a Lie algebra over k, because A is an ideal in B. More precisely, if a ∈ A, w ∈ B, and y ∈ B_{0}, then one can verify that

[a, [w, y]_{B}]_{B} = 0, (13.10.4)

using the Jacobi identity and the fact that [a, w]_{B} ∈ A, because A is an ideal in B. We also have that B corresponds to the direct sum of A and B_{0}, as a Lie algebra over k, because of (13.10.2).

13.11 Another approach

Let us mention another approach to the statement in the previous section, using an argument like the one in the proof of the second theorem on p23 of [14]. Let k be a field, and let (B, [·, ·]_{B}) be a finite-dimensional Lie algebra over k. If x ∈ B, then ad_{B,x}(z) = [x, z]_{B} (13.11.1) defines a linear mapping from B into itself, as in Section 2.4. Put

b(x, y) = tr_{B}(ad_{B,x} ∘ ad_{B,y}) (13.11.2)

for every x, y ∈ B, which is the Killing form on B, as in Section 7.9. Remember that b([x, w]_{B}, y) = b(x, [w, y]_{B}) (13.11.3)

for every w, x, y ∈ B, which is to say that (13.11.2) is associative as a bilinear form on B, or equivalently that (13.11.2) is invariant with respect to the adjoint representation on B.

Let A be an ideal in B, as a Lie algebra over k. If x ∈ A, then let ad_{A,x} be the restriction of (13.11.1) to z ∈ A, which defines a linear mapping from A into itself. Put b_{A}(x, y) = tr_{A}(ad_{A,x} ∘ ad_{A,y}) (13.11.4)

for every x, y ∈ A, which is the Killing form on A, as a Lie algebra over k. If x, y ∈ A, then b_{A}(x, y) = b(x, y), (13.11.5)

as in Section 7.10, because A is an ideal in B.

Put (13.11.6) A^{⊥} = {x ∈ B : b(x, y) = 0 for every y ∈ A}.

It is easy to see that this is an ideal in B, using (13.11.3) and the hypothesis that A is an ideal in B, as in Section 10.14. Let us suppose from now on in this section that (13.11.4) is nondegenerate on A. Remember that this holds when A is semisimple as a Lie algebra and k has characteristic 0, as in Section 10.13. It follows that A ∩ A^{⊥} = {0} (13.11.7)

in this situation. Note that [A, A^{⊥}] ⊆ A ∩ A^{⊥}, because A and A^{⊥} are ideals in A. Hence [A, A^{⊥}] = {0}, (13.11.8)

by (13.11.7).

If x ∈ B and y ∈ A, then put

λ_{x}(y) = b(x, y), (13.11.9)

which defines λ_{x} as a linear functional on A. Using the nondegeneracy of (13.11.4), we get that there is a w ∈ A such that

λ_{x}(y) = b_{A}(w, y) (13.11.10)

for every y ∈ A. This means that

b(x, y) = b_{A}(w, y) = b(w, y) (13.11.11)

for every y ∈ A, so that x − w ∈ A^{⊥}. Thus

A + A^{⊥} = B. (13.11.12)

This shows that B corresponds to the direct sum of A and A^{⊥}, as a Lie algebra over k.

One can check that A^{⊥} is the same as the centralizer of A in B, as in the previous section. More precisely, A^{⊥} is contained in the centralizer of A, by (13.11.8). The nondegeneracy of (13.11.4) on A implies that the center of A is trivial, and in fact that A is semisimple as a Lie algebra, as in Section 10.13. One can use this and (13.11.12) to get that A^{⊥} is the centralizer of A, as before.

13.12 Inner derivations

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. If x ∈ A, then let ad x be the mapping from A into itself that sends y ∈ A to [x, y]_{A}, as usual. Remember that the space Der(A) of derivations on A is a Lie algebra over k with respect to the commutator bracket associated to composition of mappings, as in Section 2.5. If x ∈ A, then ad x ∈ Der(A), and x ↦ ad x is a Lie algebra homomorphism from A into Der(A), as in Section 2.4. Let

ad A = {ad x : x ∈ A} (13.12.1)

be the image of this mapping, whose elements are said to be inner derivations on A.

Let x, y ∈ A and δ ∈ Der(A) be given, and observe that

([δ, ad x])(y) = δ((ad x)(y)) − (ad x)(δ(y))
(13.12.2)  = δ([x, y]_{A}) − [x, δ(y)]_{A} = [δ(x), y]_{A} = (ad δ(x))(y).

This means that [δ, ad x] = ad δ(x), (13.12.3) which implies that ad A is an ideal in Der(A). This corresponds to Exercise 2.1 on p9 of [14], and some remarks on p23 of [14]. This also comes up in the proof of Corollary 2 on p48 of [25].

Suppose from now on in this section that k is a field, and that A is a finitedimensional semisimple Lie algebra over k. Remember that the kernel of the adjoint representation of A is the center of A, which is trivial, because A is semisimple. Thus the adjoint representation is a Lie algebra isomorphism from A onto ad A, so that ad A is semisimple as a Lie algebra over k in particular. If k has characteristic 0, then it follows that Der(A) corresponds to the direct sum of ad A and the centralizer of ad A in Der(A), as in Section 13.10. If x ∈ A and δ ∈ Der(A) is in the centralizer of ad A in Der(A), then ad δ(x) = 0, by (13.12.3). This implies that δ(x) = 0, because the kernel of the adjoint representation of A is trivial. This means that δ = 0, so that every derivation on A is an element of ad A. This corresponds to Corollary 2 on p48 of [25].

This also corresponds to the second theorem on p23 of [14]. The proof given there uses the argument in the previous section instead of the one in Section 13.10. This could be used when the Killing form on A is nondegenerate, without asking that k have characteristic 0.

Nilpotence and diagonalizability

14.1 A duality argument

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. If x ∈ A, then put ad_{x}(y) = [x, y]_{A} for every y ∈ A, as usual. Let β(·, ·) be a bilinear form on A that is associative or invariant under the adjoint representation on A, so that

β(ad_{x}(w), z) = −β(w, ad_{x}(z)) (14.1.1)

for every w, x, z ∈ A. In particular,

β(ad_{x}(w), z) = 0 (14.1.2)

when [x, z]_{A} = 0. This corresponds to part of part (a) of Exercise 2 on p54 of [25].

Suppose now that k is a field, that A has finite dimension as a vector space over k, and that β(·, ·) is also nondegenerate as a bilinear form on A. Let x ∈ A be given, and note that the image

{ad_{x}(w) : w ∈ A} (14.1.3)

of ad_{x} is a linear subspace of A. If z ∈ A, then (14.1.2) holds for every w ∈ A if and only if β(w, ad_{x}(z)) = 0 (14.1.4)

for every w ∈ A, by (14.1.1). Because β(·, ·) is nondegenerate on A, (14.1.4) holds for every w ∈ A if and only if ad_{x}(z) = 0. Thus (14.1.2) holds for every w ∈ A if and only if ad_{x}(z) = 0.

Consider

(14.1.5) {y ∈ A : β(y, z) = 0 for every z ∈ A such that [x, z]_{A} = 0}, which is a linear subspace of A. Of course, (14.1.3) is contained in (14.1.5), because (14.1.2) holds for every w ∈ A when [x, z]_{A} = 0, as in the preceding paragraph. In fact, (14.1.3) is equal to (14.1.5) in this situation, because β(·, ·) is nondegenerate on A. More precisely, this uses the fact that if z ∈ A satisfies (14.1.2) for every w ∈ A, then [x, z]_{A} = 0, and hence β(y, z) = 0 for every y in (14.1.5). This corresponds to the second part of part (a) of Exercise 2 on p54 of [25].

Let us now take β(·, ·) to be the Killing form on A, so that

β(u, v) = tr_{A}(ad_{u} ∘ ad_{v}) (14.1.6)

for every u, v ∈ A. Remember that this satisfies (14.1.1), as in Section 7.9. Suppose that x, z ∈ A commute, in the sense that [x, z]_{A} = 0. This implies that ad_{x} and ad_{z} commute as linear mappings from A into itself, because the adjoint representation of A is a representation of A as a Lie algebra over k. If ad_{x} is also nilpotent as a linear mapping from A into itself, then it follows that ad_{x} ∘ ad_{z} is nilpotent as a linear mapping from A into itself, as in Section 9.7. In this case, we get that

β(x, z) = tr_{A}(ad_{x} ∘ ad_{z}) = 0, (14.1.7)

by standard arguments. If the Killing form is nondegenerate on A, then the previous arguments imply that there is a w ∈ A such that

ad_{x}(w) = x. (14.1.8)

This corresponds to part of part (b) of Exercise 2 on p54 of [25].

14.2 A criterion for nilpotence

Let k be a field of characteristic 0, and let A be an associative algebra over k, where multiplication of a, b ∈ A is expressed as a b. Also let δ be a derivation on A, and suppose that b ∈ A is an eigenvector of δ with eigenvalue λ ∈ k, so that δ(b) = λ b. (14.2.1)

In particular, this means that b commutes with δ(b). If j is an integer with j ≥ 2, then δ(b^{j}) = j · b^{j−1} δ(b) = j · λ b^{j}, (14.2.2)

so that b^{j} is an eigenvector of δ with eigenvalue j · λ. If λ ≠ 0, then the eigenvalues j · λ with j ∈ Z_{+} are all distinct, because k has characteristic 0. If, for each j ∈ Z_{+}, b^{j} ≠ 0, then it follows that the b^{j}’s are linearly independent in A, as a vector space over k. If A has finite dimension as a vector space over k, then b^{j} = 0 for some j ≥ 1.

Remember that δ_{a}(x) = [a, x] = a x − x a (14.2.3) defines a derivation on A for every a ∈ A. Suppose that b ∈ A is an eigenvector of δ_{a} with eigenvalue λ ∈ k, which is to say that

δ_{a}(b) = [a, b] = λ b. (14.2.4)

If λ ≠ 0 and A has finite dimension as a vector space over k, then b is nilpotent in A, as in the preceding paragraph.

Now let (A, [·, ·]_{A}) be a Lie algebra over k, and suppose that u, x ∈ A satisfy

[u, x]_{A} = x. (14.2.5)

Let V be a vector space over k, and let ρ be a representation of A as a Lie algebra on V. Thus

ρ_{[u,x]_{A}} = [ρ_{u}, ρ_{x}] = ρ_{u} ∘ ρ_{x} − ρ_{x} ∘ ρ_{u} (14.2.6)

on V. This means that [ρ_{u}, ρ_{x}] = ρ_{x}, (14.2.7)

by (14.2.5). Suppose that V has finite dimension, so that the algebra L(V) of linear mappings from V into itself has finite dimension as a vector space over k too. Under these conditions, we get that ρ_{x} is nilpotent as a linear mapping from V into itself. This corresponds to part of part (b) of Exercise 2 on p54 of [25].

Suppose that A has finite dimension as a vector space over k. Let x be an ad-nilpotent element of A, so that ad_{x} is nilpotent on A. If A is semisimple as a Lie algebra over k, then the Killing form on A is nondegenerate, as in Section 10.13. In this case, there is a u ∈ A such that (14.2.5) holds, as in the previous section. This is another part of part (b) of Exercise 2 on p54 in [25].

14.3 ad-Diagonalizability

Let k be a field, let V be a finite-dimensional vector space over k, and let T be a linear mapping from V into itself. If k is algebraically closed and T is diagonalizable on V, then one may say that T is semisimple as a linear mapping on V, as on p17 of [14] and p40 of [25]. Otherwise, one may say that T is semisimple on V when T becomes diagonalizable after passing to an algebraic closure of k, as in Remark 1 after Theorem 5.1 on p50 of [25].

Let (A, [·, ·]_{A}) be a Lie algebra over k, with finite dimension as a vector space over k. If x ∈ A, then we put ad_{x}(y) = [x, y]_{A} for every y ∈ A, as usual. If ad_{x} is diagonalizable as a linear mapping from A into itself, then x is said to be ad-diagonalizable as an element of A. Similarly, x is said to be ad-semisimple as an element of A when ad_{x} is semisimple as a linear mapping from A into itself, as on p24 of [14], and Definition 5.5 on p52 of [25].

Suppose that k is an algebraically closed field of characteristic 0, and that A is semisimple as a Lie algebra over k. Let x ∈ A be given, so that ad_{x} is a linear mapping from A into itself. Remember that ad_{x} can be expressed in a unique way as a sum of two commuting linear mappings from A into itself, where one of these linear mappings is diagonalizable and the other is nilpotent, as in Section 10.8. We have also seen that ad_{x} is a derivation on A, as a Lie algebra over k. The diagonalizable and nilpotent parts of ad_{x} are derivations on A as well, as in Section 10.9. Because k has characteristic 0 and A is semisimple, every derivation on A is an inner derivation, as in Section 13.12. This means that there are x_{1}, x_{2} ∈ A such that the diagonalizable and nilpotent parts of ad_{x} are given by ad_{x_{1}} and ad_{x_{2}}, respectively. Of course, x_{1} and x_{2} are uniquely determined by ad_{x_{1}} and ad_{x_{2}}, respectively, because the center of A is trivial, by semisimplicity. Similarly, [x_{1}, x_{2}]_{A} = 0, (14.3.1)

because ad_{x_{1}} and ad_{x_{2}} commute as linear mappings on A. By construction, x_{1} is ad-diagonalizable on A, and x_{2} is ad-nilpotent on A. Note that x_{1} and x_{2} are uniquely determined by these properties, because of the analogous uniqueness property for the diagonalizable and nilpotent parts of ad_{x}. This corresponds to remarks on p24 of [14], and Theorem 5.6 on p52 of [25]. In particular, this is called the abstract Jordan decomposition of x in A in [14].

Let V be a finite-dimensional vector space over k again, and suppose that A is a Lie subalgebra of gl(V). Let x ∈ A be given, so that x is a linear mapping from V into itself. As before, x can be expressed in a unique way as y_{1} +y_{2}, where y_{1} and y_{2} are commuting linear mappings from V into itself, y_{1} is diagonalizable on V, and y_{2} is nilpotent on V. Of course, one would like to have y_{1}, y_{2} ∈ A, as on p24 of [14]. It is easy to get this when A = sl(V), as in [14]. Indeed, y_{2} has trace 0 on V, because y_{2} is nilpotent on V. This implies that y_{1} ∈ sl(V) in this case, because x ∈ A by hypothesis. Some other situations will be discussed in the next section.

If w, z ∈ gl(V), then put

ad_{gl(V),w}(z) = [w, z] = w ∘ z − z ∘ w, (14.3.2)

as usual. Let y ∈ gl(V) be given, so that y can be expressed in a unique way as y_{1} + y_{2}, where y_{1} and y_{2} are commuting linear mappings from V into itself, y_{1} is diagonalizable on V, and y_{2} is nilpotent on V, as before. It follows that

ad_{gl(V),y} = ad_{gl(V),y_{1}} + ad_{gl(V),y_{2}}, (14.3.3)

ad_{gl(V),y_{1}} commutes with ad_{gl(V),y_{2}} as linear mappings on gl(V), ad_{gl(V),y_{1}} is diagonalizable on gl(V), and ad_{gl(V),y_{2}} is nilpotent on gl(V), as in Section 10.9. Let A be a Lie subalgebra of gl(V). If w ∈ A, then ad_{gl(V),w} maps A into itself, and the restriction of ad_{gl(V),w} to A is the same as ad_{A,w}. If y_{1} ∈ A, then it follows that ad_{A,y_{1}} is diagonalizable as a linear mapping from A into itself. Similarly, if y_{2} ∈ A, then ad_{A,y_{2}} is nilpotent as a linear mapping from A into itself. Of course, ad_{A,y_{1}} commutes with ad_{A,y_{2}} on A, because ad_{gl(V),y_{1}} commutes with ad_{gl(V),y_{2}} on gl(V). Under these conditions, we also have that y ∈ A, and ad_{A,y} = ad_{A,y_{1}} + ad_{A,y_{2}}. (14.3.4)

This corresponds to some more of the remarks on p24 of [14].

14.4 Semisimple subalgebras of gl(V)

Let k be an algebraically closed field of characteristic 0, and let V be a finitedimensional vector space over k. Also let A be a Lie subalgebra of gl(V), and suppose that A is semisimple as a Lie algebra over k. Let x ∈ A be given, and remember that x can be expressed in a unique way as y_{1} + y_{2}, where y_{1} and y_{2} are commuting linear mappings from V into itself, y_{1} is diagonalizable on V, and y_{2} is nilpotent on V. Under these conditions, y_{1}, y_{2} ∈ A, as in the theorem on p29 of [14], and Corollary 5.4 on p52 of [25]. We shall follow the argument in [14] here, with some help from Section 13.10.

If w ∈ gl(V), then let ad_{gl(V),w} be as in (14.3.2). Thus ad_{gl(V),x} maps A into itself, because x ∈ A. Remember that ad_{gl(V),y_{1}} and ad_{gl(V),y_{2}} are the corresponding diagonalizable and nilpotent parts of ad_{gl(V),x}, as a linear mapping from gl(V) into itself, as in Section 10.9. It follows that ad_{gl(V),y_{1}} and ad_{gl(V),y_{2}} map A into itself as well, as in Section 10.8. Let N = N_{gl(V)}(A) be the normalizer of A in gl(V), as in Section 9.8. The previous statement says that y_{1}, y_{2} ∈ N. (14.4.1)

If N were equal to A, then the proof would be finished, but this does not work, because constant multiples of the identity mapping on V are automatically in N. Thus we need some additional conditions on y_{1}, y_{2}.

Note that V may be considered as a module over A, as a Lie algebra over k. Let W be an A-submodule of V, which is to say a linear subspace of V such that every element of A maps W into itself. Let B_{W} be the collection of z ∈ gl(V) such that z(W) ⊆ W, and the restriction of z to W has trace equal to 0, as a linear mapping from W into itself. Let us check that

A ⊆ B_{W} . (14.4.2)

If z ∈ A, then z(W) ⊆ W, by our hypothesis about W. Remember that A = [A, A], because A is semisimple and k has characteristic 0, as in Section 10.15. One can use this to show that the trace of the restriction of z to W is 0, as desired. Note that B_{W} is a Lie subalgebra of gl(V). In particular, we can take W = V, for which we get B_{V} = sl(V).

Let us check that y_{1}, y_{2} ∈ B_{W} (14.4.3)

for every A-submodule W of V. Remember that x maps W into itself, because x ∈ A. This implies that y_{1} and y_{2} map W into itself, as in Section 10.8. The restriction of y_{2} to W is nilpotent on W, because y_{2} is nilpotent on V. Hence the trace of the restriction of y_{2} to W is equal to 0. The trace of the restriction of x to W is equal to 0 too, as in the preceding paragraph. It follows that the trace of the restriction of y_{1} to W is equal to 0 as well. This gives (14.4.3), as desired.

Let B be the intersection of N with all the subalgebras B_{W}, over all Asubmodules W of V. Thus B is a Lie subalgebra of gl(V), and

A ⊆ B, (14.4.4) by (14.4.2). More precisely, A is an ideal in B, as a Lie algebra over k, because B ⊆ N, by construction. We also have that

y_{1}, y_{2} ∈ B, (14.4.5)

by (14.4.1) and (14.4.3). We would like to show that A = B.

Note that there is a unique ideal B_{0} in B such that B corresponds to the direct sum of A and B_{0}, as in Section 13.10. In particular, [A, B_{0}] = {0}, as before.

Let W be an irreducible A-submodule of V. If z ∈ B_{0}, then z maps W into itself, and z commutes with every element of A. Because k is algebraically closed, Schur’s lemma implies that z is equal to a constant multiple of the identity mapping on W, as in Section 6.14. However, we also have that the trace of the restriction of z to W is equal to 0, because z ∈ B. This implies that z ≡ 0 on W, because k has characteristic 0.

By Weyl’s theorem, V can be expressed as the direct sum of irreducible Asubmodules, as in Section 6.14. If z ∈ B_{0}, then it follows that z ≡ 0 on V, using the remarks in the preceding paragraph. This means that B_{0} = {0}, so that A = B, as desired.

14.5 Tensors

Let k be a field, and let V be a vector space over k of positive finite dimension n. If p is a positive integer, then T^{p}V is the pth tensor power of V, as in Section 7.15, which is to say the tensor product of p copies of V. This is a vector space over k of dimension n^{p}. More precisely, if v_{1}, …, v_{n} is a basis for V, then we can get a basis for T^{p}V using elements of the form

v_{j_{1}} ⊗ ⋯ ⊗ v_{j_{p}}, (14.5.1)

where j_{1}, …, j_{p} ∈ {1, …, n}. If p = 0, then one can interpret T^{p}V as being k, as a one-dimensional vector space over itself.

The dual space V′ of V is a vector space over k of dimension n as well. If q is a nonnegative integer, then T^{q} V′ is defined as a vector space over k of dimension n^{q}, as before. Let λ_{1}, …, λ_{q} be q linear functionals on V for some q ∈ Z_{+}, and consider

∏_{j=1}^{q} λ_{j}(u_{j}) (14.5.2)

as a k-valued function of (u_{1}, …, u_{q}) in the Cartesian product of q copies of V. This defines a multilinear mapping from the Cartesian product of q copies of V into k, which is to say a q-linear form on V. Of course, the space of q-linear forms on V is a vector space over k with respect to pointwise addition and scalar multiplication of functions. The mapping from (λ_{1}, …, λ_{q}) to (14.5.2) is multilinear over k as a mapping from the Cartesian product of q copies of V′ into the space of q-linear forms on V. This leads to a linear mapping from T^{q}V′ into the space of q-linear forms on V, which is in fact a vector space isomorphism.

If p and q are nonnegative integers, then put

T^{p,q}V = T^{p}V ⨂ T^{q}V′, (14.5.3)

which may also be denoted V_{p,q}, as on p40 of [25]. This reduces to T^{q}V′ when p = 0, and to T^{p}V when q = 0. Note that there is a natural isomorphism from T^{1,1}V = V ⨂ V′ onto the space L(V) of linear mappings from V into itself, as vector spaces over k, as in Section 13.4. Let v_{1}, …, v_{n} be a basis for V again, and let μ_{1}, …, μ_{n} be the corresponding dual basis for V′, so that μ_{j}(v_{l}) is equal to 1 when j = l and to 0 when j ≠ l. We can get a basis for T^{p,q}V, as a vector space over k, using elements of the form

(v_{j_{1}} ⊗ ⋯ ⊗ v_{j_{p}}) ⊗ (μ_{l_{1}} ⊗ ⋯ ⊗ μ_{l_{q}}), (14.5.4)

where j_{1}, …, j_{p}, l_{1}, …, l_{q} ∈ {1, …, n}.

Let A be a Lie algebra over k, and let ρ^{V} be a representation of A on V. If a ∈ A and λ ∈ V′, then put

ρ_{a}^{V′}(λ) = −λ ∘ ρ_{a}^{V} , (14.5.5)

which is a linear functional on V. This defines ρ^{V′} as a representation of A on V′, as in Section 7.2, with W = k. Equivalently, if a ∈ A, then ρ_{a}^{V} is a linear mapping from V into itself, which leads to a dual linear mapping from V′ into itself, as in Section 2.13. By construction, ρ_{a}^{V′} is −1 times this dual linear mapping on V′.

Using ρ^{V} and ρ^{V′}, we can make T^{p,q}V into a module over A, as in Section 7.12. More precisely, we have p + q actions of A on T^{p,q}V, using the actions of A on the individual factors on V and V′ in T^{p,q}V. These p + q actions of A on T^{p,q}V commute with each other, and T^{p,q}V is considered as a module over A as a Lie algebra with respect to the sum of these p + q actions, as before. If p = 0, then T^{p,q}V = T^{q}V′ can be identified with the space of q-linear forms on V, as mentioned earlier. In this case, the action of A corresponds to making the space of q-linear forms on V a module over A as a Lie algebra as in Section 7.5, using the trivial action of A on k, and combining the q actions of A corresponding to the q variables of a q-linear form on V.

14.6 Induced linear mappings

Let k be a field again, and let V be a finite-dimensional vector space over k of dimension n ≥ 1. Also let p and q be nonnegative integers, at least one of which is positive. Of course, V may be considered as a module over gl(V), as a Lie algebra over k. As in the previous section, T^{p,q}V may be considered as a module over gl(V) as well, by summing the actions on the various factors of V and V′. More precisely, let A be a linear mapping from V into itself, and let A′ be the corresponding dual linear mapping from the dual space V′ into itself, as in Section 2.13. Let w_{1}, …, w_{p} ∈ V and λ_{1}, …, λ_{q} ∈ V′ be given, so that

(w_{1} ⊗ ⋯ ⊗ w_{p}) ⊗ (λ_{1} ⊗ ⋯ ⊗ λ_{q}) (14.6.1)

is an element of T^{p,q}V. If 1 ≤ h ≤ p, then the action of A on the hth factor of V in T^{p,q}V sends (14.6.1) to

(w_{1} ⊗ ⋯ ⊗ w_{h−1} ⊗ A(w_{h}) ⊗ w_{h+1} ⊗ ⋯ ⊗ w_{p}) ⊗ (λ_{1} ⊗ ⋯ ⊗ λ_{q}). (14.6.2)

Similarly, if 1 ≤ i ≤ q, then A′ acts on the ith factor of V′ in T^{p,q}V, sending (14.6.1) to

(w_{1} ⊗ ⋯ ⊗ w_{p}) ⊗ (λ_{1} ⊗ ⋯ ⊗ λ_{i−1} ⊗ A′(λ_{i}) ⊗ λ_{i+1} ⊗ ⋯ ⊗ λ_{q}). (14.6.3)

If we consider T^{p,q}V as a module over gl(V) as in the previous section, then A corresponds to a linear mapping A_{p,q} from T^{p,q}V into itself, as on p40 of [25]. By construction, A_{p,q} sends (14.6.1) to the sum of (14.6.2) over h = 1, …, p, minus the sum of (14.6.3) over i = 1, …, q.

Suppose for the moment that A is diagonalizable on V, and let v_{1}, …, v_{n} be a basis of V consisting of eigenvectors of A. If μ_{1}, …, μ_{n} is the corresponding dual basis for V′, then one can check that μ_{j} is an eigenvector for A′ for each j = 1, …, n. Under these conditions, the collection of elements of T^{p,q}V of the form (14.5.4) is a basis for T^{p,q}V, as in the previous section. It is easy to see that each of these elements (14.5.4) is an eigenvector for A_{p,q}, so that A_{p,q} is diagonalizable on T^{p,q}V. This corresponds to part of the proof of Lemma 6.3 on p41 of [25].

Suppose now that A is nilpotent on A, and observe that A′ is nilpotent on V′. It is easy to see that each of the p+q linear mappings on T^{p,q}V corresponding to A as in (14.6.2) and (14.6.3) are nilpotent on T^{p,q}V. These p+q linear mappings on T^{p,q}V corresponding to A also commute with each other. It follows that A_{p,q} is nilpotent on T^{p,q} too, because A_{p,q} is defined by adding and subtracting the p + q linear mappings on T^{p,q}V, as appropriate. This corresponds to another part of the proof of Lemma 6.3 on p41 of [25].

As in the previous section, T^{p,q}V is a module over gl(V), as a Lie algebra over k, with respect to this action of gl(V) on T^{p,q}V. Equivalently, the mapping from A ∈ gl(V) to A_{p,q} ∈ gl(T^{p,q}V) is a Lie algebra homomorphism. This means that A ↦ A_{p,q} is a linear mapping from gl(V) into gl(T^{p,q}V), and that

[A_{p,q}, B_{p,q}] = ([A, B])_{p,q} (14.6.4)

for every A, B ∈ gl(V). In particular, if A and B commute as linear mappings on V, then A_{p,q} and B_{p,q} commute as linear mappings on T^{p,q}V. This corresponds to part of the proof of Lemma 6.3 on p41 of [25] again.

Remember that there is a natural isomorphism from T^{1,1}V = V ⨂ V′ onto the space L(V) of linear mappings from V into itself, as vector spaces over k, as in the previous section. One can check that A_{1,1} corresponds to ad_{A} on L(V), as in Section 10.9.

14.7 Invariant tensors

Let k be a field, and let V be a finite-dimensional vector space over k again. Also let p and q be nonnegative integers, at least one of which is positive, and let t be an element of T^{p,q}V. Consider

{A ∈ gl(V) : A_{p,q}(t) = 0}, (14.7.1)

where A_{p,q} is as in the previous section. It is easy to see that (14.7.1) is a Lie subalgebra of gl(V), because A ↦ A_{p,q} is a Lie algebra homomorphism from gl(V) into gl(T^{p,q}V).

Suppose from now on in this section that k is algebraically closed. Let A be a linear mapping from V into itself, and let A_{1}, A_{2} be as in Section 10.8. Thus A_{1} and A_{2} are commuting linear mappings from V into itself, A = A_{1} + A_{2}, A_{1} is diagonalizable on V, and A_{2} is nilpotent on V. Let A_{p,q}, (A_{1})_{p,q}, and (A_{2})_{p,q} be the corresponding linear mappings from T^{p,q}V into itself, as in the previous section. It follows that (A_{1})_{p,q} and (A_{2})_{p,q} commute on T^{p,q}V,

A_{p,q} = (A_{1})_{p,q} + (A_{2})_{p,q}, (14.7.2)

(A_{1})_{p,q} is diagonalizable on T^{p,q}V, and (A_{2})_{p,q} is nilpotent on T^{p,q}V, as before. This means that (A_{1})_{p,q} and (A_{2})_{p,q} are the diagonalizable and nilpotent parts (A_{p,q})_{1} and (A_{p,q})_{2} of A_{p,q}, respectively, as linear mappings from T^{p,q}V into itself, by uniqueness. This corresponds to Lemma 6.3 on p41 of [25].

In particular, (A_{1})_{p,q} = (A_{p,q})_{1} and (A_{2})_{p,q} = (A_{p,q})_{2} can be expressed as polynomials in A_{p,q} with no constant term, as before. If A_{p,q}(t) = 0, then it follows that (A_{1})_{p,q}(t) = (A_{2})_{p,q}(t) = 0. (14.7.3)

Equivalently, if A is an element of (14.7.1), then A_{1} and A_{2} are elements of (14.7.1) as well. This argument is used in the proof of Corollary 5.4 on p52 of [25].

Let A be a Lie subalgebra of gl(V), and suppose that A is semisimple as a Lie algebra over k. If k has characteristic 0, then Theorem 5.2 on p51 of [25] says that A can be characterized by its tensor invariants. This means that A can be expressed as the intersection of subalgebras of gl(V) of the form (14.7.1). Corollary 5.4 on p52 of [25] says that under these conditions, A contains the semisimple and nilpotent parts of its elements. Remember that another approach to the latter was discussed in Section 14.4.

Now let A be any finite-dimensional semisimple Lie algebra over k. The abstract Jordan decomposition for elements of A is given in Theorem 5.6 on p52 of [25]. This is obtained from Corollary 5.4 in [25], applied to the image of A under the adjoint representation.

14.8 Diagonalizability and quotients

Let k be a field, let V be a vector space over k, and let W be a linear subspace of V. Consider the collection L_{W} (V) of all linear mappings T from V into itself such that T (W) ⊆ W. (14.8.1)

This is a subalgebra of the algebra L(V) of all linear mappings from V into itself. Let q be the canonical quotient mapping from V onto the quotient vector space V/W. If T ∈ L_{W} (V), then there is a unique linear mapping T^{V/W} from V/W into itself such that

T^{V/W} (q(v)) = q(T (v)) (14.8.2)

for every v ∈ V, as usual. It is easy to see that T ↦ T^{V/W} is an algebra homomorphism from L_{W} (V) into L(V/W), with respect to composition of mappings. More precisely, this homomorphism maps L_{W} (V) onto L(V/W). We shall normally be concerned with finite-dimensional vector spaces here, in which case the previous statement is more elementary.

In particular, if T ∈ L_{W} (V) is nilpotent as a linear mapping on V, then T^{V/W} is nilpotent as a linear mapping on V/W. If R, T ∈ L_{W} (V) commute as linear mappings on V, then R^{V/W}, T^{V/W} commute as linear mappings on V/W.

Let T be an element of L_{W} (V), and suppose that v ∈ V is an eigenvector of T, with eigenvalue λ. Observe that

T^{V/W} (q(v)) = q(T (v)) = q(λ v) = λ q(v), (14.8.3)

so that q(v) is an eigenvector of T^{V/W} with eigenvalue λ as well. If V is spanned by the eigenvectors of T, then it follows that V/W is spanned by the images under q of the eigenvectors of T, which are eigenvectors of T^{V/W}.

Let (A, [·, ·]_{A}) and (B, [·, ·]_{B}) be Lie algebras over k. Put

ad_{A,x}(y) = [x, y]_{A}, ad_{B,w}(z) = [w, z]_{B} (14.8.4)

for every x, y ∈ A and w, z ∈ B, as usual. Suppose that ϕ is a Lie algebra homomorphism from A into B. This implies that

ϕ(ad_{A,x}(y)) = ϕ([x, y]_{A}) = [ϕ(x), ϕ(y)]_{B} = ad_{B,ϕ(x)}(ϕ(y)) (14.8.5)

for every x, y ∈ A. Suppose for the moment that ϕ maps A onto B. If x ∈ A and ad_{A,x} is nilpotent as a linear mapping from A into itself, then it follows that ad_{B,ϕ(x)} is nilpotent as a linear mapping from B into itself. Equivalently, if x ∈ A is ad-nilpotent in A, then ϕ(x) is ad-nilpotent in B.

If x, y ∈ A and y is an eigenvector of ad_{A,x}, then ϕ(y) is an eigenvector of ad_{B,ϕ(x)}, by (14.8.5). If A is spanned by the eigenvectors of ad_{A,x} and ϕ maps A onto B, then it follows that B is spanned by the eigenvectors of ad_{B,ϕ(x)}. These remarks correspond to part of the proof of the corollary on p30 of [14]. This also corresponds to Theorem 5.7 on p52 of [25], in the simpler case of surjective Lie algebra homomorphisms.

14.9 Diagonalizability and representations

Let k be an algebraically closed field of characteristic 0, and let V be a finitedimensional vector space over k. Also let B be a Lie subalgebra of gl(V), and suppose that B is semisimple as a Lie algebra over k. Let x ∈ B be given, and remember that there are unique commuting linear mappings y_{1} and y_{2} on V such that x = y_{1} + y_{2}, y_{1} is diagonalizable on V, and y_{2} is nilpotent on V, as in Section 10.8. In fact, y_{1}, y_{2} ∈ B, because B is semisimple as a Lie algebra, as in Sections 14.4 and 14.7. Remember that y_{1} is ad-diagonalizable on gl(V), and y_{2} is ad-nilpotent on gl(V), as in Section 10.9. Using this, one can check that y_{1} is ad-diagonalizable on B, and that y_{2} is ad-nilpotent on B, because B is a Lie subalgebra of gl(V). More precisely, if y ∈ B, then ad_{B,y} is the same as the restriction of ad_{gl(V),y} to B, because B is a Lie subalgebra of gl(V). This means that y_{1} and y_{2} are the same as the ad-diagonalizable and ad-nilpotent parts of the abstract Jordan decomposition of x in B, by the uniqueness of the abstract Jordan decomposition, as in Section 14.3. This is the second part of the theorem on p29 of [14].

Let (A, [·, ·]_{A}) be a finite-dimensional semisimple Lie algebra over k, and let ϕ be a Lie algebra homomorphism from A into gl(V). Put B = ϕ(A), which is a Lie subalgebra of gl(V). Note that B is semisimple as a Lie algebra over k too, as in Section 10.14. Let w ∈ A be given, and let w = w_{1} + w_{2} be the abstract Jordan decomposition of w in A, as in Section 14.3. Thus w_{1} is ad-diagonalizable on A, w_{2} is ad-nilpotent on A, and [w_{1}, w_{2}]_{A} = 0, as before. It follows that ϕ(w) = ϕ(w_{1}) + ϕ(w_{2}), where ϕ(w_{1}) is ad-diagonalizable on B and ϕ(w_{2}) is ad-nilpotent on B, as in the previous section. Note that ϕ(w_{1}) and ϕ(w_{2}) commute as linear mappings on V, because [w_{1}, w_{2}]_{A} = 0 and ϕ is a Lie algebra homomorphism. This means that ϕ(w_{1}) and ϕ(w_{2}) satisfy the requirements of the abstract Jordan decomposition of ϕ(w) in B. Put x = ϕ(w), and let y_{1}, y_{2} ∈ B be as in the preceding paragraph. It follows that

ϕ(w_{1}) = y_{1}, ϕ(w_{2}) = y_{2}, (14.9.1)

by uniqueness of the abstract Jordan decomposition of x in B. This corresponds to the corollary on p30 of [14], and to Theorem 7 on p7 of [24].

Let (A_{2}, [·, ·]_{A_{2}}) be a finite-dimensional semisimple Lie algebra over k, and let A_{1} be a Lie subalgebra of A_{2} that is semisimple as a Lie algebra over k as well. If x, y ∈ A_{2}, then put ad_{A_{2},x}(y) = [x, y]_{A_{2}}, as usual. If x ∈ A_{1}, then ad_{A_{1},x} is the same as the restriction of ad_{A_{2},x} to A_{1}, because A_{1} is a Lie subalgebra of A_{2}. Of course, A_{2} is a finite-dimensional vector space over k in particular, so that gl(A_{2}) is a Lie algebra over k with respect to the commutator bracket associated to composition of linear mappings. If x ∈ A_{1}, then put ϕ(x) = ad_{A_{2},x}, which defines a Lie algebra homomorphism from A_{1} into gl(A_{2}). Let w ∈ A_{1} be given, and let w = w′ + w′′ be the abstract Jordan decomposition of w in A_{1}. Thus

ad_{A_{2},w} = ad_{A_{2},w′} + ad_{A_{2},w′′} (14.9.2)

corresponds to the ordinary Jordan decomposition of ad_{A_{2},w}, as a linear mapping from A_{2} into itself, as before. This means that w = w′+w′′ also corresponds to the abstract Jordan decomposition of w in A_{2}, by uniqueness. This is the same as Exercise 9 on p31 of [14].

Let B_{1}, B_{2} be finite-dimensional semisimple Lie algebras over k, and let ψ be a Lie algebra homomorphism from B_{1} into B_{2}. Theorem 5.7 on p52 of [25] basically says that ψ maps abstract Jordan decompositions in B_{1} to the corresponding abstract Jordan decompositions in B_{2}. Of course, this includes the case of subalgebras, as in the preceding paragraph. If ψ maps B_{1} onto B_{2}, then this can be verified directly, as in the previous section. Otherwise, one can use this to reduce to the case of subalgebras.

14.10 Exponentiating nilpotents

Let T_{1} and T_{2} be commuting indeterminates, and remember that Z[T_{1}, T_{2}] is the space of formal polynomials in T_{1} and T_{2} with coefficients in Z, as in Section 5.8. This is a commutative associative algebra over Z with respect to formal multiplication of polynomials, as before. If n is a nonnegative integer, then

(T_{1} + T_{2})^{n} = ∑_{j=0}^{n} \binom{n}{j} T_{1}^{j} T_{2}^{n−j}, (14.10.1)

where \binom{n}{j} = n!/j! (n − j)! is the usual binomial coefficient, as in the binomial theorem. Thus

(T_{1} + T_{2})^{n+1} = ∑_{j=0}^{n+1} \binom{n + 1}{j} T_{1}^{j} T_{2}^{n+1−j} (14.10.2)

is the same as

(T_{1} + T_{2})^{n} (T_{1} + T_{2}) = ∑_{j=0}^{n} \binom{n}{j} T_{1}^{j+1} T_{2}^{n−j} + ∑_{j=0}^{n} \binom{n}{j} T_{1}^{j} T_{2}^{n−j+1}

(14.10.3)  = ∑_{j=1}^{n+1} \binom{n}{j − 1} T_{1}^{j} T_{2}^{n+1−j} + ∑_{j=0}^{n} \binom{n}{j} T_{1}^{j} T_{2}^{n+1−j}.

This implies the well-known identity

\binom{n + 1}{j} = \binom{n}{j − 1} + \binom{n}{j} (14.10.4)

for j = 1, …, n.

Let k be a field of characteristic 0. Alternatively, one can consider commutative rings k with multiplicative identity elements which are algebras over Q. In this case, a module or algebra over k could also be considered as a vector space over Q. Let A be an associative algebra over k with a multiplicative identity element e. Suppose that x ∈ A is nilpotent, so that x^{l} = 0 for some positive integer l. Under these conditions, the exponential of x is defined as an element of A by

exp x = ∑_{j=0}^{∞}(1/j!) x^{j}, (14.10.5)

as usual. More precisely, x^{j} = 0 for all sufficiently large j, so that the infinite series reduces to a finite sum, and thus defines an element of A. Of course, x^{j} is interpreted as being equal to e when j = 0.

Let y be another nilpotent element of A, and suppose that x commutes with y. This implies that x + y is nilpotent in A, so that exp(x + y) can be defined as before. Observe that

exp(x + y) = ∑_{n=0}^{∞}(1/n!) (x + y)^{n} = ∑_{n=0}^{∞}(1/n!) ∑_{j=0}^{n} \binom{n}{j} x^{j} y^{n−j}

(14.10.6)  = ∑_{n=0}^{∞} ∑_{j=0}^{n}((1/j!) x^{j}) ((1/(n − j)!) y^{n−j}) = (exp x) (exp y).

More precisely, the double sum is the same as the Cauchy product of the series defining exp x and exp y, as in Section 4.2. The nilpotency conditions on x and y ensure that all of these sums reduce to finite sums. In particular, we can take y = −x, to get that

(14.10.7) (exp x) exp(−x) = exp(−x) (exp x) = exp 0 = e.

This means that exp x is invertible in A, with

(exp x)^{−1} = exp(−x). (14.10.8)

Let V be a module over k, and let T be a mapping from V into itself that is linear over k. Suppose that T is nilpotent on V, so that the lth power T^{l} of T with respect to composition is equal to 0 on V for some positive integer l. Thus the exponential exp T of T can be defined as a linear mapping from V into itself, using the algebra of linear mappings from V into itself with respect to composition of mappings in the previous remarks. Let W be a submodule of V, and suppose that T maps W into itself. It follows that T^{j}(W) ⊆ W for every nonnegative integer j, so that

(exp T)(W) ⊆ W. (14.10.9)

More precisely, we have that

(exp T)(W) = W (14.10.10)

in this situation. This can be obtained from (14.10.9) and the analogous statement for −T. Alternatively, the restriction of exp T to W is the same as the exponential of the restriction of T to W, which is invertible on W.

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. If δ is a derivation on A, then it is well known that

δ^{n}(a b) = ∑_{j=0}^{n} \binom{n}{j} · δ^{j}(a) δ^{n−j}(b) (14.10.11)

for every nonnegative integer n and a, b ∈ A. More precisely, if this holds for some n ≥ 0, then

δ^{n+1}(a b) = δ(δ^{n}(a b)) = ∑_{j=0}^{n} \binom{n}{j} (δ^{j+1}(a) δ^{n−j}(b) + δ^{j}(a) δ^{n−j+1}(b))

(14.10.12)  = ∑_{j=1}^{n+1} \binom{n}{j − 1} δ^{j}(a) δ^{n+1−j}(b) + ∑_{j=0}^{n} \binom{n}{j} δ^{j}(a) δ^{n+1−j}(b).

This implies the analogue of (14.10.11) for n + 1, by (14.10.4). It follows that (14.10.11) holds for all n ≥ 0, by induction.

14.11 Nilpotent derivations

Let k be a commutative ring with a multiplicative identity element that is an algebra over Q, or simply a field of characteristic 0, as in the previous section. Also let A be an algebra over k in the strict sense again, and let δ be a derivation on A that is nilpotent as a mapping from A into itself, so that δ^{l} = 0 for some positive integer l. Thus exp δ can be defined as a mapping from A into itself that is linear over k, as in the previous section. If a, b ∈ A, then

(exp δ)(a b) = ∑_{n=0}^{∞}(1/n!) δ^{n}(a b) = ∑_{n=0}^{∞}(1/n!) ∑_{j=0}^{n} \binom{n}{j} δ^{j}(a) δ^{n−j}(b)

(14.11.1)  = ∑_{n=0}^{∞} ∑_{j=0}^{n}((1/j!) δ^{j}(a)) ((1/(n − j)!) δ^{n−j}(b))
= ((exp δ)(a)) ((exp δ)(b)).

More precisely, the double sum is the same as the Cauchy product of the series defining (exp δ)(a) and (exp δ)(b), and the nilpotency condition on δ ensures that all of these sums reduce to finite sums. Note that exp δ is invertible as a linear mapping from A into itself, as in the previous section. Thus (14.11.1) implies that exp δ is an algebra automorphism on A, as on p9 of [14].

Let (A, [·, ·]_{A}) be a Lie algebra over k, and put ad_{x}(y) = [x, y]_{A} for every x, y in A, as usual. Remember that ad_{x} is a derivation on A, as in Section 2.5. If x ∈ A is ad-nilpotent, then exp ad_{x} defines a Lie algebra automorphism on A, as in the preceding paragraph. Let Int A be the subgroup of the group of all Lie algebra automorphisms of A generated by these automorphisms. The elements of Int A are called inner automorphisms of A, as on p9 of [14]. If ϕ is any Lie algebra automorphism on A, then it is easy to see that

ϕ ∘ ad_{x} ∘ ϕ^{−1} = ad_{ϕ(x)} (14.11.2)

for every x ∈ A. In particular, if x ∈ A is ad-nilpotent, then ϕ(x) is ad-nilpotent as well. In this case, we get that

(14.11.3) ϕ ∘ (exp ad_{x}) ∘ ϕ^{−1} = exp ad_{ϕ(x)},

by (14.11.2). This implies that Int A is a normal subgroup of the group of all Lie algebra automorphisms on A, as on p9 of [14].

Let B be a Lie subalgebra of A, and let x be an element of the normalizer of B in A, so that ad_{x} maps B into itself. If x is ad-nilpotent on A, then exp ad_{x} can be defined as a linear mapping from A into itself as before. In fact, we have that (exp ad_{x})(B) = B, (14.11.4)

as in (14.10.10). In particular, this holds when x ∈ B, in which case the restriction of ad_{x} to B is the same as ad_{B,x}.

Now let A be an associative algebra over k with a multiplicative identity element e, where multiplication of a, b ∈ A is expressed as a b again. If a ∈ A, then let L_{a} and R_{a} be the usual operators of left and right multiplication by a on A, so that L_{a}(x) = a x and R_{a}(x) = x a for every a ∈ A. Note that

(L_{a})^{j} = L_{a^{j}}, (R_{a})^{j} = R_{a^{j}} (14.11.5)

for every nonnegative integer j. Suppose that a is nilpotent in A, which implies that L_{a} and R_{a} are nilpotent as linear mappings from A into itself. It is easy to see that (14.11.6) exp L_{a} = L_{exp a}, exp R_{a} = R_{exp a}

under these conditions.

Remember that A may be considered as a Lie algebra over k with respect to the corresponding commutator bracket [a, b] = a b − b a. If a ∈ A, then

ad_{a} = L_{a} − R_{a} = L_{a} + R_{−a}, (14.11.7)

as a linear mapping from A into itself. Suppose that a is nilpotent in A again, and remember that L_{a} and R_{−a} commute as linear mappings from A into itself. This implies that ad_{a} is nilpotent on A, and that

(14.11.8)  exp ad_{a} = exp(L_{a} + R_{−a}) = (exp L_{a}) ∘ (exp R_{−a})
= L_{exp a} ∘ R_{exp(−a)} = L_{exp a} ∘ R_{(exp a)^{−1}}.

Equivalently, (14.11.9) (exp ad_{a})(x) = (exp a) x (exp a)^{−1}

for every x ∈ A, as on p9 of [14].

14.12 Exponentials and homomorphisms

Let k be a commutative ring with a multiplicative identity element that is an algebra over Q, or simply a field of characteristic 0, as before. Also let A and B be associative algebras over k with multiplicative identity elements e_{A} and e_{B}, respectively. Suppose that ϕ is an algebra homomorphism from A into B such that ϕ(e_{A}) = e_{B}. Let x be a nilpotent element of A, which implies that ϕ(x) is nilpotent in B. Under these conditions, it is easy to see that

ϕ(exp_{A} x) = exp_{B} ϕ(x), (14.12.1)

where exp_{A} x and exp_{B} ϕ(x) are the exponentials of x and ϕ(x) in A and B, respectively.

Now let (A, [·, ·]_{A}) and (B, [·, ·]_{B}) be Lie algebras over k, and let ψ be a Lie algebra homomorphism from A into B.. As usual, we put ad_{A,x}(y) = [x, y]_{A} and ad_{B,w}(z) = [w, z]_{B} for every x, y ∈ A and w, z ∈ B. If x, y ∈ A, then

ψ(ad_{A,x}(y)) = ad_{B,ψ(x)}(ψ(y)), (14.12.2)

as in Section 14.8. It follows that

ψ((ad_{A,x})^{j}(y)) = (ad_{B,ψ(x)})^{j}(ψ(y)) (14.12.3)

for every positive integer j. Suppose that x is ad-nilpotent in A, and that ψ(x) is ad-nilpotent in B. Thus exp ad_{A,x} and exp ad_{B,ψ(x)} can be defined as linear mappings from A and B into themselves, as before. Using (14.12.3), we get that

(14.12.4) ψ((exp ad_{A,x})(y)) = (exp ad_{B,ψ(x)})(ψ(y))

for every y ∈ A.

If x is ad-nilpotent in A, and ψ maps A onto B, then ψ(x) is ad-nilpotent in B, by (14.12.3), as in Section 14.8. Similarly, if ψ(x) is ad-nilpotent in B, and ψ is injective, then x is ad-nilpotent in A.

Let V be a module over k, and let ρ be a representation of A on V. If R and T are mappings from V into itself that are linear over k, then put ad_{R}(T) = [R, T] = R ∘ T − T ∘ R, as usual. Suppose that x ∈ A has the property that ρ_{x} is nilpotent as a linear mapping from V into itself. Thus exp ρ_{x} is defined as an invertible linear mapping from V onto itself, as before. We also have that ad_{ρ_{x}} is nilpotent as a linear mapping on the space of linear mappings from V into itself, so that exp ad_{ρ_{x}} is defined as an invertible linear mapping on the space of linear mappings from V into itself. In fact,

(14.12.5) (exp ad_{ρ_{x}})(T) = (exp ρ_{x}) ∘ T ∘ (exp ρ_{x})^{−1}

for every linear mapping T from V into itself, as in the previous section. In particular, (14.12.6) (exp ad_{ρ_{x}})(ρ_{y}) = (exp ρ_{x}) ∘ ρ_{y} ∘ (exp ρ_{x})^{−1}

for every y ∈ A.

Suppose that x is ad-nilpotent in A, in addition to ρ_{x} being nilpotent on V. Thus exp ad_{A,x} is defined as a linear mapping from A into itself, and

(14.12.7) ρ_{(exp ad_{A,x})(y)} = (exp ad_{ρ_{x}})(ρ_{y})

for every y ∈ A. This follows from (14.12.4), with B taken to be the space of linear mappings from V into itself, considered as a Lie algebra over k with respect to the commutator bracket associated to composition of linear mappings, and with ψ taken to be the Lie algebra homomorphism from A into B corresponding to ρ. This implies that

(14.12.8) ρ_{(exp ad_{A,x})(y)} = (exp ρ_{x}) ∘ ρ_{y} ∘ (exp ρ_{x})^{−1}

for every y ∈ A.

14.13 Representations and structure constants

Let k be a commutative ring with a multiplicative identity element, and let m and n be positive integers. The spaces k^{m} and k^{n} of m and n-tuples of elements of k are (free) modules over k with respect to coordinatewise addition and scalar multiplication, as usual. Let ρ be a bilinear action of k^{m} on k^{n}, which corresponds to a mapping from k^{m} × k^{n} into k^{n} that is bilinear over k. If x = (x_{1}, …, x_{m}) ∈ k^{m} and v = (v_{1}, …, v_{n}) ∈ k^{n}, then ρ_{x}(v) ∈ k^{n} can be expressed as

(ρ_{x}(v))_{r} = ∑_{j=1}^{m} ∑_{l=1}^{n} a_{j,l}^{r} x_{j} v_{l}, (14.13.1)

where the left side is the rth coordinate of ρ_{x}(v) for each r = 1, …, n. The coefficients a_{j,l}^{r} are elements of k for each j = 1, …, m and l, r = 1, …, n, and do not depend on x or v. It is easy to see that these coefficients are uniquely determined by ρ. Conversely, any coefficients a_{j,l}^{r} ∈ k determine a bilinear action of k^{m} on k^{n} in this way.

Let K be a commutative associative algebra over k, where multiplication of y, z ∈ K is expressed as y z. The spaces K^{m} and K^{n} of m and n-tuples of elements of K may be considered as modules over k with respect to coordinatewise addition and scalar multiplication. If x ∈ K^{m} and v ∈ K^{n}, then let ρ_{x}^{K}(v) be the element of K^{n} whose rth coordinate is equal to

(ρ_{x}^{K}(v))_{r} = ∑_{j=1}^{m} ∑_{l=1}^{n} a_{j,l}^{r} x_{j} v_{l} (14.13.2)

for each r = 1, …, n. This uses both multiplication on K and scalar multiplication of elements of K by elements of k to define the terms of the sum on the right. This defines ρ^{K} as a bilinear action of K^{m} on K^{n}.

Suppose that k^{m} is a Lie algebra over k, with respect to a Lie bracket [·, ·]_{k^{m}}. Using the structure constants for [·, ·]_{k^{m}}, we get a Lie bracket [·, ·]_{K^{m}} on K^{m}, as in Section 9.14.

Suppose also that ρ is a representation of k^{m}, as a Lie algebra over k, on k^{n}. This means that ρ_{[w,x]_{k^{m}}} (v) = ρ_{w}(ρ_{x}(v)) − ρ_{x}(ρ_{w}(v)) (14.13.3)

for every w, x ∈ k^{m} and v ∈ k^{n}. This can be characterized by suitable conditions on the coefficients a_{j,l}^{r} associated to ρ and the structure constants for [·, ·]_{k^{m}}. It follows that ρ_{[w,x]_{K^{m}}}^{K} (v) = ρ_{w}^{K}(ρ_{x}^{K}(v)) − ρ_{x}^{K}(ρ_{w}^{K}(v)) (14.13.4)

for every w, x ∈ K^{m} and v ∈ K^{n}. Thus ρ^{K} is a representation of K^{m}, as a Lie algebra over k, on K^{n} under these conditions.

If T is a homomorphism from k^{n} into itself, as a module over k, then T corresponds to an n × n matrix with entries in k in the usual way. Using the same matrix, we get a homomorphism T_{K} from K^{n} into itself, as a module over k. This defines a homomorphism from Hom_{k}(k^{n}, k^{n}) into Hom_{k}(K^{n}, K^{n}), as associative algebras over k with respect to composition of mappings.

Suppose that K has a multiplicative identity element e, so that K^{m} and K^{n} may be considered as free modules over K with respect to coordinatewise addition and scalar multiplication. In this case, K^{m} may be considered as a Lie algebra over K with respect to [·, ·]_{K^{m}}, as in Section 9.14. Similarly, ρ^{K} may be considered as a representation of K^{m}, as a Lie algebra over K, on K^{n}, as a module over K. If T is a homomorphism from k^{n} into itself as a module over k, then T_{K} is a homomorphism from K^{n} into itself as a module over K.

As before, t ↦ t e defines a ring homomorphism from k into K. This leads to homomorphisms from k^{m} and k^{n} into K^{m} and K^{n}, as modules over k. More precisely, this gives a homomorphism from k^{m} into K^{m} as Lie algebras over k. Similarly, if x ∈ k^{m} and v ∈ k^{n}, then the image of ρ_{x}(v) in K^{n} is the same as taking the images of x and v in K^{m} and K^{n}, respectively, and then using ρ^{K}. If T is a homomorphism from k^{n} into itself, as a module over k, and v ∈ k^{n}, then the image of T (v) in K^{n} is the same as first taking the image of v in K^{n}, and then taking the image of that under T_{K}.

Suppose that t ↦ t e is injective as a mapping from k into K, so that the corresponding mappings from k^{m} and k^{n} into K^{m} and K^{n}, respectively, are injective as well. If T is a homomorphism from k^{n} into itself, as a module over k, and T_{K} = 0 on K^{n}, then T = 0 on k^{n}. Similarly, if T_{K} is nilpotent on K^{n}, then T is nilpotent on k^{n}.

Let x ∈ k^{m} be given, and let x_{K} be its image in K^{m}. If T = ρ_{x}, then T_{K} = ρ_{x_{K}}^{K}. In particular, if ρ_{x_{K}}^{K} is nilpotent as a mapping from K^{n} into itself, then ρ_{x} is nilpotent on k^{n}.

14.14 Lie’s theorem and nilpotence

Let k be a field of characteristic 0, and let (A, [·, ·]_{A}) be a solvable Lie algebra over k with positive finite dimension m as a vector space over k. Also let V be a vector space over k of positive finite dimension n, and let ρ be a representation of A on V. Remember that [A, A] ⊆ A is as defined in Section 9.2.

Suppose for the moment that k is algebraically closed. Under these conditions, there is a flag F = {V_{j}}_{j=0}^{n} in V such that

ρ_{a}(V_{j}) ⊆ V_{j} (14.14.1)

for every a ∈ A and j = 0, 1, …, n, as in Section 9.13. If a ∈ [A, A], then it follows that ρ_{a}(V_{j}) ⊆ V_{j−1} (14.14.2)

for every j = 1, …, n, as in Section 9.11. In particular, this implies that

(ρ_{a})^{n} = 0 (14.14.3)

on V for every a ∈ [A, A].

Otherwise, let K be an algebraically closed field that contains k as a subfield. We may as well suppose that A = k^{m} as a vector space over k, and that V = k^{n}. As in Section 9.14, [·, ·]_{A} can be extended to a Lie bracket [·, ·]_{A_{K}} on A_{K} = K^{m}, so that A_{K} becomes a solvable Lie algebra over K. Similarly, we can use ρ to get a representation ρ^{K} of A_{K}, as a Lie algebra over K, on K^{n}, as a vector space over K, as in the previous section.

Of course, [A_{K}, A_{K}] can be defined as a subset of A_{K} as in Section 9.2 too. If a ∈ [A_{K}, A_{K}], then it follows that

(ρ_{a}^{K})^{n} = 0 (14.14.4)

on K^{n}, as before. In particular, this holds when a ∈ [A, A]. This implies that (14.14.3) holds on V = k^{n} when a ∈ [A, A].

Complexifications and sl_{2}(k) modules

15.1 Basic properties of sl_{2}(k) modules

Let k be a commutative ring with a multiplicative identity element. Remember that sl_{2}(k) is freely generated, as a module over k, by

x = \binom{0 1}{0 0}, y = \binom{0 0}{1 0}, and h = \binom{1 0}{0 − 1}, (15.1.1)

as in Section 10.2. Thus every element of sl_{2}(k) can be expressed in a unique way as a linear combination of x, y, and h, with coefficients in k. We also have that [h, x] = 2 · x, [h, y] = −2 · y, and [x, y] = h, (15.1.2)

as before, using the commutator bracket associated to matrix multiplication.

Let V be a module over k, and let ρ = ρ^{V} be a representation of sl_{2}(k) on V, as a Lie algebra over k. Thus ρ_{x}, ρ_{y}, and ρ_{h} are homomorphisms from V into itself, as a module over k, such that

[ρ_{h}, ρ_{x}] = ρ_{[h,x]} = 2 · ρ_{x}, [ρ_{h}, ρ_{y}] = ρ_{[h,y]} = −2 · ρ_{y}, (15.1.3)

and [ρ_{x}, ρ_{y}] = ρ_{[x,y]} = ρ_{h}.

Equivalently, we may consider V as a module over sl_{2}(k), as a Lie algebra over k, with a · v = ρ_{a}(v) (15.1.4)

for every a ∈ sl_{2}(k) and v ∈ V.

312 CHAPTER 15. COMPLEXIFICATIONS AND SL_{2}(K) MODULES

Let us now take k to be a field, so that V is a vector space over k. If λ ∈ k, then put V_{λ} = {v ∈ V : h · v = λ v}, (15.1.5) which is the linear subspace of V consisting of eigenvectors of ρ_{h} with eigenvalue λ. Thus V_{λ} ≠ {0} exactly when λ is an eigenvalue of ρ_{h} on V. In this case, λ may be called a weight of h on V, and V_{λ} may be called a weight space, as on p31 of [14]. An element of V_{λ} may be said to have weight λ, as on p17 of [24].

Suppose that v ∈ V_{λ} for some λ ∈ k. Observe that

(15.1.6)  h · (x · v) = ([h, x]) · v + x · (h · v)
= (2 · x) · v + x · (λ v) = (λ + 2) (x · v).

More precisely, λ + 2 means λ + 2 · 1 = λ + 1 + 1 as an element of k, where 1 is the multiplicative identity element in k. It follows that

x · v ∈ V_{λ+2}. (15.1.7)

Similarly,

(15.1.8)  h · (y · v) = ([h, y]) · v + y · (h · v)
= (−2 · y) · v + y · (λ v) = (λ − 2) (y · v),

where λ − 2 = λ − 2 · 1 = λ − 1 − 1, as an element of k. This means that

y · v ∈ V_{λ−2}. (15.1.9)

Of course, if k has characteristic 0, then k may be considered as containing the rational numbers as a subfield.

Remember that nonzero eigenvectors of ρ_{h} with distinct eigenvalues are automatically linearly independent. If V has finite dimension as a vector space over k, then it follows that there are only finitely many weights λ of h on V.

15.2 Maximal or primitive vectors

Let us continue with the same notation and hypotheses as in the previous section. Suppose for the moment that k is an algebraically closed field of characteristic 0, and that V has positive finite dimension. This implies that ρ_{h} is diagonalizable on V, as in Section 14.9. This also uses the fact that h is ad-diagonalizable in sl_{2}(k).

Under these conditions, there is a weight λ ∈ k of h on V, because V ≠ {0}. In fact, there is a weight λ of h on V such that V_{λ+2} = {0}, because there are only finitely many weights of h on V. In this case, if v ∈ V_{λ}, then

x · v = 0, (15.2.1)

by (15.1.7).

If v is a nonzero element of V_{λ} for some λ ∈ k that satisfies (15.2.1), then v may be called a maximal vector of weight λ, as on p32 of [14]. In this situation, v may also be said to be primitive of weight λ, as in Definition 1 on p18 of [24]. Note that V does not need to have finite dimension for this definition.

If k is an algebraically closed field of characteristic 0 and V has positive finite dimension, then one can get a maximal vector of some weight λ ∈ k by choosing λ so that V_{λ+2} = {0}, as before. Alternatively, if v ∈ V is a nonzero eigenvector of ρ_{h} with eigenvalue λ ∈ k, then

ρ_{h}((ρ_{x})^{j}(v)) = (λ + 2 j) (ρ_{x})^{j}(v) (15.2.2)

for every nonnegative integer j, as before. We also have that (ρ_{x})^{j}(v) = 0 for some positive integer j, because V has finite dimension. Let j_{0} be the largest nonnegative integer such that (ρ_{x})^{j_{0}}(v) ≠ 0, so that (ρ_{x})^{j_{0}+1}(v) = 0. This means that (ρ_{x})^{j_{0}}(v) is a maximal vector of weight λ + 2 j_{0}, as in the alternate proof of Proposition 3 on p18 of [24].

Let B be the linear subspace of sl_{2}(k) spanned by x and h. This is a Lie subalgebra of sl_{2}(k), which is solvable as a Lie algebra over k. Let v be a nonzero element of V, so that {t v : t ∈ k} (15.2.3)

is a one-dimensional linear subspace of V. If v is primitive, then (15.2.3) is mapped into itself by ρ_{h} and ρ_{x}. Equivalently, this means that (15.2.3) is a submodule of V, as a module over B, as a Lie algebra over k.

Conversely, suppose that (15.2.3) is a submodule of V, as a module over B. This is the same as saying that v is an eigenvector of both ρ_{h} and ρ_{x}. It follows that 2 · ρ_{x}(v) = ([ρ_{h}, ρ_{x}])(v) = 0. (15.2.4)

If k does not have characteristic 2, then we get that ρ_{x}(v) = 0, so that v is primitive. This corresponds to Proposition 2 on p18 of [24].

Suppose that k is an algebraically closed field of characteristic 0 again, and that V has positive finite dimension. Because B is solvable, there is a v ∈ V such that v ≠ 0 and v is an eigenvector of ρ_{b} for every b ∈ B, as in Section 9.13. This implies that v is primitive, as in the preceding paragraph. This corresponds to the proof of Proposition 3 on p18 of [24], and to Exercise 1 on p34 of [14].

15.3 Related submodules

Let k be a field of characteristic 0, and let V be a vector space over k. Also let ρ = ρ^{V} be a representation of sl_{2}(k) on V, as a Lie algebra over k, so that V may be considered as a module over sl_{2}(k). Suppose that v is a primitive element of V of weight λ ∈ k. Put

v_{j} = (1/j!) (ρ_{y})^{j}(v) (15.3.1)

for every positive integer j, as on p32 of [14], and Theorem 3 on p18 of [24]. More precisely, (ρ_{y})^{j} is the jth power of ρ_{y} with respect to composition, as a linear mapping from V into itself. We can define v_{j} as in (15.3.1) when j = 0 too, with the usual interpretations, so that v_{0} = v. It is convenient to put v_{−1} = 0 as well.

314 CHAPTER 15. COMPLEXIFICATIONS AND SL_{2}(K) MODULES

Under these conditions, we have that h · v_{j} = (λ − 2 j) v_{j} (15.3.2) for every j ≥ 0, which is to say that v_{j} ∈ V_{λ−2 j}. This follows by using (15.1.9) repeatedly, or using induction. We also have that y · v_{j} = (j + 1) · v_{j+1} (15.3.3) for every j ≥ −1, by definition of v_{j}. Let us check that x · v_{j} = (λ − j + 1) v_{j−1} (15.3.4) for every j ≥ 0. If j = 0, then this uses the hypothesis that v_{0} = v be primitive, so that x · v = 0. Otherwise, if j ≥ 1, then we can use (15.3.3) to get that j x · v_{j} = x · (y · v_{j−1}) = ([x, y]) · v_{j−1} + y · (x · v_{j−1}). (15.3.5) Using induction, we may suppose that the analogue of (15.3.4) for j − 1 holds, so that j x · v_{j} = h · v_{j−1} + (λ − (j − 1) + 1) y · v_{j−2}. (15.3.6) It follows that j x · v_{j} = (λ − 2 (j − 1)) v_{j−1} + (λ − j + 2) ((j − 2) + 1) v_{j−1}, (15.3.7) by (15.3.2) and (15.3.3). This reduces to

(15.3.8) j x · v_{j} = ((λ − 2 j + 2) + (λ − j + 2) (j − 1)) v_{j−1}
= (λ + (λ − j) (j − 1)) v_{j−1} = j (λ − j + 1) v_{j−1}.

This implies (15.3.4), as desired. This corresponds to the lemma on p32 of [14], and Theorem 3 on p18 of [24].

Remember that v_{0} = v ≠ 0, because v is primitive, by hypothesis. As in Corollary 1 on p19 of [24], there are two cases to consider. In the first case, (15.3.9) for each j ≥ 0, we have that v_{j} ≠ 0. This means that the v_{j}’s are linearly independent in V, because they are eigenvectors for ρ_{h} with distinct eigenvalues, by (15.3.2). In particular, this implies that V has infinite dimension as a vector space over k.

In the second case, v_{l} = 0 for some l ≥ 1. (15.3.10) This implies that v_{j} = 0 for every j ≥ l, by the definition of v_{j}. In this case, let m be the largest nonnegative integer such that v_{m} ≠ 0. Thus v_{j} ≠ 0 when j ≤ m, and v_{j} = 0 when j > m. Observe that x · v_{m+1} = (λ − m) v_{m}, (15.3.11)

15.4. CONSTRUCTING MODULES W (M) 315

by (15.3.4) with j = m + 1. It follows that

λ = m, (15.3.12)

because v_{m} ≠ 0 and v_{m+1} = 0. This is also related to some of the remarks on p32 of [14].

In this second case, let W be the linear span of v_{0}, v_{1}, …, v_{m} in V. It is easy to see that W is a submodule of V, as a module over sl_{2}(k), by (15.3.2), (15.3.3), and (15.3.4). Let W_{0} be a nontrivial linear subspace of W such that ρ_{h}(W_{0}) ⊆ W_{0}. Because ρ_{h} is diagonalizable on W, we get that the restriction of ρ_{h} to W_{0} is diagonalizable as well. In particular, W_{0} contains a nonzero eigenvector of ρ_{h}. In this situation, this means that v_{j_{0}} ∈ W_{0} for some j_{0}, 0 ≤ j_{0} ≤ m, because the corresponding eigenvalues for ρ_{h} on W are distinct. If W_{0} is a submodule of W, as a module over sl_{2}(k), then it follows that v_{j} ∈ W_{0} for every j = 0, 1, …, m, by (15.3.3) and (15.3.4). This implies that W_{0} = W, so that W is irreducible as a module over sl_{2}(k). This corresponds to Corollary 2 on p19 of [24], as well as some of the remarks on p32-3 and part of Exercise 3 on p34 of [14].

15.4 Constructing modules W (m)

Let k be a commutative ring with a multiplicative identity element, and let m be a nonnegative integer. Also let W (m) be a module over k freely generated by m + 1 nonzero distinct elements v_{0}, v_{1}, …, v_{m}. This corresponds to V (m) on p33 of [14], and to W_{m} on p19 of [24]. More precisely, one can take W (m) to be the space k^{m+1} of (m + 1)-tuples of elements of k, as a module over k with coordinatewise addition and scalar multiplication, and v_{0}, v_{1}, …, v_{m} to be the standard basis vectors in k^{m+1}. It will be convenient to put v_{−1} = v_{m+1} = 0, as elements of W (m).

Put

(15.4.1)  H(v_{j}) = (m − 2 j) · v_{j},
(15.4.2)  Y (v_{j}) = (j + 1) · v_{j+1},
(15.4.3)  X(v_{j}) = (m − j + 1) · v_{j−1}

for j = 0, 1, …, m. More precisely, there are unique module homomorphisms H, Y, and X from W (m) into itself that satisfy these conditions. Note that (15.4.1) holds trivially when j = −1 or m + 1. Similarly, (15.4.2) holds trivially when j = −1, and (15.4.3) holds trivially when j = m + 1.

Observe that

H(X(v_{j})) − X(H(v_{j})) = (m − j + 1) · H(v_{j−1}) − (m − 2 j) · X(v_{j})
(15.4.4)  = (m − j + 1) (m − 2 (j − 1)) · v_{j−1}
−(m − 2 j) · X(v_{j})
= ((m − 2 (j − 1)) − (m − 2 j)) · X(v_{j}) = 2 · X(v_{j})

for each j = 0, 1, …, m. Similarly,

H(Y (v_{j})) − Y (H(v_{j})) = (j + 1) · H(v_{j+1}) − (m − 2 j) · Y (v_{j})
(15.4.5)  = (j + 1) (m − 2 (j + 1)) · v_{j+1} − (m − 2 j) · Y (v_{j})
= ((m − 2 (j + 1)) − (m − 2 j)) · Y (v_{j}) = −2 · Y (v_{j})

for every j = 0, 1, ⋯, m. We also have that

X(Y (v_{j})) − Y (X(v_{j})) = (j + 1) · X(v_{j+1}) − (m − j + 1) · Y (v_{j−1})
(15.4.6)  = (j + 1) (m − (j + 1) + 1) · v_{j}
−(m − j + 1) ((j − 1) + 1) · v_{j}
= ((j + 1) (m − j) − (m − j + 1) j) · v_{j}
= ((m − j) − j) · v_{j} = H(v_{j})

for each j = 0, 1, …, n. This shows that

[H, X] = 2 · X, [H, Y] = −2 · Y, and [X, Y] = H, (15.4.7)

using the commutator bracket associated to composition of module homomorphisms from W (m) into itself.

Let x, y, and h be the usual elements of sl_{2}(k), as in (15.1.1). Thus every element of sl_{2}(k) can be expressed in a unique way as a linear combination of x, y, and h with coefficients in k, as in Section 10.2. The Lie brackets of x, y, and h in sl_{2}(k) satisfy (15.1.2), as before. Put

ρ_{x} = X, ρ_{y} = Y, and ρ_{h} = H. (15.4.8)

If a ∈ sl_{2}(k), then we can define ρ_{a} as a module homomorphism from W (m) into itself that satisfies (15.4.8) and is linear over k in a. This defines a representation ρ = ρ^{W (m)} of sl_{2}(k), as a Lie algebra over k, on W (m), because of (15.4.7). This corresponds to the statement before Theorem 2 on p20 of [24], as well as remarks on p33 and part of Exercise 3 on p34 of [14].

If m = 0, then ρ corresponds to the trivial representation of sl_{2}(k) on k. If m = 1, then ρ corresponds to the standard representation of sl_{2}(k) on k^{2}. If m = 2, then one can check that ρ is isomorphic to the adjoint representation on sl_{2}(k). More precisely, if one takes v_{0} = x, v_{1} = −h, and v_{2} = y, then X, Y, and H correspond to ad_{x}, ad_{y}, and ad_{h} on sl_{2}(k), respectively. This corresponds to the examples mentioned on p20 of [24], and some remarks on p33 of [14].

15.5 More on W (m)

Let k be a field of characteristic 0, and let m be a nonnegative integer. Also let W (m) be as in the previous section, so that W (m) is a vector space over k of dimension m + 1, and ρ = ρ^{W (m)} is a representation of sl_{2}(k) on W (m). In this situation, v_{0} is a primitive element of V = W (m) with weight λ = m, by 15.5. MORE ON W (M) 317

construction. Of course, (15.3.2), (15.3.3), and (15.3.4) correspond to (15.4.1), (15.4.2), and (15.4.3) here, respectively. Thus v_{1}, …, v_{m} in the previous section correspond to the analogous vectors in Section 15.3, and v_{j} = 0 when j > m in the notation of Section 15.3. This means that we are in the second case (15.3.10), and W (m) is the same as W in Section 15.3. In particular, W (m) is irreducible as a module over sl_{2}(k), as before. This corresponds to part (a) of Theorem 2 on p20 of [24], as well as remarks on p33 and part of Exercise 3 on p34 of [14].

Let V be a vector space over k, and let ρ^{V} be a representation of sl_{2}(k) on V. Suppose that v ∈ V is primitive of weight λ ∈ k, as in Section 15.3, and that we are in the second case (15.3.10), which holds automatically when V has finite dimension. If W ⊆ V and m ≥ 0 are as in Section 15.3, then W is isomorphic to W (m) in the previous section, as modules over sl_{2}(k). If V is irreducible as a module over sl_{2}(k), then V = W, so that V is isomorphic to W (m) as a module over sl_{2}(k).

Suppose now that k is algebraically closed, and that V has positive finite dimension. Under these conditions, there is a v ∈ V that is primitive of some weight λ ∈ k, as in Section 15.1. In this situation, we are automatically in the second case (15.3.10), as in the preceding paragraph. This corresponds to part (b) of Theorem 2 on p20 of [24], and the theorem on p33 of [14].

Let k be a field, and let W (m) be as in the previous section for each nonnegative integer m again. Note that W (0) is automatically irreducible as a module over sl_{2}(k). Similarly, W (1) is irreducible as a module over sl_{2}(k). Suppose from now on in this section that k has positive characteristic, and that m ≥ 2.

Suppose for the moment that m is strictly less than the characteristic of k, which is thus greater than 2. Under these conditions, W (m) is irreducible as a module over sl_{2}(k), as in Exercise 5 on p34 of [14]. To see this, let W_{0} be a nontrivial linear subspace of W (m) such that H(W_{0}) ⊆ W_{0}. The restriction of H to W_{0} is diagonalizable on W_{0}, because H is diagonalizable on W (m). This implies that W_{0} contains a nonzero eigenvector of H. One can check that the eigenvalues of H on W (m) are distinct as elements of k in this situation. It follows that v_{j_{0}} ∈ W_{0} for some j_{0}, 0 ≤ j_{0} ≤ m.

Suppose that W_{0} is a submodule of W (m), as a module over sl_{2}(k). One can verify that v_{j} ∈ W_{0} for every j = 0, 1, …, m, using (15.4.2) and (15.4.3). More precisely, this uses the fact that the relevant coefficients on the right sides of (15.4.2) and (15.4.3) correspond to nonzero elements of k. This implies that W_{0} = W (m), as desired.

If k has characteristic equal to m, then W (m) is reducible as a module over sl_{2}(k), as in Exercise 5 on p34 of [14]. To see this, consider the linear span W_{1} of v_{1}, …, v_{m−1} in W (m). Clearly H(W_{1}) ⊆ W_{1}, because the v_{j}’s are eigenvectors for H. It is easy to see that Y (W_{1}) ⊆ W_{1}, because Y (v_{m−1}) = 0 when the characteristic of k is equal to m, by (15.4.2). Similarly, one can check that X(W_{1}) ⊆ W_{1}, because X(v_{1}) = 0 when the characteristic of k is m, by (15.4.3). This implies that W_{1} is a submodule of W (m), as a module over sl_{2}(k). Of course, W_{1} ≠ {0}, W (m).

15.6 Another construction

Let k be a commutative ring with a multiplicative identity element, and let T_{1}, T_{2} be commuting indeterminates. Remember that k[T_{1}, T_{2}] is the space of formal polynomials in T_{1}, T_{2} with coefficients in k, as in Section 5.8. This is a commutative associative algebra over k with respect to the usual formal multiplication of polynomials, as before. If d is a nonnegative integer, then the space k_{d}[T_{1}, T_{2}] of formal polynomials in T_{1}, T_{2} with coefficients in k that are homogeneous of degree d is a submodule of k[T_{1}, T_{2}], as a module over k, as in Section 5.13. More precisely, k[T_{1}, T_{2}] corresponds to the direct sum of the submodules k_{d}[T_{1}, T_{2}] over d ≥ 0, as a module over k, as before.

We would like to make k[T_{1}, T_{2}] into a module over sl_{2}(k), as a Lie algebra over k, as in Exercise 4 on p34 of [14], and the remark on p20 of [24]. Thus, if a ∈ sl_{2}(k) and f(T) ∈ k[T_{1}, T_{2}], then we need to define a · f(T) as an element of k[T_{1}, T_{2}]. More precisely, if f(T) ∈ k_{d}[T_{1}, T_{2}] for some d ≥ 0, then a · f(T) will be an element of k_{d}[T_{1}, T_{2}]. This means that k_{d}[T_{1}, T_{2}] will be a submodule of k[T_{1}, T_{2}], as a module over sl_{2}(k), for each d ≥ 0. Indeed, k[T_{1}, T_{2}] will correspond to the direct sum of k_{d}[T_{1}, T_{2}] over d ≥ 0, as a module over sl_{2}(k).

Of course, k_{0}[T_{1}, T_{2}] corresponds to constant formal polynomials with coefficients in k, and is isomorphic to k in an obvious way. If a ∈ sl_{2}(k) and f(T) ∈ k_{0}[T_{1}, T_{2}], then we put a · f(T) = 0.

An element of k_{1}[T_{1}, T_{2}] can be expressed in a unique way as

f(T) = f_{1} T_{1} + f_{2} T_{2}, (15.6.1)

where f_{1}, f_{2} ∈ k. In the notation of Section 5.8, f_{1} and f_{2} correspond to f_{α} with α = (1, 0) and (0, 1), respectively. If a ∈ sl_{2}(k), then we can define a · f(T) as an element of k_{1}[T_{1}, T_{2}] using the standard action of sl_{2}(k) on (f_{1}, f_{2}) as an element k^{2}. In particular, if x, y, h ∈ sl_{2}(k) are as in (15.1.1), as usual, then

x · f(T) = f_{2} T_{1}, y · f(T) = f_{1} T_{2}, and h · f(T) = f_{1} T_{1} − f_{2} T_{2}. (15.6.2)

This makes k_{1}[T_{1}, T_{2}] into a module over sl_{2}(k), as a Lie algebra over k.

If a ∈ sl_{2}(k), then we would like to define the action of a on k[T_{1}, T_{2}] so that it is a derivation, as in Exercise 4 on p34 of [14]. Thus we would like to have that a · (f(T) g(T)) = (a · f(T)) g(T) + f(T) (a · g(T)) (15.6.3)

for every f(T), f(T) ∈ k[T_{1}, T_{2}]. If α = (α_{1}, α_{2}) is an ordered pair of nonnegative integers, then we should have that

a · (T_{1}^{α_{1}} T_{2}^{α_{2}}) = α_{1} · (a · T_{1}) T_{1}^{α_{1}−1} T_{2}^{α_{2}} + α_{2} · (a · T_{2}) T_{1}^{α_{1}} T_{2}^{α_{2}−1}. (15.6.4)

More precisely, if α_{1} or α_{2} is 0, then the corresponding term on the right should be interpreted as being equal to 0. One can use this to define the left side of the equation, using the definition of a · T_{1} and a · T_{2} from the preceding paragraph. This can be used to define a·f(T) for every f(T) ∈ k[T_{1}, T_{2}], by linearity. Using this definition, one can check that (15.6.3) holds for every f(T), g(T) ∈ k[T_{1}, T_{2}].

Let ρ_{a} be the action of a ∈ sl_{2}(k) on k[T_{1}, T_{2}] defined in the previous paragraph. Thus ρ_{a} is a derivation on k[T_{1}, T_{2}] for every a ∈ sl_{2}(k), and it is easy to see that ρ_{a} is linear over k in a. We also have that ρ_{a} maps k_{d}[T_{1}, T_{2}] into itself for every a ∈ sl_{2}(k) and d ≥ 0, by construction. In order to show that this defines a representation of sl_{2}(k), as a Lie algebra over k, on k[T_{1}, T_{2}], we should verify that [ρ_{a}, ρ_{b}] = ρ_{[a,b]} (15.6.5)

for every a, b ∈ sl_{2}(k). Note that the left side is a derivation on k[T_{1}, T_{2}] as well, as in Section 2.5. We already have that ρ defines a representation of sl_{2}(k) on k_{0}[T_{1}, T_{2}] and k_{1}[T_{1}, T_{2}], so that (15.6.5) holds on these subspaces of k[T_{1}, T_{2}]. One can use this to check that (15.6.5) holds on all of k[T_{1}, T_{2}], because both sides of (15.6.5) are derivations on k[T_{1}, T_{2}].

Let α = (α_{1}, α_{2}) be an ordered pair of nonnegative integers again. Observe that

(15.6.6)  x · (T_{1}^{α_{1}} T_{2}^{α_{2}}) = α_{2} · T_{1}^{α_{1}+1} T_{2}^{α_{2}−1},
(15.6.7)  y · (T_{1}^{α_{1}} T_{2}^{α_{2}}) = α_{1} · T_{1}^{α_{1}−1} T_{2}^{α_{2}+1},
(15.6.8)  h · (T_{1}^{α_{1}} T_{2}^{α_{2}}) = (α_{1} − α_{2}) · T_{1}^{α_{1}} T_{2}^{α_{2}}.

If a ∈ sl_{2}(k), then ρ_{a} corresponds to the formal differential operator

(a · T_{1}) ∂_{1} + (a · T_{2}) ∂_{2} = (a · T_{1}) \frac{∂}{∂T_{1}} + (a · T_{2}) \frac{∂}{∂T_{2}} (15.6.9)

on k[T_{1}, T_{2}], in the notation of Section 5.11. More precisely, this is a first-order differential operator, as in Section 5.12, which is homogeneous of degree 0, as in Section 5.14.

15.7 Some exponentials

Let k be a field of characteristic 0, or a commutative ring with a multiplicative identity element that is an algebra over Q, for simplicity. As in (15.1.1), we put x = \binom{0 1}{0 0} and y = \binom{0 0}{1 0}. Note that x^{2} = y^{2} = 0, so that x and y are nilpotent as elements of the algebra M_{2}(k) of 2 × 2 matrices with entries in k. Thus the exponentials of x and y are defined in M_{2}(k) as in Section 14.10, with

exp x = I + x = \binom{1 1}{0 1}, exp y = I + y = \binom{1 0}{1 1}, (15.7.1)

where I = I_{2} = \binom{1 0}{0 1} is the identity matrix in M_{2}(k). Similarly,

exp(−y) = I − y = \binom{1 0}{−1 1}. (15.7.2)

Observe that

(exp x) (exp −y) = \binom{1 1}{0 1} \binom{1 0}{−1 1} = \binom{0 1}{−1 1}, (15.7.3)

320 CHAPTER 15. COMPLEXIFICATIONS AND SL_{2}(K) MODULES and hence (15.7.4) (exp x) (exp −y) (exp x) = \binom{0 1}{−1 1} \binom{1 1}{0 1} = \binom{0 1}{−1 0}.

Remember that [x, y] = h = \binom{1 0}{0 −1}, [h, x] = 2 · x, and [h, y] = −2 · y.

Put ad_{x}(z) = [x, z] and ad_{y}(z) = [y, z] for every z ∈ sl_{2}(k), as usual. Thus ad_{x}(x) = ad_{y}(y) = 0, (15.7.5) ad_{x}(y) = −ad_{y}(x) = h, ad_{x}(h) = −2 · x, ad_{y}(h) = 2 · y. This implies that (ad_{x})^{2}(h) = (ad_{y})^{2}(h) = 0. (15.7.6) Similarly, (15.7.7) (ad_{x})^{2}(y) = ad_{x}(h) = −2 · x, (ad_{y})^{2}(x) = ad_{y}(−h) = −2 · y. It follows that (ad_{x})^{3} = (ad_{y})^{3} = 0 as mappings from sl_{2}(k) into itself, so that x and y are ad-nilpotent in sl_{2}(k).

This means that the exponentials of ad_{x} and ad_{y} are defined as linear mappings from sl_{2}(k) into itself, as in Section 14.10. In this situation, we have that

(15.7.8)  exp ad_{x} = I + ad_{x} + (1/2) (ad_{x})^{2},
(15.7.9)  exp ad_{y} = I + ad_{y} + (1/2) (ad_{y})^{2},

where I is the identity mapping on sl_{2}(k). Similarly, (15.7.10) exp(−ad_{y}) = I − ad_{y} + (1/2) (ad_{y})^{2}. Observe that

(15.7.11) (exp ad_{x})(x) = x, (exp ad_{x})(h) = h − 2 · x,
and  (exp ad_{x})(y) = y + h − x.

We also have that

(15.7.12) (exp −ad_{y})(y) = y, (exp −ad_{y})(h) = h − 2 · y,
and  (exp −ad_{y})(x) = x + h − y.

Put (15.7.13) σ = (exp ad_{x}) ∘ (exp −ad_{y}) ∘ (exp ad_{x}), which defines a linear mapping from sl_{2}(k) into itself. More precisely, this is an inner automorphism of sl_{2}(k), as in Section 14.11. Observe that σ(x) = (exp ad_{x})(x + h − y) = x + h − 2 · x − y − h + x = −y. (15.7.14)

15.8. EXPONENTIALS AND W (M) 321

Similarly,

σ(y) = (exp ad_{x})((exp −ad_{y})(y + h − x))
(15.7.15) = (exp ad_{x})(y + h − 2 · y − x − h + y) = (exp ad_{x})(−x) = −x.

We also have that

(15.7.16) σ(h) = (exp ad_{x})((exp −ad_{y})(h − 2 · x))
= (exp ad_{x})(h − 2 · y − 2 · x − 2 · h + 2 · y)
= (exp ad_{x})(−h − 2 · x) = −h + 2 · x − 2 · x = −h.

It follows from (15.7.14), (15.7.15), and (15.7.16) that

σ(z) = −z^{t} (15.7.17)

for every z ∈ sl_{2}(k), where z^{t} is the transpose of z. This is the same as conjugating z ∈ sl_{2}(k) by (15.7.4), as in Section 14.11. This corresponds to some remarks on p9 and Exercise 10 on p10 of [14].

15.8 Exponentials and W (m)

Let k be a commutative ring with a multiplicative identity element that is an algebra over Q, or simply a field of characteristic 0, and let m be a nonnegative integer. Remember that W (m) is defined as a module over k as in Section 15.4, with corresponding linear mappings X, Y, and H. Note that X and Y are nilpotent as linear mappings on W (m), so that their exponentials are defined as invertible linear mappings on W (m), as in Section 14.10. Put

(15.8.1) θ = (exp X) ∘ (exp −Y) ∘ (exp X),

which is an invertible linear mapping from W (m) into itself.

Let ρ be the representation of sl_{2}(k) on W (m) corresponding to X, Y, and H as before. Thus ρ_{x} = X, ρ_{y} = Y, and ρ_{h} = H, where x, y, and h are the usual elements of sl_{2}(k). Remember that exp ad_{x} and exp ad_{y} are defined as invertible linear mappings on sl_{2}(k), because x and y are ad-nilpotent in sl_{2}(k). If z ∈ sl_{2}(k), then

(15.8.2) ρ_{(exp ad_{x})(z)} = (exp ρ_{x}) ∘ ρ_{z} ∘ (exp ρ_{x})^{−1},

as in Section 14.12. Similarly,

(15.8.3) ρ_{(exp −ad_{y})(z)} = (exp −ρ_{y}) ∘ ρ_{z} ∘ (exp −ρ_{y})^{−1}

for every z ∈ sl_{2}(k).

If σ is as in (15.7.13), then we get that

ρ_{σ(z)} = θ ∘ ρ_{z} ∘ θ^{−1} (15.8.4) for every z ∈ sl_{2}(k). In particular,

−Y = ρ_{σ(x)} = θ ∘ ρ_{x} ∘ θ^{−1} = θ ∘ X ∘ θ^{−1}, (15.8.5)

using (15.7.14) in the first step. Similarly,

−X = ρ_{σ(y)} = θ ∘ ρ_{y} ∘ θ^{−1} = θ ∘ Y ∘ θ^{−1}, (15.8.6)

using (15.7.15) in the first step. We also have that

−H = ρ_{σ(h)} = θ ∘ ρ_{h} ∘ θ^{−1} = θ ∘ H ∘ θ^{−1}, (15.8.7)

using (15.7.16) in the first step. This corresponds to remarks on p33-4 of [14], and Remark 1 on p21 of [24].

15.9 Additional properties of sl_{2}(k) modules

Let K be a field, and let k be a subfield of K. Also let n be a positive integer, so that V = k^{n} is an n-dimensional vector space over k, and V_{K} = K^{n} is an n-dimensional vector space over K. If T is a linear mapping from V into itself, as a vector space over k, then there is a unique extension T_{K} of T to a linear mapping from V_{K} into itself, as a vector space over K. If T is invertible on V, then T_{K} is invertible on V_{K}, with inverse equal to the extension of T^{−1} on V to V_{K} as before. If λ ∈ k is an eigenvalue of T on V, then λ is an eigenvalue of T_{K} on V_{K}. However, if λ ∈ k is not an eigenvalue of T on V, then λ is not an eigenvalue of T_{K} on V_{K}. More precisely, T − λ I_{V} is invertible on V in this case, where I_{V} is the identity mapping on V. This implies that T_{K} −λ I_{V_{K}} is invertible on V_{K}, where I_{V_{K}} is the identity mapping on V_{K}. Thus the eigenvalues of T on V are the same as the eigenvalues of T_{K} on V_{K} that are elements of k.

Let ρ^{V} be a representation of sl_{2}(k), as a Lie algebra over k, on V = k^{n}. This leads to a representation ρ^{V_{K}} of sl_{2}(K), as a Lie algebra over K, on V_{K} = K^{n}, in a natural way. More precisely, if z ∈ sl_{2}(k), then ρ_{z}^{V_{K}} is the K-linear mapping on K^{n} that corresponds to ρ_{z}^{V} as a k-linear mapping on k^{n} as in the preceding paragraph. In particular, this can be applied to the usual elements x, y, and h of sl_{2}(k), as in (15.1.1). If z ∈ sl_{2}(K), then z can be expressed as a linear combination of x, y, and h with coefficients in K, and one can take ρ_{z}^{V_{K}} to be the corresponding linear combination of ρ_{x}^{V_{K}}, ρ_{y}^{V_{K}}, and ρ_{h}^{V_{K}}.

Suppose now that k has characteristic 0, and that K is algebraically closed. Of course, we may consider Q as a subfield of k, and thus as a subfield of K. Remember that V_{K} can be expressed as the direct sum of finitely many irreducible sl_{2}(K) modules, by Weyl’s theorem. Each of these irreducible sl_{2}(K) modules is isomorphic to the analogue of W (m) in Section 15.4 for K, and for some nonnegative integer m, as in Section 15.5. It follows that ρ_{h}^{V_{K}} is diagonalizable on V_{K}, with eigenvalues in Z.

In particular, ρ_{h}^{V_{K}} has an eigenvalue λ in Z, which implies that λ is an eigenvalue for ρ_{h}^{V} on V, as before. One can use this to show that there is a maximal or primitive vector in V of some weight, as in Section 15.2. If V is irreducible as an sl_{2}(k) module, then it follows that V is isomorphic as an sl_{2}(k) module to W (m) for some nonnegative integer m, as in Section 15.5. Otherwise, one can use Weyl’s theorem to reduce to this case. This corresponds to Exercise 2 on p62 of [25], with n = 2.

15.10 Complexifying real vector spaces

Let V_{1} be a vector space over the real numbers. The complexification of V_{1} may be defined as a vector space V_{2} over the complex numbers as follows. We start by taking V_{2} to be the direct sum V_{1} ⊕ V_{1} of V_{1} with itself, as a vector space over the real numbers. Equivalently, V_{2} is the Cartesian product V_{1} × V_{1} of V_{1} with itself, where addition and scalar multiplication by real numbers is defined coordinatewise. If (v, w) ∈ V_{1} × V_{1}, then i (v, w) is defined as an element of V_{1} × V_{1} by i (v, w) = (−w, v). (15.10.1)

It is easy to see that this makes V_{2} into a vector space over the complex numbers. We shall normally identify v ∈ V_{1} with (v, 0) ∈ V_{1} × V_{1} = V_{2}, so that V_{1} may be considered as a subset of V_{2}. Note that any basis for V_{1} as a vector space over the real numbers may be considered as a basis for V_{2} as a vector space over the complex numbers.

Let V be a vector space over the complex numbers, so that V may be considered as a vector space over the real numbers too. Suppose that V_{0} is a real-linear subspace of V, which is to say that V_{0} is a linear subspace of V as a vector space over the real numbers. In this case,

i V_{0} = {i v : v ∈ V_{0}} (15.10.2)

is a real-linear subspace of V as well. Note that V_{0} + i V_{0} is a complex-linear subspace of V, which is to say a linear subspace of V as a vector space over the complex numbers. Let us say that V_{0} is totally real in V if

V_{0} ∩ (i V_{0}) = {0}. (15.10.3)

This means that V_{0} + i V_{0} is isomorphic to the direct sum of of V_{0} and i V_{0}, as a vector space over the real numbers. Under these conditions, there is a natural isomorphism from the complexification of V_{0} onto V_{0} + i V_{0}, as vector spaces over the complex numbers, which is the identity mapping on V_{0}.

Let V_{1} be a vector space over the real numbers again, and let W be a vector space over the complex numbers. Thus W may be considered as a vector space over the real numbers, and a linear mapping from V_{1} into W, as a vector space over R, may be called a real-linear mapping. One can verify that a real-linear mapping from V_{1} into W has a unique extension to a complex-linear mapping from the complexification V_{2} of V_{1} into W.

Let W_{1} be a vector space over the real numbers, and let W_{2} be the complexification of W_{1}. If ϕ_{1} is a linear mapping from V_{1} into W_{1}, as vector spaces over the real numbers, then ϕ_{1} may be considered as a real-linear mapping from V_{1} into W_{2}, because W_{1} is identified with a real-linear subspace of W_{2}. Thus ϕ_{2} has a unique extension to a complex-linear mapping ϕ_{2} from V_{2} into W_{2}, as in the preceding paragraph. If ϕ_{1} is injective as a mapping from V_{1} into W_{1}, then one can check that ϕ_{2} is injective as a mapping from V_{2} into W_{2}. Similarly, if ϕ_{1} maps V_{1} onto W_{1}, then ϕ_{2} maps V_{2} onto W_{2}.

Let W be a vector space over the complex numbers again, let ϕ_{1} be a reallinear mapping from V_{1} into W, and let ϕ_{2} be the extension of ϕ_{1} to a complexlinear mapping from V_{2} into W. If ϕ_{1} is injective as a mapping from V_{1} into W, and if ϕ_{1}(V_{1}) is totally real as a real-linear subspace of W, then one can verify that ϕ_{2} is injective as a mapping from V_{2} into W.

Let W_{1} be a vector space over the real numbers again, and let Z_{1} be another vector space over the real numbers. Also let ϕ_{1} be a linear mapping from V_{1} into W_{1}, and let ψ_{1} be a linear mapping from W_{1} into Z_{1}, as vector spaces over R. As before, ϕ_{1} and ψ_{1} have unique complex-linear extensions ϕ_{2} from V_{2} into W_{2} and ψ_{2} from W_{2} into the complexification Z_{2} of Z_{1}, respectively. Of course, ψ_{1} ∘ ϕ_{1} is a linear mapping from V_{1} into Z_{1}, as vector spaces over R. Note that ψ_{2} ∘ ϕ_{2} is the unique extension of ψ_{1} ∘ ϕ_{1} to a complex-linear mapping from V_{2} into Z_{2}.

Let Z be a vector space over the complex numbers again. A mapping from V_{1} × W_{1} into Z that is bilinear over R, where Z is considered as a vector space over the real numbers, may be called a real-bilinear mapping. One can check that such a mapping has a unique extension to a mapping from V_{2} × W_{2} into Z that is bilinear over C. If Z_{1} is a vector space over the real numbers, then a mapping from V_{1} × W_{1} into Z_{1} that is bilinear over R may be considered as a real-bilinear mapping from V_{1} × W_{1} into the complexification Z_{2} of Z_{1}. This can be extended to a mapping from V_{2} × W_{2} into Z_{2} that is bilinear over C, as before.

15.11 Spaces of linear mappings

Let V_{1} be a vector space over the real numbers, and let V_{2} be its complexification, as in the previous section. If W is a vector space over the complex numbers, then we let L_{R}(V_{1}, W) be the space of real-linear mappings from V_{1} into W, and L_{C}(V_{2}, W) be the space of complex-linear mappings from V_{2} into W. If ϕ_{1} is a real-linear mapping from V_{1} into W, then there is a unique extension ϕ_{2} of ϕ_{1} to a complex-linear mapping from V_{2} into W, as before. Conversely, if ϕ_{2} is any complex-linear mapping from V_{2} into W, then the restriction of ϕ_{2} to V_{1}, considered as a real-linear subspace of V_{2}, is a real-linear mapping from V_{1} into W. This defines a one-to-one correspondence between L_{R}(V_{1}, W) and L_{C}(V_{2}, W).

Note that L_{R}(V_{1}, W) may be considered as a vector space over the complex numbers, with respect to pointwise addition and scalar multiplication by complex numbers of mappings into W. It is easy to see that this corresponds to pointwise addition and scalar multiplication by complex numbers on L_{C}(V_{2}, W). Thus the one-to-one correspondence between L_{R}(V_{1}, W) and L_{C}(V_{2}, W) mentioned in the preceding paragraph is an isomorphism between complex vector spaces.

Let W_{1} be another vector over the real numbers, and let W_{2} be its complexification, as before. The space L(V_{1}, W_{1}) of linear mappings from V_{1} into W_{1} may be considered as a real-linear subspace of the space L_{R}(V_{1}, W_{2}) of real-linear mappings from V_{1} into W_{2}, because W_{1} is identified with a real-linear subspace of W_{2}. Similarly, i W_{1} may be considered as a real-linear subspace of W_{2}, so that the space L(V_{1}, i W_{1}) of linear mappings from V_{1} into i W_{1}, as vector spaces over R, may be considered as a real-linear subspace of L_{R}(V_{1}, W_{2}). Observe that L(V_{1}, i W_{1}) = i L(V_{1}, W_{1}), (15.11.1)

as real-linear subspaces of L_{R}(V_{1}, W_{2}). It is easy to see that

L(V_{1}, W_{1}) ∩ (i L(V_{1}, W_{1})) = {0}, (15.11.2)

because W_{1} ∩ (i W_{1}) = {0} in W_{2}. We also have that

L(V_{1}, W_{1}) + L(V_{1}, i W_{1}) = L_{R}(V_{1}, W_{2}). (15.11.3)

More precisely, L_{R}(V_{1}, W_{2}) corresponds to the direct sum of L(V_{1}, W_{1}) and L(V_{1}, i W_{1}) as a vector space over R, because W_{2} corresponds to the direct sum of W_{1} and i W_{1} as a vector space over R. Equivalently, this means that

L(V_{1}, W_{1}) + i L(V_{1}, W_{1}) = L_{R}(V_{1}, W_{2}). (15.11.4)

Thus the complexification of L(V_{1}, W_{2}) can be identified with L_{R}(V_{1}, W_{2}), as a vector space over the complex numbers, as in the previous section.

Remember that there is a natural isomorphism between L_{R}(V_{1}, W_{2}) and L_{C}(V_{2}, W_{2}), as before. This leads to a natural embedding of L(V_{1}, W_{2}) into L_{C}(V_{2}, W_{2}). More precisely, this embedding takes a linear mapping ϕ_{1} from V_{1} into W_{1}, and associates to it the unique extension ϕ_{2} of ϕ_{1} to a complexlinear mapping from V_{2} into W_{2}, where V_{1}, W_{1} are identified with real-linear subspaces of V_{2}, W_{2}, as usual. The image of L(V_{1}, W_{1}) in L_{C}(V_{2}, W_{2}) consists of the complex-linear mappings ϕ_{2} from V_{2} into W_{2} such that

ϕ_{2}(V_{1}) ⊆ W_{1}. (15.11.5)

Using this embedding, we get an isomorphism between the complexification of L(V_{1}, W_{1}) and L_{C}(V_{2}, W_{2}).

Let n be a positive integer, so that R^{n} is a vector space over the real numbers, whose complexification can be identified with C^{n}. Remember that the space L(R^{n}) of linear mappings from R^{n} into itself is isomorphic to the space M_{n}(R) of n×n matrices with entries in R in the usual way. Similarly, the space L_{C}(C^{n}) of complex-linear mappings from C^{n} into itself is isomorphic to the space M_{n}(C) of n×n matrices with entries in C in essentially the same way. Of course, M_{n}(R) may be considered as a real-linear subspace of M_{n}(C), and M_{n}(C) can be identified with the complexification of M_{n}(R). This corresponds to identifying a linear mapping from R^{n} into itself with a complex-linear mapping from C^{n} into itself that takes R^{n} into itself, as in the preceding paragraph.

15.12 Complexifying algebras over R

Let A_{1} be an algebra over the real numbers in the strict sense, and let A_{2} be the complexification of A_{1} as a vector space over R, as in Section 15.10. The algebra structure on A_{1} corresponds to a mapping from A_{1} × A_{1} into A_{1} that is bilinear over R, which has a unique extension to a mapping from A_{2} × A_{2} into A_{2} that is bilinear over C, as before. This makes A_{2} into an algebra over C in the strict sense. If A_{1} is commutative, associative, or a Lie algebra, then one can check that A_{2} has the analogous property, as an algebra over C. Similarly, if A_{1} has a multiplicative identity element e, then e is the multiplicative identity element in A_{2} as well.

Let B be an algebra over the complex numbers in the strict sense, which may be considered as an algebra over the real numbers in the strict sense too. If ϕ_{1} is a real-linear mapping from A_{1} into B, then there is a unique extension of ϕ_{1} to a complex-linear mapping ϕ_{2} from A_{2} into B, as in Section 15.10. If ϕ_{1} is an algebra homomorphism from A_{1} into B, then it is easy to see that ϕ_{2} is an algebra homomorphism from A_{2} into B. Similarly, if ϕ_{1} is an opposite algebra homomorphism from A_{1} into B, then ϕ_{2} is an opposite algebra homomorphism from A_{2} into B.

If B_{1} is an algebra over the real numbers in the strict sense, then its complexification B_{2} is an algebra over the complex numbers in the strict sense, as before. If ϕ_{1} is a linear mapping from A_{1} into B_{1}, as vector spaces over the real numbers, then ϕ_{1} may be considered as a real-linear mapping from A_{1} into B_{2}. Thus there is a unique extension of ϕ_{1} to a complex-linear mapping ϕ_{2} from A_{2} into B_{2}, as before. If ϕ_{1} is an algebra homomorphism or an opposite algebra homomorphism from A_{1} into B_{1}, then ϕ_{2} has the analogous property as a mapping from A_{2} into B_{2}.

Let A be an algebra over the complex numbers in the strict sense, which may also be considered as an algebra over the real numbers in the strict sense. Suppose that A_{0} is a real-linear subspace of A that is totally real in A. This implies that there is a natural isomorphism between the complexification of A_{0}, as a vector space over R, and A_{0}+i A_{0}, as in Section 15.10. If A_{0} is a subalgebra of A too, as an algebra over R in the strict sense, then it is easy to see that A_{0} + i A_{0} is a subalgebra of A, as an algebra over C in the strict sense. In this case, A_{0} + i A_{0} is isomorphic to the complexification of A_{0}, as an algebra over R in the strict sense.

Let V_{1} be a vector space over the real numbers, with complexification V_{2}, as in Section 15.10. Remember that the space L(V_{1}) of linear mappings from V_{1} into itself is an associative algebra over R with respect to composition of mappings. Similarly, the space L_{C}(V_{2}) of complex-linear mappings from V_{2} into itself is an associative algebra over C with respect to composition of mappings. We have also seen that L_{C}(V_{2}) can be identified with the complexification of L(V_{1}), as a vector space over R, as in the previous section. More precisely, this uses the correspondence between linear mappings from V_{1} into itself and complex-linear mappings from V_{2} into itself that map V_{1} into itself, where V_{1} is considered as a real-linear subspace of V_{2}, as usual. This correspondence sends compositions of linear mappings on V_{1} to the analogous compositions of complex-linear mappings on V_{2}, as in Section 15.10. Thus L(V_{1}) corresponds to a subalgebra of L_{C}(V_{2}), as an algebra over R. It follows that L_{C}(V_{2}) can be identified with the complexification of L(V_{1}) as an algebra over R in the same way, as in the preceding paragraph.

Similarly, gl(V_{1}) is a Lie algebra over the real numbers, and the space gl(V_{2}) = gl_{C}(V_{2}) of complex-linear mappings from V_{2} into itself is a Lie algebra over C, with respect to the usual commutator brackets. As before, gl_{C}(V_{2}) can be identified with the complexification of gl(V_{1}), as a vector space over R, and in fact as a Lie algebra over R.

Suppose that V_{1} has finite dimension as a vector space over R, so that V_{2} has the same dimension as a vector space over C. Let ϕ_{1} be a linear mapping from V_{1} into itself, and let ϕ_{2} be the extension of ϕ_{1} to a complex-linear mapping from V_{2} into itself. It is easy to see that

tr_{V_{1}}ϕ_{1} = tr_{V_{2}}ϕ_{2}, (15.12.1)

where the left side is the trace of ϕ_{1} on V_{1}, and the right side is the trace of ϕ_{2} on V_{2}. This uses the fact that a basis for V_{1} as a vector space over R may be considered as a basis for V_{2} as a vector space over C. It follows that the space sl(V_{2}) = sl_{C}(V_{2}) of complex-linear mappings from V_{2} into itself with trace 0 corresponds to the complexification of sl(V_{1}) as a vector space over R, and hence as a Lie algebra over R.

Let n be a positive integer, and remember that the spaces M_{n}(R), M_{n}(C) of n × n matrices with entries in R, C, respectively, are associative algebras over R, C with respect to matrix multiplication. As before, M_{n}(C) can be identified with the complexification of M_{n}(R) as a vector space over R, and more precisely as an algebra over R. Similarly, gl_{n}(C) can be identified with the complexification of gl_{n}(R) as a Lie algebra over R. We can also identify sl_{n}(C) with the complexification of sl_{n}(R) as a Lie algebra over R.

15.13 Conjugate-linear involutions

Let A be a vector space over the complex numbers, and let

a ↦ a^{∗} (15.13.1)

be a conjugate-linear mapping from A into itself. Put

A_{sa} = {a ∈ A : a^{∗} = a} (15.13.2)

and A_{asa} = {a ∈ A : a^{∗} = −a}, (15.13.3)

which are the collections of vectors in A that are self-adjoint and anti-self-adjoint with respect to (15.13.1), respectively. Clearly A_{sa} and A_{asa} are reallinear subspaces of A, with A_{asa} = i A_{sa}. (15.13.4)

We also have that A_{sa} ∩ A_{asa} = {0}, (15.13.5)

so that A_{sa} and A_{asa} are totally real in A. Thus A_{sa}+A_{asa} can be identified with the complexifications of A_{sa} and A_{asa}, as vector spaces over the real numbers, as in Section 15.10.

Suppose that (a^{∗})^{∗} = a (15.13.6)

for every a ∈ A, so that a ↦ a^{∗} is a conjugate-linear involution on A. If a ∈ A, then a_{sa} = (a + a^{∗})/2 ∈ A_{sa}, a_{asa} = (a − a^{∗})/2 ∈ A_{asa}, (15.13.7)

and a = a_{sa} + a_{asa}. This means that A = A_{sa} + A_{asa}, so that A can be identified with the complexifications of A_{sa} and A_{asa}, as vector spaces over the real numbers.

Suppose now that A is an associative algebra over the complex numbers, and that (15.13.1) is a conjugate-linear algebra involution on A. Let A_{Lie,C} be A considered as a Lie algebra over the complex numbers with respect to the corresponding commutator bracket, and let A_{Lie,R} be A_{Lie,C} considered as a Lie algebra over the real numbers. In this situation, A_{asa} is a Lie subalgebra of A_{Lie,R}, and A_{Lie,C} can be identified with the complexification of A_{asa}, as a Lie algebra over the real numbers.

Let V be a finite-dimensional vector space over the complex numbers, and let β(·, ·) be a nondegenerate Hermitian form on V. If T is a linear mapping from V into itself, then there is a unique adjoint linear mapping T^{∗,β} from V into itself such that β(T (v), w) = β(v, T^{∗,β}(w)) (15.13.8)

for every v, w ∈ V. Under these conditions,

T ↦ T^{∗,β} (15.13.9)

defines a conjugate-linear algebra involution on the algebra L(V) of all linear mappings from V into itself. Remember that gl(V) is the same as L(V), considered as a Lie algebra over the complex numbers with respect to the corresponding commutator bracket. Let u_{β}(V) be the collection of linear mappings T from V into itself that are anti-self-adjoint with respect to β. Thus u_{β}(V) is a real-linear subspace of gl(V), and a Lie subalgebra of gl(V), considered as a Lie algebra over the real numbers. As in the preceding paragraph, gl(V) can be identified with the complexification of u_{β}(V), as a Lie algebra over the real numbers.

If T ∈ L(V), then tr_{V} T^{∗,β} = \overline{tr_{V} T} , (15.13.10)

where tr_{V} T denotes the trace of T on V. This can be seen by choosing a basis for V to reduce to the case where V = C^{n} for some positive integer n, and expressing β(·, ·) and T^{∗,β} in terms of matrices, as in Section 3.13. In particular, this implies that sl(V) is invariant under (15.13.9). Note that tr_{V} T is real when T is self-adjoint with respect to β, and imaginary when T is anti-self-adjoint with respect to β, by (15.13.10). Put

su_{β}(V) = sl(V) ∩ u_{β}(V), (15.13.11)

which is a real-linear subspace of sl(V), and in fact a Lie subalgebra of sl(V), considered as a Lie algebra over the complex numbers. If T ∈ sl(V), then T^{∗,β} ∈ sl(V), and hence the self-adjoint and anti-self-adjoint parts of T with respect to β are contained in sl(V) too. Thus sl(V) can be identified with the complexification of su_{β}(V), as a Lie algebra over the real numbers.

Module homomorphisms

16.1 Bilinear actions and homomorphisms

Let k be a commutative ring with a multiplicative identity element, and let V, W be modules over k. Remember that Hom(V, W) = Hom_{k}(V, W) is the space of homomorphisms from V into W, as modules over k. This is a module over k too, with respect to pointwise addition and scalar multiplication of mappings. Let A be another module over k, and let ρ^{V}, ρ^{W} be bilinear actions of A on V and W, as in Section 6.1. Also let ϕ be a homomorphism from V into W, as modules over k. Remember that ϕ is said to intertwine ρ^{V}, ρ^{W} when

ϕ ∘ ρ_{a}^{V} = ρ_{a}^{W} ∘ ϕ (16.1.1)

for every a ∈ A, as in Section 6.2. In this case, we may say that ϕ is a homomorphism from V into W, with respect to the bilinear actions ρ^{V}, ρ^{W}. Let

Hom^{A}(V, W) = Hom_{k}^{A}(V, W) (16.1.2)

be the space of these homomorphisms from V into W with respect to ρ^{V}, ρ^{W}. It is easy to see that this is a submodule of Hom_{k}(V, W), as a module over k.

Let Z be another module over k. If ϕ is a homomorphism from V into W, and ψ is a homomorphism from W into Z, as modules over k, then ψ ∘ ϕ is a homomorphism from V into Z, as modules over k. Suppose that ρ^{Z} is a bilinear action of A on Z. If ϕ is a homomorphism from V into W with respect to ρ^{V}, ρ^{W}, and if ψ is a homomorphism from W into Z with respect to ρ^{W}, ρ^{Z}, then ψ ∘ ϕ is a homomorphism from V into Z with respect to ρ^{V}, ρ^{Z}. Indeed, if a ∈ A, then (ψ ∘ ϕ) ∘ ρ_{a}^{V} = ψ ∘ (ρ_{a}^{W} ∘ ϕ) = ρ_{a}^{Z} ∘ (ψ ∘ ϕ). (16.1.3)

Let us now take V = W, and remember that Hom_{k}(V, V) is an associative algebra over k, with respect to composition of mappings. More precisely, let us use the same bilinear action ρ^{V} of A on V on both the domain and range of these homomorphisms. Observe that Hom_{k}^{A}(V, V) is a subalgebra of Hom_{k}(V, V), as in the preceding paragraph. Of course, the identity mapping I = I_{V} on V is a homomorphism from V into itself, with respect to ρ^{V}. Similarly, if t ∈ k, then t I_{V} is an element of Hom_{k}^{A}(V, V).

Suppose for the moment that k is a field, so that V is a vector space over k, and the algebra of linear mappings from V into itself may be denoted L(V). Remember that L^{ρ}(V) may be used to denote the subalgebra of L(V) consisting of linear mappings from V into itself that are homomorphisms from V into itself with respect to ρ, as in Section 6.14. If V is irreducible with respect to ρ, then Schur’s lemma says that every nonzero element of L^{ρ}(V) is invertible in L^{ρ}(V). If k is also algebraically closed, and V has finite dimension as a vector space over k, then L_{ρ}(V) consists exactly of the multiples of I_{V} by elements of k. This is related to Exercises 4 and 5 on p54-5 of [25], for irreducible modules over a Lie algebra over k.

Now let (A, [·, ·]_{A}) be a Lie algebra over k, and let us consider A as a module over itself, with respect to the adjoint representation. If x, y ∈ A, then put ad_{x}(y) = [x, y]_{A}, as usual. Let ϕ be a homomorphism from A into itself, as a module over k. Thus ϕ is a homomorphism from A into itself, as a module over itself with respect to the adjoint representation, if and only if

ϕ(ad_{x}(y)) = ad_{x}(ϕ(y)) (16.1.4)

for every x, y ∈ A. This is the same as saying that

ϕ([x, y]_{A}) = [x, ϕ(y)]_{A} (16.1.5)

for every x, y ∈ A. If k is a field, then we may use L^{ad}(A) to denote the algebra of linear mappings from A into itself, as a vector space over k, that are homomorphisms from A into itself, as a module over itself with respect to the adjoint representation.

16.2 Homomorphisms and semisimplicity

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let A_{1}, …, A_{n} be ideals in A, and suppose that A corresponds to the direct sum of the A_{j}’s. This means that every z ∈ A can be expressed in a unique way as

z = ∑_{j=1}^{n} π_{j}(z), (16.2.1)

where π_{j}(z) ∈ A_{j} for each j = 1, …, n. In particular, this implies that A_{j} ∩A_{l} = {0} when j ≠ l, so that [A_{j}, A_{l}] = {0} when j ≠ l, as in Section 10.15. It follows that π_{j} is a Lie algebra homomorphism from A into A_{j} for each j = 1, …, n. The restriction of π_{j} to A_{j} is the same as the identity mapping on A_{j} for each j = 1, …, n, so that π_{j} maps A onto A_{j}. Observe that

π_{j}([x, y]_{A}) = [x, π_{j}(y)]_{A} (16.2.2) for every x, y ∈ A and j = 1, …, n. Thus π_{j} may be considered as a homomorphism from A into itself, as a module over itself with respect to the adjoint representation, for each j = 1, …, n.

Let ϕ be a linear mapping from A into itself, and suppose that ϕ is a homomorphism from A into itself, as a module over itself with respect to the adjoint representation. Let x ∈ A_{j} and y ∈ A_{l} be given, for some j, l ∈ {1, …, n} such that j ≠ l. This implies that [x, y]_{A} = 0, so that

[x, ϕ(y)]_{A} = ϕ([x, y]_{A}) = 0, (16.2.3)

using (16.1.5) in the first step. It follows that

[x, π_{j}(ϕ(y))]_{A} = 0, (16.2.4)

because [x, π_{r}(ϕ(y))]_{A} = 0 automatically when r ≠ j. Equivalently, this means that π_{j}(ϕ(y)) is an element of the center Z(A_{j}) of A_{j}, as a Lie algebra over k. If Z(A_{j}) = {0}, then we get that

π_{j}(ϕ(y)) = 0. (16.2.5)

If Z(A_{j}) = {0} for every j = 1, …, n, then

ϕ(A_{l}) ⊆ A_{l} (16.2.6)

for every l = 1, …, n. In this situation, one can check that the restriction of ϕ to A_{l} is a homomorphism from A_{l} into itself, as a module over itself with respect to the adjoint representation.

Suppose now that ϕ_{l} is a homomorphism from A_{l} into itself, as a module over itself with respect to the adjoint representation, for each l = 1, …, n. Under these conditions, one can verify that

ϕ = ∑_{l=1}^{n} ϕ_{l} ∘ π_{l} (16.2.7)

defines a homomorphism from A into itself, as a module over itself with respect to the adjoint representations. If Z(A_{j}) = {0} for each j = 1, …, n, then we get that L^{ad}(A) corresponds to the direct sum of L^{ad}(A_{l}), 1 ≤ l ≤ n. This is related to parts (a) and (b) of Exercise 5 on p55 of [25].

Suppose from now on in this section that A_{j} is simple as a Lie algebra over k for each j = 1, …, n. In particular, this means that Z(A_{j}) = {0} for every j = 1, …, n. We also get that A_{j} is irreducible as a module over itself with respect to the adjoint representation for every j = 1, …, n. Suppose that k is algebraically closed, and that A_{j} has positive finite dimension as a vector space over k for each j = 1, …, n. Schur’s lemma implies that L^{ad}(A_{j}) consists of multiples of the identity mapping on A_{j} by elements of k for each j = 1, …, n, as in the previous section. It follows that L^{ad}(A) is isomorphic to the direct sum of n copies of k, as an associative algebra over k, as in the preceding paragraph. This corresponds to part (a) of Exercise 5 on p55 of [25].

16.3 Linear mappings and dimensions

Let k be a field, and let n, r be positive integers. Thus the spaces k^{n}, k^{r} of n, r-tuples of elements of k may be considered as vector spaces over k with respect to coordinatewise addition and scalar multiplication, respectively. Let T be a linear mapping from k^{n} into k^{r}, and let λ_{l}(v) be the lth component of T (v) ∈ k^{r} for each l = 1, …, r and v ∈ k^{n}. Equivalently, λ_{l} is a linear functional on k^{n} for each l = 1, …, r, and any collection of r linear functionals λ_{1}, …, λ_{r} on k^{n} determine a linear mapping T from k^{n} into k^{r} in this way. Let us suppose that T ≠ 0, to avoid trivialities, so that λ_{l} ≠ 0 for some l. Let r_{0} be the maximal number of λ_{l}’s that are linearly independent as linear functionals on k^{n}. We can rearrange the λ_{l}’s, if necessary, to get that λ_{1}, …, λ_{r_{0}} are linearly independent as linear functionals on k^{n}, and that λ_{l} can be expressed as a linear combination of λ_{1}, …, λ_{r_{0}} when l > r_{0}.

Similarly, let T_{0} the linear mapping from k^{n} into k^{r_{0}} that corresponds to λ_{1}, …, λ_{r_{0}}. By construction, the kernel of T_{0} is the same as the kernel of T. It is well known that T_{0}(k^{n}) = k^{r_{0}} (16.3.1)

under these conditions, and in particular that r_{0} ≤ n. The dimension of the kernel of T_{0} is n − r_{0}, as a vector space over k.

Let k_{1} be a field that contains k as a subfield. Thus k_{1}^{n}, k_{1}^{r}, and k_{1}^{r_{0}} may be considered as vector spaces over k_{1}, which contain k^{n}, k^{r}, and k^{r_{0}} as subsets, respectively. Let T_{k_{1}} be the natural extension of T to a mapping from k_{1}^{n} into k_{1}^{r} that is linear over k_{1}, and let λ_{l,k_{1}} be the natural extension of λ_{l} to a linear functional on k_{1}^{n} for each l = 1, …, r. Thus λ_{l,k_{1}} corresponds to the lth coordinate of T_{k_{1}} for each l = 1, …, r, as before. If l > r_{0}, then λ_{l,k_{1}} can be expressed as a linear combination of λ_{1,k_{1}}, …, λ_{r_{0},k_{1}} with coefficients in k, because of the analogous property of λ_{l}.

Let T_{0,k_{1}} be the natural extension of T_{0} to a mapping from k_{1}^{n} into k_{1}^{r_{0}} that is linear over k_{1}. This corresponds to λ_{1,k_{1}}, …, λ_{r_{0},k_{1}} in the usual way. The kernel of T_{0,k_{1}} in k_{1}^{n} is the same as the kernel of T_{k_{1}}, because of the property of λ_{l,k_{1}} when l > r_{0} mentioned in the preceding paragraph. It is easy to see that

T_{0,k_{1}}(k_{1}^{n}) = k_{1}^{r_{0}}, (16.3.2)

using (16.3.1). This implies that λ_{1,k_{1}}, …, λ_{r_{0},k_{1}} are linearly independent as linear functionals on k_{1}^{n}, and that the dimension of the kernel of T_{0,k_{1}} in k_{1}^{n} is n − r_{0}, as a vector space over k_{1}.

16.4 Homomorphisms and dimensions

Let k be a field, and let (A, [·, ·]) be a Lie algebra over k, with positive finite dimension n as a vector space over k. We may as well take A = k^{n}, by choosing a basis for A as a vector space over k. As in Section 9.14, the Lie bracket on A can be given by

([x, y])_{r} = ∑_{j=1}^{n} ∑_{l=1}^{n} c_{j,l}^{r} x_{j} y_{l} (16.4.1)

for every x, y ∈ k^{n}, where the left side is the rth coordinate of [x, y] as an element of k^{n}, and the structure constants c_{j,l}^{r} are elements of k for each j, l, r = 1, …, n. More precisely, the structure constants satisfy (9.14.5), (9.14.7), and (9.14.8), as before.

Let u_{1}, …, u_{n} be the standard basis vectors in k^{n}, so that the jth coordinate of u_{l} is equal to 1 when j = l, and to 0 when j ≠ l. Also let ϕ be a linear mapping from k^{n} into itself, as a vector space over k. It is easy to see that ϕ is a homomorphism from A into itself, as a module over itself with respect to the adjoint representation, if and only if

ϕ([u_{j}, u_{l}]) = [u_{j}, ϕ(u_{l})] (16.4.2)

for every j, l = 1, …, n. Of course, the space L(A) of linear mappings from A into itself, as a vector space over k, corresponds to the space M_{n}(k) of n × n matrices with entries in k in the usual way. The space L^{ad}(A) of homomorphisms from A into itself, as a module over itself with respect to the adjoint representation, corresponds to a linear subspace of M_{n}(k) that can be described in terms of linear equations for the matrix entries, using (16.4.2).

Let k_{1} be a field that contains k as a subfield, so that k^{n} ⊆ k_{1}^{n}. Let us take A_{k_{1}} to be k_{1}^{n}, as a vector space over k_{1} with respect to coordinatewise addition and scalar multiplication. If x, y ∈ A_{k_{1}}, then [x, y] can be defined as an element of A_{k_{1}} as in (16.4.1), where the right side is now an element of k_{1} for each r = 1, …, n. This makes A_{k_{1}} a Lie algebra over k_{1}, as before.

Of course, u_{1}, …, u_{n} may be considered as the standard basis vectors in k_{1}^{n} as well. Let ϕ be a linear mapping from k_{1}^{n} into itself, as a vector space over k_{1}. As before, ϕ is a homomorphism from A_{k_{1}} into itself, as a module over itself with respect to the adjoint representation, if and only if (16.4.2) holds for every j, l = 1, …, n. As usual, the space L(A_{k_{1}}) of linear mappings from A_{k_{1}} into itself, as a vector space over K, corresponds to the space M_{n}(k_{1}) of n×n matrices with netries in k_{1}. The space L^{ad}(A_{k_{1}}) of homomorphisms from A_{k_{1}} into itself, as a module over itself with respect to the adjoint representation, corresponds to a linear subspace of M_{n}(k_{1}) that can be described in terms of essentially the same linear equations for the matrix entries as for k, using (16.4.2) again.

This brings us to the same type of situation as discussed in the previous section. It follows that the dimension of L^{ad}(A_{k_{1}}), as a vector space over k_{1}, is the same as the dimension of L^{ad}(A), as a vector space over k. This is related to part of part (b) of Exercise 5 on p55 of [25].

16.5 Semisimplicity and dimensions

Let k be a field of characteristic 0, and let (A, [·, ·]) be a Lie algebra over k of positive finite dimension n. As in the previous section, we may as well take A = k^{n}, with Lie bracket as in (16.4.1). Let k_{1} be an algebraically closed field that contains k as a subfield. As before, A_{k_{1}} = k_{1}^{n} is a Lie algebra over k_{1}, with Lie bracket as in (16.4.1). In this section, we suppose that A is semisimple as a Lie algebra over k, which is equivalent to asking that A_{k_{1}} be semisimple as a Lie algebra over k_{1}, as in Section 11.5.

As in Section 10.15, A_{k_{1}} is isomorphic to the direct sum of h simple Lie algebras over k_{1}, for some positive integer h. Remember that the space L^{ad}(A_{k_{1}}) of module homomorphisms from A_{k_{1}} into itself, as a module over itself with respect to the adjoint representation, is an associative algebra over k_{1} with respect to composition of mappings. In this situation, L^{ad}(A_{k_{1}}) is isomorphic to the direct sum of h copies of k_{1}, as in Section 16.2. In particular, the dimension of L^{ad}(A_{k_{1}}) is h, as a vector space over k_{1}. We also get that L^{ad}(A_{k_{1}}) is commutative as an algebra over k_{1}.

Similarly, the space L^{ad}(A) of module homomorphisms from A into itself, as a module over itself with respect to the adjoint representation, is an associative algebra over k with respect to composition of mappings. As in the previous section, the dimension of L^{ad}(A), as a vector space over k, is the same as the dimension of L^{ad}(A_{k_{1}}) as a vector space over k_{1}, which is equal to h. This corresponds to the first part of part (b) of Exercise 5 on p55 of [25].

If ϕ is a linear mapping from A into itself, as a vector space over k, then ϕ has a natural extension to a linear mapping from A_{k_{1}} into itself, as a vector space over k_{1}. If ϕ is a homomorphism from A into itself, as a module over itself with respect to the adjoint representation, then the extension of ϕ to A_{k_{1}} is a homomorphism from A_{k_{1}} into itself, as a module over itself with respect to the adjoint representation. This follows from the characterization of module homomorphisms as linear mappings that satisfy (16.4.2) in both cases. Using this, we get that L^{ad}(A) is commutative as an algebra over k with respect to composition of mappings, because L^{ad}(A_{k_{1}}) is commutative, as before. This is related to the second part of part (b) of Exercise 5 on p55 of [25].

Suppose that A is simple as a Lie algebra over k, which implies that A is irreducible as a module over itself, with respect to the adjoint representation. In this case, nonzero elements of L^{ad}(A) are invertible in L^{ad}(A), by Schur’s lemma, as in Section 16.1. This means that L^{ad}(A) is a field, because L^{ad}(A) is commutative, as in the preceding paragraph. This corresponds to the second part of part (b) of Exercise 5 on p55 of [25], with m = 1. Otherwise, A is isomorphic to the direct sum of simple Lie algebras A_{1}, …, A_{m} over k for some positive integer m, as in Section 10.15. This implies that L^{ad}(A) is isomorphic to the direct sum of L^{ad}(A_{1}), …, L^{ad}(A_{m}) as an associative algebra over k, as in Section 16.2. It follows that L^{ad}(A) is isomorphic to the direct sum of m fields, as in the second part of part (b) of Exercise 5 on p55 of [25].

16.6 Absolutely simple Lie algebras

Let k be a field, and let (A, [·, ·]) be a Lie algebra over k of positive finite dimension n, as a vector space over k. As before, we may as well take A to be k^{n}, with Lie bracket as in (16.4.1). Let k_{1} be an algebraically closed field that contains k as a subfield, and take A_{k_{1}} = k_{1}^{n} as a Lie algebra over k_{1}, with Lie bracket defined as in (16.4.1). If A_{k_{1}} is simple as a Lie algebra over k_{1}, then A is said to be absolutely simple as a Lie algebra over k, as in part (c) of Exercise 5 on p55 of [25]. This implies that A is simple as a Lie algebra over k, as in Section 11.4.

Suppose that A is absolutely simple, so that A_{k_{1}} is simple as a Lie algebra over k_{1}. This implies that A_{k_{1}} is irreducible as a module over itself with respect to the adjoint representation. Remember that L^{ad}(A_{k_{1}}) is the algebra of homomorphisms from A_{k_{1}} into itself, as a module over itself with respect to the adjoint representation. Under these conditions, L^{ad}(A_{k_{1}}) consists exactly of multiples of the identity mapping on A_{k_{1}} by elements of k_{1}, by Schur’s lemma, as in Section 16.1. Equivalently, this means that the dimension of L^{ad}(A_{k_{1}}), as a vector space over k_{1}, is equal to one. It follows that the space L^{ad}(A) of homomorphisms from A into itself, as a module over itself with respect to the adjoint representation, is equal to one as a vector space over k, as in Section 16.4. Of course, L^{ad}(A) automatically contains all multiples of the identity mapping on A by elements of k. This means that L^{ad}(A) consists exactly of multiples of the identity mapping on A by elements of k in this situation. This corresponds to part of the first part of part (c) of Exercise 5 on p55 of [25].

Conversely, suppose that L^{ad}(A) consists exactly of multiples of the identity mapping on A by elements of k. This is the same as saying that L^{ad}(A) has dimension one as a vector space over k, as before. It follows that L^{ad}(A_{k_{1}}) has dimension one as a vector space over k_{1}, as in Section 16.4. Suppose that k has characteristic 0, so that k_{1} has characteristic 0 too. If A_{k_{1}} is semisimple as a Lie algebra over k_{1}, then A_{k_{1}} is isomorphic to the direct sum of h simple Lie algebras over k_{1} for some positive integer h, as in Section 10.15. This implies that the dimension of L^{ad}(A_{k_{1}}) is equal to h as a vector space over k_{1}, as in the previous section. Under these conditions, we get that h = 1, so that A_{k_{1}} is simple as a Lie algebra over k_{1}. This corresponds to the other part of the first part of part (c) of Exercise 5 on p55 of [25].

Suppose now that A is simple as a Lie algebra over k, and that k has characteristic 0. In particular, this means that A is semisimple as a Lie algebra over k. Put K = L^{ad}(A), which is a field in this situation, as in the previous section. By construction, the elements of K are linear mappings from A into itself, as a vector space over k. We may consider A as a vector space over K, where scalar multiplication by an element of K is defined by the corresponding linear mapping on A. It is easy to see that the Lie bracket on A is bilinear over K, as a mapping from A × A into A, because the elements of K are homomorphisms from A into itself, as a module over itself as a Lie algebra over k, with respect to the adjoint representation. Thus A may be considered as a Lie algebra over K, with respect to the Lie bracket already defined on A as a Lie algebra over k.

Let us use A_{K} to refer to A as a Lie algebra over K in this way. As before, L^{ad}(A_{K}) denotes the space of linear mappings from A_{K} into itself, as a vector space over K, that are homomorphisms from A_{K} into itself as a module over itself, as a Lie algebra over K, and with respect to the adjoint representation.

Of course, L^{ad}(A_{K}) contains the multiples of the identity mapping on A by elements of K, which correspond to linear mappings from A into itself as a vector space over k. Conversely, if ϕ is any element of L^{ad}(A_{K}), then ϕ is linear as a mapping from A into itself, as a vector space over k. This follows from the fact that K contains the multiples of the identity mapping on A by elements of k, by construction. We also have that ϕ is a homomorphism from A into itself, as a module over itself as a Lie algebra over k, and with respect to the adjoint representation, because of the analogous property of ϕ as a mapping on A_{K}. This means that ϕ is an element of L^{ad}(A) = K. Thus L^{ad}(A_{K}) consists exactly of multiples of the identity mapping on A by elements of K. This implies that A_{K} is absolutely simple as a Lie algebra over K, as before. This is the second part of part (c) of Exercise 5 on p55 of [25].

16.7 Algebras over subrings

Let k be a commutative ring with a multiplicative identity element, and let k_{0} be a subring of k that contains the multiplicative identity element. If A is a module over k, then A may be considered as a module over k_{0} as well. Let A_{0} be A considered as a module over k_{0}. If B is a submodule of A, as a module over k, then B may be considered as a submodule of A_{0} too.

Similarly, if A is an algebra over k in the strict sense, then A may be considered as an algebra over k_{0} in the strict sense. Let A_{0} be A considered as an algebra over k_{0} in the strict sense. If B is a subalgebra of A, as an algebra over k in the strict sense, then B may be considered as a subalgebra of A_{0}, as an algebra over k_{0} in the strict sense. If B is a one or two-sided ideal in A, as an algebra over k in the strict sense, then B has the analogous property in A_{0} as well.

Now let (A, [·, ·]) be a Lie algebra over k. As before, let A_{0} be A, considered as a Lie algebra over k_{0}. It is easy to see that A is commutative, solvable, or nilpotent as a Lie algebra over k if and only if A_{0} has the same property as a Lie algebra over k_{0}.

Let B be an ideal in A, as a Lie algebra over k. Thus B may be considered as an ideal in A_{0}, as a Lie algebra over k_{0}. Let B_{0} be B, considered as an ideal in A_{0}, and as a Lie algebra over k_{0} in particular. If B is solvable, as a Lie algebra over k, then B_{0} is solvable, as a Lie algebra over k_{0}, as in the preceding paragraph. If A_{0} is semisimple as a Lie algebra over k_{0}, then it follows that A is semisimple as a Lie algebra over k.

Of course, k may be considered as a module over k_{0}. Let us suppose from now on in this section that k is free as a module over k_{0}, of rank n_{0} for some positive integer n_{0}. This means that k is isomorphic to k_{0}^{n_{0}} as a module over k_{0}, where k_{0}^{n_{0}} is the space of n_{0}-tuples of elements of k_{0}, considered as a module over k_{0} with respect to pointwise addition and scalar multiplication.

If x ∈ k, then M_{x}(y) = x y defines a homomorphism from k into itself, as a module over k_{0}. Let tr_{k/k_{0}} x be the trace of M_{x}, as a homomorphism from k into itself, as a free module over k_{0} of rank n_{0}. This defines a homomorphism from k into k_{0}, as modules over k_{0}. This uses the fact that x ↦ M_{x} is linear over k_{0}, as a mapping from k into the space homomorphisms from k into itself, as a module over k_{0}. If x ∈ k_{0}, then

tr_{k/k_{0}}x = n_{0} · x. (16.7.1)

Let n be a positive integer, so that V = k^{n} is a free module over k with respect to coordinatewise additional and scalar multiplication. We may also consider V as a free module over k_{0} of rank n_{0} n, because k is a free module over k_{0} of rank n_{0}. If T is a homomorphism from V into itself as a module over k, then T may be considered as a homomorphism from V into itself as a module over k_{0} as well. Let tr_{V,k} T ∈ k be the trace of T as a homomorphism from V into itself as a free module over k of rank n, and let tr_{V,k_{0}} T ∈ k_{0} be the trace of T as a homomorphism from V into itself as a free module over k_{0} of rank n_{0} n. One can check that tr_{V,k_{0}}T = tr_{k/k_{0}}(tr_{V,k}T). (16.7.2)

16.8 Lie algebras over subfields

Let k be a field, and let k_{0} be a subfield of k. Also let (A, [·, ·]) be a Lie algebra over k, and let A_{0} be A considered as a Lie algebra over k_{0}. Suppose that k has finite dimension n_{0} as a vector space over k_{0}, and that A has positive finite dimension n as a vector space over k. Thus A_{0} has dimension n_{0} n as a vector space over k_{0}. If x ∈ A, then put ad_{x}(z) = [x, z] for every z ∈ A, as usual. This defines ad_{x} as a linear mapping from A into itself, as a vector space over k. Of course, ad_{x} may be considered as a linear mapping from A_{0} into itself as well, as a vector space over k_{0}.

The Killing form for A, as a Lie algebra over k, is defined by

b(x, y) = tr_{A}(ad_{x} ∘ ad_{y}) ∈ k (16.8.1)

for every x, y ∈ A, using the trace of ad_{x} ∘ ad_{y} on A as a vector space over k. Similarly, the Killing form for A_{0} is defined by

b_{0}(x, y) = tr_{A_{0}}(ad_{x} ∘ ad_{y}) ∈ k_{0} (16.8.2)

for every x, y ∈ A_{0}, using the trace of ad_{x} ∘ ad_{y} on A_{0} as a vector space over k. Let tr_{k/k_{0}} be the trace mapping from k into k_{0} mentioned in the previous section. If x, y ∈ A, then

b_{0}(x, y) = tr_{k/k_{0}}b(x, y), (16.8.3)

as in (16.7.2).

Suppose that b(·, ·) is nondegenerate on A. Let x ∈ A with x ≠ 0 be given, so that there is a y ∈ A such that b(x, y) ≠ 0. Put y_{1} = (1/b(x, y)) y, so that

b(x, y_{1}) = 1. (16.8.4)

This implies that b_{0}(x, y_{1}) = tr_{k/k_{0}}1 = n_{0} · 1, (16.8.5)

using (16.8.3) in the first step, and (16.7.1) in the second step.

Suppose from now on in this section that k_{0} has characteristic 0, which is the same as saying that k has characteristic 0. In this case, (16.8.5) implies that b_{0}(x, y_{1}) ≠ 0 in k_{0}. This means that b_{0}(·, ·) is nondegenerate on A_{0} in this situation.

Suppose that A is semisimple as a Lie algebra over k, so that b(·, ·) is nondegenerate on A, as in Section 10.13. This implies that b_{0}(·, ·) is nondegenerate on A_{0}, as in the preceding paragraphs. It follows that A_{0} is semisimple as a Lie algebra over k_{0}, as in Section 10.13 again.

As in Section 10.15, there are finitely many ideals A_{0,1}, …, A_{0,r} in A_{0}, as a Lie algebra over k_{0}, such that A_{0,j} is simple as a Lie algebra over k_{0} for each j = 1, …, r, and A_{0} corresponds to the direct sum of A_{0,1}, …, A_{0,r} as a Lie algebra over k_{0}. If t ∈ k, then

ϕ_{t}(x) = t x (16.8.6)

defines a linear mapping from A_{0} into itself as a vector space over k_{0}, and in fact a homomorphism from A_{0} into itself as a module over itself, with respect to the adjoint representation. It follows that

ϕ_{t}(A_{0,j}) ⊆ A_{0,j} (16.8.7)

for each j = 1, …, r, as in Section 16.2. This means that A_{0,j} is a linear subspace of A, as a vector space over k, for each j = 1, …, r. Thus A_{0,j} is an ideal in A, as a Lie algebra over k, for each j = 1, …, r.

In particular, A_{0,j} is a Lie subalgebra of A, as a Lie algebra over k, for each j = 1, …, r. It is easy to see that A_{0,j} is simple as a Lie algebra over k for each j = 1, …, r, because A_{0,j} is simple as a Lie algebra over k_{0}.

If A is simple as a Lie algebra over k, then A is semisimple. In this case, we get that r = 1 in the previous argument, so that A_{0} = A_{0,1} is simple as a Lie algebra over k_{0}. This corresponds to part (d) of Exercise 5 on p55 of [25].

Part III

Toral subalgebras and roots Subalgebras and diagonalizability

17.1 Toral subalgebras

Let k be a field, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. Suppose that y, z are ad-diagonalizable elements of A such that

[y, z]_{A} = 0. (17.1.1)

This implies that ad_{y} and ad_{z} commute as linear mappings from A into itself, as in Section 2.4. It follows that ad_{y} + ad_{z} is diagonalizable as a linear mapping from A into itself, as in Section 10.6. This means that y +z is ad-diagonalizable, as an element of A.

If every element of A is ad-nilpotent, then A is nilpotent as a Lie algebra over k, as in Section 9.10. Suppose that k is an algebraically closed field of characteristic 0, and that A is semisimple as a Lie algebra over k. If A ≠ {0}, then A is not nilpotent, and hence there is an x ∈ A that is not ad-nilpotent. Using the abstract Jordan decomposition, as in Section 14.3, we get that there are x_{1}, x_{2} ∈ A such that x = x_{1} + x_{2}, x_{1} is ad-diagonalizable, and x_{2} is adnilpotent. In particular, x_{1} ≠ 0, because x is not ad-nilpotent.

A Lie subalgebra B of A is said to be toral if every element of B is addiagonalizable, as an element of A, at least when k is algebraically closed, as on p35 of [14]. Under the conditions mentioned in the preceding paragraph, there are nonzero toral subalgebras of A, as on p35 of [14]. More precisely, one can take the linear span of a nonzero ad-diagonalizable element of A.

Let k be any field again, and let A be a finite-dimensional Lie algebra over k. If B is a Lie subalgebra of A, and if every element of B is ad-diagonalizable as an element of A, then B is commutative as a Lie algebra over k, as in the lemma on p35 of [14]. To see this, let x ∈ B be given, and note that ad_{x} maps B into itself, because B is a Lie subalgebra of A. The restriction of ad_{x} to B is the same as ad_{B,x}. We would like to show that ad_{B,x} = 0.

Of course, ad_{x} is diagonalizable as a linear mapping from A into itself, by hypothesis. This implies that ad_{B,x} is diagonalizable as a linear mapping from B into itself, as in Section 10.6. Let λ ∈ k be an eigenvalue of ad_{B,x}, so that there is a y ∈ B such that y ≠ 0 and

ad_{B,x}(y) = [x, y]_{A} = λ y. (17.1.2)

We would like to show that λ = 0.

Equivalently, ad_{B,y}(x) = −λ y, (17.1.3)

by (17.1.2). Remember that y is ad-diagonalizable as an element of A, by hypothesis. This implies that ad_{B,y} is diagonalizable as a linear mapping from B into itself, as before. Thus x can be expressed as a sum of eigenvectors of ad_{B,y}. It follows that ad_{B,y}(x) is a sum of eigenvectors of ad_{B,y} corresponding to nonzero eigenvalues of ad_{B,y}, if there are any.

If ad_{B,y}(x) ≠ 0, then the previous statement implies that ad_{B,y}(ad_{B,y}(x)) ≠ 0. However,

ad_{B,y}(ad_{B,y}(x)) = −λ ad_{B,y}(y) = −λ [y, y]_{A} = 0, (17.1.4)

using (17.1.3) in the first step. Thus ad_{B,y}(x) = 0, so that λ = 0, by (17.1.3). This means that ad_{B,x} = 0, because ad_{B,x} is diagonalizable on B, and 0 is its only eigenvalue. This shows that B is commutative as a Lie algebra, as desired.

Note that we only used the diagonalizability of ad_{B,z} on B for each z ∈ B, rather than the diagonalizability of ad_{z} on A. This also corresponds to simply taking A = B.

17.2 Simultaneous diagonalizability

Let k be a field, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. Also let B be a Lie subalgebra of A, and suppose that every element of B is ad-diagonalizable as an element of A. Thus B is commutative as a Lie algebra over k, as in the previous section. Let B′ be the dual of B, as a vector space over k. Remember that B′ has the same dimension as B.

Suppose that x ∈ A is an eigenvector for ad_{w} for each w ∈ B, so that

ad_{w}(x) = [w, x]_{A} = α(w) x (17.2.1)

for some α(w) ∈ k. If x ≠ 0, then α(w) is linear in w, and hence defines an element of B′.

If α ∈ B′, then put

(17.2.2) A_{α} = {x ∈ A : for each w ∈ B, ad_{w}(x) = [w, x]_{A} = α(w) x}.

This is a linear subspace of A, as a vector space over k. An element of A_{α} is said to have weight α, as on p43 of [24].

In particular, if we take α = 0, then we get

(17.2.3) A_{0} = {x ∈ A : for each w ∈ B, ad_{w}(x) = [w, x]_{A} = 0}.

This is the same as the centralizer C_{A}(B) of B in A, as in Section 7.6. Note that B ⊆ A_{0} = C_{A}(B), (17.2.4)

because B is commutative as a Lie algebra, as before.

Let us suppose that B ≠ {0}, to avoid trivialities. This implies that A_{0} ≠ {0}, by (17.2.4). Put

Φ_{B} = {α ∈ B′ : α ≠ 0 and A_{α} ≠ {0}}. (17.2.5)

Thus Φ_{B} ∪ {0} is the same as the set of α ∈ B′ such that A_{α} ≠ {0}.

If u, v ∈ B, then [u, v]_{A} = 0, as before. This implies that ad_{u} and ad_{v} commute as linear mappings from A into itself, as in Section 2.4. By hypothesis, ad_{u} is diagonalizable as a linear mapping from A into itself for each u ∈ B. It follows that the linear mappings ad_{u}, u ∈ B, are simultaneously diagonalizable on A, by standard arguments.

This means that A corresponds to the direct sum of the subspaces A_{α} with α ∈ Φ_{B} ∪ {0}, as a vector space over k. In particular, the number of elements of Φ_{B} is strictly less than the dimension of A. This corresponds to some of the remarks on p35 of [14], and to Theorem 1 on p43 of [24].

Let α, β ∈ B′ be given, and suppose that x ∈ A_{α}, y ∈ A_{β}. If w ∈ B, then

(17.2.6)  [w, [x, y]_{A}]_{A} = [[w, x]_{A}, y]_{A} + [x, [w, y]_{A}]_{A}
= α(w) [x, y]_{A} + β(w) [x, y]_{A}.

This uses the Jacobi identity in the first step, or, equivalently, the fact that ad_{w} is a derivation on A. Thus [x, y]_{A} ∈ A_{α+β} (17.2.7)

under these conditions. This corresponds to the first part of the proposition near the top of p36 in [14], and to the statement 2.1 on p45 of [24].

If α ∈ B′ and x ∈ A_{α}, then

ad_{x}(A_{γ}) ⊆ A_{α+γ} (17.2.8)

for every γ ∈ B′, by (17.2.7). This implies that

(ad_{x})^{n}(A_{γ}) ⊆ A_{n·α+γ} (17.2.9)

for every positive integer n, where (ad_{x})^{n} is the nth power of ad_{x} on A, with respect to composition of mappings. If α ≠ 0 and k has characteristic 0, then one can use this to get that x is ad-nilpotent as an element of A, as in the second part of the proposition at the top of p36 in [14]. This could also be obtained as in Section 14.2.

Similarly, if x ∈ A_{α} and y ∈ A_{β} for some α, β ∈ B′, then

(17.2.10) (ad_{x} ∘ ad_{y})(A_{γ}) = ad_{x}(ad_{y}(A_{γ})) ⊆ A_{α+β+γ}

for every γ ∈ B′, by (17.2.8).

17.3 Related bilinear forms

Let us continue with the same notation and hypotheses as in the previous section. Let b(·, ·) be a bilinear form on A that is associative, or equivalently invariant with respect to the adjoint representation on A. Thus

b([w, x]_{A}, y) = −b(x, [w, y]_{A}) (17.3.1)

for every w, x, y ∈ A, as in Sections 6.10 and 7.7. In particular, the Killing form

b_{A}(x, y) = tr_{A}(ad_{x} ∘ ad_{y}) (17.3.2)

on A has this property, as in Section 7.9.

Let α, β ∈ B′, x ∈ A_{α}, y ∈ A_{β}, and w ∈ B be given. Using (17.3.1), we get that α(w) b(x, y) = −β(w) b(x, y). (17.3.3) If α + β ≠ 0, then there is a w ∈ B such that α(w) ≠ −β(w), and hence

b(x, y) = 0. (17.3.4)

This corresponds to the third part of the proposition on the top of p36 in [14], and the first part of Theorem 3 (i) on p44 of [24]. Alternatively, if b(·, ·) is the Killing form (17.3.2) on A, then (17.3.4) follows from (17.2.10).

Let α ∈ B′ and w ∈ B be given again. If x ∈ A_{α} and y ∈ A, then

b(w, [x, y]_{A}) = b([w, x]_{A}, y) = α(w) b(x, y). (17.3.5)

Similarly, if x ∈ A and y ∈ A_{−α}, then

b([x, y]_{A}, w) = b(x, [y, w]_{A}) = α(w) b(x, y). (17.3.6)

This corresponds to Theorem 3 (ii) on p44 of [24], and is related to part (c) of the proposition on p37 of [14]. Note that

[x, y]_{A} ∈ A_{0} (17.3.7)

when x ∈ A_{α} and y ∈ A_{−α}, as in (17.2.7).

Suppose now that b(·, ·) is also nondegenerate on A. In this case, the restriction of b(·, ·) to A_{0} is nondegenerate on A_{0}, because of (17.3.4). This corresponds to the corollary on p36 of [14], and to the third part of Theorem 3 (i) on p44 of [24].

Let α ∈ B′ with α ≠ 0 be given. If x ∈ A_{α} and x ≠ 0, then there is a y ∈ A such that b(x, y) ≠ 0, (17.3.8) because b(·, ·) is nondegenerate on A. More precisely, we can take y ∈ A_{−α}, because of (17.3.4). In particular, this shows that A_{−α} ≠ {0} when A_{α} ≠ {0}. Equivalently, if α ∈ Φ_{B}, then −α ∈ Φ_{B}. This corresponds to part (b) of the proposition on p37 of [14], and part of Theorem 2 (a) on p43 of [24]. Similarly, the restriction of b(·, ·) to A_{α} + A_{−α} (17.3.9) is nondegenerate. This corresponds to the second part of Theorem 3 (i) on p44 of [24].

17.4 Maximal toral subalgebras

Let k be a field, and let V be a finite-dimensional vector space over k. Also let R and T be commuting linear mappings from V into itself. If T is nilpotent on V, then R ∘ T is nilpotent on V as well. In particular, this implies that

tr_{V} (R ∘ T) = 0. (17.4.1)

This is the lemma on p36 of [14].

Let k be an algebraically closed field of characteristic 0, and let (A, [·, ·]_{A}) be a finite-dimensional semisimple Lie algebra over k. Suppose that B is a toral subalgebra of A, as in Section 17.1, and that B is maximal with respect to inclusion. Under these conditions,

C_{A}(B) = B, (17.4.2)

where C_{A}(B) is the centralizer of B in A, as in Section 7.6. This is the second proposition on p36 of [14].

Remember that C_{A}(B) is a Lie subalgebra of A, as in Section 7.6. In this situation, B ⊆ C_{A}(B), because B is commutative as a Lie algebra, as in (17.2.4). Thus we would like to show that

C_{A}(B) ⊆ B. (17.4.3)

Let x ∈ C_{A}(B) be given, and let

x = x_{1} + x_{2} (17.4.4)

be the abstract Jordan decomposition of x in A, as in Section 14.3. Thus x_{1}, x_{2} ∈ A, ad_{x} = ad_{x_{1}} + ad_{x_{2}}, (17.4.5)

and ad_{x_{1}}, ad_{x_{2}} are the diagonalizable and nilpotent parts of ad_{x}, as a linear mapping from A into itself, as in Section 10.8. The condition that x ∈ C_{A}(B) means exactly that ad_{x} maps B into {0}. This implies that ad_{x_{1}} and ad_{x_{2}} map B into {0} as well, as in Section 10.8. It follows that

x_{1}, x_{2} ∈ C_{A}(B), (17.4.6)

as in Step (1) of the proof on p36 of [14].

If x ∈ C_{A}(B), then the linear span

B(x) = {w + t x : w ∈ B, t ∈ k} (17.4.7)

of B and x in A is a Lie subalgebra of A that is commutative as a Lie algebra. More precisely, this uses the fact that B is commutative as a Lie algebra, and that [w, x]_{A} = 0 for every w ∈ B. If x is ad-diagonalizable as an element of A too, then B(x) is a toral subalgebra of A, by the remark at the beginning of Section 17.1. This implies that B(x) = B, because B is supposed to be maximal in A, so that x ∈ B. (17.4.8)

This is Step (2) of the proof on p36 of [14].

Let b_{A}(·, ·) be the Killing form on A, as in (17.3.2). Remember that b_{A}(·, ·) is nondegenerate on A, because A is semisimple and k has characteristic 0, as in Section 10.13. This implies that the restriction of b_{A}(·, ·) to A_{0} = C_{A}(B) is nondegenerate, as in the previous section. Step (3) of the proof on p36 of [14] states that the restriction of b_{A}(·, ·) to B is nondegenerate.

To see this, let x ∈ B be given, and suppose that

b_{A}(x, y) = 0 (17.4.9)

for every y ∈ B. We would like to show that x = 0. If (17.4.9) holds for every y ∈ C_{A}(B), then x = 0, because b_{A}(·, ·) is nondegenerate on C_{A}(B), as in the preceding paragraph.

Let z ∈ C_{A}(B) be given, and so that [x, z]_{A} = 0, because x ∈ B. This implies that ad_{x} and ad_{z} commute as linear mappings from A into itself, as in Section 2.4. If ad_{z} is nilpotent on A, then it follows that

b_{A}(x, z) = tr_{A}(ad_{x} ∘ ad_{z}) = 0, (17.4.10)

as in (17.4.1).

Let y ∈ C_{A}(B) be given, and let y = y_{1} + y_{2} be the abstract Jordan decomposition of y. Thus y_{1}, y_{2} ∈ C_{A}(B), as before. We also have that y_{1} ∈ B, because y_{1} is ad-diagonalizable as an element of A. It follows that b_{A}(x, y_{1}) = 0, by hypothesis. Observe that b_{A}(x, y_{2}) = 0, because y_{2} is ad-nilpotent as an element of A, as in (17.4.10). Combining thise two staements, we get that (17.4.9) holds. This implies that x = 0, because y is an arbitrary element of C_{A}(B), as desired.

Step (4) of the proof on p36 of [14] states that C_{A}(B) is nilpotent as a Lie algebra over k. Let x ∈ C_{A}(B) be given. We would like to show that x is ad-nilpotent as an element of C_{A}(B), which is to say that ad_{C_{A}(B),x} is nilpotent as a linear mapping from C_{A}(B) into itself. Let (17.4.4) be the abstract Jordan decomposition of x in A again. Thus x_{1} ∈ B, by the first two steps, which implies that ad_{C_{A}(B),x_{1}} = 0. We also have that ad_{C_{A}(B),x_{2}} is nilpotent on C_{A}(B), because ad_{x_{2}} is nilpotent on A by construction, and ad_{C_{A}(B),x_{2}} is the same as the restriction of ad_{x_{2}} to C_{A}(B). This implies that

ad_{C_{A}(B),x} = ad_{C_{A}(B),x_{2}} (17.4.11)

is nilpotent on C_{A}(B). It follows that C_{A}(B) is nilpotent as a Lie algebra over k, as in Section 9.10.

Step (5) of the continuation of the proof on p37 of [14] states that

B ∩ ([C_{A}(B), C_{A}(B)]) = {0}. (17.4.12)

To see this, observe that

b_{A}([x_{1}, x_{2}]_{A}, y) = b_{A}(x_{1}, [x_{2}, y]_{A}) = 0 (17.4.13)

for every x_{1}, x_{2} ∈ C_{A}(B) and y ∈ B. This uses the associativity of b_{A}(·, ·) on A in the first step, and the fact that [x_{2}, y]_{A} = 0 in the second step. It follows that b_{A}(x, y) = 0 for every x ∈ [C_{A}(B), C_{A}(B)] and y ∈ B. If x ∈ B as well, then we get that x = 0, because b_{A}(·, ·) is nondegenerate on B.

Step (6) of the proof on p37 of [14] states that C_{A}(B) is commutative as a Lie algebra over k. Suppose for the sake of a contradiction that

[C_{A}(B), C_{A}(B)] ≠ {0}. (17.4.14)

This implies that

([C_{A}(B), C_{A}(B)]) ∩ Z(C_{A}(B)) ≠ {0}, (17.4.15)

because [C_{A}(B), C_{A}(B)] is an ideal in C_{A}(B), and C_{A}(B) is nilpotent as a Lie algebra, as in Section 9.10. Let z be an element of the left side of (17.4.15) with z ≠ 0. Note that z ∉ B, by the previous step.

It follows that z is not ad-diagonalizable as an element of A, as before. Let z = z_{1} + z_{2} be the abstract Jordan decomposition of z in A. Thus z_{2} is adnilpotent as an element of A, z_{2} ∈ C_{A}(B), as before, and z_{2} ≠ 0, because z is not ad-diagonalizable. Because z ∈ Z(C_{A}(B)), ad_{z} commutes with ad_{x} for every x ∈ C_{A}(B), as in Section 2.4. This implies that ad_{z_{2}} commutes with ad_{x} for every x ∈ C_{A}(B), as in Section 10.8. Hence

b_{A}(x, z_{2}) = tr_{A}(ad_{x} ∘ ad_{z_{2}}) = 0 (17.4.16)

for every x ∈ C_{A}(B), as in (17.4.1), because ad_{z_{2}} is nilpotent on A. This means that z_{2} = 0, because b_{A}(·, ·) is nondegenerate on C_{A}(B).

Step (7) of the proof on p37 of [14] states that (17.4.2) holds. Otherwise, there is an element x of C_{A}(B) not in B, and we can take x to be ad-nilpotent as an element of A. If y ∈ C_{A}(B), then [x, y]_{A} = 0, by the previous step, so that ad_{x} commutes with ad_{y} on A, as in Section 2.4. It follows that

b_{A}(x, y) = tr_{A}(ad_{x} ∘ ad_{y}) = 0, (17.4.17)

as in (17.4.1), because ad_{x} is nilpotent on A. This implies that x = 0, because b_{A}(·, ·) is nondegenerate on C_{A}(B).

If k = C, then Lie subalgebras B of A with the same types of properties are given in Theorem 3 on p15 of [24].

17.5 Self-centralizability and diagonalizability

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let B be a Lie subalgebra of A that is commutative as a Lie algebra. This implies that

B ⊆ C_{A}(B), (17.5.1) where C_{A}(B) is the centralizer of B in A, as in Section 7.6. If B_{1} is another Lie subalgebra of A that is commutative as a Lie algebra, and if B ⊆ B_{1}, then

B_{1} ⊆ C_{A}(B_{1}) ⊆ C_{A}(B). (17.5.2)

If C_{A}(B) = B, (17.5.3)

then it follows that B_{1} ⊆ B. This means that B is maximal as a commutative Lie subalgebra of A under these conditions. This corresponds to Corollary 1 on p15 of [24].

Conversely, let x ∈ C_{A}(B) be given, and let B(x) be the linear span of B and x in A, as in (17.4.7). This is a commutative Lie subalgebra of A that contains B, as in the previous section. If B is maximal as a commutative Lie subalgebra of A, then x ∈ B, and hence (17.5.3) holds.

Suppose now that k is a field, and that A is a finite-dimensional Lie algebra over k. Let B be a Lie subalgebra of A such that every element of B is addiagonalizable as an element of A. This implies that B is commutative as a Lie algebra, as in Section 17.1, so that (17.5.1) holds. If α is an element of the dual B′ of B, then we let A_{α} be the set of x ∈ A such that

ad_{w}(x) = [w, x]_{A} = α(w) x (17.5.4)

for every w ∈ B, as in Section 17.2. Let Φ_{B} be the set of α ∈ B′ such that α ≠ 0 and A_{α} ≠ {0}, as before.

Suppose that w ∈ B satisfies α(w) = 0 for every α ∈ Φ_{B}. This implies that [w, x]_{A} = 0 for every x ∈ A_{α} when α ∈ Φ_{B}, which holds automatically when α = 0. It follows that [w, x]_{A} = 0 for every x ∈ A, because A is spanned by the A_{α}’s with α ∈ Φ_{B} ∪ {0}. This means that w is an element of the center Z(A) of A as a Lie algebra. Of course, if Z(A) = {0}, then w = 0. In this case, we get that the linear span of Φ_{B} in B′ is equal to B′. This corresponds to part (a) of the proposition on p37 of [14], and to the statement 2.2 on p45 of [24].

Let b(·, ·) be a bilinear form on A that is associative, or equivalently invariant under the adjoint representation on A, as in Section 17.3. Suppose that b(·, ·) is nondegenerate on A, so that the restriction of b(·, ·) to A_{0} = C_{A}(B) is nondegenerate, as before. Let us suppose from now on in this section that (17.5.3) holds, which means that the restriction of b(·, ·) to B is nondegenerate. If α ∈ B′, then it follows that there is a unique t_{b,α} ∈ B such that

α(w) = b(w, t_{b,α}) (17.5.5)

for every w ∈ B.

Let α ∈ B′, x ∈ A_{α}, and y ∈ A_{−α} be given, so that

[x, y]_{A} ∈ A_{0} = C_{A}(B) = B. (17.5.6)

Remember that b(w, [x, y]_{A}) = α(w) b(x, y) (17.5.7)

17.6. CHARACTERISTIC 0, Z(A) = {0} 349

for every w ∈ B, as in Section 17.3. It follows that

[x, y]_{A} = b(x, y) t_{b,α}, (17.5.8)

because of (17.5.5) and the nondegeneracy of b(·, ·) on B. This corresponds to part (c) of the proposition on p37 of [14], and to Theorem 3 (iii) on p44 of [24].

Put B_{α} = [A_{α}, A_{−α}] (17.5.9)

for each α ∈ B′, using the notation in Section 9.2 on the right side. If α ∈ Φ_{B}, then there is an x ∈ A_{α} with x ≠ 0, and hence a y ∈ A_{−α} such that b(x, y) ≠ 0, as in Section 17.3. This implies that B_{α} is the same as the one-dimensional linear subspace of B spanned by t_{b,α}, because of (17.5.8). This corresponds to part (d) of the proposition on p37 in [24], and to the part of Theorem 2 (b) on p43 of [24] shown in statement 2.3 on p45 of [24].

17.6 Characteristic 0, Z(A) = {0}

Let us continue with the same notation and hypotheses as in the previous section. Let us also suppose in this section that k has characteristic 0, and that Z(A) = {0} again. Let α ∈ Φ_{B} be given, and let us show that

α(t_{b,α}) = b(t_{b,α}, t_{b,α}) ≠ 0. (17.6.1)

This corresponds to part (e) of the proposition on p37 of [14], and to the proof of statement 2.4 on p45 of [24].

Suppose for the sake of a contradiction that α(t_{b,α}) = 0. This implies that

[t_{b,α}, x]_{A} = α(t_{b,α}) x = 0 (17.6.2)

for every x ∈ A_{α}, and that

[t_{b,α}, y]_{A} = −α(t_{b,α}) y = 0 (17.6.3)

for every y ∈ A_{−α}. As before, there are x ∈ A_{α} and y ∈ A_{−α} such that b(x, y) ≠ 0, and we can choose them so that b(x, y) = 1. Thus

[x, y]_{A} = t_{b,α}, (17.6.4)

by (17.5.8).

Let C be the linear span of x, y, and t_{b,α} in A. This is a Lie subalgebra of A, which is nilpotent and hence solvable as a Lie algebra over k, by (17.6.2), (17.6.3), and (17.6.4). The restriction of the adjoint representation on A to C defines a representation of C, as a Lie algebra over k, on A, as a vector space over k. Because t_{b,α} ∈ [C, C], by (17.6.4), we get that ad_{t_{b,α}} is nilpotent as a linear mapping from A into itself, as in Section 14.14. However, ad_{t_{b,α}} is diagonalizable as a linear mapping from A into itself, because t_{b,α} ∈ B. It follows that ad_{t_{b,α}} = 0, so that t_{b,α} ∈ Z(A). This means that t_{b,α} = 0, because Z(A) = {0}, by hypothesis. This contradicts the fact that α ≠ 0, by definition of Φ_{B}.

Let α ∈ Φ_{B} be given again, and put

h_{α} = 2 b(t_{b,α}, t_{b,α})^{−1} t_{b,α} = 2 α(t_{b,α})^{−1} t_{b,α}. (17.6.5)

Thus α(h_{α}) = 2 α(t_{b,α})^{−1} α(t_{b,α}) = 2. (17.6.6)

Note that h_{α} is uniquely determined as an element of (17.5.9) by (17.6.6), as in Theorem 2 (b) on p43-4 of [24].

Let x_{α} be a nonzero element of A_{α}. As before, there is a y ∈ A_{−α} such that b(x_{α}, y) ≠ 0. More precisely, we can choose y_{α} ∈ A_{−α} such that

b(x_{α}, y_{α}) = 2 b(t_{b,α}, t_{b,α})^{−1}. (17.6.7)

This implies that [x_{α}, y_{α}]_{A} = h_{α}, (17.6.8)

by (17.5.8).

Observe that [h_{α}, x_{α}]_{A} = α(h_{α}) x_{α} = 2 x_{α}, (17.6.9)

using the facts that h_{α} ∈ B and x_{α} ∈ A_{α} in the first step. Similarly,

[h_{α}, y_{α}]_{A} = −α(h_{α}) y_{α} = −2 y_{α}, (17.6.10)

because y_{α} ∈ A_{−α}. This shows that the linear span of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A, which is isomorphic to sl_{2}(k) as a Lie algebra over k. More precisely, x_{α}, y_{α}, and h_{α} correspond to the usual basis elements \binom{0 1}{0 0}, \binom{0 0}{1 0}, and \binom{1 0}{0 −1} of sl_{2}(k) under this isomorphism, as in Section 10.2. This corresponds to part (f) of the proposition on p37 of [14], and to part of Theorem 2 (c) on p44 of [24], as in statement 2.5 on p45 of [24].

It is easy to see that t_{b,−α} = −t_{b,α}, (17.6.11)

by the definition (17.5.5) of t_{b,α}. Remember that −α ∈ Φ_{B}, because α ∈ Φ_{B}, as in Section 17.3. Using (17.6.11), we get that

h_{−α} = −h_{α}, (17.6.12)

as in part (g) of the proposition on p37 of [14].

17.7 The dimension of A_{α}

Let us continue with the situation considered in the previous two sections. Thus k is a field of characteristic 0, and (A, [·, ·]_{A}) is a finite-dimensional Lie algebra over k. Let B be a Lie subalgebra of A such that every element of B is addiagonalizable as an element of A, so that B is commutative as a Lie algebra, 17.7. THE DIMENSION OF A_{α} 351

as in Section 17.1. If α is an element of the dual B′ of B, then A_{α} is the set of x ∈ A such that ad_{w}(x) = [w, x]_{A} = α(w) x (17.7.1)

for every w ∈ B, as before. In particular, A_{0} is the same as the centralizer C_{A}(B) of B in A. We suppose here that this is equal to B, so that

A_{0} = C_{A}(B) = B, (17.7.2)

and that Z(A) = {0}. Remember that Φ_{B} is the set of α ∈ B′ such that α ≠ 0 and A_{α} ≠ 0.

Suppose that b(·, ·) is a nondegenerate bilinear form on A that is associative, or equivalently invariant under the adjoint representation on A. It follows that b(·, ·) is nondegenerate on (17.7.2), as before. If α ∈ B′, then we take t_{b,α} to be the unique element of B such that

α(w) = b(w, t_{b,α}) (17.7.3)

for every w ∈ B, as in Section 17.5.

Let α ∈ Φ_{B} be given, so that −α ∈ Φ_{B} too, and the restriction of b(·, ·) to A_{α} + A_{−α} is nondegenerate, as in Section 17.3. Remember that the restriction of b(·, ·) to each of A_{α} and A_{−α} is equal to 0. One can use this and the nondegeneracy of b(·, ·) on A_{α} + A_{−α} to get that the dimensions of A_{α} and A_{−α} are the same, as vector spaces over k.

Let h_{α} be as in (17.6.5), let x_{α} be a nonzero element of A_{α}, and let y_{α} ∈ A_{−α} be as in (17.6.7). Suppose for the sake of a contradiction that the dimension of A_{α} is strictly larger than 1, which means that the dimension of A_{−α} is strictly larger than 1 too. This implies that there is a y ∈ A_{−α} such that y ≠ 0 and

b(x_{α}, y) = 0. (17.7.4)

It follows that [x_{α}, y]_{A} = 0, (17.7.5)

by (17.5.8). Note that

[h_{α}, y]_{A} = −α(h_{α}) y = −2 y, (17.7.6)

because h_{α} ∈ B, y ∈ A_{α}, and α(h_{α}) = 2.

Remember that the linear span of x_{α}, y_{α},, and h_{α} in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k) as a Lie algebra over k. Thus A may be considered as a module over sl_{2}(k), using the restriction of the adjoint representation on A to this Lie subalgebra of A, acting on A as a vector space over k. With respect to this representation, y is a maximal or primitive vector of weight −2, as in Section 15.2. This contradicts the fact that the weight of y should be nonnegative, as in Section 15.3. This shows that A_{α} has dimension one as a vector space over k, as in Theorem 2 (b) on p43 of [24], and statement 2.6 on p45 of [24].

Of course, this means that the dimension of A_{−α} is one as well. Remember that B_{α} = [A_{α}, A_{−α}] is the same as the one-dimensional linear subspace of B spanned by h_{α}, as in the previous two sections. It follows that

A_{α} + A_{−α} + B_{α} (17.7.7)

is the same as the linear span in A of x_{α}, y_{α}, and h_{α}, which is a Lie subalgebra of A isomorphic to sl_{2}(k), as before. This corresponds to part of Theorem 2 (c) on p44 of [24], and to statement 2.7 on p46 of [24]. Because A_{−α} has dimension one, y_{α} ∈ A_{−α} is uniquely determined by x_{α} ∈ A_{α} and (17.6.7), as in Theorem 2 (c) on p44 of [24] and statement 2.8 on p46 of [24].

Now let us consider the argument given on the bottom of p38 of [14], which can also be used to get some of the same properties as before. Let C be the linear span in A of B and the linear subspaces of the form A_{c α}, where c ∈ k and c α ∈ Φ_{B}, (17.7.8)

which implies that c ≠ 0. This is a Lie subalgebra of A, because of (17.2.7). In particular, C is a submodule of A, as a module over the Lie subalgebra spanned by x_{α}, y_{α}, and h_{α}, because x_{α}, y_{α}, h_{α} ∈ C. Thus C may be considered as a module over sl_{2}(k).

Of course, [h_{α}, w]_{A} = 0 for every w ∈ B, because B is commutative as a Lie algebra over k. If z ∈ A_{c α} for some c ∈ k, then

[h_{α}, z]_{A} = c α(h_{α}) z = 2 c z, (17.7.9)

because α(h_{α}) = 2. This means that the weights of h_{α} on C consist of 0 and 2 c for each c ∈ k satisfying (17.7.8), as in Section 15.1. In particular, the action of h_{α} on C is diagonalizable in this situation.

Remember that C corresponds to the direct sum of finitely many irreducible submodules, as a module over sl_{2}(k), by Weyl’s theorem, as in Section 13.2. Any submodule of C, as a module over sl_{2}(k), is mapped into itself by the action of h_{α}, and hence the action of h_{α} on this submodule is diagonalizable, as in Section 10.6. This implies that any nonzero submodule of C has a maximal or primitive vector, as in Section 15.2. It follows that any nonzero irreducible submodule of C is as in Section 15.3.

The weights of h_{α} on any nonzero irreducible submodule of C correspond to integers, under the usual embedding of Q into k, as in Section 15.3. This means that the weights of h_{α} on C correspond to integers. If c ∈ k satisfies (17.7.8), then we get that 2 c corresponds to an integer.

If w ∈ B satisfies α(w) = 0, then [w, x_{α}]_{A} = α(w) x_{α} = 0 and [w, y_{α}]_{A} = −α(w) y_{α} = 0. Of course, [w, h_{α}]_{A} = 0, because B is commutative as a Lie algebra. Thus the linear span of x_{α}, y_{α}, and h_{α} acts trivially on the kernel of α in B.

Note that the linear span of x_{α}, y_{α}, and h_{α} is a submodule of C. More precisely, this is an irreducible submodule of C, because sl_{2}(k) is simple as a Lie algebra over k, as in Section 11.1.

17.8. OTHER ELEMENTS OF Φ_{B} 353

Of course, B is spanned by h_{α} and the kernel of α in B, so that the linear span in A of the kernel of α in B, x_{α}, y_{α}, and h_{α} is the same as the linear span of B, x_{α}, and y_{α}. This is a submodule of C, and the proof of Weyl’s theorem shows that C corresponds to the direct sum of this submodule and finitely many irreducible submodules of C, if necessary.

The elements of C of weight 0 with respect to h_{α} are in B. Thus nonzero elements of irreducible submodules of C complementary to the linear span of B, x_{α}, and y_{α} cannot have weight 0 with respect to h_{α}. This implies that the weights of h_{α} on these complementary irreducible submodules of C correspond to odd integers, as in Section 15.3. It follows that the only even integers which can correspond to weights of h_{α} on C are 0 and ±2, which are the weights of h_{α} on B, x_{α}, and y_{α}.

In particular, 4 does not correspond to a weight of h_{α} on C, which implies that 2 α ∉ Φ_{B}. (17.7.10)

If α/2 were in Φ_{B}, then we could apply the same argument to it, to get that α ∉ Φ_{B}. Thus α/2 ∉ Φ_{B}. (17.7.11)

This shows that 1 is not a weight of h_{α} on C.

Suppose for the sake of a contradiction that C_{1} is a nonzero irreducible submodule of C complementary to the linear span of B, x_{α}, and y_{α}. Thus C_{1} does not contain any nonzero elements with weight 0 or 1 with respect to h_{α}, as in the previous two paragraphs. This is a contradiction, as in Section 15.3. This means that C is the same as the linear span of B, x_{α}, and y_{α}.

It follows in particular that A_{α} and A_{−α} are spanned by x_{α} and y_{α}, respectively, so that the dimensions of A_{α} and A_{−α} are both equal to one, as vector spaces over k. We also get that ±α are the only multiples of α by elements of k in Φ_{B}. This corresponds to parts (a) and (b) of the proposition on p39 of [14]. The second statement corresponds to the second part of Theorem 2 (a) on p43 of [24] as well.

17.8 Other elements of Φ_{B}

Let us continue with the same notation and hypotheses as in the previous section. Let α, β ∈ Φ_{B} be given, and remember that h_{α} ∈ B is as in (17.6.5). Let x_{α} be a nonzero element of A_{α} again, and let y_{α} ∈ A_{−α} be as in (17.6.7), so that the linear span of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k), as a Lie algebra over k. Thus A may be considered as a module over the linear span of x_{α}, y_{α}, and h_{α}, or over sl_{2}(k). Weyl’s theorem implies that A corresponds to the direct sum of finitely many irreducible submodules. Any submodule of A is mapped into itself by ad_{h_{α}}, and ad_{h_{α}} is diagonalizable on the submodule. This implies that any nonzero submodule of A has a maximal or primitive vector, as in Section 15.2, and hence a nonzero irreducible submodule is as in Section 15.3. In particular, the eigenvalues of ad_{h_{α}} on A correspond to integers, with respect to the usual embedding of Q in k.

Let y be a nonzero element of A_{β}, so that

ad_{h_{α}}(y) = [h_{α}, y]_{A} = β(h_{α}) y. (17.8.1)

Thus β(h_{α}) corresponds to an integer, with respect to the standard embedding of Q in k, as in the preceding paragraph. This is part of statement 2.9 on p46 of [24].

Let n be the integer corresponding to β(h_{α}), and put

(17.8.2)  z = (ad_{y_{α}})^{n}(y) when n ≥ 0
= (ad_{x_{α}})^{−n}(y) when n ≤ 0.

One can check that z ≠ 0, because A corresponds to the direct sum of finitely many irreducible submodules, each of which is as in Section 15.3. We also have that z ∈ A_{β−β(h_{α}) α}, (17.8.3)

because y ∈ A_{β}, x_{α} ∈ A_{α}, and y_{α} ∈ A_{−α}, and using (17.2.7). In particular,

A_{β−β(h_{α}) α} ≠ {0}, (17.8.4)

because z ≠ 0.

Let us check that β − β(h_{α}) α ≠ 0. (17.8.5)

Otherwise, if β − β(h_{α}) α = 0, then

0 = β(h_{α}) − β(h_{α}) α(h_{α}) = β(h_{α}) − 2 β(h_{α}) = −β(h_{α}), (17.8.6)

because α(h_{α}) = 2. In this case, we get that β = 0, contradicting the hypothesis that β ∈ Φ_{B}. Thus (17.8.5) holds, and hence

β − β(h_{α}) α ∈ Φ_{B}, (17.8.7)

by (17.8.4). This corresponds to statement 2.9 on p46 of [24].

If β = α, then

β − β(h_{α}) α = α − α(h_{α}) α = α − 2 α = −α, (17.8.8)

because α(h_{α}) = 2, as before. Similarly, if β = −α, then

β − β(h_{α}) α = −α − (−α(h_{α})) α = −α + 2 α = α. (17.8.9)

This is part of statement 2.10 on p46 of [24].

Here is another proof of the fact that c ∈ k satisfies c α ∈ Φ_{B} only when c = ±1, as in statement 2.11 on p4 of [24]. If β = c α ∈ Φ_{B}, then β(h_{α}) = c α(h_{α}) = 2 c corresponds to an integer, as before. Similarly, we can interchange 17.9. ADDING ELEMENTS OF Φ_{B} 355 the roles of α and β, to get that 2/c corresponds to an integer. Thus the only possibilities for c are ±1, ±1/2, and ±2.

It suffices to show that 2 α ∉ Φ_{B}, which is to say that c ≠ 2. This will imply that α/2 ∉ Φ_{B}, since otherwise we would have that α ∉ Φ_{B}. Similarly, this will show that c ≠ −1/2, −2, because −α ∈ Φ_{B}, as in Section 17.3.

Suppose for the sake of a contradiction that 2 α ∈ Φ_{B}, and let y be a nonzero element of A_{2 α}. Thus [h_{α}, y]_{A} = 2 α(h_{α}) y = 4 y. (17.8.10) Note that ad_{x_{α}}(y) ∈ A_{3 α}, by (17.2.7), and because x_{α} ∈ A_{α}. However, A_{3 α} = {0}, because 3 α ∉ Φ_{B}, as before. This means that ad_{x_{α}}(y) = 0. (17.8.11)

Observe that (17.8.12) ad_{h_{α}}(y) = ad_{x_{α}}(ad_{y_{α}}(y)) − ad_{y_{α}}(ad_{x_{α}}(y)) = ad_{x_{α}}(ad_{y_{α}}(y)), using (17.6.8) in the first step, and (17.8.11) in the second step. Using (17.2.7) again, we get that ad_{y_{α}}(y) ∈ A_{α}, because y_{α} ∈ A_{−α} and y ∈ A_{2 α}. This means that ad_{y_{α}}(y) is a multiple of x_{α}, because A_{α} has dimension one as a vector space over k, as in the previous section. It follows that ad_{x_{α}}(ad_{y_{α}}(y)) = 0, (17.8.13) because ad_{x_{α}}(x_{α}) = 0. This contradicts (17.8.10), because of (17.8.12) and the fact that y ≠ 0.

17.9 Adding elements of Φ_{B}

Let us continue with the same notation and hypotheses as in the previous two sections. Let α, β ∈ Φ_{B} be given again, and suppose that β ≠ ±α. (17.9.1) Thus, for each j ∈ Z, β + j α ≠ 0, (17.9.2) as in the previous sections. Let E be the linear span in A of the subspaces A_{β+j α}, (17.9.3) where j ∈ Z. More precisely, we may restrict our attention to j ∈ Z such that β + j α ∈ Φ_{B}, (17.9.4) since otherwise (17.9.3) is equal to {0}.

Remember that h_{α} ∈ B is as in (17.6.5), and let x_{α} be a nonzero element of A_{α} again. Also let y_{α} ∈ A_{−α} be as in (17.6.7), so that the linear span of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k), as a Lie algebra over k. As before, A may be considered as a module over the linear span of x_{α}, y_{α}, and h_{α}, or over sl_{2}(k). In fact, E is a submodule of A, because of (17.2.7). The weights of h_{α} on E are given by

β(h_{α}) + j α(h_{α}) (17.9.5)

for j ∈ Z such that (17.9.4) holds, as in Section 15.1. Of course, (17.9.5) is the same as β(h_{α}) + 2 j, (17.9.6)

because α(h_{α}) = 2. Note that j = 0 satisfies (17.9.4) automatically.

Weyl’s theorem implies that E corresponds to the direct sum of finitely many irreducible submodules, as in Section 13.2. Any submodule of E is mapped into itself by the action of h_{α}, and the action of h_{α} on this submodule is diagonalizable, as in Section 10.6. Hence any nonzero submodule of E has a maximal or primitive vector, as in Section 15.2. This implies that any nonzero irreducible submodule of E is as in Section 15.3.

Remember that (17.9.3) has dimension one as a vector space over k when (17.9.4) holds, as in Section 17.7. One can use this to check that E is irreducible as a module over sl_{2}(k). More precisely, note that at most one of 0 and 1 can be of the form (17.9.6), as mentioned near the top of p39 of [14].

Thus there is a nonnegative integer m such that E is as in Section 15.3, as a module over sl_{2}(k). In particular, the dimension of E is m + 1, as a vector space over k.

Let q and r be the largest integers such that (17.9.4) holds with j = q and j = −r, respectively. Note that q, r ≥ 0, because (17.9.4) holds when j = 0. The maximal and minimal weights of h_{α} on E are

m = β(h_{α}) + 2 q (17.9.7)

and −m = β(h_{α}) − 2 r, (17.9.8)

as in Section 15.3. In particular,

β(h_{α}) = r − q. (17.9.9)

Suppose that j ∈ Z satisfies

−r ≤ j ≤ q, (17.9.10)

so that β(h_{α}) − 2 r ≤ β(h_{α}) + 2 j ≤ β(h_{α}) + 2 q. (17.9.11)

Under these conditions, β(h_{α}) + 2 j is a weight of h_{α} on E, as in Section 15.3. This means that (17.9.4) holds, as in part (e) of the proposition on p39 of [14].

17.10. THE DUAL OF B 357

Similarly, if j ∈ Z satisfies

−r ≤ j ≤ q − 1, (17.9.12)

then the restriction of ad_{x_{α}} to A_{β+j α} is a one-to-one mapping onto

A_{β+(j+1) α}. (17.9.13)

This follows from Section 15.3 again. These properties of E, q, and r correspond to statement 2.12 on p46 of [24].

The same type of arguments are discussed starting at the bottom of p38 and continuing on p39 of [14], as before. In particular, these arguments can also be used to get that β(h_{α}) corresponds to an integer for which (17.8.7) holds, as in part (c) of the proposition on p39 of [14].

Suppose now that α + β ∈ Φ_{B}. This means that (17.9.4) holds with j = 1, and in particular that q ≥ 1. Thus j = 0 satisfies (17.9.12), so that ad_{x_{α}} maps A_{β} onto A_{β+α}, as before. This implies that

[A_{α}, A_{β}] = A_{α+β}, (17.9.14)

as in Theorem 2 (d) on p44 of [24], and statement 2.13 on p47 of [24]. This corresponds to part (d) of the proposition on p39 of [14] as well.

17.10 The dual of B

Let k be a field of characteristic 0 again, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. Also let B be a Lie subalgebra of A such that every element of B is ad-diagonalizable as an element of A, so that B is commutative as a Lie algebra, as in Section 17.1. If α is an element of the dual B′ of B, then put

(17.10.1) A_{α} = {x ∈ A : for every w ∈ B, ad_{w}(x) = [w, x]_{A} = α(w) x},

as before. Thus A_{0} is the same as the centralizer C_{A}(B) of B in A, and we suppose again that this is equal to B, so that

A_{0} = C_{A}(B) = B. (17.10.2)

We suppose that the center Z(A) of A as a Lie algebra is equal to {0} again too.

Suppose that b(·, ·) is a nondegenerate bilinear form on A that is associative, or equivalently invariant under the adjoint representation on A, as before. This implies that that the restriction of b(·, ·) to (17.10.2) is nondegenerate, as in Section 17.3. If α ∈ B′, then there is a unique t_{b,α} ∈ B such that

α(w) = b(w, t_{b,α}) (17.10.3)

for every w ∈ B, as in Section 17.5. Note that α ↦ t_{b,α} is a one-to-one linear mapping from B′ onto B.

If α, β ∈ B′, then put

b′(α, β) = b(t_{b,α}, t_{b,β}). (17.10.4)

This defines a bilinear form on B′. Equivalently,

b′(α, β) = β(t_{b,α}) (17.10.5)

for every α, β ∈ B′. Observe that (17.10.4) is nondegenerate on B′.

Let Φ_{B} be the set of α ∈ B′ such that α ≠ 0 and A_{α} ≠ {0}, as before. If α ∈ Φ_{B}, then b′(α, α) = b(t_{b,α}, t_{b,α}) ≠ 0, (17.10.6)

as in Section 17.6. In this case, we put

h_{α} = 2 b(t_{b,α}, t_{b,α})^{−1} t_{b,α} = 2 b′(α, α)^{−1} t_{b,α}, (17.10.7)

as before, so that α(h_{α}) = 2. If β ∈ B′, then

2 b′(α, β) b′(α, α)^{−1} = 2 β(t_{b,α}) b′(α, α)^{−1} = β(h_{α}). (17.10.8)

If β ∈ Φ_{B}, then (17.10.8) corresponds to an integer, with respect to the standard embedding of Q into k, as in Section 17.8.

Remember that the linear span of Φ_{B} in B′ is equal to B′, as in Section 17.5. Let α_{1}, …, α_{n} be a basis for B′ consisting of elements of Φ_{B}. Let β ∈ Φ_{B} be given, so that β can be expressed in a unique way as

β = ∑_{j=1}^{n} c_{j} α_{j}, (17.10.9)

where c_{j} ∈ k for each j = 1, …, n. We would like to show that c_{j} corresponds to an element of Q for every j = 1, …, n, with respect to the standard embedding of Q into k, as discussed near the bottom of p39 of [14].

Of course,

b′(α_{l}, β) = ∑_{j=1}^{n} c_{j} b′(α_{l}, α_{j}) (17.10.10)

for each l = 1, …, n. This implies that

2 b′(α_{l}, β) b′(α_{l}, α_{l})^{−1} = ∑_{j=1}^{n} 2 b′(α_{l}, α_{j}) b′(α_{l}, α_{l})^{−1} c_{j} (17.10.11)

for every l = 1, …, n. The left side corresponds to an integer for every l = 1, …, n, because β ∈ Φ_{B}, as before. Similarly,

2 b′(α_{l}, α_{j}) b′(α_{l}, α_{l})^{−1} (17.10.12)

corresponds to an integer for every j, l = 1, …, n.

Note that (b′(α_{l}, α_{j})) is invertible as an n × n matrix with entries in k, because b′(·, ·) is nondegenerate as a bilinear form on B′. This implies that the n × n matrix with entries (17.10.12) is invertible as well. Equivalently, the determinant of this matrix is not 0. As in the preceding paragraph, this corresponds to an n × n matrix of integers, whose determinant is not 0.

Thus the entries of the inverse of the matrix in (17.10.12) correspond to rational numbers. This permits us to solve (17.10.11), to get that c_{j} corresponds to an element of Q for each j = 1, …, n, as desired.

17.11 Using the Killing form

Let us continue with the same notation and hypotheses as in the previous section, with some additional conditions, as follows. Suppose that A is semisimple, and let b_{A}(x, y) = tr_{A}(ad_{x} ∘ ad_{y}) (17.11.1)

be the Killing form on A. Thus b_{A} is associative on A, or equivalently invariant under the adjoint representation on A, and b_{A} is nondegenerate on A, as in Sections 7.9 and 10.13. In particular, we can take b = b_{A} in the previous section.

Remember that the restriction of b_{A} to (17.10.2) is nondegenerate, as before. If α ∈ B′, then there is a unique t_{α} = t_{b_{A},α} ∈ B such that

α(w) = b_{A}(w, t_{α}) (17.11.2)

for every w ∈ B, as in (17.10.3). Put

b′_{A}(α, β) = b_{A}(t_{α}, t_{β}) (17.11.3)

for every α, β ∈ B′, as in (17.10.4). Equivalently,

b′_{A}(α, β) = β(t_{α}) (17.11.4)

for every α, β ∈ B′, as in (17.10.5).

If x, y ∈ B, then

b_{A}(x, y) = ∑_{α∈Φ_{B}} α(x) α(y). (17.11.5)

To see this, remember that A corresponds to the direct sum of the subspaces A_{α}, α ∈ Φ_{B} ∪ {0}, as a vector space over k, as in Section 17.2. These subspaces are mapped into themselves by ad_{x} and ad_{y}, so that the trace of ad_{x} ∘ ad_{y} over A is the same as the sum of the traces over A_{α}, α ∈ Φ_{B} ∪ {0}. The trace over A_{0} is equal to 0, because ad_{x} = ad_{y} = 0 on A_{0}, and so it suffices to consider the sum over α ∈ Φ_{B}. By construction, ad_{x} and ad_{y} correspond to multiplication by α(x) and α(y) on A_{α}. If α ∈ Φ_{B}, then the dimension of A_{α} is equal to one, as in Section 17.7. This implies (17.11.5).

If λ, μ ∈ B′, then we get that

b′_{A}(λ, μ) = b_{A}(t_{λ}, t_{μ}) = ∑_{α∈Φ_{B}} α(t_{λ}) α(t_{μ}). (17.11.6)

It follows that

b′_{A}(λ, μ) = ∑_{α∈Φ_{B}} b′_{A}(λ, α) b′_{A}(μ, α), (17.11.7)

by (17.11.4). In particular,

b′_{A}(λ, λ) = ∑_{α∈Φ_{B}} b′_{A}(λ, α)^{2}. (17.11.8)

Suppose that β ∈ Φ, so that b′_{A}(β, β) ≠ 0, as in (17.10.6). Using (17.11.8) with λ = β and multipliying both sides by 1/b′_{A}(β, β)^{2}, we get that

1/b′_{A}(β, β) = ∑_{α∈Φ_{B}} b′_{A}(β, α)^{2}/b′_{A}(β, β)^{2}. (17.11.9)

If α ∈ Φ_{B}, then 2 b′_{A}(β, α)/b′_{A}(β, β) (17.11.10)

corresponds to an integer under the standard embedding of Q into k, as mentioned in the previous section. This implies that the right side of (17.11.9) corresponds to a rational number. Thus b′_{A}(β, β) corresponds to a rational number. It follows that b′_{A}(β, α) (17.11.11)

corresponds to a rational number as well when α ∈ Φ_{B}, because (17.11.10) corresponds to an integer. This corresponds to some of the remarks near the top of p40 of [14].

17.12 Vector spaces over Q

Let us return for the moment to the situation considered in Section 17.10. We may consider the dual B′ of B as a vector space over the rational numbers, using the standard embedding of Q into k. Let E_{Q} be the linear subspace of B′, as a vector space over Q, spanned by Φ_{B}. Equivalently, E_{Q} consists of linear combinations of elements of Φ_{B} whose coefficients in k correspond to rational numbers. If α_{1}, …, α_{n} are elements of Φ_{B} that form a basis for B′ as a vector space over k, then α_{1}, …, α_{n} forms a basis for E_{Q} as a vector space over Q, as in Section 17.10. Of course, linear independence over k automatically implies linear independence over Q. In particular, the dimension of E_{Q} as a vector space over Q is the same as the dimension of B′ as a vector space over k, which is equal to the dimension of B as a vector space over k.

Suppose now that we are in the more particular situation considered in the previous section. If α, β ∈ Φ_{B}, then b′_{A}(α, β) corresponds to a rational number with respect to the standard embedding of Q into k, as before. It follows that b′_{A}(α, β) corresponds to a rational number when α, β ∈ E_{Q}. Let (α, β)_{E_{Q}} be the rational number that corresponds to b′_{A}(α, β) when α, β ∈ E_{Q}. This defines a bilinear form on E_{Q}, as a vector space over Q. If λ, μ ∈ E_{Q}, then

(λ, μ)_{E_{Q}} = ∑_{α∈Φ_{B}}(λ, α)_{E_{Q}} (μ, α)_{E_{Q}}, (17.12.1)

as in (17.11.7).

It follows that

(λ, λ)_{E_{Q}} = ∑_{α∈Φ_{B}}(λ, α)_{E_{Q}}^{2} (17.12.2)

for every λ ∈ E_{Q}. This implies that

(λ, λ)_{E_{Q}} ≥ 0 (17.12.3)

for every λ ∈ E_{Q}. More precisely, (λ, λ)_{E_{Q}} = 0 if and only if

(17.12.4) (λ, α)_{E_{Q}} = 0 for every α ∈ Φ_{B}.

Let us check that this can only happen when λ = 0.

Of course, (17.12.4) is the same as saying that

b′_{A}(λ, α) = 0 (17.12.5)

for every α ∈ Φ_{B}. This implies that (17.12.5) holds for every α ∈ B′, because the linear span of Φ_{B} in B′, as a vector space over k, is equal to B′. It follows that λ = 0, because b′_{A} is nondegenerate on B′. This corresponds to some more of the remarks near the top of p40 in [14].

17.13 Vector spaces over R

Let us return again for the moment to the situation considered in Section 17.10, and the beginning of the previous section. Thus E_{Q} is the linear subspace of B′, as a vector space over Q, spanned by Φ_{B}. Let n be the dimension of B and B′ as vector spaces over k, which is the same as the dimension of E_{Q} as a vector space over Q. Using any basis for E_{Q}, we get that E_{Q} is isomorphic to Q^{n}, as a vector space over Q. We can get a vector space E_{R} over the real numbers with the same basis, using formal linear combinations with coefficients in R. This amounts to taking R^{n} as a vector space over R that contains Q^{n}. This does not depend on the choice of basis for E_{Q}, up to suitable equivalence. In particular, E_{Q} corresponds to a subset of E_{R}.

Suppose now, for the rest of the section, that we are in the more particular situation considered in Section 17.11 again, and as in the previous section. Thus (α, β)_{E_{Q}} is defined as a bilinear form on E_{Q}, as a vector space over Q, as before.

This has a natural extension to a bilinear form (α, β)_{E_{R}} on E_{R}, as a vector space over R. If λ, μ ∈ E_{R}, then

(λ, μ)_{E_{R}} = ∑_{α∈Φ_{B}}(λ, α)_{E_{R}} (μ, α)_{E_{R}}, (17.13.1)

as in (17.12.1). This implies that

(λ, λ)_{E_{R}} = ∑_{α∈Φ_{B}}(λ, α)_{E_{R}}^{2} ≥ 0 (17.13.2)

for every λ ∈ E_{R}.

Let us verify that λ ∈ E_{R} satisfies (λ, λ)_{E_{R}} = 0 only when λ = 0, so that (·, ·)_{E_{R}} defines an inner product on E_{R}. One way to do this is to use an orthogonal basis for E_{Q} with respect to (·, ·)_{E_{Q}}, which is an orthogonal basis for E_{R} with respect to (·, ·)_{E_{R}} as well. If λ ∈ E_{R} and λ ≠ 0, then one can get that (λ, λ)_{E_{R}} > 0 from the analogous condition for the basis vectors in E_{Q}.

Alternatively, let α_{1}, …, α_{n} be elements of Φ_{B} that form a basis for B′ as a vector space over k, so that α_{1}, …, α_{n} forms a basis for E_{Q} as a vector space over Q too, as before. If μ ∈ E_{Q} satisfies

(μ, α_{j})_{E_{Q}} = 0 (17.13.3)

for each j = 1, …, n, then b′_{A}(μ, α_{j}) = 0 (17.13.4) for every j = 1, …, n. This implies that

b′_{A}(μ, α) = 0 (17.13.5)

for every α ∈ B′, and hence that μ = 0, because b′_{A} is nondegenerate on B′. Of course, μ ↦ (μ, α_{j})_{E_{Q}} (17.13.6) defines a linear functional on E_{Q} for each j = 1, …, n. Using these linear functionals, we get a mapping from E_{Q} into Q^{n} that is linear over Q. The kernel of this linear mapping is trivial, by the previous remarks. It follows that this linear mapping sends E_{Q} onto Q^{n}, because E_{Q} has dimension n as a vector space over Q.

Similarly, μ ↦ (μ, α_{j})_{E_{R}} (17.13.7) is a linear functional on E_{R} for each j = 1, …, n. Using these linear functionals, we get a mapping from E_{R} into R^{n} that is linear over R. The image of this linear mapping contains Q^{n}, as in the preceding paragraph. This implies that this linear mapping sends E_{R} onto R^{n}, because the image is a linear subspace of R^{n}, as a vector space over R. It follows that this linear mapping is injective, because E_{R} has dimension n, as a vector space over R. Thus, if λ ∈ E_{R} satisfies

(λ, α_{j})_{E_{R}} = 0 (17.13.8)

for every j = 1, …, n, then λ = 0. If (λ, λ)_{E_{R}} = 0, then (17.13.8) holds for each j = 1, …, n, by (17.13.2), so that λ = 0. This corresponds to some of the remarks near the top of p40 of [14] again.

17.14 Normalizers and centralizers

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let B be a Lie subalgebra of A. Remember that the centralizer C_{A}(B) of B in A consists of the x ∈ A such that [x, w]_{A} = 0 for every w ∈ B, as in Section 7.6. The normalizer N_{A}(B) of B in A is the set of x ∈ A such that [x, w]_{A} ∈ B for every w ∈ B, as in Section 9.8. Thus B ⊆ N_{A}(B), (17.14.1)

because B is a Lie subalgebra of A. Note that

C_{A}(B) ⊆ N_{A}(B). (17.14.2)

Both C_{A}(B) and N_{A}(B) are Lie subalgebras of A, as before. Of course, if B is commutative as a Lie algebra over k, then B ⊆ C_{A}(B).

Suppose from now on in this section that k is a field, and that A is a finitedimensional Lie algebra over k. Let B be a Lie subalgebra of A such that every element of B is ad-diagonalizable as an element of A. Thus B is commutative as a Lie algebra over k, as in Section 17.1. Let B′ be the dual of B, as a vector space over k, and for each α ∈ B′, let A_{α} be the set of x ∈ A such that ad_{w}(x) = [w, x]_{A} = α(w) x for every w ∈ B, as usual. If α = 0, then this is the same as C_{A}(B), as before.

In particular, B ⊆ C_{A}(B) = A_{0}. Let us suppose that B ≠ {0}, to avoid trivialities, so that A_{0} ≠ {0}. Let Φ_{B} be the set of α ∈ B′ such that α ≠ 0 and A_{α} ≠ {0} again. Remember that A corresponds to the direct sum of A_{α}, α ∈ Φ_{B} ∪ {0}, as a vector space over k, as in Section 17.2.

Let us check that A_{0} = N_{A}(B) (17.14.3)

under these conditions. Of course, A_{0} ⊆ N_{A}(B), as in (17.14.2). Let x ∈ A be given, so that x can be expressed in a unique way as

x = ∑_{α∈Φ_{B}∪{0}} x_{α}, (17.14.4)

where x_{α} ∈ A_{α} for each α ∈ Φ_{B} ∪ {0}. If w ∈ B, then

[w, x]_{A} = ∑_{α∈Φ_{B}∪{0}}[w, x_{α}]_{A} = ∑_{α∈Φ_{B}} α(w) x_{α}. (17.14.5)

This is in B, or even A_{0}, only when it is equal to 0. If this happens for every w ∈ B, then it follows that x ∈ A_{0} = C_{A}(B). Equivalently, this means that x_{α} = 0 for every α ∈ Φ_{B}. This shows that N_{A}(B) ⊆ A_{0}, so that (17.14.3) holds.

If C_{A}(B) = B, then we get that N_{A}(B) = B. This corresponds to Exercise 5 on p40 of [14].

17.15 Semisimplicity and diagonalizability

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k with positive finite dimension, as a vector space over k. Also let B be a Lie subalgebra of A such that every element of B is ad-diagonalizable as an element of A. Thus B is commutative as a Lie algebra over k, as in Section 17.1. Let B′ be the dual of B, as a vector space over k, and for each α ∈ B′, let A_{α} be the set of x ∈ A such that ad_{w}(x) = [w, x]_{A} = α(w) x for every w ∈ B, as before. Remember that A_{0} is the same as the centralizer C_{A}(B) of B in A, which contains B, because B is commutative as a Lie algebra.

Suppose that B = C_{A}(B) = A_{0}. (17.15.1)

This implies that B ≠ {0}, because A ≠ {0}, by hypothesis. Let Φ_{B} be the set of α ∈ B′ such that α ≠ 0 and A_{α} ≠ {0} again, so that A corresponds to the direct sum of A_{α}, α ∈ Φ_{B} ∪ {0}, as a vector space over k, as in Section 17.2. Note that the center Z(A) of A as a Lie algebra is contained in A_{0}. Let us suppose also that Z(A) = {0}. (17.15.2)

Suppose in addition that for each α ∈ Φ_{B} and x_{α} ∈ A_{α} with x_{α} ≠ 0 there is a y_{α} ∈ A_{−α} with the following property. Put

z_{α} = [x_{α}, y_{α}]_{A}, (17.15.3)

so that z_{α} ∈ A_{0}, as in Section 17.2. This means that z_{α} ∈ B, by (17.15.1), and we ask that α(z_{α}) ≠ 0. (17.15.4)

We have seen that this holds under some additional conditions, as in Section 17.6.

We would like to show that A is semisimple as a Lie algebra over k in this situation. This corresponds to Step (13) on p100 of [14], and to (k) on p55 of [24]. Let C be an ideal in A such that C is commutative as a Lie algebra. It suffices to show that C = {0}, as in Section 9.4.

If w ∈ B, then ad_{w} maps C into itself, because C is an ideal in A. The restriction of ad_{w} to C is diagonalizable, as in Section 10.6, because ad_{w} is diagonalizable on A, by hypothesis. Remember that the mappings ad_{w}, with w ∈ B, commute on A, because B is commutative as a Lie algebra, as in Section 17.2. This implies that the linear mappings ad_{w}, w ∈ B, are simultaneously diagonalizable on C, as before. If C ≠ {0}, then it follows that C is spanned by its intersections with the subspaces A_{α}, with α ∈ Φ_{B} ∪ {0}.

Suppose for the sake of a contradiction that there is an α ∈ Φ_{B} such that A_{α} ∩ C ≠ {0}. Let x_{α} be a nonzero element of A_{α} ∩ C, and let y_{α} ∈ A_{−α} and z_{α} ∈ B be as before. Observe that z_{α} ∈ C, because of (17.15.3) and the hypothesis that C be an ideal in A. Remember that

[z_{α}, y_{α}]_{A} = −α(z_{α}) y_{α}, (17.15.5) because y_{α} ∈ A_{−α} and z_{α} ∈ B. This implies that y_{α} ∈ C too, because of (17.15.4). It follows that z_{α} = 0, by (17.15.3) and the commutativity of C as a Lie algebra. This contradicts (17.15.4), as desired.

This shows that A_{α} ∩ C = {0} for every α ∈ Φ_{B}, so that

C ⊆ A_{0} = B. (17.15.6)

Let α ∈ Φ_{B} and x_{α} ∈ A_{α} be given, so that

[w, x_{α}]_{A} = α(w) x_{α} (17.15.7)

for every w ∈ B. If w ∈ C, then [w, x_{α}]_{A} ∈ C, because C is an ideal in A. Combining this with (17.15.7), we get that

[w, x_{α}]_{A} = 0, (17.15.8)

because A_{0} ∩ A_{α} = {0} when α ≠ 0. If x_{0} ∈ A_{0}, then [w, x_{0}]_{A} = 0 for every w ∈ B, and hence every w ∈ C, by (17.15.6). This implies that C is contained in the center of A, because A is spanned by A_{α}, α ∈ Φ_{B} ∪ {0}. It follows that C = {0}, by (17.15.2), as desired.

Cartan subalgebras

18.1 Subspaces related to ad_{x}

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let x ∈ A be given, so that ad_{x}(w) = [x, w]_{A} defines a linear mapping from A into itself, as usual. If α ∈ k, then put

(18.1.1) A_{x,α} = {y ∈ A : (ad_{x} − α I)^{l}(y) = 0 for some l ∈ Z_{+}},

where I is the identity mapping on A. This is a linear subspace of A, as in Section 10.7. Note that A_{x,α} ≠ {0} if and only if α is an eigenvalue of ad_{x} on A, as before.

Let y, z ∈ A and α, β ∈ k be given. Remember that ad_{x} is a derivation on A, as in Section 2.5. It follows that

(ad_{x} − (α + β) I)([y, z]_{A}) = [ad_{x}(y), z]_{A} + [y, ad_{x}(z)]_{A} − α [y, z]_{A} − β [y, z]_{A}
(18.1.2)  = [(ad_{x} − α I)(y), z]_{A} + [y, (ad_{x} − β I)(z)]_{A},

as in Section 10.9. If y ∈ A_{x,α} and z ∈ A_{x,β}, then one can use this repeatedly to get that (ad_{x} − (α + β) I)^{j}([y, z]_{A}) = 0 (18.1.3)

when j is sufficiently large. This means that

[y, z]_{A} ∈ A_{x,α+β}, (18.1.4)

as in Section 10.9. In particular, A_{x,0} is a Lie subalgebra of A. This corresponds to the first two assertions in the lemma at the bottom of p78 of [14], and to parts (b) and (c) of Proposition 2 on p11 of [24].

Let α_{1} …, α_{n} ∈ k be finitely many distinct eigenvalues of ad_{x}, and suppose that y_{j} ∈ A_{x,α_{j}} for each j = 1, …, n. If ∑_{j=1}^{n} y_{j} = 0, then y_{j} = 0 for each j = 1, …, n, as in Section 10.7.

Suppose from now on in this section that k is algebraically closed, and that A has positive finite dimension as a vector space over k. In this case, A corresponds 18.2. SOME REMARKS ABOUT CHARACTERISTIC POLYNOMIALS 367

to the direct sum of the subspaces A_{x,α} with α ∈ k an eigenvalue of ad_{x}, as a vector space over k. This is part (a) of Proposition 2 on p11 of [24], when k = C, and is mentioned on p78 of [14].

Suppose that k has characteristic 0. If y ∈ A_{x,α} for some α ∈ k with α ≠ 0, then one can check that y is ad-nilpotent as an element of A, using (18.1.4) and the fact that there are only finitely many eigenvalues of ad_{x} on A. This is the third assertion of the lemma on p78 of [14].

18.2 Some remarks about characteristic polynomials

Let k be a field, and let V be a vector space over k of positive finite dimension. Also let a be a linear mapping from V into itself, and let T be an indeterminate. As usual, the characteristic polynomial of a is defined as a formal polynomial in T with coefficients in k by

p_{a}(T) = det(a − T I), (18.2.1)

where I is the identity mapping on V. More precisely, one can choose a basis for V, so that a corresponds to a matrix with entries in k, and a − T I corresponds to a matrix with entries in the algebra k[T] of formal polynomials in T with coefficients in k. The determinant of this matrix is an element of k[T], which does not depend on the choice of basis for V.

Let V_{0} be a proper nontrivial linear subspace of V, and suppose that

a(V_{0}) ⊆ V_{0}. (18.2.2)

Let V_{1} be the quotient space V/V_{0}, so that a induces a linear mapping a_{1} from V_{1} into itself. If a_{0} is the restriction of a to V_{0}, then

p_{a}(T) = p_{a_{0}}(T) p_{a_{1}}(T), (18.2.3)

where p_{a_{0}}(T), p_{a_{1}}(T) are the characteristic polynomials of a_{0}, a_{1} as linear mappings on V_{0}, V_{1}, respectively. This can be seen using a basis for V that contains a basis for V_{0}. The basis vectors not in V_{0} can be mapped into V_{1} by the canonical quotient mapping, to get a basis for V_{1}.

Remember that

(18.2.4) V_{0} = {v ∈ V : a^{l}(v) = 0 for some l ∈ Z_{+}}

is a linear subspace of V, as in Section 10.7. Clearly V_{0} satisfies (18.2.2). If v ∈ V and a(v) ∈ V_{0}, then it is easy to see that v ∈ V_{0} too. This implies that the kernel of the induced mapping a_{1} on V_{1} = V/V_{0} is trivial. Note that V_{0} ≠ {0} exactly when 0 is an eigenvalue of a on V, as in Section 10.7.

Let n_{0} be the dimension of V_{0}, as a vector space over k. If n_{0} ≥ 1, then it is well known that p_{a_{0}}(T) = (−1)^{n_{0}} T^{n_{0}}. (18.2.5)

Suppose for the moment that V_{0} ≠ V, so that V_{1} ≠ {0}. Of course, the constant term in p_{a_{1}}(T) is the same as the determinant of a_{1} on V. This is a nonzero element of k, because a_{1} is invertible on V_{1}, as in the preceding paragraph.

If V_{0} = V, then p_{a}(T) = p_{a_{0}}(T) is as in (18.2.5). If V_{0} = {0}, then 0 is not an eigenvalue of a on V, and the constant term in p_{a}(T) is nonzero.

18.3 Characteristic polynomials and ad_{x}

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k, with positive finite dimension n as a vector space over k. Also let T be an indeterminate, so that the characteristic polynomial of a linear mapping from A into itself is defined as a formal polynomial in T with coefficients in k, as in the previous section. If x ∈ A, then let Q_{x}(T) = det(ad_{x} − T I) (18.3.1)

be the characteristic polynomial of ad_{x}, as a linear mapping from A into itself. This is the same as (−1)^{n} times the polynomial P_{x}(T) defined on p10 of [24].

Using a basis for A, we can identify A with the space k^{n} of n-tuples of elements of k, as a vector space over k. Of course, ad_{x} depends linearly on x, and can be expressed in terms of the coordinates of x and the corresponding structure constants for the Lie bracket on A. Similarly, (18.3.1) can be expressed as

Q_{x}(T) = ∑_{j=0}^{n} q_{j}(x) T^{j}, (18.3.2)

where q_{j}(x) corresponds to a homogeneous polynomial of degree n − j in the coordinates of x for each j = 0, 1, …, n, as on p11 of [24]. More precisely, the coefficients for q_{j} can be given in terms of the structure constants for the Lie bracket on A. By construction, q_{n}(x) corresponds to the constant polynomial (−1)^{n}.

Suppose for the moment that A is nilpotent as a Lie algebra over k. In this case, we can choose a basis for A such that ad_{x} is always corresponds to a strictly upper-triangular matrix. This implies that the coefficients of q_{j}(x) are all equal to 0 when j < n.

Let x ∈ A be given, and let A_{x,0} be as in (18.1.1). This is the same as (18.2.4), with V = A and a = ad_{x}. Let n_{0}(x) be the dimension of A_{x,0}. Note that x ∈ A_{x,0}, and that A_{x,0} = A when x = 0, so that n_{0}(x) ≥ 1. We also have that q_{j}(x) = 0 when j < n_{0}(x), (18.3.3)

and q_{n_{0}(x)}(x) ≠ 0, (18.3.4)

as in the previous section.

Of course, n_{0}(x) = n exactly when ad_{x} is nilpotent on A. If this happens for every x ∈ A, then A is nilpotent as a Lie algebra, as in Section 9.10. These remarks correspond to some of those on p11 in [24].

18.4 Engel subalgebras

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k. If x ∈ A, then

(18.4.1) A_{x,0} = {y ∈ A : (ad_{x})^{l}(y) = 0 for some l ∈ Z_{+}}

is a Lie subalgebra of A, as in Section 18.1. A Lie subalgebra of A of this form is called an Engel subalgebra, following Barnes, as on p79 of [14].

Let us suppose from now on in this section that A has positive finite dimension n as a vector space over k, and that the number of elements of k is strictly larger than n, as in Exercise 5 on p81 of [14]. Let C be a Lie subalgebra of A, and let z be an element of C such that A_{z,0} is minimal with respect to inclusion among the A_{x,0}’s with x ∈ C. Of course, it suffices to choose z ∈ C so that the dimension of A_{z,0}, as a vector space over k, is minimal. If

C ⊆ A_{z,0}, (18.4.2)

then A_{z,0} ⊆ A_{x,0} (18.4.3)

for every x ∈ C. This is Lemma A on p79 of [14].

We may as well suppose that A_{z,0} ≠ {0}, since otherwise (18.4.3) is trivial. We may as well suppose too that A_{z,0} ≠ A, because (18.4.3) would follow from the minimality of A_{z,0} otherwise. Let r be the dimension of A_{z,0} as a vector space over k, so that 1 ≤ r < n.

Let x ∈ C and c ∈ k be given, and observe that

z + c x ∈ C. (18.4.4)

Thus z + c x ∈ A_{z,0}, by (18.4.2). This implies that ad_{z+c x} maps A_{z,0} into itself, because A_{z,0} is a Lie subalgebra of A. The quotient A/A_{z,0} is defined as a vector space over k, and ad_{z+c x} induces a mapping from A/A_{z,0} into itself in the usual way.

Note that A_{z,0} corresponds to (18.2.4) with V = A and a = ad_{z}. Thus the mapping from A/A_{z,0} into itself induced by ad_{z} is injective, as before.

Let T be an indeterminate, and let h(T, c) be the characteristic polynomial of ad_{z+c x} on A. As in Section 18.2, this can be expressed as

h(T, c) = f(T, c) g(T, c), (18.4.5)

where f(T, c) is the characteristic polynomial of the restriction of ad_{z+c x} to A_{z,0}, and g(T, c) is the characteristic polynomial of the linear mapping from A/A_{z,0} into itself induced by ad_{z+c x}. These polynomials can be expressed as

f(T, c) = (−1)^{r} T^{r} + f_{1}(c) T^{r−1} + ⋯ + f_{r}(c) (18.4.6)

and g(T, c) = (−1)^{n−r} T^{n−r} + g_{1}(c) T^{n−r−1} + ⋯ + g_{n−r}(c), (18.4.7) as on p79 of [14]. More precisely, f_{j}(c) and g_{j}(c) can be given by polynomials with coefficients in k, of degree less than or equal to j.

Observe that g_{n−r}(0) ≠ 0, because the mapping induced on A/A_{z,0} by ad_{z} is invertible. In particular, g_{n−r} ≢ 0 on k, so that g_{n−r} has at most n − r zeros in k. Remember that k has at least n + 1 elements, by hypothesis. It follows that there are r + 1 distinct elements c_{1}, …, c_{r+1} of k such that

g_{n−r}(c_{l}) ≠ 0 (18.4.8)

for each l = 1, …, r + 1.

By construction, g_{n−r}(c) is the determinant of the linear mapping from A/A_{z,0} into itself induced by ad_{z+c x}. Suppose for the moment that c ∈ k has the property that g_{n−r}(c) ≠ 0, so that the mapping from A/A_{z,0} into itself induced by ad_{z+c x} is injective. Of course, this implies that all positive powers of this mapping are injective. Remember that the positive powers of the mapping from A/A_{z,0} induced by ad_{z+c x} correspond to the mappings from A/A_{z,0} into itself induced by the positive powers of ad_{z+c x}. It follows that the kernels of the positive powers of ad_{z+c x} on A are contained A_{z,0}. This implies that

A_{z+c x,0} ⊆ A_{z,0}. (18.4.9)

Under these conditions, we get that

A_{z+c x,0} = A_{z,0}, (18.4.10)

because of the minimality condiiton on A_{z,0}.

As before, A_{z+c x,0} corresponds to (18.2.4) with V = A and a = ad_{z+c x}. If (18.4.10) holds, then the characteristic polynomial f(T, c) of the restriction of ad_{z+c x} to (18.4.10) is equal to (−1)^{r} T^{r}. This means that

f_{j}(c) = 0 (18.4.11)

for each j = 1, …, r. In particular, this holds for r + 1 distinct elements of k, as in (18.4.8). This implies that

f_{j} ≡ 0 (18.4.12)

on k for each j = 1, …, r, because f_{j} is given by a polynomial of degree less than or equal to j.

Equivalently, f(T, c) = (−1)^{r} T^{r} (18.4.13)

for every c ∈ k. Let c ∈ k be given again, and consider

A_{z+c x,0} ∩ A_{z,0}, (18.4.14)

which is a linear subspace of A_{z,0}. If V = A_{z,0} and a = ad_{z+c x}, then (18.4.14) corresponds to (18.2.4). Using (18.4.13), we get that (18.4.14) is the same as A_{z,0}, as in Section 18.2. This means that

A_{z,0} ⊆ A_{z+c x,0} (18.4.15) for every c ∈ k.

Remember that this holds for every x ∈ C. Thus we can replace x with x−z and take c = 1 to get (18.4.3), as desired.

18.5 Self-normalizing subalgebras

Let k be a field again, and let (A, [·, ·]) be a Lie algebra over k. Also let C be a Lie subalgebra of A, and remember that the normalizer N_{A}(C) of C in A is the set of w ∈ A such that [w, z]_{A} ∈ C for every z ∈ C. This is the largest Lie subalgebra of A that contains C as an ideal, as in Section 9.8. If N_{A}(C) = C, then C is said to be self-normalizing in A, as before.

Let x ∈ A be given, and let A_{x,0} be the set of y ∈ A such that (ad_{x})^{l}(y) = 0 for some positive integer l, as before. This is a Lie subalgebra of A, as in Section 18.1. Equivalently, if V = A and a = ad_{x}, then A_{x,0} is the same as (18.2.4). Remember that the mapping from A/A_{x,0} into itself induced by ad_{x} is injective, as in Section 18.2.

Let w be an element of the normalizer N_{A}(A_{x,0}) of A_{x,0} in A. Note that x ∈ A_{x,0}, so that [w, x]_{A} ∈ A_{x,0}. (18.5.1)

This means that (ad_{x})^{l}([w, x]_{A}) = 0 for some positive integer l, and hence (ad_{x})^{l+1}(w) = 0. It follows that w ∈ A_{x,0}, so that

N_{A}(A_{x,0}) = A_{x,0}. (18.5.2)

This corresponds to the last part of the proof of Theorem 1 on p12 of [24].

Suppose from now on in this section that the dimension of A is finite, as a vector space over k. Thus the dimension of the quotient space A/A_{x,0} is finite too. It follows that the mapping on A/A_{x,0} induced by ad_{x} is invertible, because it is injective, as before.

Suppose that C is a Lie subalgebra of A with

A_{x,0} ⊆ C. (18.5.3)

In particular, x ∈ C, because x ∈ A_{x,0}. Thus ad_{x} maps C into itself, so that the mapping on A/A_{x,0} induced by ad_{x} maps C/A_{x,0} into itself. More precisely, the mapping induced on A/A_{x,0} maps C/A_{x,0} onto itself, because this mapping is injective, and the dimension of C/A_{x,0} is finite.

The normalizer N_{A}(C) of C in A is mapped into C by ad_{x}, because x ∈ C. This implies that the mapping induced on A/A_{x,0} by ad_{x} maps N_{A}(C)/A_{x,0} into C/A_{x,0}. It follows that

N_{A}(C)/A_{x,0} = C/A_{x,0}, (18.5.4)

because the mapping induced on A/A_{x,0} by ad_{x} is injective and maps C/A_{x,0} onto itself.

This means that N_{A}(C) = C, (18.5.5)

because of (18.5.3) and the fact that C ⊆ N_{A}(C). This corresponds to Lemma B on p79 of [14].

18.6 Orthogonality conditions

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let b(·, ·) be a bilinear form on A that is associative, or equivalently invariant under the adjoint representation. This means that

b([w, y]_{A}, z) = −b(y, [w, z]_{A}) (18.6.1)

for every w, y, z ∈ A, as in Sections 6.10 and 7.7. Let x ∈ A be given, so that

b(ad_{x}(y), z) + b(y, ad_{x}(z)) = 0 (18.6.2)

for every y, z ∈ A. If α, β ∈ k, then we get that

b((ad_{x} − α I)(y), z) + b(y, (ad_{x} − β I)(z)) = −(α + β) b(y, z) (18.6.3)

for every y, z ∈ A.

Let l be a positive integer, and consider

(−1)^{l} (α + β)^{l} b(y, z). (18.6.4)

If l_{1} and l_{2} are nonnegative integers, then consider

b((ad_{x} − α I)^{l_{1}}(y), (ad_{x} − β I)^{l_{2}}(z)). (18.6.5)

If l_{1} or l_{2} is equal to 0, then (ad_{x} −α I)^{l_{1}} or (ad_{x} −β I)^{l_{2}} is interpreted as being the identity operator on A, as appropriate. We can express (18.6.4) as a sum of terms of the form (18.6.5), using (18.6.3) repeatedly. More precisely, we have that l_{1} + l_{2} = l (18.6.6)

in each of the terms of the form (18.6.5).

Remember that A_{x,α} consists of the y ∈ A such that (ad_{x} −α I)^{j}(y) = 0 for all sufficiently large positive integers j, as in Section 18.1, and similarly for β. If y ∈ A_{x,α} and z ∈ A_{x,β}, then (18.6.5) is equal to 0 when l_{1} or l_{2} is sufficiently large. This implies that (α + β)^{l} b(y, z) = 0 (18.6.7)

when l is sufficiently large. It follows that

b(y, z) = 0 (18.6.8)

when α + β ≠ 0. This corresponds to a remark in the proof of part (d) of Theorem 3 on p15 of [24].

Remember that A_{x,α} ≠ {0} exactly when α ∈ k is an eigenvalue of ad_{x} on A. Suppose now that k is algebraically closed, and that A has positive finite dimension as a vector space over k. In this case, A corresponds to the direct sum of the nonzero subspaces A_{x,α}, as a vector space over k.

Suppose that b(·, ·) is also nondegenerate on A. Under these conditions, it is easy to see that the restriction of b(·, ·) to A_{x,0} is nondegenerate, using (18.6.8). This corresponds to part (d) of Theorem 3 on p15 of [24].

18.7 Nilpotence and normalizing elements

Let k be a field, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. Also let C_{1}, C_{2} be Lie subalgebras of A, with

C_{1} ⊆ C_{2}. (18.7.1)

Thus the quotient C_{2}/C_{1} is defined as a vector space over k, and we let q be the canonical quotient mapping from C_{2} onto C_{2}/C_{1}.

If x ∈ C_{1}, then ad_{x} maps C_{1} and C_{2} into themselves. This leads to a linear mapping ad_{x}^{C_{2}/C_{1}} from C_{2}/C_{1} into itself, with

ad_{x}^{C_{2}/C_{1}} ∘ q = q ∘ ad_{x}. (18.7.2)

It is easy to see that x ↦ ad_{x}^{C_{2}/C_{1}} defines a representation of C_{1}, as a Lie algebra over k, on C_{2}/C_{1}. In particular,

{ad_{x}^{C_{2}/C_{1}} : x ∈ C_{1}} (18.7.3)

is a Lie subalgebra of the Lie algebra gl(C_{2}/C_{1}) of linear mappings from C_{2}/C_{1} into itself, with respect to the usual commutator bracket.

Observe that (ad_{x}^{C_{2}/C_{1}})^{l} = 0 (18.7.4)

for some positive integer l if and only if

q ∘ (ad_{x})^{l} = 0. (18.7.5)

Equivalently, this means that

(ad_{x})^{l}(C_{2}) ⊆ C_{1}. (18.7.6)

In this case, if ad_{x} is nilpotent on C_{1}, then it follows that ad_{x} is nilpotent on C_{2}. Of course, if ad_{x} is nilpotent on C_{2}, then ad_{x}^{C_{2}/C_{1}} is nilpotent on C_{2}/C_{1}.

Suppose that C_{1} ≠ C_{2}, so that C_{2}/C_{1} ≠ {0}, and that ad_{x}^{C_{2}/C_{1}} is nilpotent on C_{2}/C_{1} for each x ∈ C_{1}. Under these conditions, there is a nonzero element of C_{2}/C_{1} that is mapped to 0 by ad_{x}^{C_{2}/C_{1}} for every x ∈ C_{1}, as in Section 9.9. This means that there is a y ∈ C_{2} such that y ∉ C_{1} and

ad_{x}^{C_{2}/C_{1}}(q(y)) = 0 (18.7.7) for every x ∈ X_{1}, and hence q(ad_{x}(y)) = 0 (18.7.8)

for every x ∈ C_{1}. This is the same as saying that

ad_{x}(y) = [x, y]_{A} ∈ C_{1} (18.7.9)

for every x ∈ C_{1}. Equivalently, this means that y is an element of the normalizer N_{A}(C_{1}) of C_{1} in A, as in Section 9.8.

If C_{2} is nilpotent as a Lie algebra over k, then ad_{x} is nilpotent on C_{2} for every x ∈ C_{2}. In particular, this implies that ad_{x}^{C_{2}/C_{1}} is nilpotent on C_{2}/C_{1} for every x ∈ C_{1}, as before. If C_{1} is a proper Lie subalgebra of C_{2}, then it follows that C_{1} is a proper subset of its normalizer N_{C_{2}}(C_{1}) in C_{2}, as in the preceding paragraph. This corresponds to Exercise 7 on p14 of [14].

18.8 Minimal Engel subalgebras

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k. A Lie subalgebra C of A is said to be a Cartan subalgebra if C is nilpotent as a Lie algebra over k, and C is self-normalizing in A, as on p80 of [14], and Definition 1 on p10 of [24]. Remember that the normalizer N_{A}(C) of C in A is the largest Lie subalgebra of A that contains C as an ideal, and that C is self-normalizing in A when N_{A}(C) = C, as in Section 9.8.

Let us suppose from now on in this section that A has finite dimension as a vector space over k, and that the number of elements of k is strictly larger than the dimension of A as a vector space over k. If x ∈ A, then A_{x,0} is the set of y ∈ A such that (ad_{x})^{l}(y) = 0 for some positive integer l, as in Section 18.1. This is a Lie subalgebra of A, which is known as an Engel subalgebra of A, as before. Remember that Engel subalgebras of A are self-normalizing in A, as in Section 18.5.

Suppose that z ∈ A has the property that A_{z,0} is minimal with respect to inclusion among Engel subalgebras of A. In particular, one could take A_{z,0} to be an Engel subalgebra of minimal dimension, as a vector space over k. Under these conditions, A_{z,0} is a Cartan subalgebra of A, as in the theorem on p80 of [14]. This is related to Theorem 1 on p12 of [24] when k = C.

It suffices to show that A_{z,0} is nilpotent as a Lie algebra over k, because A_{z,0} is self-normalizing in A, as before. Note that C = A_{z,0} satisfies the conditions in Section 18.4, because A_{z,0} is minimal, by hypothesis. If x ∈ A_{z,0}, then we get that A_{z,0} ⊆ A_{x,0}, (18.8.1)

as before. Thus, for each y ∈ A_{z,0}, we have that (ad_{x})^{l}(y) = 0 for some positive integer l. This implies that the restriction of ad_{x} to A_{z,0} is nilpotent as a linear mapping from A_{z,0} into itself. Equivalently, this means that x is ad-nilpotent as an element of A_{z,0}, considered as a Lie algebra over k. It follows that A_{z,0} is nilpotent as a Lie algebra over k, as in Section 9.10.

Let C be a Lie subalgebra of A. If x ∈ C, then ad_{C,x} is the same as the restriction of ad_{x} to C. Suppose that C is nilpotent as a Lie algebra over k, so that ad_{C,x} is nilpotent on C for every x ∈ C. This implies that

C ⊆ A_{x,0} (18.8.2)

for every x ∈ C. In particular, if C is an Engel subalgebra of A, then it follows that C is minimal among Engel subalgebras of A.

If C is a Cartan subalgebra of A, then the other part of the theorem on p80 of [14] states that C is a minimal Engel subalgebra of A. This corresponds to Corollary 2 on p13 of [24] when k = C. Of course, (18.8.2) holds for every x ∈ C in this case, because C is nilpotent. Suppose for the sake of a contradiction that for each x ∈ C, C ≠ A_{x,0}. Let z be an element of C be such that A_{z,0} is minimal with respect to inclusion. As usual, one can take z ∈ C such that A_{z,0} has minimal dimension, as a vector space over k. Note that

C ⊆ A_{z,0}, (18.8.3)

by (18.8.2). Under these conditions, we get that (18.8.1) holds for every x ∈ C, as in Section 18.4 again.

Let x ∈ C be given. Of course, ad_{x} maps C into itself, because C is a Lie subalgebra of A. Observe that x ∈ A_{z,0}, by (18.8.3). Thus ad_{x} maps A_{z,0} into itself, because A_{z,0} is a Lie subalgebra of A too. The restriction of ad_{x} to A_{z,0} is nilpotent, because of (18.8.1).

Consider the quotient A_{z,0}/C, as a finite-dimensional vector space over k. If x ∈ C, then ad_{x} induces a linear mapping from A_{z,0}/C into itself, by the remarks in the preceding paragraph. This defines a representation of C, as a Lie algebra over k, on A_{z,0}/C. The mapping induced on A_{z,0}/C by ad_{x} is nilpotent for each x ∈ C, because ad_{x} is nilpotent on A_{z,0}.

By hypothesis, C ≠ A_{z,0}, so that A_{z,0}/C ≠ {0}. It follows that there is a nonzero element of A_{z,0}/C that is mapped to 0 by the mapping induced by ad_{x} for every x ∈ C, as in Section 9.9. Equivalently, this means that there is a y ∈ A_{z,0} such that y ∉ C and

ad_{x}(y) = [x, y]_{A} ∈ C (18.8.4)

for every x ∈ C. This means that y ∈ N_{A}(C), which is a contradiction, because C is supposed to be self-normalizing in A. This is another instance of the argument in the previous section.

18.9 Nilpotent vectors and sums

Let k be a field, and let V be a vector space over k. If T is a linear mapping from V into itself, then let E_{0}(T) be the set of v ∈ V such that T^{l}(v) = 0 for some positive integer l, as in Section 10.7. Remember that this is a linear subspace of 376 CHAPTER 18. CARTAN SUBALGEBRAS

V, as before. Let R be another linear mapping from V into itself, and suppose that R commutes with T. Under these conditions, one can check that E_{0}(R) ∩ E_{0}(T) ⊆ E_{0}(R + T). (18.9.1) More precisely, if v ∈ V and l ∈ Z_{+}, then (R + T)^{l}(v) (18.9.2) can be expressed as a sum of terms of the form R^{l_{1}}(T^{l_{2}}(v)) = T^{l_{2}}(R^{l_{1}}(v)), (18.9.3) where l_{1}, l_{2} are nonnegative integers such that l_{1} + l_{2} = l. If v is an element of the left side of (18.9.1), then (18.9.3) is equal to 0 when l_{1} or l_{2} is sufficiently large. This implies that (18.9.2) is equal to 0 when l is sufficiently large, as desired.

Suppose for the moment that E_{0}(R) = V, (18.9.4) so that E_{0}(T) ⊆ E_{0}(R + T), (18.9.5) by (18.9.1). Similarly, E_{0}(R + T) ⊆ E_{0}(T), (18.9.6) by considering T as (R + T) + (−R). Thus E_{0}(T) = E_{0}(T + R) (18.9.7) under these conditions. If T is diagonalizable on V, then one can check that E_{0}(T) is the same as the kernel of T.

Let (A, [·, ·]_{A}) be a Lie algebra over k. If x ∈ A, then A_{x,0} = E_{0}(ad_{x}) (18.9.8) in the notation of Section 18.1, with V = A as a vector space over k on the right side. Suppose that y ∈ A satisfies [x, y]_{A} = 0, so that ad_{x} and ad_{y} commute on A, as in Section 2.4. This implies that A_{x,0} ∩ A_{y,0} ⊆ A_{x+y,0}, (18.9.9) as in (18.9.1). If A_{y,0} = A, then we get that A_{x,0} = A_{x+y,0}, (18.9.10) as in (18.9.7). If ad_{x} is diagonalizable on A, then A_{x,0} is the same as the kernel of ad_{x} on A, as before. This means that A_{x,0} = C_{A}(x) = C_{A}({x}) (18.9.11) is the centralizer of x in A, as in Section 7.6.

Suppose that k is an algebraically closed field of characteristic 0, and that A is a finite-dimensional semisimple Lie algebra over k. Let x ∈ A be given, and let x = x_{1} + x_{2} be the abstract Jordan decomposition of x in A, as in Section 14.3. Thus x_{1} ∈ A is ad-diagonalizable, x_{2} ∈ A is ad-nilpotent, and [x_{1}, x_{2}]_{A} = 0. It follows that

A_{x,0} = A_{x_{1},0} = C_{A}(x_{1}), (18.9.12)

as in (18.9.10) and (18.9.11). This corresponds to some of the arguments in the proof of the corollary on p80 in [14].

18.10 Cartan subalgebras and semisimplicity

Let k be an algebraically closed field of characteristic 0, and let (A, [·, ·]_{A}) be a finite-dimensional semisimple Lie algebra over k. If B is a maximal toral subalgebra of A, then B is commutative as a Lie algebra, and the centralizer C_{A}(B) of B in A is equal to B, as in Sections 17.1 and 17.4. The normalizer N_{A}(B) of B in A is equal to B too, as in Section 17.14. It follows that B is a Cartan subalgebra of A, as mentioned at the top of p80 of [14].

Conversely, let C be any Cartan subalgebra of A. In particular, C is an Engel subalgebra of A, as in Section 18.8. Thus there is an x ∈ A such that

C = A_{x,0}, (18.10.1)

and one can take x to be ad-diagonalizable, as in the previous section. This means that the linear span of x in A is a toral subalgebra of A. Let B be a maximal toral subalgebra of A that contains x, which can be obtained by taking B to have maximal dimension. Observe that

B ⊆ C_{A}(x) = C, (18.10.2)

because B is commutative as a Lie algebra, as in Section 17.1. However, B is a Cartan subalgebra of A, as in the preceding paragraph, and hence an Engel subagebra of A, as in Section 18.8. It follows that

B = C, (18.10.3)

because C is a minimal Engel subalgebra of A, as in Section 18.8. This is the corollary on p80 of [14].

Alternatively, suppose for the moment that k is any field of characteristic 0, and that A is a finite-dimensional semisimple Lie algebra over k. Suppose also that (18.10.1) is a Cartan subalgebra of A for some x ∈ A. Remember that the Killing form on A is nondegenerate, as in Section 10.13. This implies that the restriction of the Killing form on A to C is nondegenerate, as in Section 18.6, and part (d) of Theorem 3 on p15 of [24].

Remember that the collection of linear mappings on A of the form ad_{w}, w ∈ A, is a Lie subalgebra of the Lie algebra gl(A) of all linear mappings from A into itself, with respect to the usual commutator bracket. This is the same as the image of A under the adjoint representation. Similarly,

{ad_{w} : w ∈ C} (18.10.4)

is a Lie subalgebra of the Lie algebra just mentioned, because C is a Lie subalgebra of A. More precisely, (18.10.4) is nilpotent as a Lie algebra, because C is nilpotent, by hypothesis. In particular, (18.10.4) is solvable as a Lie algebra over k.

If w, y, z ∈ C, then it follows that

tr_{A}(ad_{w} ∘ ad_{[y,z]_{A}}) = 0, (18.10.5)

as in Section 10.5. This uses the fact that ad_{[y,z]_{A}} = [ad_{y}, ad_{z}], as in Section 2.4. This implies that [y, z]_{A} = 0 (18.10.6)

for every y, z ∈ C, because the restriction of the Killing form on A to C is nondegenerate, as before. This shows that C is commutative as a Lie algebra over k, as in part (a) of Theorem 3 on p15 of [24].

It follows that C is contained in its centralizer in A. The centralizer of C in A is automatically contained in the normalizer N_{A}(C) of C in A, as in Section 17.14. In this situation, N_{A}(C) = C, so that C is equal to its centralizer in A. This is part (b) of Theorem 3 on p80 of [24]. This implies that C is maximal among commutative Lie subalgebras of A, as in Corollary 1 on p15 of [24].

Suppose now that k is also algebraically closed again. Let w ∈ C be given, and let w = w_{1} + w_{2} (18.10.7)

be the abstract Jordan decomposition of w in A, as in Section 14.3. This means that w_{1} ∈ A is ad-diagonalizable, w_{2} ∈ A is ad-nilpotent, and [w_{1}, w_{2}]_{A} = 0. Let y ∈ C be given, so that [w, y]_{A} = 0, and hence ad_{w} commutes with ad_{y} on A, as in Section 2.4. This implies that ad_{y} commutes with ad_{w_{1}} and ad_{w_{2}}, as in Section 10.8. It follows that

[w_{1}, y]_{A} = [w_{2}, y]_{A} = 0. (18.10.8)

More precisely, this uses the fact that the kernel of the adjoint representation of A is the center of A as a Lie algebra, which is trivial in this case, because A is semisimple by hypothesis.

Thus ad_{w_{2}} commutes with ad_{y} on A, as in Section 2.4. This implies that ad_{w_{2}} ∘ ad_{y} is nilpotent on A, because ad_{w_{2}} is nilpotent on A. It follows that

tr_{A}(ad_{w_{2}} ∘ ad_{y}) = 0. (18.10.9)

This shows that w_{2} = 0, because the restriction of the Killing form on A to C is nondegenerate, as before. This means that w = w_{1} is ad-diagonalizable as an element of A, as in part (c) of Theorem 3 on p15 of [24].

The hypothesis that (18.10.1) be a Cartan subalgebra of A really means that C is nilpotent as a Lie algebra over k, because Engel subalgebras of A are self-normalizing, as in Section 18.5. In the previous arguments, it is enough to ask that C be solvable as a Lie algebra over k, so that (18.10.4) is solvable as a Lie algebra. The same conclusion is mentioned in Exercise 2 on p81 of [14].

18.11 Subalgebras and homomorphisms

Let k be a field, and let ([·, ·]_{A}) be a Lie algebra over k. If C is a Cartan subalgebra of A, then C is maximal among nilpotent Lie subalgebras of A. This is the first part of Exercise 4 on p81 in [14], and it follows from the statement mentioned at the end of Section 18.7.

If C_{1}, C_{2} are Cartan subalgebras of A with

C_{1} ⊆ C_{2}, (18.11.1)

then C_{1} = C_{2}, (18.11.2)

because C_{1} is maximal among nilpotent Lie subalgebras of A, as in the preceding paragraph. Alternatively, if C_{1}, C_{2} are Engel subalgebras of A, and C_{2} is minimal among Engel subalgebras of A, then (18.11.1) implies (18.11.2).

Let Ã and C be Lie subalgebras of A, with

C ⊆ Ã. (18.11.3)

It is easy to see that

N_{A}(C) = N_{A}(C) ∩ Ã, (18.11.4)

where N_{A}(C), N_{A}(C) are the normalizers of C in A and Ã, respectively. In particular, if C is self-normalizing in A, then C is self-normalizing in Ã. If C is a Cartan subalgebra of A, then it follows that C is a Cartan subalgebra of Ã.

Let (A′, [·, ·]_{A′}) be another Lie algebra over k, and let ϕ be a Lie algebra homomorphism from A onto A′. If x ∈ A and x′ ∈ A′, then we put

ad_{A,x}(y) = [x, y]_{A}, ad_{A′,x′}(y′) = [x′, y′]_{A′} (18.11.5)

for every y ∈ A, y′ ∈ A′, as usual. Thus

ϕ(ad_{A,x}(y)) = ϕ([x, y]_{A}) = [ϕ(x), ϕ(y)]_{A′} = ad_{A′,ϕ(x)}(ϕ(y)) (18.11.6)

for every x, y ∈ A.

Let C′ be a Lie subalgebra of A′. If x ∈ A, then it is easy to see that

ad_{A′,ϕ(x)}(C′) ⊆ C′ (18.11.7)

if and only if ad_{A,x}(ϕ^{−1}(C′)) ⊆ ϕ^{−1}(C′), (18.11.8) using (18.11.6). This means that ϕ(x) is an element of the normalizer N_{A′}(C′) of C′ in A′ if and only if x is an element of the normalizer N_{A}(ϕ^{−1}(C′)) of ϕ^{−1}(C′) in A. It follows that C′ is self-normalizing in A′ if and only if ϕ^{−1}(C′) is self-normalizing in A.

Suppose from now on in this section that A has finite dimension as a vector space over k, so that the dimension of A′ is finite as well. If ϕ^{−1}(C′) contains an Engel subalgebra of A, then ϕ^{−1}(C′) is self-normalizing in A, as in Section 18.5.

Let C be a Lie subalgebra of A again, so that ϕ(C) is a Lie subalgebra of A′. Note that ϕ^{−1}(ϕ(C)) is a Lie subalgebra of A, which is spanned by C and the kernel of ϕ. If C contains an Engel subalgebra of A, then ϕ^{−1}(ϕ(C)) contains the same Engel subalgebra of A.

Suppose for the rest of the section that the number of elements of k is strictly larger than the dimension of A as a vector space over k. If C is a Cartan subalgebra of A, then C is an Engel subalgebra of A, as in Section 18.8. It follows that ϕ(C) is self-normalizing in A′, as in the previous paragraphs. We also have that C is nilpotent as a Lie algebra over k, so that ϕ(C) is nilpotent too, as in Section 9.5. This shows that ϕ(C) is a Cartan subalgebra of A′, as in Lemma A on p81 of [14].

Suppose now that C′ is a Cartan subalgebra of A′, and that C is a Cartan subalgebra of ϕ^{−1}(C′), as a Lie algebra over k. We would like to show that C is a Cartan subalgebra of A too, as in Lemma B on p81 of [14]. Observe that ϕ(C) is a Cartan subalgebra of ϕ(ϕ^{−1}(C′)) = C′, as in the preceding paragraph. We also have that C′ is a Cartan subalgebra of itself, because C′ is nilpotent as a Lie algebra. This implies that

ϕ(C) = C′, (18.11.9)

as in (18.11.2).

Suppose that x is an element of the normalizer N_{A}(C) of C in A, and let us check that x ∈ C. It is easy to see that ϕ(x) is in the normalizer N_{A′}(ϕ(C)) of ϕ(C) in A′. This implies that ϕ(x) ∈ ϕ(C), because of (18.11.9) and the hypothesis that C′ be a Cartan subalgebra of A′. Thus

x ∈ ϕ^{−1}(ϕ(C)) = ϕ^{−1}(C′). (18.11.10)

It follows that x ∈ C, as desired, because C is a Cartan subalgebra of ϕ^{−1}(C′), by hypothesis.

18.12 Unions of hyperplanes

Let k be a field, and let V be a nonzero vector space over k. By a hyperplane in V we mean a linear subspace W of codimension 1 in V, so that V is spanned by W and a single vector in V that is not in W. Equivalently, this means that W is the kernel of a nonzero linear functional on V. If W_{1}, …, W_{r} are finitely many hyperplanes in V, and if r is less than or equal to the number of elements of k, then we would like to check that ⋃_{j=1}^{r} W_{j} is a proper subset of V. Note that ⋂_{j=1}^{r} W_{j} is a linear subspace of V with codimension less than or equal to r. Thus the quotient space V/ ⋂_{j=1}^{r} W_{j} is a vector space over k of dimension less than or equal to r. We may as well suppose that the dimension of V is finite, since otherwise we could replace V with this quotient.

Of course, k^{2} = k × k is a two-dimensional vector space over k, with respect to coordinatewise addition and scalar multiplication. The lines in k^{2} are characterized by their slope in k, except for the second coordinate axis. The number of distinct lines in k^{2} is the number of elements of k plus 1, which is to say that there are infinitely many distinct lines in k^{2} when k has infinitely many elements. These lines are pairwise disjoint away from 0. It follows that k^{2} cannot be expressed as the union of finitely many lines, where the number of lines is less than or equal to the number of elements of k.

If V has dimension one, then {0} is the only hyperplane in V, and the statement under consideration is trivial. If V has dimension at least two, then one can find at least as many distinct hyperplanes in V as lines in k^{2}. In particular, there is a hyperplane Z in V that is different from W_{j} for each j = 1, …, r. Thus Z ∩ W_{j} is a hyperplane in Z for each j = 1, …, r. Note that the dimension of Z is one less than the dimension of V, which is strictly less than the dimension of V, because the dimension of V is finite. This permits us to use induction to get that the union of Z ∩ W_{j}, 1 ≤ j ≤ r, is a proper subset of Z. This implies that the union of W_{j}, 1 ≤ j ≤ r, is a proper subset of V, as desired.

If a is a nonzero element of V, then

V′_{a} = {λ ∈ V′ : λ(a) = 0} (18.12.1)

is a hyperplane in the dual space V′ of linear functionals on V. If A is a nonempty finite set of nonzero elements of V, and if the number of elements of A is less than or equal to the number of elements in k, then it follows that the union of V′_{a}, a ∈ A, is a proper subset of V′. This means that there is a linear functional λ on V such that λ(a) ≠ 0 (18.12.2)

for every a ∈ A.

18.13 Centralizers and diagonalizability

Let k be a field, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. Also let B be a nonzero Lie subalgebra of A such that every element of B is ad-diagonalizable as an element of A. This implies that B is commutative as a Lie algebra over k, as in Section 17.1. Let B′ be the dual of B, as a vector space over k, as usual. If α ∈ B′, then we let A_{α} be the set of x ∈ A such that ad_{w}(x) = [w, x]_{A} = α(w) x for every w ∈ B, as before.

Thus A_{0} is the same as the centralizer C_{A}(B) of B in A. Of course, B is contained in C_{A}(B), because B is commutative as a Lie algebra. Let Φ_{B} be the 382 CHAPTER 18. CARTAN SUBALGEBRAS set of α ∈ B′ such that α ≠ 0 and A_{α} ≠ {0}, as in Section 17.2. Remember that A corresponds to the direct sum of A_{α}, α ∈ Φ_{B} ∪ {0}, as before.

Let w ∈ B be given, and let C_{A}(w) = C_{A}({w}) be the centralizer of w in A, which is the set of x ∈ A such that [w, x]_{A} = 0. This corresponds to the direct sum of A_{α}, where α ∈ Φ_{B} ∪ {0} satisfies α(w) = 0. (18.13.1) Of course, this holds automatically when α = 0. If α(w) ≠ 0 (18.13.2) for every α ∈ Φ_{B}, then we get that C_{A}(w) = A_{0} = C_{A}(B). (18.13.3) This is related to the second part of Exercise 7 on p41 of [14].

Root systems

19.1 Symmetries and reflections

Let V be a finite-dimensional vector space over the real numbers, and let α be a nonzero element of V. A one-to-one linear mapping σ from V onto itself is said to be a symmetry with vector α if it satisfies the following two conditions. First, σ(α) = −α. (19.1.1) Second, the linear subspace H = {v ∈ V : σ(v) = v} (19.1.2) of V is a hyperplane, which is to say that it has codimension one in V. Let L_{α} = {t α : t ∈ R} (19.1.3) be the line in V passing through α. Note that L_{α} + H = V, L_{α} ∩ H = {0}, (19.1.4) so that L_{α} and H are complementary linear subspaces of V. It is easy to see that σ is uniquely determined by L_{α} and H. One can also get a symmetry on V from any line and complementary hyperplane in V, where the symmetry is the same as multiplication by −1 on the line, and the identity mapping on the hyperplane.

Let V′ be the dual space of linear functionals on V, as a vector space over R, as usual. In this situation, there is a unique λ_{α} ∈ V′ such that λ_{α}(α) = 2 (19.1.5) and (19.1.6) λ_{α}(v) = 0 for every v ∈ H.

Observe that σ(v) = v − λ_{α}(v) α (19.1.7)

for every v ∈ V. Conversely, if α ∈ V and λ_{α} ∈ V′ satisfy (19.1.5), then (19.1.7) defines a symmetry on V with vector α, for which the kernel of λ_{α} is the fixed-point set. These remarks correspond to the discussion on p24 of [24].

Suppose now that (v, w) is an inner product on V. Let σ be a symmetry on V with vector α and fixed hyperplane H as in (19.1.2) again. Suppose that α is orthogonal to H, so that (v, α) = 0 (19.1.8)

for every v ∈ H. Let us say that σ is a reflection on V with respect to (·, ·) in this case, as on p42 of [14]. One can check that σ is an orthogonal transformation on V with respect to (·, ·) under these conditions, which is to say that

(σ(v), σ(w)) = (v, w) (19.1.9)

for every v, w ∈ V.

Conversely, suppose that σ is a symmetry on V with vector α, and that σ is an orthogonal transformation on V with respect to (·, ·). If v is an element of the hyperplane H in (19.1.2), then

(v, α) = (σ(v), σ(α)) = −(v, α). (19.1.10)

This implies (19.1.8), so that σ is a reflection on V. This is related to some of the remarks on p28 of [24].

Let α ∈ V, α ≠ 0, be given, and let

H_{α} = {v ∈ V : (v, α) = 0} (19.1.11)

be the hyperplane in V orthogonal to α. Put

μ_{α}(v) = 2 (v, α) (α, α)^{−1} (19.1.12)

for every v ∈ V, which defines a linear functional on V. By construction,

μ_{α}(α) = 2, (19.1.13)

and (19.1.11) is the kernel of μ_{α}. The reflection associated to α on V is given by σ(v) = v − μ_{α}(v) α = v − 2 (v, α) (α, α)^{−1} α (19.1.14)

for every v ∈ V, as in (19.1.7). This corresponds to some of the remarks on p42 of [14], and p28 of [24].

19.2 Invariant inner products

Let V be a vector space over the real numbers of positive finite dimension, and let G be a finite subgroup of the general linear group GL(V) of invertible linear mappings on V. Under these conditions, it is well known that there is an inner product (v, w) on V that is invariant under G, so that the elements of G are orthogonal transformations on V with respect to (v, w). More precisely, if (v, w)_{0} is any inner product on V, then one can verify that

(v, w) = ∑_{g∈G}(g(v), g(w))_{0} (19.2.1)

is an inner product on V that is invariant under the elements of G.

Let A be a nonempty subset of V, and let G_{A} be the collection of one-to-one linear mappings T from V onto itself such that

T (A) = A. (19.2.2)

Of course, G_{A} is a subgroup of GL(V). The mapping from T ∈ G_{A} to the restriction of T to A defines a group homomorphism from G_{A} into the group of one-to-one mappings from A onto itself. Suppose that the linear span of A in V is equal to V. If T ∈ G_{A} is equal to the identity mapping on A, then it follows that T is the identity mapping on V. If A has only finitely many elements, then G_{A} has only finitely many elements too. This implies that there is an inner product on V that is invariant under G_{A}, as in the preceding paragraph.

Let α be a nonzero element of V, and let A be a finite subset of V whose linear span in V is equal to V again. Under these conditions, there is at most one symmetry σ on V with vector α that maps A onto itself, as in the lemma on the top of p25 in [24]. To see this, let (v, w) be an inner product on V that is invariant under G_{A}, as in the preceding paragraph. Thus σ is an orthogonal transformation on V with respect to (v, w), because σ ∈ G_{A}, by hypothesis. It follows that σ is the reflection on V associated to α with respect to (v, w), as in the previous section.

Another version of this lemma is stated on p42 of [14], in terms of reflections. In this version, V is equipped with an inner product (v, w)_{V}, and A is a finite subset of V whose linear span in V is equal to V again. Let σ be a symmetry on V with vector α ∈ A, which is not necessarily a reflection on V with respect to (v, w)_{V}. Suppose that σ(A) = A, and that the reflection σ_{α} on V associated to α with respect to (v, w)_{V} maps A onto itself. Under these conditions, σ = σ_{α}.

This follows from the previous version, because σ_{α} is a symmetry on V with vector α. Note however that (v, w)_{V} is not necessarily invariant under G_{A}. Of course, any symmetry on V with vector α is the reflection associated to α with respect to some inner product on V.

19.3 Defining root systems

Let V be a vector space over the real numbers of positive finite dimension. A subset A of V is said to be a root system in V if it satisfies the following three conditions, as in Definition 1 on p25 of [24]. First, A has only finitely many elements, the linear span of A in V is equal to V, and 0 ∉ A. Second, for each α ∈ A, there is a symmetry σ_{α} on V with vector α such that

σ_{α}(A) = A. (19.3.1)

Note that σ_{α} is uniquely determined by these properties, as in the previous section. Third, for every α, β ∈ A,

(19.3.2) σ_{α}(β) − β is an integer multiple of α.

The dimension of V may be called the rank of A, and the elements of A may be called roots of V with respect to A.

If α ∈ A, then there is a unique linear functional λ_{α} on V such that λ_{α}(α) = 2 and λ_{α} = 0 on the hyperplane in V consisting of vectors fixed by σ_{α}, as in Section 19.1. Remember that σ_{α}(v) = v − λ_{α}(v) α (19.3.3)

for every v ∈ V, as in (19.1.7). Using this, the third condition in the preceding paragraph can be reformulated as saying that for every α, β ∈ A,

λ_{α}(β) ∈ Z. (19.3.4)

If α ∈ A, then we also have that

−α ∈ A, (19.3.5)

by (19.3.1) and the fact that s_{α}(α) = −α.

A root system A in V is said to be reduced if for every α ∈ A, the only elements of A that are proportional to α are α and −α, as in Definition 2 on p25 of [24]. Otherwise, suppose that α ∈ A and t α ∈ A for some t ∈ R with t ≠ ±1. Of course, t ≠ 0, because 0 ∉ A, and we may as well suppose that |t| < 1, since otherwise we could interchange the roles of α and t α. Using (19.3.4), we get that 2 t ∈ Z, because λ_{α}(α) = 2. This means that t = ±1/2.

In the definition of a root system on p42 of [14], one supposes that V is already equipped with an inner product (v, w), and that the symmetries σ_{α} are reflections with respect to this inner product. This means that

λ_{α}(v) = 2 (v, α) (α, α)^{−1} (19.3.6)

for every α ∈ A and v ∈ V, as in Section 19.1. In this situation, (19.3.4) is the same as saying that 2 (β, α) (α, α)^{−1} ∈ Z (19.3.7)

for every α, β ∈ A. The condition that the root system be reduced is also included in the definition, as discussed further on p43 of [14]. If A is a root system in V as before, then one can find a compatible inner product on V, as in the next section.

Of course, we have seen properties like these in connection with Lie algebras, in Chapter 17. This corresponds to the remarks following the theorem on p40 of [14], and part (a) of Theorem 2 on p43 of [24]. This will be discussed further in Section 22.1. Some classical examples of root systems will be discussed starting in Section 20.10.

19.4 The Weyl group

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. The Weyl group of A is defined to be the subgroup W of GL(V) generated by the symmetries σ_{α}, α ∈ A, as in Definition 3 on p27 of [24]. Let G_{A} be the subgroup of GL(V) consisting of invertible linear mappings T on V that map A onto itself, as in Section 19.2. Thus

W ⊆ G_{A}, (19.4.1)

because σ_{α} ∈ G_{A} for every α ∈ A, by definition of a root system. Remember that G_{A} has only finitely many elements in this situation, because A is a finite subset of V that spans V.

More precisely, W is a normal subgroup of G_{A}, as mentioned on p27 of [24]. To see this, let α ∈ A and T ∈ G_{A} be given. It is easy to see that T ∘ σ_{α} ∘ T^{−1} is a symmetry on V with vector T (α). We also have that

(T ∘ σ_{α} ∘ T^{−1})(A) = A, (19.4.2)

because σ_{α} and T both map A onto itself. This implies that

T ∘ σ_{α} ∘ T^{−1} = σ_{T (α)}, (19.4.3)

because T (α) ∈ A, as in the previous section. It follows that W is invariant under conjugation by T, as desired. Note that (19.4.3) corresponds to the first part of the lemma on p43 of [14], just after the definition of the Weyl group.

Let α ∈ A be given again, and let λ_{α} ∈ V′ be as in the previous section, so that σ_{α} can be expressed as in (19.3.3). If T ∈ GL(V), then it follows that

(T ∘ σ_{α} ∘ T^{−1})(v) = v − λ_{α}(T^{−1}(v)) T (α) (19.4.4)

for every v ∈ V. Suppose that T ∈ G_{A}, and let λ_{T (α)} be the linear functional on V corresponding to T (α) ∈ A in the same way. Thus

σ_{T (α)}(v) = v − λ_{T (α)}(v) T (α) (19.4.5)

for every v ∈ V, as before. This means that

λ_{T (α)}(v) = λ_{α}(T^{−1}(v)) (19.4.6)

for every v ∈ V, because of (19.4.3).

Because the Weyl group W has only finitely many elements, one can find an inner product on V that is invariant under W as in Section 19.2. This corresponds to Proposition 1 on p27 of [24]. The choice of such an inner product is sometimes included in the definition of a root system, as in [14], and mentioned on p28 of [25].

Let (v, w) be an inner product on V that is invariant under W. If α ∈ A, then λ_{α} ∈ V′ can be expressed in terms of (·, ·) as in (19.3.6). Similarly, if T ∈ G_{A}, then T (α) ∈ A, and

λ_{T (α)}(v) = 2 (v, T (α)) (T (α), T (α))^{−1} (19.4.7) for every v ∈ V. Combining this with (19.4.6), we get that

(v, T (α)) (T (α), T (α))^{−1} = (T^{−1}(v), α) (α, α)^{−1} (19.4.8)

for every v ∈ V. Equivalently, this means that

(T (u), T (α)) (T (α), T (α))^{−1} = (u, α) (α, α)^{−1} (19.4.9)

for every u ∈ V. This corresponds to the second part of the lemma on p43 of [14]. Note that this holds automatically when (·, ·) is invariant under G_{A}.

19.5 Isomorphic root systems

Let V and Ṽ be vector spaces over the real numbers with the same positive finite dimension, and let A and Ã be root systems in V and Ṽ, respectively. Let us say that a one-to-one linear mapping T from V onto Ṽ is an isomorphism between these root systems if T (A) = Ã. (19.5.1)

In particular, an invertible linear mapping on V is an automorphism of the root system A if it maps A onto itself, as on p27 of [24].

Let T : V → Ṽ be an isomorphism between the root systems A and Ã, and let α ∈ A be given. If σ_{α} is the corresponding symmetry on V with vector α, then it is easy to see that T ∘ σ_{α} ∘ T^{−1} is a symmetry on Ṽ with vector T (α). Observe that (T ∘ σ_{α} ∘ T^{−1})(Ã) = Ã, (19.5.2)

because of (19.5.1) and the fact that σ_{α}(A) = A. It follows that

T ∘ σ_{α} ∘ T^{−1} = σ̃_{T (α)}, (19.5.3)

where the right side is the symmetry on Ṽ corresponding to T (α) ∈ Ã as in Section 19.3.

Of course, if τ is a one-to-one linear mapping from V onto itself, then

T ∘ τ ∘ T^{−1} (19.5.4)

is a one-to-one linear mapping from Ṽ onto itself. More precisely,

τ ↦ T ∘ τ ∘ T^{−1} (19.5.5)

defines a group isomorphism from GL(V) onto GL(Ṽ). This isomorphism sends the Weyl group W of A onto the Weyl group W̃ of Ã, because of (19.5.3).

Let λ_{α} be the linear functional on V corresponding to σ_{α} as before, so that σ_{α} can be expressed as in (19.3.3). Thus

(T ∘ σ_{α} ∘ T^{−1})(ṽ) = ṽ − λ_{α}(T^{−1}(ṽ)) T (α) (19.5.6)

19.6. DUALITY AND SYMMETRIES 389

for every ṽ ∈ Ṽ. Let λ̃_{T (α)} be the linear functional on Ṽ that corresponds to T (α) ∈ Ã in the same way, so that σ̃_{T (α)}(ṽ) = ṽ − λ̃_{T (α)}(ṽ) T (α) (19.5.7) for every ṽ ∈ Ṽ. It follows that λ̃_{T (α)}(ṽ) = λ_{α}(T^{−1}(ṽ)) (19.5.8) for every ṽ ∈ Ṽ, by (19.5.3). Equivalently, λ̃_{T (α)}(T (v)) = λ_{α}(v) (19.5.9) for every v ∈ V.

Suppose that (·, ·)_{V} and (·, ·)_{V} are inner products on V and Ṽ that are compatible with the root systems A and Ã, respectively, in the sense that σ_{α} and σ̃_{α} are reflections on V and Ṽ with respect to (·, ·)_{V} and (·, ·)_{V} for every α ∈ A and α̃ ∈ Ã, respectively. Thus λ_{α}(v) = 2 (v, α)_{V} (α, α)_{V}^{−1} (19.5.10) for every α ∈ A and v ∈ V, as in (19.3.6), and similarly

λ̃_{α}(ṽ) = 2 (ṽ, α̃)_{V} (α̃, α̃)−1 ; _{V} (19.5.11)

for every α̃ ∈ Ã and ṽ ∈ Ṽ. In this case, (19.5.9) is the same as saying that

(T (v), T (α))_{V} (T (α), T (α))−1 ; _{V} = (v, α)_{V} (α, α)_{V}^{−1} (19.5.12)

for every α ∈ A and v ∈ V. This is related to some of the remarks on p43 of [14]. In particular, (19.5.12) holds automatically when

(T (u), T (v))_{V} = (u, v)_{V} (19.5.13)

for every u, v ∈ V.

19.6 Duality and symmetries

Let V be a vector space over the real numbers of positive finite dimension, and let V′ be the dual space of all linear functionals on V. Remember that V′ has the same dimension as V. If v ∈ V and μ ∈ V′, then put L_{v}(μ) = μ(v). (19.6.1) This defines L_{v} as a linear functional on V′, so that L_{v} is an element of the dual space V′′ = (V′)′ of V′. More precisely, v ↦ L_{v} is a one-to-one linear mapping from V onto V′′.

390 CHAPTER 19. ROOT SYSTEMS

If T is a linear mapping from V into itself and μ ∈ V′, then T′(μ) = μ ∘ T (19.6.2) defines a linear functional on V too. Equivalently, (T′(μ))(v) = μ(T (v)) (19.6.3) for every v ∈ V. This defines T′ as a linear mapping from V′ into itself, which is the dual linear mapping associated to T. If T_{1}, T_{2} are linear mappings from V into itself, then (T_{1} ∘ T_{2})′ = T′_{2} ∘ T′_{1}, (19.6.4) as linear mappings from V′ into itself. Note that (I_{V})′ = I_{V′}, (19.6.5) where I_{V} and I_{V′} are the identity mappings on V and V′, respectively.

If T is a linear mapping from V into itself again, then T′ is a linear mapping from V′ into itself, which leads to a dual mapping T′′ = (T′)′ from V′′ into itself. Thus, if L ∈ V′′, then T′′(L) = L ∘ T′, (19.6.6) as a linear functional on V′. This means that (T′′(L))(μ) = L(T′(μ)) = L(μ ∘ T) (19.6.7) for every μ ∈ V′. If v ∈ V and L_{v} ∈ V′′ is as in (19.6.1), then it follows that (T′′(L_{v}))(μ) = L_{v}(μ ∘ T) = μ(T (v)) = L_{T (v)}(μ) (19.6.8) for every λ ∈ V′. This means that T′′(L_{v}) = L_{T (v)} (19.6.9) for every v ∈ V.

Let z ∈ V and τ ∈ V′ be given, so that T_{z,τ} (v) = τ(v) z (19.6.10) defines a linear mapping from V into itself. If μ ∈ V′, then (T′_{z,τ} (μ))(v) = μ(T_{z,τ} (v)) = τ(v) μ(z) = τ(v) L_{z}(μ) (19.6.11) for every v ∈ V. Equivalently, this means that T′_{z,τ} (μ) = L_{z}(μ) τ (19.6.12) for every μ ∈ V′.

Let σ be a symmetry on V with vector α ∈ V. Thus there is a linear functional λ_{α} on V such that λ_{α}(α) = 2 and σ(v) = v − λ_{α}(v) α for every v ∈ V, as in Section 19.1. This is the same as saying that

σ = I_{V} − T_{α,λ_{α}}, (19.6.13)

where T_{α,λ_{α}} is as in (19.6.10). The corresponding dual linear mapping is given by σ′(μ) = μ − L_{α}(μ) λ_{α} (19.6.14)

for every μ ∈ V′, by (19.6.5) and (19.6.12). Note that

L_{α}(λ_{α}) = λ_{α}(α) = 2. (19.6.15)

It follows that σ′ is a symmetry on V′ with vector λ_{α}, as in Section 19.1. Alternatively, one can check more directly that

σ′(λ_{α}) = λ_{α} ∘ σ = −λ_{α}, (19.6.16)

because σ(α) = −α and the kernel of λ_{α} is the fixed-point set of σ. If μ ∈ V′ satisfies μ(α) = 0, then one can verify that

σ′(μ) = μ ∘ σ = μ, (19.6.17)

because σ is equal to the identity on a hyperplane in V complementary to the line passing through α.

19.7 Adjoints and reflections

Let V be a vector space over the real numbers of positive finite dimension again, and suppose that (v, w) is an inner product on V. If T is a linear mapping from V into itself, then there is a unique adjoint linear mapping T^{∗} from V into itself such that (T (v), w) = (v, T^{∗}(w)) (19.7.1)

for every v, w ∈ V, as before. Remember that

(T_{1} ∘ T_{2})^{∗} = T_{2}^{∗} ∘ T_{1}^{∗} (19.7.2)

for all linear mappings T_{1} and T_{2} from V into itself, and that I_{V}^{∗} = I_{V}. If T is any linear mapping from V into itself, then (T^{∗})^{∗} = T. We also have that T is an orthogonal transformation on V with respect to (·, ·), in the sense that

(T (v), T (w)) = (v, w) (19.7.3)

for every v, w ∈ V, if and only if T is invertible on V, with T^{−1} = T^{∗}.

If u ∈ V, then μ_{u}(v) = (v, u) (19.7.4) defines a linear functional on V, and u ↦ μ_{u} is a one-to-one linear mapping from V onto V′. If T is a linear mapping from V into itself, then it is easy to see that T′(μ_{u}) = μ_{T^{∗}(u)} (19.7.5)

for every u ∈ V. Let u, z ∈ V be given, and put

T_{z,u}(v) = (v, u) z (19.7.6)

for every v ∈ V. This defines a linear mapping from V into itself, which corresponds to (19.6.10), with τ = μ_{u}. Observe that

(T_{z,u}(v), w) = (v, u) (z, w) = (v, T_{u,z}(w)) (19.7.7)

for every v, w ∈ V, so that (T_{z,u})^{∗} = T_{u,z}. (19.7.8)

Let α ∈ V with α ≠ 0 be given, and put

α̂ = 2 α (α, α)^{−1}. (19.7.9)

Thus

μ_{α}(v) = (v, α̂) = 2 (v, α) (α, α)^{−1} (19.7.10)

for every v ∈ V, in the notation of (19.7.4). Note that

μ_{α}(α) = 2, (19.7.11)

and that

σ_{α}(v) = v − μ_{α}(v) α = v − 2 (v, α) (α, α)^{−1} α (19.7.12)

is the reflection on V associated to α with respect to (·, ·), as in Section 19.1. It is easy to see that σ_{α}^{∗} = σ_{α}, (19.7.13)

using (19.7.8). This also corresponds to the fact that σ_{α} is an orthogonal transformation on V, and its own inverse.

Observe that σ_{α} may be considered as the reflection on V associated to α̂ with respect to (·, ·) as well, so that

σ_{α} = σ_{α}. (19.7.14)

Indeed, σ_{α} only depends on the line in V passing through α. Put

ˆ(α̂) = 2 α̂ (α̂, α̂)^{−1}, (19.7.15)

as in (19.7.9). Clearly

(α̂, α̂) = 4 (α, α) (α, α)^{−2} = 4 (α, α)^{−1}, (19.7.16)

so that ˆ(α̂) = 4 α (α, α)^{−1} (4 (α, α)^{−1})^{−1} = α. (19.7.17)

Using (19.7.12), we have that

σ_{α}(v) = v − μ_{α}(v) α̂ = v − μ_{(_{α})}(v) α̂ (19.7.18)

for every v ∈ V, which is another way to look at (19.7.14).

Let β ∈ V with β ≠ 0 be given, and put β̂ = 2 β (β, β)^{−1}, as in (19.7.9). If T is an orthogonal transformation on V with respect to (·, ·), then

T (β̂) = 2 T (β) (β, β)^{−1} = 2 T (β) (T (β), T (β))^{−1} = T̂ (β), (19.7.19)

where T̂ (β) is defined as in (19.7.9) too.

19.8 Inverse roots

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. If α ∈ A, then there is a unique symmetry σ_{α} on V with vector α that maps A onto itself, as in Section 19.3. Let λ_{α} be the linear functional on V such that λ_{α}(α) = 2 and λ_{α} = 0 on the hyperplane of vectors in V fixed by σ_{α}, so that

σ_{α}(v) = v − λ_{α}(v) α (19.8.1)

for every v ∈ V, as before. As on p25 of [24], λ_{α} may be called the inverse root associated to α in the dual space V′ of V. Let

A′ = {λ_{α} : α ∈ A} (19.8.2)

be the set of these inverse roots, which is a finite subset of V′ that does not contain 0.

Let (v, w) be an inner product on V that is invariant under the Weyl group W of A, as in Section 19.4. Thus σ_{α} is the reflection on V associated to α with respect to (v, w), as in Section 19.1. In this situation,

λ_{α}(v) = μ_{α}(v) = (v, α̂) = 2 (v, α) (α, α)^{−1} (19.8.3)

for every v ∈ V, using the notation in (19.7.4) and (19.7.9). Put

Â = {α̂ : α ∈ A} = {2 α (α, α)^{−1} : α ∈ A}, (19.8.4)

which is a finite subset of V that does not contain 0. It is easy to see that the linear span of (19.8.4) in V is equal to V, because of the corresponding property of A. Note that (19.8.4) corresponds to (19.8.2) under the isomorphism u ↦ μ_{u} from V onto V′ associated to the inner product. It follows that the linear span of A′ in V′ is equal to V′.

If β ∈ V, β ≠ 0, and β̂ is as in (19.7.9), then

σ_{α}(β̂) = ˆ (σ_{α}(β)), (19.8.5)

394 CHAPTER 19. ROOT SYSTEMS where ˆ (σ_{α}(β)) is defined as in (19.7.9) too. This follows from (19.7.19), because σ_{α} is an orthogonal transformation on V with respect to (·, ·). This implies that σ_{α}(Â) = Â, (19.8.6) because σ_{α}(A) = A. This is the same as saying that

σ_{α}(Â) = Â, (19.8.7)

because σ_{α} = σ_{α}, as in (19.7.14). Using (19.8.6), we get that σ′_{α}(A′) = A′, (19.8.8) where σ′_{α} is the dual linear mapping on V′ associated to σ_{α}. This also uses the correspondence between (19.8.2) and (19.8.4) mentioned earlier, the correspondence between adjoints and dual linear mappings in (19.7.5), and the self-adjointness of σ_{α}, as in (19.7.13). Remember that σ′_{α} is a symmetry on V′ with vector λ_{α}, as in Section 19.6.

If μ ∈ V′, then σ′_{α}(μ) = μ − L_{α}(μ) λ_{α} = μ − μ(α) λ_{α}, (19.8.9) as in (19.6.1) and (19.6.14). Let β ∈ A be given, and let λ_{β} be the corresponding element of (19.8.2). We would like to check that (19.8.10) σ′_{α}(λ_{β}) − λ_{β} is an integer multiple of λ_{α}. This is the same as saying that λ_{β}(α) ∈ Z, (19.8.11) by (19.8.9). This condition holds because A is a root system in V, as in Section 19.3. This shows that A′ is a root system in V′, as in Proposition 2 on p28 of [24]. This is called the inverse system or dual system of A in V′.

Alternatively, let us check that (19.8.12) σ_{α}(β̂) − β̂ is an integer multiple of α̂. This means that (α, β̂) ∈ Z, (19.8.13) by (19.7.14) and (19.7.18). This condition holds because A is a root system in V, as before. This shows that Â is a root system in V, which may be called the inverse system or dual system of A in V with respect to (·, ·). This corresponds to the formulation on p43 of [14].

If T is an invertible linear mapping on V, then the dual linear mapping T′ is invertible on V′, with (T′)^{−1} = (T^{−1})′. (19.8.14)

It is well known and easy to see that the mapping from T to (19.8.14) defines a group isomorphism from GL(V) onto CL(V′). This isomorphism takes the Weyl group of A onto the Weyl group of A′.

Similarly, if T is an invertible linear mapping on V, then the adjoint T^{∗} of T with respect to (·, ·) is invertible on V as well, with

(T^{∗})^{−1} = (T^{−1})^{∗}. (19.8.15)

The mapping from T to (19.8.15) defines a group automorphism on GL(V). The Weyl group of Â is the same as the Weyl group of A, because it is generated by the same reflections on V.

19.9 Angles between roots

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. If α ∈ A, then let σ_{α} be the unique symmetry on V with vector α that maps A onto itself, as in Section 19.3. Also let λ_{α} be the linear functional on V such that λ_{α}(α) = 2 and λ_{α} = 0 on the hyperplane of vectors in V fixed by σ_{α}, as before. If β ∈ A too, then put

n(β, α) = λ_{α}(β), (19.9.1)

as on p29 of [24]. Remember that n(β, α) ∈ Z, as in Section 19.3.

Let (v, w) be an inner product on V that is invariant under the Weyl group of A, as in Section 19.4. If v ∈ V, then put

∥v∥ = (v, v)^{1/2}, (19.9.2)

which is the norm on V associated to (·, ·). Note that

n(β, α) = 2 (β, α) (α, α)^{−1} (19.9.3)

for every α, β ∈ A, as in Section 19.3. Of course,

(β, α) = ∥α∥ ∥β∥ cos θ, (19.9.4)

where θ ∈ [0, π] is the angle between α and β with respect to (·, ·). Thus

n(β, α) = 2 ∥α∥^{−1} ∥β∥ cos θ. (19.9.5)

It follows that n(β, α) n(α, β) = 4 cos^{2} θ (19.9.6)

for every α, β ∈ A. The left side is an integer, and the right side is a nonnegative real number less than or equal to 4. This means that 4 cos^{2} θ can only take the values 0, 1, 2, 3, 4. Of course, 4 cos^{2} θ = 4 (19.9.7)

396 CHAPTER 19. ROOT SYSTEMS

exactly when θ = 0 or π, which means that α and β are proportional in V. Let us suppose now that α and β are not proportional in V, so that 0 < θ < π.

Similarly, 4 cos^{2} θ = 0 (19.9.8) exactly when θ = π/2, in which case n(β, α) = n(α, β) = 0. (19.9.9) Otherwise, if θ ≠ π/2, then cos θ ≠ 0, (α, β) ≠ 0, and n(α, β)/n(β, α) = ∥α∥^{2}/∥β∥^{2}. (19.9.10)

Remember that cos^{2} θ + sin^{2} θ = 1. Note that 4 cos^{2} θ = 2 (19.9.11) if and only if θ = π/4 or 3 π/4. If θ = π/4, then we either have that n(α, β) = 1, n(β, α) = 2, and ∥β∥ = \sqrt{2} ∥α∥, (19.9.12) or the analogous conditions with α and β interchanged. If θ = 3 π/4, then we either have that n(α, β) = −1, n(β, α) = −2, and ∥β∥ = \sqrt{2} ∥α∥, (19.9.13) or the analogous conditions with α and β interchanged.

We also have that 4 cos^{2} θ = 1 (19.9.14) if and only of θ = π/3 or 2 π/3. More precisely, n(α, β) = n(β, α) = 1 (19.9.15) when θ = π/3, and n(α, β) = n(β, α) = −1 (19.9.16) when θ = 2 π/3. In both cases, ∥α∥ = ∥β∥. (19.9.17)

Similarly, 4 cos^{2} θ = 3 (19.9.18) if and only if θ = π/6 or 5 π/6. If θ = π/6, then we either have that n(α, β) = 1, n(β, α) = 3, and ∥β∥ = \sqrt{3} ∥α∥, (19.9.19) or the analogous conditions with α and β interchanged. If θ = 5 π/6, then we either have that n(α, β) = −1, n(β, α) = −3, and ∥β∥ = \sqrt{3} ∥α∥, (19.9.20) or the analogous conditions with α and β interchanged. This corresponds to remarks on p44 of [14], and p29 of [24].

If α and β are not proportional in V and

n(α, β) > 0, (19.9.21)

then α − β ∈ A, (19.9.22)

as in Proposition 3 on p29 of [24]. Note that (19.9.21) is the same as saying that θ < π/2, which is symmetric in α and β. In this situation, we have that n(α, β) = 1 or n(β, α) = 1, as before. Suppose that n(β, α) = 1, so that λ_{α}(β) = 1, as in (19.9.1). In this case,

α − β = −(β − λ_{α}(β) α) = −σ_{α}(β), (19.9.23)

by the usual expression for σ_{α} in terms of λ_{α}. This implies (19.9.22), because σ_{α}(β) ∈ A, and A is symmetric about 0 in V. If n(α, β) = 1, then α − β = σ_{β}(α) ∈ A.

Of course, (19.9.21) is equivalent to

(α, β) > 0, (19.9.24)

by (19.9.3). The lemma on p45 of [14] is stated in terms of (19.9.24), where it is also mentioned that α + β ∈ A (19.9.25)

when α and β are not proportional and

(α, β) < 0. (19.9.26)

This follows from the previous statement, applied to β in place of −β.

19.10 A criterion for linear independence

Let V be a vector space over the real numbers, and let (v, w) be an inner product on V. Also let τ be a linear functional on V, and let B be a nonempty subset of V. Suppose that τ(β) > 0 (19.10.1)

for every β ∈ B, and that (β, γ) ≤ 0 (19.10.2)

for every β, γ ∈ B with β ≠ γ. Under these conditions, the elements of B are linearly independent in V. This corresponds to Lemma 4 on p31 of [24], and Step (3) on p48 of [14].

To see this, observe that a linear relation between elements of B can be expressed as

∑_{β∈B_{1}} b_{β} β = ∑_{γ∈B_{2}} c_{γ} γ, (19.10.3)

where B_{1} and B_{2} are nonempty disjoint finite subsets of B, and the coefficients b_{β} and c_{γ} are nonnegative real numbers. Let v ∈ V be the vector determined by (19.10.3), so that

(v, v) = ∑_{β∈B_{1}} ∑_{γ∈B_{2}} b_{β} c_{γ}(β, γ). (19.10.4)

It follows that (v, v) ≤ 0, by (19.10.2). Of course, this means that v = 0. Using this, we get that

0 = τ(v) = ∑_{β∈B_{1}} b_{β} τ(β). (19.10.5)

This implies that b_{β} = 0 for every β ∈ B_{1}, because of (19.10.1). Similarly, c_{γ} = 0 for every γ ∈ B_{2}, as desired.

19.11 Bases for root systems

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. A subset B of A is said to be a base for A if it satisfies the following two conditions. The first condition is that B is a basis for V as a vector space over R. The second condition is that every α ∈ A can be expressed as

α = ∑_{β∈B} m_{β} β, (19.11.1)

where m_{β} ∈ Z for every β ∈ B, and either m_{β} ≥ 0 for every β ∈ B, or m_{β} ≤ 0 for every β ∈ B. This is Definition 4 on p30 of [24], which corresponds to the definition of a base on p47 of [14]. A base for a root system may also be called a simple root system or a fundamental root system, as mentioned on p30 of [24]. Similarly, the elements of the base may be called simple roots.

Of course, the second condition in the preceding paragraph implies in particular that A is contained in the linear span of B in V. Remember that the linear span of A in V is equal to V, by definition of a root system. Thus the second condition implies that the linear span of B in V is equal to V too.

Theorem 1 on p30 of [24] states that bases exist, which corresponds to the theorem at the top of p48 of [14]. More precisely, if τ is a linear functional on V, then put A_{τ}^{+} = {α ∈ A : τ(α) > 0}. (19.11.2)

Let us choose τ ∈ V′ so that for every α ∈ A,

τ(α) ≠ 0. (19.11.3)

One can find such a τ ∈ V′ as in Section 18.12. This implies that

A = A_{τ}^{+} ∪ (−A_{τ}^{+}). (19.11.4)

Let us say that α ∈ A_{τ}^{+} is decomposable if it can be expressed as the sum of two elements of A_{τ}^{+}, and indecomposable otherwise. The first part of Proposition 4 on p30 of [24] states that

(19.11.5) B_{τ} = {α ∈ A_{τ}^{+} : α is indecomposable}

is a base for A. This corresponds to the first part of the second theorem on p48 of [14].

The first step in the proof is Lemma 2 on p30 of [24], which corresponds to Step (1) on p48 of [14]. This states that every element of A_{τ}^{+} can be expressed as a linear combination of elements of B_{τ}, with coefficients that are nonnegative integers. Otherwise, there would be an α ∈ A_{τ}^{+} that does not have this property, and for which τ(α) is as small as possible. In particular, α ∉ B_{τ}, so that α is decomposable. Thus there are α_{1}, α_{2} ∈ A_{τ}^{+} whose sum is equal to α. This implies that τ(α) = τ(α_{1}) + τ(α_{2}), (19.11.6)

and hence τ(α_{1}), τ(α_{2}) < τ(α), (19.11.7)

because τ(α_{1}), τ(α_{2}) > 0. It follows that α_{1} and α_{2} can be expressed as linear combinations of elements of B_{τ} with coefficients that are nonnegative integers, by the minimality of τ(α). This means that α has the same property, which is a contradiction.

Let (v, w) be an inner product on V that is invariant under the Weyl group of A, as in Section 19.4. If β_{1} and β_{2} are distinct elements of B_{τ}, then

(β_{1}, β_{2}) ≤ 0. (19.11.8)

This corresponds to Lemma 3 on p30 of [24], and Step (2) on p48 of [14]. Suppose for the sake of a contradiction that

(β_{1}, β_{2}) > 0. (19.11.9)

If β_{1} and β_{2} are proportional in V, then the proportionality constant is positive, by (19.11.9). This implies that one of β_{1} and β_{2} is equal to 2 times the other, as in Section 19.3, because β_{1} ≠ β_{2}, by hypothesis. However, this would contradict the fact that the elements of B_{τ} are indecomposable. Otherwise, if β_{1} and β_{2} are not proportional in V, then γ = β_{1} − β_{2} ∈ A, as in Section 19.9. If γ ∈ A_{τ}^{+}, then β_{1} = β_{2} + γ is decomposable, contradicting the hypothesis that β_{1} ∈ B_{τ}. Similarly, if −γ ∈ A_{τ}^{+}, then β_{2} = β_{1} + (−γ) is decomposable, contradicting the hypothesis that β_{2} ∈ B_{τ}.

Using (19.11.8) and the fact that B_{τ} ⊆ A_{τ}^{+}, by construction, we get that the elements of B_{τ} are linearly independent in V, as in the previous section. Remember that the elements of A_{τ}^{+} can be expressed as linear combinations of elements of B_{τ}, with coefficients that are nonnegative integers. Thus the elements of −A_{τ}^{+} can be expressed as linear combinations of elements of B_{τ} with coefficients that are integers less than or equal to 0. This shows that B_{τ} satisfies the second condition in the definition of a base for A. In particular, the linear span of B_{τ} in V is equal to V, as before. It follows that B_{τ} is a basis for V as a vector space over R. This implies that B_{τ} is a base for A, as on p31 of [24], and Step (4) on p48 of [14].

Conversely, suppose that B is any base for A. Let τ be a linear functional on V such that τ(β) > 0 (19.11.10)

for every β ∈ B. It is easy to find such a τ ∈ V′, because B is a basis for V. Let A^{+} be the set of α ∈ A that can be expressed as a linear combination of elements of B with coefficients that are nonnegative integers. Observe that

A^{+} ⊆ A_{τ}^{+}, (19.11.11)

where A_{τ}^{+} is as in (19.11.2). This implies that

−A^{+} ⊆ −A_{τ}^{+}. (19.11.12)

It follows that A^{+} = A_{τ}^{+}, (19.11.13)

because A = A^{+} ∪ (−A^{+}), by hypothesis. We also get that (19.11.3) holds, so that the remarks in the previous paragraphs hold for τ. Using (19.11.13), we obtain that the elements of B are indecomposable in A_{τ}^{+}, so that B ⊆ B_{τ}. This implies that B = B_{τ} , (19.11.14)

because B and B_{τ} have the same number of elements, which is the dimension of V. This corresponds to the second part of Proposition 4 on p30 of [24], and the second part of the second theorem on p48 of [14].

If β_{1} and β_{2} are distinct elements of B, then it is easy to see that their difference is not a root, using the second condition in the definition of a base for A. We also have that β_{1} and β_{2} are not proportional in V, because B is a basis for V. This implies that (19.11.8) holds, as in Section 19.9. This corresponds to the lemma on p47 of [14]. This could also be obtained from (19.11.14) and the previous proof, but this argument is more direct.

19.12 Positive roots

Let V be a vector space over the real numbers of positive finite dimension again, and let A be a root system in V. Also let B be a base for A, and let A^{+} be the set of α ∈ A such that α can be expressed as a linear combination of elements of B with coefficients that are nonnegative integers, as in the previous section. The elements of A^{+} are called positive roots with respect to B, as on p31 of [24], and p47 of [14].

Let τ be the linear functional on V such that

τ(β) = 1 (19.12.1) for every β ∈ B. If γ ∈ A^{+}, then

τ(γ) ∈ Z_{+}. (19.12.2)

Let (v, w) be an inner product on V that is invariant under the Weyl group of A, as in Section 19.4. If γ ∈ A^{+}, then we would like to check that there is a β ∈ B such that (β, γ) > 0. (19.12.3)

Suppose for the sake of a contradiction that (β, γ) ≤ 0 for every β ∈ B. Remember that (β_{1}, β_{2}) ≤ 0 for all β_{1}, β_{2} ∈ B with β_{1} ≠ β_{2}, as in the previous section. Under these conditions, we get that B ∪ {γ} is linearly independent in V, as in Section 19.10, which is a contradiction.

Let γ ∈ A^{+} be given, and put r = τ(γ) ∈ Z_{+}. Proposition 5 on p32 of [24] states that γ can be expressed as

γ = ∑_{j=1}^{r} α_{j}, (19.12.4)

where α_{j} ∈ B for each j = 1, …, r, and

∑_{j=1}^{l} α_{j} ∈ A (19.12.5)

for every l = 1, …, r. This corresponds to the corollary to Lemma A on p50 of [14]. If r = 1, then γ ∈ B, and this is trivial. Otherwise, we can use induction on r, as follows.

As before, there is a β ∈ B such that (19.12.3) holds. If β and γ are proportional in V, then one can check that either γ = β or γ = 2 β, using remarks in Section 19.3. The statement in the preceding paragraph obviously holds in both cases, and so we suppose now that β and γ are not proportional in V. This implies that γ − β ∈ A, (19.12.6)

as in Section 19.9.

If γ − β ∈ −A^{+}, so that β − γ ∈ A^{+}, then β = γ + (β − γ) would be the sum of two elements of A^{+}. This is not possible, because β ∈ B. Otherwise, γ − β ∈ A^{+}, and of course

τ(γ − β) = τ(γ) − τ(β) = r − 1. (19.12.7)

This means that the induction hypothesis can be applied to γ − β. Thus the desired statement for γ can be obtained from the analogous statement for γ −β, with α_{r} = β.

Suppose now that A is reduced, as in Section 19.3. Let α ∈ B be given, and let σ_{α} be the symmetry on V with vector α that maps A onto itself. Under these conditions, σ_{α}(A^{+} \ {α}) = A^{+} \ {α}, (19.12.8) as in Proposition 6 on p32 of [24], and Lemma B on p50 of [14]. To see this, it suffices to verify that σ_{α}(A^{+} \ {α}) ⊆ A^{+} \ {α}, (19.12.9)

because σ_{α} is its own inverse on V.

Let γ ∈ A^{+} \ {α} be given. Thus γ can be expressed as

γ = ∑_{β∈B} m_{β} β, (19.12.10)

where m_{β} is a nonnegative integer for each β ∈ B, because γ ∈ A^{+}. Observe that γ is not proportional to α in this situation, because γ ≠ α by hypothesis, and A is reduced. This means that there is a β ∈ B such that β ≠ α and m_{β} ≠ 0.

Remember that σ_{α}(γ) is equal to γ minus a multiple of α. If we express σ_{α}(γ) as a linear combination of the basis vectors in B, we get that the coefficient of β in the expression for σ_{α}(γ) is equal to m_{β} too. This implies that σ_{α}(γ) ∈ A^{+}, because σ_{α}(γ) ∈ A, m_{β} > 0, and B is a base for A. We also have that σ_{α}(γ) ≠ α, because m_{β} ≠ 0, as desired.

Put

ρ = \frac{1}{2} ∑_{γ∈A^{+}} γ, (19.12.11)

which is an element of V. Let us check that

σ_{α}(ρ) = ρ − α, (19.12.12)

as in the corollary on p32 of [24], and the corollary to Lemma B on p50 of [14]. Put

ρ_{α} = \frac{1}{2} ∑_{γ∈A^{+}\\{α\}} γ, (19.12.13)

which is interpreted as being 0 when A^{+} = {α}. Using (19.12.8), we get that σ_{α}(ρ_{α}) = ρ_{α}. Note that ρ = ρ_{α} + α/2, by construction. We also have that σ_{α}(α) = −α. Thus

σ_{α}(ρ) = σ_{α}(ρ_{α}) + σ_{α}(α)/2 = ρ_{α} − α/2 = ρ − α, (19.12.14)

as desired.

19.13 Bases and duality

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. If α ∈ A, then we let σ_{α} be the symmetry on V with vector α that maps A onto itself, as usual. Let λ_{α} ∈ V′ be the corresponding linear functional on V, so that λ_{α}(α) = 2 and λ_{α} = 0 on the hyperplane of vectors in V fixed by σ_{α}. Remember that

A′ = {λ_{α} : α ∈ A} (19.13.1) is a root system in the dual space V′, as in Section 19.8.

If α ∈ A and 2 α ∈ A, then

σ_{2 α} = σ_{α}, (19.13.2)

because σ_{α} may also be considered as a symmetry on V with vector 2 α. In particular, this means that λ_{α} and λ_{2 α} have the same kernel in V. One can check that λ_{2 α} = λ_{α}/2, (19.13.3)

using the condition that λ_{2 α}(2 α) = 2.

Let B be a base for A, and put

(19.13.4)  B_{1} = {β ∈ B : 2 β ∉ A},
(19.13.5)  B_{2} = {β ∈ B : 2 β ∈ A}.

Of course, B_{2} = ∅ when A is reduced. We would like to show that

{λ_{β} : β ∈ B_{1}} ∪ {λ_{β}/2 : β ∈ B_{2}} (19.13.6)

is a base for A′. This corresponds to Proposition 7 on p32 and the remark on p33 of [24]. Note that (19.13.6) is contained in A′, by (19.13.3).

Let (v, w) be an inner product on V that is invariant under the Weyl group of A, as in Section 19.4. Using (·, ·), we get an isomorphism between V and V′ in the usual way, as in Section 19.7. If α ∈ V and α ≠ 0, then put

α̂ = 2 α (α, α)^{−1}, (19.13.7)

as before. Clearly ˆ(2 α) = α̂/2 (19.13.8)

in this case.

If α ∈ A, then we have seen that α̂ corresponds to λ_{α} under the isomorphism between V and V′ associated to (·, ·). Thus

Â = {α̂ : α ∈ A} (19.13.9)

corresponds to A′ under this isomorphism, as before. We also have that Â is a root system in V, as in Section 19.8.

We would like to show that

{β̂: β ∈ B_{1}} ∪ {β̂/2 : β ∈ B_{2}} (19.13.10)

is a base for Â, as in [24]. This corresponds to Exercise 1 on p54 of [14], in the reduced case. Observe that (19.13.10) is contained in Â, because of (19.13.8). It will follow that (19.13.6) is a base for A′, because of the isomorphism between V and V′ associated to (·, ·). It is easy to see that (19.13.10) is a basis for V, because B is a basis for V.

Let A^{+} be the set of α ∈ A that can be expressed as a linear combination of elements of B with coefficients that are nonnegative integers, as before. Also let τ be a linear functional on V such that τ(β) > 0 for every β ∈ B. If α ∈ A^{+}, then τ(α) > 0, and hence τ(α̂) > 0. (19.13.11)

Similarly, if α ∈ −A^{+}, then τ(α) < 0, so that

τ(α̂) < 0. (19.13.12)

In particular, τ is nonzero at every element of Â.

Let (Â)_{τ}^{+} be the set of elements of Â on which τ is positive, as in Section 19.11. Equivalently, (Â)_{τ}^{+} = {α̂ : α ∈ A^{+}}, (19.13.13)

as in the preceding paragraph. Let us say that an element of (Â)_{τ}^{+} is decomposable if it can be expressed as the sum of two elements of (Â)_{τ}^{+}, and indecomposable otherwise, as in Section 19.11. The set of indecomposable elements of (Â)_{τ}^{+} is a base for Â, as before. Note that (19.13.10) is contained in (Â)_{τ}^{+}.

It suffices to show that (19.13.10) is the same as the set of indecomposable elements of (Â)_{τ}^{+}. If an element of (19.13.10) is decomposable as an element of (Â)_{τ}^{+}, then it could be expressed as the sum of two proportional elements of (Â)_{τ}^{+}. One can check that this is not possible, because of the way that B_{1} and B_{2} are defined. Thus (19.13.10) is contained in the set of indecomposable elements of (Â)_{τ}^{+}. It follows that these two sets are the same, because they have the same number of elements, which is the dimension of V.

The argument in [24] considers the associated convex cones in V. More precisely, let C be the set of vectors in V that can be expressed as linear combinations of elements of B with coefficients that are nonnegative real numbers. This is the same as the closed convex cone in V generated by A^{+}. This is also the same as the closed convex cone in V generated by (Â)_{τ}^{+}, by (19.13.13). The extremal rays in C correspond exactly to half-lines through elements of B. Similarly, the extremal rays in C correspond to half-lines through the indecomposable elements of (Â)_{τ}^{+}. This means that these collections of half-lines in V are the same. It follows that the indecomposable elements of (Â)_{τ}^{+} are all proportional to elements of B. One can use this to check that the set of indecomposable elements of (Â)_{τ}^{+} is the same as (19.13.10), as desired.

19.14 Bases and the Weyl group

Let V be a vector space over the real numbers of positive finite dimension, and let A be a reduced root system in V. Also let σ_{α} be the symmetry on V with vector α ∈ A that maps A onto itself, as usual. Thus the Weyl group W of A is the subgroup of GL(V) generated by σ_{α}, α ∈ A. Let B be a base for A, and let W_{B} be the subgroup of W generated by σ_{β}, β ∈ B. It is well known that

W_{B} = W, (19.14.1)

19.14. BASES AND THE WEYL GROUP 405 as in Theorem 2 on p33 of [24], and the theorem on p51 of [14].

To show this, we first consider some other properties of W_{B}, as in [14, 24]. Let τ be a linear functional on V. If σ ∈ W, then let σ′ be the corresponding dual linear mapping on the dual space V′ of V, so that σ′(τ) = τ ∘ σ. (19.14.2) We would like to find a σ ∈ W_{B} such that (σ′(τ))(β) = τ(σ(β)) ≥ 0 (19.14.3) for every β ∈ B. This corresponds to part (a) of Theorem 2 on p33 of [24], as mentioned in its proof. Part (a) of the theorem on p51 of [14] is very similar, but formulated a bit differently. See also Exercise 14 on p55 of [14].

Let A^{+} be the set of α ∈ A that can be expressed as a linear combination of elements of B with coefficients that are nonnegative integers, as usual. Also let ρ be the element of V that is one-half the sum of the elements of A^{+}. We can choose σ ∈ W_{B} such that (σ′(τ))(ρ) = τ(σ(ρ)) (19.14.4) is maximal in R. If β ∈ B, then σ ∘ σ_{β} ∈ W_{B}, so that (σ′(τ))(ρ) ≥ ((σ ∘ σ_{β})′(τ))(ρ). (19.14.5) Of course, ((σ ∘ σ_{β})′(τ))(ρ) = τ(σ(σ_{β}(ρ))). (19.14.6) Under these conditions, we have that σ_{β}(ρ) = ρ − β, (19.14.7) as in (19.12.12). Thus ((σ ∘ σ_{β})′(τ))(ρ) = τ(σ(ρ − β)) = (σ′(τ))(ρ − β). (19.14.8) It follows that (σ′(τ))(ρ) ≥ (σ′(τ))(ρ − β), (19.14.9) by (19.14.5). This implies (19.14.3), as desired.

If B̃ is another base for A, then there is a σ ∈ W_{B} such that σ(B) = B̃ (19.14.10) This corresponds to part (b) of Theorem 2 on p33 of [24], and part (b) of the theorem on p51 of [14], as mentioned in their proofs again. To see this, let τ̃ be a linear functional on V such that τ̃(β̃) > 0 (19.14.11) for every β̃ ∈ B̃, which is possible because B̃ is a basis for V. As in the previous paragraphs, there is a σ ∈ W_{B} such that

(σ′(τ̃))(β) ≥ 0 (19.14.12)

for every β ∈ B. Put τ = σ′(τ̃) = τ̃ ∘ σ ∈ V′, (19.14.13)

so that τ(β) ≥ 0 (19.14.14)

for every β ∈ B.

If α ∈ A can be expressed as a linear combination of elements of B̃ with coefficients that are nonnegative integers, then τ̃(α) > 0, as usual. Otherwise, α̃ < 0, so that τ̃(α) ≠ 0 for every α ∈ A. This implies that

τ(α) = τ̃(σ(α)) ≠ 0 (19.14.15)

for every α ∈ A, because σ(A) = A. It follows that τ(β) > 0 for every β ∈ B, because of (19.14.14).

Let A_{τ}^{+} and A+ ; _{τ} be the collections of α ∈ A such that

τ̃(σ(α)) = τ(α) > 0 (19.14.16)

and τ̃(α) > 0, respectively. It is easy to see that

σ(A_{τ}^{+}) = A+ ; _{τ}, (19.14.17)

because σ(A) = A. The elements of B and B̃ may be characterized as the indecomposable elements of A_{τ}^{+} and A+ ; _{τ}, respectively, as in Section 19.11. Thus (19.14.10) follows from (19.14.17), as desired.

If γ ∈ A, then there is a σ ∈ W_{B} such that

σ(γ) ∈ B. (19.14.18)

This corresponds to part (c) of Theorem 2 on p33 of [24], and part (c) of the theorem on p51 of [14], as mentioned in their proofs. In this argument, it will be convenient to let V′_{a} = {λ ∈ V′ : λ(a) = 0} (19.14.19)

be the hyperplane in V′ dual to a nonzero element a of V. If b is another nonzero element of V, then V′_{a} = V′_{b} exactly when a and b are proportional in V. In particular, if α ∈ A and α ≠ ±γ, then

V′_{α} ≠ V′_{γ}, (19.14.20)

because A is reduced, by hypothesis.

Thus, if α ∈ A and α ≠ ±γ, then

V′_{α} ∩ V′_{γ} (19.14.21) is a hyperplane in V′_{γ}. The union of (19.14.21) over α ∈ A with α ≠ ±γ is not all of V′_{γ}, as in Section 18.12. This implies that there is a τ_{0} ∈ V′ such that τ_{0}(γ) = 0 and τ_{0}(α) ≠ 0 for every α ∈ A with α ≠ ±γ. Of course, there is also a linear functional on V that is positive on γ. By adding a sufficiently small multiple of such a linear functional on V to τ_{0}, we can get τ ∈ V′ such that

τ(γ) > 0 (19.14.22)

and |τ(α)| > τ(γ) (19.14.23)

for every α ∈ A with α ≠ ±γ.

In particular, τ(α) ≠ 0 for every α ∈ A. Thus we can get a base B_{τ} for A using τ as in Section 19.11. One can check that

γ ∈ B_{τ} (19.14.24)

in this situation, which is to say that γ is indecomposable as an element of the set A_{τ}^{+} of α ∈ A with τ(α) > 0. As in (19.14.10), there is an element of W_{B} that maps B onto B_{τ}. Using this, we can get (19.14.18) from (19.14.24), as desired.

Let us now use (19.14.18) to obtain (19.14.1), as in [14, 24]. If γ ∈ A, then we would like to show that σ_{γ} ∈ W_{B}. (19.14.25)

Let σ be an element of W_{B} such that σ(γ) ∈ B, which implies that σ_{σ(γ)} ∈ W_{B}. It is easy to see that σ ∘ σ_{γ} ∘ σ^{−1} (19.14.26)

is a symmetry on V with vector σ(γ), because σ_{γ} is a symmetry on V with vector γ. We also have that (19.14.26) maps A onto itself, because σ(A) = σ_{γ}(A) = A. This implies that (19.14.26) is equal to σ_{σ(γ)}. Equivalently, this means that

σ_{γ} = σ^{−1} ∘ σ_{σ(γ)} ∘ σ. (19.14.27)

It follows that (19.14.25) holds, as desired, because σ, σ_{σ(γ)} ∈ W_{B}.

19.15 Weyl chambers

Let V be a vector space over the real numbers of positive finite dimension n, and let A be a root system in V. Let us say that a linear functional τ on V is regular with respect to A if for every α ∈ A, τ(α) ≠ 0. This corresponds to terminology on p48 of [14], although in [14] an inner product on V is used, so that the dual space V′ of V can be identified with V. The set of linear functionals on V that are regular with respect to A is the same as

V′ \ ( ⋃_{α∈A} V′_{α}), (19.15.1)

where V′_{α} is the hyperplane in V′ dual to α ∈ A, as in (19.14.19). Of course, V′ is isomorphic to R^{n}, as a vector space over R. This leads to a natural topology on V′, corresponding to the standard topology on R^{n}. This topology does not depend on the particular isomorphism with R^{n}, because linear mappings on R^{n} are continuous with respect to the standard topology. Note that (19.15.1) is an open set with respect to this topology. Remember that (19.15.1) is nonempty, as in Section 18.12.

Let B be a base for A, and let A^{B,+} be the set of α ∈ A that can be expressed as a linear combination of elements of B with coefficients that are nonnegative integers. Thus A = A^{B,+} ∪ (−A^{B,+}), (19.15.2)

by the definition of a base. If τ ∈ V′ satisfies

(19.15.3) τ(β) > 0 for every β ∈ B,

then (19.15.4) τ(α) > 0 for every α ∈ A^{B,+},

as before. The converse holds trivially, because B ⊆ A^{B,+}. Observe that τ is regular with respect to A when (19.15.4) holds, because of (19.15.2). The set of τ ∈ V′ that satisfy (19.15.3) is called the Weyl chamber associated to B, as in Remark 2 on p34 of [24]. This corresponds to the fundamental Weyl chamber relative to B defined on p49 of [14], using an inner product on V, as before.

Let τ be a linear functional on V that is regular with respect to A, and let A_{τ}^{+} be the set of α ∈ A such that τ(α) > 0, as in Section 19.11. If τ satisfies (19.15.3) and hence (19.15.4), then A^{B,+} ⊆ A_{τ}^{+}, which implies that

A^{B,+} = A_{τ}^{+}, (19.15.5)

because of (19.15.2), as in Section 19.11. Conversely, (19.15.5) implies (19.15.4), and thus (19.15.3). Let B_{τ} be the set of indecomposable elements of A_{τ}^{+}, which is a base for A, as in Section 19.11. If τ satisfies (19.15.3), then B = B_{τ}, as before. Conversely, if B = B_{τ}, then B ⊆ A_{τ}^{+}, which means that (19.15.3) holds. This shows that the Weyl chamber associated to B consists exactly of the linear functionals τ on V that are regular with respect to A and satisfy B = B_{τ}. In particular, if τ is any linear functional on V that is regular with respect to A, then τ is an element of the Weyl chamber associated to B_{τ}.

It is easy to see that the Weyl chamber associated to B is an open convex cone in V, so that it is connected in particular. If B̃ is another base for A, B̃ ≠ B, then the Weyl chambers associated to B and B̃ are disjoint, by the remarks in the preceding paragraph. The union of the Weyl chambers associated to bases for A is equal to the set of linear functionals on V that are regular with respect to A, as before. It follows that the Weyl chambers associated to bases for A are the same as the connected components of (19.15.1).

If B̃ is another base for A and

B̃ ⊆ A^{B,+}, (19.15.6) then B̃ = B. (19.15.7)

Indeed, if τ ∈ V′ satisfies (19.15.3) and hence (19.15.4), then τ(β̃) > 0 for every β̃ ∈ B̃. This implies that B̃ = B_{τ}, as before, so that (19.15.7) holds, because B = B_{τ} too.

Root systems, 2

20.1 Products in the Weyl group

Let V be a vector space over the real numbers of positive finite dimension, and let A be a reduced root system in V. If α ∈ A, then we let σ_{α} be the symmetry on V with vector α that maps A onto itself, as before. Let B be a base for A, and remember that the Weyl group W of A is generated by σ_{β}, β ∈ B, as in Section 19.14.

Let r ≥ 2 be an integer, and let γ_{1}, …, γ_{r} be r elements of B, possibly with repetitions. Put σ_{j} = σ_{γ_{j}} for each j = 1, …, r, and consider

(σ_{1} ∘ ⋯ ∘ σ_{r−1})(γ_{r}), (20.1.1)

which is an element of A. Thus (20.1.1) can be expressed as a linear combination of elements of B with integer coefficients, and where the coefficients are either all greater than or equal to 0, or all less than or equal to 0. If the coefficients are all less than or equal to 0, then there is a positive integer n < r such that

σ_{1} ∘ ⋯ ∘ σ_{r} = σ_{1} ∘ ⋯ σ_{n−1} ∘ σ_{n+1} ∘ ⋯ ∘ σ_{r−1}. (20.1.2)

This corresponds to Lemma C on p50 of [14]. More precisely, in the first part of the right side of (20.1.2), σ_{1} ∘ ⋯ ∘ σ_{n−1} is interpreted as being the identity mapping on V when n = 1. Similarly, if n = r − 1, then σ_{n+1} ∘ ⋯ ∘ σ_{r−1} is interpreted as being the identity mapping on V in the second part of the right side of (20.1.2).

To see this, put δ_{j} = (σ_{j+1} ∘ ⋯ ∘ σ_{r−1})(γ_{r}) (20.1.3)

for j = 0, …, r − 2, and δ_{r−1} = γ_{r}. Note that δ_{0} is the same as (20.1.1), whose coefficients with respect to B are all less than or equal to 0, by hypothesis. Of course, the coefficients of δ_{r−1} = γ_{r} with respect to B are all greater than or equal to 0, because γ_{r} ∈ B. Let n be the smallest positive integer less than or equal to 0 such that the coefficients of δ_{n} with respect to B are all greater than or equal to 0, so that the coefficients of

δ_{n−1} = σ_{n}(δ_{n}) (20.1.4)

with respect to B are all less than or equal to 0. This implies that

γ_{n} = δ_{n}, (20.1.5)

by (19.12.8).

If α ∈ A and T is an invertible linear mapping on V that maps A onto itself, then T ∘ σ_{α} ∘ T^{−1} = σ_{T (α)}, as in Section 19.4. Let us take α = γ_{r} and T = σ_{n+1} ∘ ⋯ ∘ σ_{r−1}, so that T (α) = δ_{n}. Combining this with (20.1.5), we obtain that

σ_{n} = (σ_{n+1} ∘ ⋯ ∘ σ_{r−1}) ∘ σ_{r} ∘ (σ_{r−1} ∘ ⋯ ∘ σ_{n+1}). (20.1.6)

It is easy to get (20.1.2) from (20.1.6).

Put σ = σ_{1} ∘ ⋯ ∘ σ_{r}, (20.1.7)

and suppose now that r is the smallest positive integer such that σ can be expressed as the composition of r symmetries associated to elements of B. The previous statement implies that the coefficients of (20.1.1) with respect to B are all greater than or equal to 0. Remember that σ_{r}(γ_{r}) = −γ_{r}, so that

σ(γ_{r}) = −(σ_{1} ∘ ⋯ ∘ σ_{r−1})(γ_{r}). (20.1.8)

It follows that the coefficients of σ(γ_{r}) with respect to B are all less than or equal to 0, as in the corollary to Lemma C on p50 of [14]. Note that this holds automatically when r = 1.

If σ is any element of the Weyl group W of A, then σ can be expressed as the composition of finitely many symmetries associated to elements of B, as before. In particular, if σ is not the identity mapping on V, then one needs at least one of these symmetries. This means that σ can be expressed as in (20.1.7) for some positive integer r, and we may as well take r to be as small as possible. Under these conditions, σ cannot map B onto itself, as in the preceding paragraph.

Equivalently, if σ ∈ W satisfies

σ(B) = B, (20.1.9)

then σ is the identity mapping on V. This corresponds to part (e) of the theorem on p51 of [14]. This also corresponds to Proposition 4 on p62 of [24].

20.2 The Cartan matrix

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. If α ∈ A, then let σ_{α} be the symmetry on V with vector α that maps A onto itself, and let λ_{α} be the corresponding linear functional on V. Thus λ_{α}(α) = 2, and λ_{α} = 0 on the hyperplane of vectors in V that are fixed by σ_{α}. If β ∈ A too, then we put

n(α, β) = λ_{β}(α), (20.2.1)

as in Section 19.9. Of course, this is an integer, by the definition of a root system.

Let (·, ·) be an inner product on V that is invariant under the Weyl group W of A. This means that σ_{α} is the reflection on V with respect to (·, ·) associated to α ∈ A, so that λ_{α}(v) = 2 (v, α) (α, α)^{−1} for every v ∈ V. In particular,

n(α, β) = 2 (α, β) (β, β)^{−1} (20.2.2)

for every α, β ∈ A.

Let B be a base for A. The Cartan matrix of A with respect to B is n(α, β) as a function of α, β ∈ B, as in Definition 5 on p34 of [24], and p55 of [14]. Note that n(α, α) = λ_{α}(α) = 2 for every α ∈ A, and in particular for α ∈ B. If α, β ∈ B and α ≠ β, then n(α, β) ≤ 0, by (19.11.8) and (20.2.2). Of course, α and β are not proportional in this situation. It follows that

n(α, β) ∈ {0, −1, −2, −3} (20.2.3)

when α ≠ β, as in Section 19.9, and mentioned on p34 of [24]. If A is reduced, then A is determined up to isomorphism by the Cartan matrix, as in Proposition 8 on p34 of [24].

More precisely, let Ṽ be another vector space over the real numbers of positive finite dimension, and let Ã be a reduced root system in Ṽ. If α̃ ∈ Ã, then let σ̃_{α} on Ṽ with vector α̃ that maps Ã onto itself, as usual. Also let λ̃_{α} be the corresponding linear functional on Ṽ, so that λ̃_{α}(α̃) = 2 and λ̃_{α} = 0 on the hyperplane of vectors in Ṽ that are fixed by σ̃_{α}. If β̃ ∈ Ã too, then we put

ñ(α̃, β̃) = λ̃_{β}(α̃), (20.2.4)

as before. Let B̃ be a base for Ã, so that the restriction of (20.2.4) to α̃, β̃ ∈ B̃ is the Cartan matrix of Ã with respect to B̃.

Suppose that ϕ is a one-to-one mapping from B onto B̃ such that

ñ(ϕ(α), ϕ(β)) = n(α, β) (20.2.5)

for every α, β ∈ B. If A is reduced, then ϕ extends to a unique one-to-one linear mapping f from V onto Ṽ such that

f(A) = Ã. (20.2.6)

This is Proposition 8’ on p35 of [24], which corresponds to the proposition on p55 of [14]. Remember that B and B̃ are bases for V and Ṽ, respectively, as vector spaces over R, by the definition of a base for a root system. Thus ϕ automatically extends to a unique one-to-one linear mapping f from V onto Ṽ, and in particular V and Ṽ have the same dimension as vector spaces over R.

If α, β ∈ B, then

(σ̃_{ϕ(α)} ∘ f)(β) = σ̃_{ϕ(α)}(ϕ(β)) = ϕ(β) − ñ(ϕ(β), ϕ(α)) ϕ(α) (20.2.7)

and (f ∘ σ_{α})(β) = f(β − n(β, α) α) = ϕ(β) − n(β, α) ϕ(α). (20.2.8)

Using our hypothesis (20.2.5), we get that the right sides of (20.2.7) and (20.2.8) are the same. This implies that

σ̃_{ϕ(α)} ∘ f = f ∘ σ_{α} (20.2.9)

on V for every α ∈ B, because V is spanned by β ∈ B. Equivalently,

σ̃_{ϕ(α)} = f ∘ σ_{α} ∘ f^{−1} (20.2.10)

on Ṽ for every α ∈ B.

As usual, GL(V) and GL(Ṽ) denote the general linear groups of invertible linear mappings on V and Ṽ, respectively. Of course,

σ ↦ f ∘ σ ∘ f^{−1} (20.2.11)

is a group isomorphism from GL(V) onto GL(Ṽ). Remember that the Weyl groups W and W̃ of A and Ã are generated by the symmetries associated to elements of B and B̃, respectively, as in Section 19.14. Using this and (20.2.10), we get that (20.2.11) maps W onto W̃.

Remember that A is the same as the set of images of elements of B under elements of W, and similarly for Ã, as in Section 19.14. Using this and the remarks in the preceding paragraph, it is easy to see that (20.2.6) holds, as desired.

Of course, a one-to-one linear mapping f from V onto Ṽ that satisfies (20.2.6) is the same as an isomorphism between the root systems A and Ã, as in Section 19.5. In this case, we have that

ñ(f(α), f(β)) = n(α, β) (20.2.12)

for every α, β ∈ A, by (19.5.9).

Let Aut(A) be the group of automorphisms of A, which is the subgroup of GL(V) consisting of invertible linear mappings on V that send A onto itself. Remember that the Weyl group W of A is a normal subgroup of Aut(A), as in Section 19.4. If T ∈ Aut(A), then

n(T (α), T (β)) = n(α, β) (20.2.13)

for every α, β ∈ A, as in (20.2.12). Of course,

{T ∈ Aut(A) : T (B) = B} (20.2.14) is a subgroup of Aut(A) as well. The elements of this subgroup are uniquely determined by their restrictions to B, because B is a basis for V as a vector space over R. If A is reduced, then the restrictions of elements of (20.2.14) to B are characterized by the condition that (20.2.13) hold for every α, β ∈ B, as before. In this case, the intersection of W and (20.2.14) is the trivial subgroup of Aut(A), as in the previous section.

If T is any element of Aut(A), then it is easy to see that T (B) is a base for A too. If A is reduced, then there is a σ ∈ W such that

σ(T (B)) = B, (20.2.15)

as in Section 19.14. Equivalently, this means that σ∘T is an element of (20.2.14). This corresponds to Proposition 9 on p35 of [24], and some remarks beginning on p65 of [14].

20.3 Coxeter graphs

As on p35 of [24], a Coxeter graph is a finite graph, where each pair of distinct vertices may be joined by 0, 1, 2, or 3 edges.

Let V be a vector space over the real numbers of positive finite dimension, let A be a root system in V, and let B be a base for A. If α, β ∈ A, then λ_{α} ∈ V′ and n(α, β) ∈ Z are defined in the usual way, as in the previous section. If α, β ∈ B and α ≠ β, then α and β are not proportional, and

n(α, β) n(β, α) (20.3.1)

can only be equal to 0, 1, 2, or 3, as in Section 19.9. Of course, (20.3.1) is symmetric in α and β.

The Coxeter graph of A with respect to B is defined as follows, as on p35 of [24], and p56 of [14]. We take B to be the set of vertices of the Coxeter graph. If α and β are distinct elements of B, then the number of edges between α and β is equal to (20.3.1).

If B̃ is another base for A, then there is an element σ of the Weyl group of A that maps B onto B̃, as in Section 19.14. This leads to an isomorphism between the Coxeter graphs of A with respect to B and B̃.

Let (·, ·) be an inner product on V that is invariant under the Weyl group of A. Using (20.2.2), we get that

n(α, β) n(β, α) = 4 (α, β)^{2} ∥α∥^{−2} ∥β∥^{−2} (20.3.2)

for every α, β ∈ A, where ∥·∥ is the norm on V associated to (·, ·). Equivalently, this is the same as 4 times the square of the cosine of the angle between α and β, as in Section 19.9. In particular, (20.3.2) is equal to 0 exactly when (α, β) = 0.

The Coxeter graph of A with respect to B is determined by the Cartan matrix of A with respect to B in an obvious way. Remember that the diagonal entries of the Cartan matrix are equal to 2, and that the other entries are less than or equal to 0. It is easy to see that the Cartan metrix of A with respect to B is determined by the Coxeter graph of A with respect to B together with the ratios of the norms of the elements of B.

More precisely, let α and β be distinct elements of B. If (20.3.2) is equal to 0, then (α, β) = 0, and hence n(α, β) = 0. Otherwise, if (20.3.2) is not zero, and if one knows if ∥α∥ and ∥β∥ are the same, or which is larger, then n(α, β) can be determined from (20.3.2) as in Section 19.9. This is discussed beginning on p56 of [14], and on p38 of [24].

20.4 Reducibility

Let V be a vector space over the real numbers of positive finite dimension again, let A be a root system in V, and let (·, ·) be an inner product on V that is invariant under the Weyl group of A. Suppose for the moment that V is the direct sum of two nontrivial linear subspaces V_{1} and V_{2}, as a vector space over R, and that A ⊆ V_{1} ∪ V_{2}. (20.4.1)

Put A_{j} = A∩V_{j} for j = 1, 2. If α ∈ V_{1} and β ∈ V_{2}, then α −β is not an element of V_{1} ∪ V_{2}, and thus is not in A. This implies that (α, β) ≤ 0, as in Section 19.9. The same argument could be applied to α and −β, to get that (α, β) ≥ 0, and hence (α, β) = 0. (20.4.2)

It is easy to see that the linear span of A_{j} is equal to V_{j} for each j = 1, 2, because the linear span of A is V. It follows that the elements of V_{1} and V_{2} are orthogonal to each other with respect to (·, ·), as in part (a) of Proposition 10 on p36 of [24].

If α ∈ A, then the symmetry σ_{α} on V with vector α that maps A onto itself is the reflection associated to α with respect to (·, ·). If α ∈ A_{1}, then σ_{α} fixes every element of the hyperplane orthogonal to α, which contains V_{2}. In particular, σ_{α}(V_{2}) = V_{2}, which implies that σ_{α}(V_{1}) = V_{1}. One also can check the latter more directly. It follows that A_{1} is a root system in V_{1}, and similarly A_{2} is a root system in V_{2}. This is part (b) of Proposition 10 on p36 of [24]. Under these conditions, A is considered to be the sum of A_{1} and A_{2}, as on p36 of [24]. If A cannot be expressed as the sum of two root systems in this way, then A is said to be irreducible, as on p36 of [24], and p52 of [14].

Irreducibility of A is equivalent to the connectedness of the Coxeter graph of A, as in Proposition 12 on p36 of [24], and mentioned on p57 of [14]. More precisely, the connectedness of the Coxeter graph of A with respect to a base B for A does not depend on the choice of B, because other choices lead to isomorphic graphs, as before.

Suppose that A corresponds to the sum of subsystems A_{1} and A_{2}, as before. If B_{1} and B_{2} are bases for A_{1} and A_{2}, then it is easy to see that B_{1} ∪ B_{2} is a base for A. If α ∈ B_{1} and β ∈ B_{2}, then (α, β) = 0, so that α and β are not connected by any edges in the Coxeter graph of A with respect to B_{1} ∪B_{2}. This means that the Coxeter graph of A with respect to B_{1} ∪ B_{2} is not connected, as desired. This is one half of the proof of Proposition 12 on p36 of [24].

Alternatively, let B be any base for A, and put B_{j} = A_{j} ∩ B for j = 1, 2. Every element of B_{1} is orthogonal to every element of B_{2}, so that there are no edges between B_{1} and B_{2} in the Coxeter graph of A with respect to B. To get that the Coxeter graph is not connected, one has to check that B_{1}, B_{2} ≠ ∅. Equivalently, one can verify that B cannot be contained in A_{1} or A_{2}, as on p52 of [14].

Conversely, suppose that the Coxeter graph of A with respect to a base B is not connected. This implies that B can be expressed as the union of disjoint nonempty subsets B_{1} and B_{2}, with no edges in the Coxeter graph between elements of B_{1} and B_{2}. Equivalently, every element of B_{1} is orthogonal to every element of B_{2} with respect to (·, ·). Let V_{1} and V_{2} be the linear subspaces of V spanned by B_{1} and B_{2}, respectively, so that every element of V_{1} is orthogonal to every element of V_{2}. Note that V corresponds to the direct sum of V_{1} and V_{2}, because V is spanned by B.

If β ∈ B_{1}, then σ_{β} fixes every element of the hyperplane orthogonal to β, which contains V_{2}. Thus σ_{β}(V_{2}) = V_{2}, which implies that σ_{β}(V_{1}) = V_{1}. Alternatively, one can check directly that σ_{β}(V_{1}) = V_{1}, which implies that σ_{β}(V_{2}) = V_{2}. Similarly, if β ∈ B_{2}, then σ_{β} maps V_{1} and V_{2} onto themselves. Remember that every element α of A can be obtained from elements of B by compositions of the reflections σ_{β}, β ∈ B, as in Section 19.14. It follows that α is contained in V_{1} or V_{2}. This implies that A is reducible, as on p37 of [24], and p52 of [14].

20.5 Root strings

Let V be a vector space over the real numbers of positive finite dimension, let A be a root system in V, and let (·, ·) be an inner product on V that is invariant under the Weyl group of A. Suppose that α and β are nonproportional elements of A. Consider the elements of A of the form

β + j α, (20.5.1)

where j is an integer. The collection of these elements of A is called the α-string through β in A, as on p45 of [14]. Let r and q be the largest integers such that

β − r α, β + q α ∈ A, (20.5.2)

respectively. Thus r, q ≥ 0, because β ∈ A, by hypothesis. We would like to show that if j ∈ Z satisfies −r < j < q, then (20.5.1) is an element of A too.

Otherwise, there are integers j_{1}, j_{2} such that −r ≤ j_{1} < j_{2} ≤ q such that

β + j_{1} α, β + j_{2} α ∈ A (20.5.3)

and β + (j_{1} + 1) α, β + (j_{2} − 1) α ∉ A. (20.5.4)

20.5. ROOT STRINGS 417 Observe that neither β + j_{1} α nor β + j_{2} α is proportional to α, because β is not proportional to α, by hypothesis. It follows that (α, β + j_{1} α) ≥ 0, (α, β + j_{2} α) ≤ 0, (20.5.5) as in Section 19.9. However, (j_{2} − j_{1}) (α, α) > 0, (20.5.6) because j_{1} < j_{2} and α ≠ 0. This is a contradiction, as on p45 of [14].

The fact that (20.5.1) is an element of A when j ∈ Z satisfies −r < j < q also came up in the Lie algebra setting in Section 17.9, as in part (e) of the proposition on p39 of [14].

Let σ_{α} be the symmetry on V with vector α that maps A onto itself, which is the reflection on V associated to α with respect to (·, ·). Remember that σ_{α}(β) − β is an integer multiple of α, by the definition of a root system. If j ∈ Z, then σ_{α}(β + j α) = σ_{α}(β) − j α (20.5.7) is the sum of β and an integer multiple of α as well. Thus σ_{α} maps the α-string through β onto itself, because σ_{α}(A) = A. Using this, it is easy to see that σ_{α}(β + q α) = β − r α. (20.5.8) Equivalently, σ_{α}(β) − q α = β − r α, so that σ_{α}(β) − β = (q − r) α. (20.5.9) This means that 2 (β, α) (α, α)^{−1} = r − q, (20.5.10) by the usual expression for σ_{α} in terms of α and (·, ·). It follows that |r − q| ≤ 3, (20.5.11) because α and β are not proportional, as in Section 19.9.

Put β_{0} = β − r α, which is an element of A that is not proportional to α. Observe that the α-string through β_{0} consists of the same elements of A as the α-string through β. Let r_{0}, q_{0} be the largest integers such that β_{0} − r_{0} α and β_{0} + q_{0} α are elements of A, as before. By construction, r_{0} = 0 and q_{0} = q + r. We also have that |r_{0} − q_{0}| ≤ 3, as in (20.5.11). This means that q + r ≤ 3. (20.5.12) Thus the α-string through β has at most 4 elements, as on p45 of [14].

20.6 Roots and linear subspaces

Let V be a vector space over the real numbers of positive finite dimension, as usual. Suppose for the moment that α is a nonzero element of V, and that σ_{α} is a symmetry on V with vector α. If V_{0} is a linear subspace of V, then we might like to know if σ_{α}(V_{0}) ⊆ V_{0}. (20.6.1)

Equivalently, this means that

σ_{α}(V_{0}) = V_{0}, (20.6.2)

because σ_{α} is its own inverse on V. If α ∈ V_{0}, then it is easy to see that (20.6.1) holds, and hence (20.6.2) holds as well. Of course, (20.6.2) holds when V_{0} is contained in the hyperplane of vectors in V that are fixed by σ_{α}. In fact, these are the only two ways in which (20.6.2) can hold, as in Exercise 1 on p45 of [14]. Indeed, if (20.6.1) holds, and if there is a v_{0} ∈ V_{0} such that σ_{α}(v_{0}) ≠ v_{0}, then σ_{α}(v_{0}) − v_{0} is a nonzero multiple of α that is contained in V_{0}.

Let A be a root system in V, and for each α ∈ A, let σ_{α} be the symmetry on V with vector α that maps A onto itself. Let V_{0} be a nontrivial linear subspace of V, and put A_{0} = A ∩ V_{0}. (20.6.3)

If V_{0} is spanned by A_{0}, then A_{0} is a root system in V_{0}. This follows from the fact that (20.6.2) holds for every α ∈ A_{0}. This is related to Exercise 7 on p46 of [14].

Let A_{1} be a nonempty subset of A that satisfies the following two conditions. First, if α ∈ A_{1}, then −α ∈ A_{1}. Second, if α, β ∈ A_{1} and α + β ∈ A, then

α + β ∈ A_{1}. (20.6.4)

Under these conditions, A_{1} is a root system in the linear subspace V_{1} of V that it spans, as in Exercise 7 on p46 of [14]. Note that this includes the situation mentioned in the preceding paragraph. One way to see this is to check that if α and β are nonproportional elements of A_{1}, then A_{1} contains the α-string through β in A, using the remarks in the previous section. This implies that σ_{α}(β) ∈ A_{1} in this case. Of course, if α, β ∈ A_{1} are proportional, then σ_{α}(β) = −β ∈ A_{1}, by the first condition. It follows that for every α ∈ A_{1}, σ_{α} maps A_{1} into itself, and hence σ_{α}(A_{1}) = A_{1}.

Let (·, ·) be an inner product on V that is invariant under the Weyl group of A. If c is a positive real number, then let A(c) be the set of α ∈ A such that (α, α) = c. If A(c) ≠ ∅, then one can verify that A(c) is a root system in the linear subspace V (c) of V that it spans. This is part of Exercise 11 on p47 of [14].

20.7 More on irreducibility

Let V be a vector space over the real numbers of positive finite dimension, let A be an irreducible root system in V, and let (·, ·) be an inner product on V that is invariant under the Weyl group W of A. We would like to check that W acts irreducibly on V, as in the first part of Lemma B on p53 of [14]. More precisely, this means that there is no nontrivial proper linear subspace of V that is mapped into itself by the elements of W. To see this, let V_{1} be a nonzero linear subspace of V that is mapped into itself by elements of W. If α ∈ A, then let σ_{α} be the reflection on V associated to α with respect to (·, ·), as usual. By hypothesis, σ_{α}(V_{1}) ⊆ V_{1}, (20.7.1)

so that either α ∈ V_{1} or V_{1} is contained in the hyperplane of vectors that are fixed by σ_{α}, as in the previous section. Remember that the hyperplane of vectors that are fixed by σ_{α} is orthogonal to α. In the second case, we get that α is contained in the orthogonal complement V_{2} of V_{1} in V. Thus A ⊆ V_{1} ∪V_{2}, which implies that V_{2} = {0}, because A is supposed to be irreducible in V. This means that V_{1} = V, as desired.

If v ∈ V, then the linear span of the set of images of v under elements of W is a linear subspace of V that is mapped into itself by elements of W. If v ≠ 0, then it follows that this linear span is equal to V, as in the second part of Lemma B on p53 of [14]. If u is another nonzero element of V, then we get that (σ(v), u) ≠ 0 (20.7.2)

for some σ ∈ W.

Suppose that α, β ∈ A satisfy

∥α∥ = ∥β∥, (20.7.3)

where ∥ · ∥ is the norm associated to (·, ·) on V. We would like to show that α is the image of β under an element of W, as in the second part of Lemma C on p53 of [14]. We may as well suppose that (α, β) ≠ 0, by replacing β by one of its images under an element of W if necessary, as in the preceding paragraph. If α and β are proportional, then α = ±β, by (20.7.3). There is nothing to do when α = β, and if α = −β, then α = σ_{β}(β), as desired. Suppose that α and β are not proportional, and let n(α, β), n(β, α) be as in Section 19.9. Observe that n(α, β) = n(β, α) = ±1 (20.7.4)

in this situation, as before. We may as well suppose that n(α, β) = n(β, α) = 1, by replacing β with σ_{β}(β) = −β if necessary. Under these conditions, we get that (σ_{α} ∘ σ_{β} ∘ σ_{α})(β) = (σ_{α} ∘ σ_{β})(β − α) = σ_{α}(−β − α + β) = α, (20.7.5)

as desired.

Let α, β ∈ A be given, and let σ be an element of W such that

(σ(α), β) ≠ 0. (20.7.6)

If σ(α) and β are proportional, then

∥α∥^{2}/∥β∥^{2} = ∥σ(α)∥^{2}/∥β∥^{2} = 1, 4, or 1/4. (20.7.7)

If σ(α) and β are not proportional, then

∥α∥^{2}/∥β∥^{2} = ∥σ(α)∥^{2}/∥β∥^{2} = 1, 2, 3, 1/2, or 1/3, (20.7.8)

as in Section 19.9. In particular, if A is reduced, then the only possible ratios are as in (20.7.8). In this case, one can check that there are at most two possible values for the norms of elements of A, as in the first part of Lemma C on p53 of [14]. If A is not reduced, then one can verify that there are at most three possible values for the norms of elements of A. In order to have three different values for the norms of elements of A, the ratio in (20.7.8) can only be 1, 2, or 1/2 when σ(α) and β are not proportional.

If α ∈ A, then put α̂ = 2 α/∥α∥^{2}, as in (19.7.9), so that

∥α̂∥ = 2/∥α∥. (20.7.9)

Remember that Â = {α̂ : α ∈ A} is the inverse system of A in V with respect to (·, ·), as in Section 19.8. It is easy to see that the irreducibility of A in V implies that Â is irreducible in V too. If the elements of A have the same norm, then Â has the same property, by (20.7.9). In this case, Â is isomorphic to A, using a dilation on V. Otherwise, the number of distinct values of the norms of elements of Â is the same as for A. This corresponds to part of Exercise 11 on p55 of [14].

If ⟨·, ·⟩ is another inner product on V, then there is a unique linear mapping T from V into itself such that

⟨u, v⟩ = (T (u), v) (20.7.10)

for every u, v ∈ V. More precisely, T is self-adjoint and positive definite with respect to (·, ·), because of the symmetry and positivity properties of ⟨·, ·⟩ on V. If ⟨·, ·⟩ is also invariant under W, then one can check that T commutes with the elements of W, because (·, ·) is invariant under W too. It is well known that there is a basis for eigenvectors of T in V that is orthonormal with respect to (·, ·), because T is self-adjoint with respect to (·, ·). The eigenspaces of T are mapped into themselves by elements of W, because T commutes with elements of W. It follows that T is a constant multiple of the identity mapping on V, because W acts irreducibly on V. This means that ⟨·, ·⟩ is a constant multiple of (·, ·) on V.

Suppose that A is reduced, so that there are at most two distinct values of the norms of elements of A. If there are two distinct values for the norms of the elements of A, then the elements of A whose norm is the larger value are called long roots, and the elements of A whose norm is the smaller value are called short roots, as on p53 of [14]. If the elements of A have the same norm, then it is customary to refer to them as long roots, as in [14].

20.8 Maximal roots

Let V be a vector space over the real numbers of positive finite dimension, let A be a root system in V, and let (·, ·) be an inner product on V that is invariant under the Weyl group of A. Also let B be a base for A, so that every element of A can be expressed as a linear combination of elements of B with integer coefficients, and where the coefficients are either all greater than or equal to 0, or all less than or equal to 0.

An element α of A is said to be maximal with respect to B if there is no element of A that can be expressed as the sum of α and some elements of B, which need not be distinct. Equivalently, this means that if γ ∈ A, and each of the coefficients of α associated to elements of B is less than or equal to the corresponding coefficient of γ, then α = γ. In particular, α is maximal when the sum of its coefficients associated to elements of B is maximal, as an element of Z.

Suppose from now on in this section that A is irreducible, and let α_{0} be an element of A that is maximal with respect to B. It is easy to see that the coefficients of α_{0} with respect to elements of B are nonnegative, by maximality. Let B_{0} be the set of β ∈ B such that the coefficient of α with respect to β is positive, so that the coefficients of α_{0} with respect to β ∈ B \ B_{0} are equal to 0. Note that B_{0} ≠ ∅, because α_{0} ≠ 0. We would like to show that

B_{0} = B, (20.8.1)

which is part of Lemma A at the bottom of p52 of [14].

If β and γ are distinct elements of B, then (β, γ) ≤ 0, as in (19.11.8). In particular, this holds when β ∈ B_{0} and γ ∈ B \ B_{0}. If B_{0} ≠ B, then there are β ∈ B_{0} and γ ∈ B \ B_{0} such that (β, γ) ≠ 0, because A is irreducible. It follows that (α_{0}, γ) < 0 under these conditions, by the definition of B_{0}. Observe that α_{0} is not proportional to γ in this situation. This implies that α_{0} + γ ∈ A, as in Section 19.9. This contradicts the maximality of α_{0} with respect to B, so that (20.8.1) holds.

Next, we would like to check that

(α_{0}, β) ≥ 0 (20.8.2)

for every β ∈ B, which is another part of Lemma A at the bottom of p52 of [14]. If the dimension of V is one, then B has only one element, α_{0} is a positive integer multiple of that element of B, and (20.8.2) follows. Suppose now that the dimension of V is at least two. Let β ∈ B be given, and note that α_{0} is not proportional to β, because all of the coefficients of α_{0} with respect to elements of B are positive. If (α_{0}, β) < 0, then α_{0} + β ∈ A, as in Section 19.9 again. This would contradict the maximality of α_{0} with respect to B, so that (20.8.2) holds. We also have that (α_{0}, β) > 0 (20.8.3)

for at least one β ∈ B, because V is spanned by B.

We would like to show that α_{0} is unique, as in Lemma A at the bottom of p52 of [14]. Let α_{1} be another element of A that is maximal with respect to B. This implies that the coefficients of α_{1} with respect to all elements of B are positive, as before. It follows that

(α_{0}, α_{1}) > 0, (20.8.4) by (20.8.2) and (20.8.3). If α_{0} and α_{1} are proportional, then it is easy to see that α_{0} = α_{1}, as desired. Otherwise, if α_{0} and α_{1} are not proportional, then α_{0} − α_{1} ∈ A, as in Section 19.9. If the coefficients of α_{0} − α_{1} with respect to the elements of B are all nonnegative, then we get a contradiction with the maximality of α_{1}. Similarly, if the coefficients of α_{0} − α_{1} are all less than or equal to 0, then we get a contradiction with the maximality of α_{0}.

If α ∈ A and β ∈ B, then

(20.8.5) the coefficient of α associated to β is less than or equal to the coefficient of α_{0} with respect to β.

To see this, one can first find an element α_{1} of A that is maximal with respect to B such that the coefficient of α associated to β is less than or equal to the coefficient of α_{1} associated to β for every β ∈ B. One can choose α_{1} ∈ A so that the sum of its coefficients associated to elements of B is maximal, among elements of A whose coefficients are greater than or equal to the corresponding coefficients of α. The uniqueness of α_{0} implies that α_{0} = α_{1}, so that (20.8.5) holds. This is implicit in an argument near the top of p54 of [14]. It follows that the sum of the coefficients of α associated to elements of B is less than or equal to the sum of the coefficients of α_{0}, with equality only when α = α_{0}. This is another part of Lemma A at the bottom of p52 of [14].

20.9 More on maximality

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. Also let (·, ·) be an inner product on V that is invariant under the Weyl group W of A, and let B be a base for A. Suppose for the moment that A is reduced. If τ is a linear functional on V, then there is an element σ of W such that τ(σ(β)) ≥ 0 for every β ∈ B, as in Section 19.14. Let α ∈ V be given, so that τ_{α}(v) = (v, α) (20.9.1)

defines a linear functional on V. As before, there is a σ ∈ W such that

(σ(β), α) = τ_{α}(σ(β)) ≥ 0 (20.9.2)

for every β ∈ B. Equivalently, this means that

(β, σ^{−1}(α)) ≥ 0 (20.9.3)

for every β ∈ B, because σ is an orthogonal transformation on V with respect to (·, ·). This corresponds to the formulation in part (a) of the theorem on p51 and Exercise 14 on p55 of [14].

Suppose now that A is irreducible as well as reduced, and let α_{0} ∈ A be maximal with respect to B, as in the previous section. We would like to show that ∥α_{0}∥ ≥ ∥α∥ (20.9.4) for every α ∈ A, where ∥ · ∥ is the norm on V associated to (·, ·), as in Lemma D on p53 of [14]. We may as well suppose that

(β, α) ≥ 0 (20.9.5)

for every β ∈ B, by replacing α with σ^{−1}(α) for σ ∈ W as in (20.9.3). Note that the coefficients of α associated to elements of B are less than or equal to the corresponding coefficients of α_{0}, as in (20.8.5). Combining this with (20.9.5), we get that (α_{0} − α, α) ≥ 0, (20.9.6) which is the same as saying that

(α_{0}, α) ≥ (α, α). (20.9.7)

This implies that ∥α∥^{2} = (α, α) ≤ ∥α_{0}∥ ∥α∥, (20.9.8) by the Cauchy–Schwarz inequality. It is easy to obtain (20.9.4) from this, as desired.

Let A be a root system in V again, that is not necessarily reduced. If α ∈ A, then put α̂ = 2 α/∥α∥^{2}, as in (19.7.9), and remember that Â = {α̂ : α ∈ A} is the inverse system of A in V with respect to (·, ·), as in Section 19.8. Let B be a base for A, as before, and let B_{1}, B_{2} be the sets of β ∈ B such that 2 β ∉ A and 2 β ∈ A, respectively, as in Section 19.13. Remember that

B̃ = {β̂: β ∈ B_{1}} ∪ {β̂/2 : β ∈ B_{2}} (20.9.9)

is a base for Â. If α, γ ∈ A satisfy

∥α∥ = ∥γ∥, (20.9.10)

then one can check that the coefficients of α associated to elements of B are less than or equal to the corresponding coefficients of γ if and only if the coefficients of α̂ associated to elements of B̃ are less than or equal to the corresponding coefficients of γ̂.

Suppose that A is reduced, so that B_{2} = ∅, and B̃ is the same as

B̂ = {β̂: β ∈ B}. (20.9.11)

Suppose that A is irreducible as well, which implies that Â is irreducible too, as in Section 20.7. Remember that there are at most two distinct values for the norms of the elements of A, as in Section 20.7. Let us suppose that there are two distinct values for the norms of elements of A, which means that there are two distinct values for the norms of elements of Â.

Let γ̂_{0} be a maximal root in Â, as in the previous section. This implies that γ̂_{0} is a long root in Â, as before. It follows that the corresponding element γ_{0} of A is a short root in A. If α ∈ A, then the coefficients of α̂ asociated to elements of B̂ are less than or equal to the corresponding coefficients of γ̂_{0}, as in (20.8.5). If α is a short root in A, then ∥α∥ = ∥γ_{0}∥, and it follows that the coefficients of α associated to elements of B are less than or equal to the coefficients of γ_{0}, as before. Note that γ_{0} is uniquely determined by this property. This corresponds to part of Exercise 11 on p55 of [14].

20.10 Root systems of type A_{n}

Let n be a positive integer, and let R^{n+1} be the usual space of (n + 1)-tuples of real numbers. The standard inner product is defined on R^{n+1} by

(u, v) = ∑_{j=1}^{n+1} u_{j} v_{j}, (20.10.1)

as usual. Let e_{1}, …, e_{n+1} be the standard basis vectors in R^{n+1}, so that the jth coordinate of e_{l} is equal to 1 when j = l, and to 0 otherwise. Of course, the set Z^{n+1} of (n + 1)-tuples of integers is a subgroup of R^{n+1}, as a commutative group with respect to addition. Put

V = {v ∈ R^{n+1} : ∑_{j=1}^{n+1} v_{j} = 0}, (20.10.2)

which is the hyperplane in R^{n+1} orthogonal to e_{1} + ⋯ + e_{n+1}.

Consider A = {α ∈ V ∩ Z^{n+1} : (α, α) = 2}. (20.10.3)

Equivalently, it is easy to see that A consists exactly of the vectors of the form e_{j} − e_{l}, where 1 ≤ j, l ≤ n + 1 and j ≠ l. Note that A is a finite set of nonzero elements of V whose linear span is equal to V.

Suppose for the moment that α ∈ R^{n+1} satisfies (α, α) = 2. In this case, the reflection on R^{n+1} associated to α with respect to the standard inner product is given by σ_{α}(v) = v − (v, α) α (20.10.4)

for every v ∈ R^{n+1}. If α ∈ V, then σ_{α} maps V onto itself, as in Section 20.6. If α ∈ Z^{n+1}, then σ_{α}(Z^{n+1}) ⊆ Z^{n+1}, and hence

σ_{α}(Z^{n+1}) = Z^{n+1}, (20.10.5)

because σ_{α} is its own inverse. It follows that σ_{α}(A) = A when α ∈ A, because σ_{α} is an orthogonal transformation on R^{n+1}. Of course, (v, α) ∈ Z when α, v ∈ Z^{n+1}, so that A defines a root system in V. This root system is said to be of type A_{n}. Clearly A is reduced, as a root system in V.

Suppose that α = e_{j} − e_{l} for some j, l ∈ {1, …, n + 1}, with j ≠ l. If v ∈ R^{n+1}, then σ_{α}(v) = v − (v_{j} − v_{l}) (e_{j} − e_{l}), (20.10.6)

which interchanges the jth and lth coordinates of v, without affecting the other coordinates of v. It follows that the Weyl group of A consists of the linear mappings on V obtained from permuting the coordinates of elements of V.

Remember that the inverse system Â of A in V with respect to (·, ·) is defined in (19.8.4). In this case, Â = A.

20.10. ROOT SYSTEMS OF TYPE A_{N} 425

If α, β ∈ A and n(α, β) ∈ Z is as in Sections 19.9 and 20.2, then

n(α, β) = (α, β) (20.10.7)

in this situation, as in (20.2.2). Thus

n(α, β) n(β, α) = (α, β)^{2}. (20.10.8)

If α = ±β, then (20.10.7) is ±2, and (20.10.8) is 4. Suppose that α = e_{j_{1}} − e_{j_{2}} and β = e_{l_{1}} − e_{l_{2}} for some j_{1}, j_{2}, l_{1}, l_{2} ∈ {1, …, n + 1} with j_{1} ≠ j_{2}, l_{1} ≠ l_{2}. If {j_{1}, j_{2}} ∩ {l_{1}, l_{2}} has exactly one element, then (20.10.7) is ±1, and (20.10.8) is 1. If {j_{1}, j_{2}} ∩ {l_{1}, l_{2}} = ∅, then (20.10.7) and (20.10.8) are equal to 0.

Put B = {e_{j} − e_{j+1} : j = 1, …, n}, (20.10.9)

which is a subset of A. It is easy to see that B is a basis for V, as a vector space over the real numbers. More precisely, one can check that B is a base for A as a root system in V. Indeed, if 1 ≤ j < l ≤ n + 1, then

e_{j} − e_{l} = ∑_{r=j}^{l−1}(e_{r} − e_{r+1}). (20.10.10)

In particular, e_{1} − e_{n+1} is the unique maximal element of A with respect to B, as in Section 20.8.

Let j, l ∈ {1, …, n} be given, so that e_{j} − e_{j+1} and e_{l} − e_{l+1} are elements of (20.10.9). The corresponding entry of the Cartan matrix of A with respect to B is given by

n(e_{j} − e_{j+1}, e_{l} − e_{l+1}) = (e_{j} − e_{j+1}, e_{l} − e_{l+1}). (20.10.11)

This is symmetric in j and l, and equal to 2 when j = l. As before, (20.10.11) is equal to −1 when |j − l| = 1, and to 0 when |j − l| ≥ 2.

Similarly, let us consider the Coxeter graph of A with respect to B, as in Section 20.3. If j, l ∈ {1, …, n} and j ≠ l, then the number of edges between e_{j} − e_{j+1} and e_{l} − e_{l+1} in the Coxeter graph is the square of (20.10.11). This is equal to 1 when |j − l| = 1, and to 0 when |j − l| ≥ 2. Note that the Coxeter graph is connected, so that A is irreducible.

Let T be a one-to-one linear mapping from V onto itself such that T (A) = A. This implies that (T (α), T (β)) = (α, β) (20.10.12)

for every α, β ∈ A, by (20.2.13) and (20.10.7). It follows that (20.10.12) holds for every α, β ∈ V, because V is spanned by A.

Suppose that T also satisfies T (B) = B. If n = 1, then T is the identity mapping on V. If n ≥ 2, then there is another such mapping T on V, which sends e_{j} − e_{j+1} to e_{n−j} − e_{n−j+1} for each j = 1, …, n. Equivalently, T is the restriction to V of the mapping on R^{n+1} that sends v = (v_{1}, …, v_{n+1}) to −(v_{n+1}, …, v_{1}).

This corresponds to some of the remarks on p64 and in Tables 1 and 2 on p66 of [14], and on p39-40 of [24].

20.11 Types B_{n} and C_{n}

Let n be a positive integer, and let us take V = R^{n}, with its standard inner product (·, ·) and basis e_{1}, …, e_{n}. Consider

A_{1} = {α ∈ Z^{n} : (α, α) = 1 or 2}. (20.11.1)

Equivalently, A_{1} consists of vectors of the form ±e_{j}, 1 ≤ j ≤ n, and ±e_{j} ± e_{l}, 1 ≤ j ≠ l ≤ n. Similarly, put

A_{2} = {±2 e_{j} : 1 ≤ j ≤ n} ∪ {±e_{j} ± e_{l} : 1 ≤ j ≠ l ≤ n}, (20.11.2)

which is a subset of Z^{n} as well. Clearly A_{1} and A_{2} are finite sets of nonzero elements of V that span V. Note that the squares of the norms of the elements of A_{2} are either 2 or 4.

If α is a nonzero element of V, then let σ_{α} be the reflection on V associated to α with respect to (·, ·), as usual. Suppose for the moment that α is a nonzero multiple of e_{j} for some j = 1, …, n. In this case, σ_{α} = σ_{e_{j}} multiplies the jth coordinate of v ∈ V by −1, and leaves the other coordinates of v unchanged. In particular, σ_{α} maps Z^{n}, A_{1}, and A_{2} onto themselves. Equivalently,

σ_{α}(v) − v = −2 v_{j} e_{j} (20.11.3)

for every v ∈ V. If α = ±e_{j} or ±2 e_{j} and v ∈ Z^{n}, then it follows that σ_{α}(v) − v is an integer multiple of α. Thus this holds for every v ∈ A_{1} or A_{2}.

If α ∈ V satisfies (α, α) = 2, then σ_{α}(v) = v − (v, α) α for every v ∈ V, as before. If we also have that α ∈ Z^{n}, then σ_{α} maps Z^{n} into itself, and hence onto itself. In this case, if v ∈ Z^{n}, then (v, α) ∈ Z, so that σ_{α}(v) − v is an integer multiple of α. If α = e_{j} − e_{l}, 1 ≤ j ≠ l ≤ n, then σ_{α}(v) interchanges the jth and lth coordinates of v, without affecting the other coordinates, as in the previous section. This implies that σ_{α} maps A_{1} and A_{2} onto themselves.

Suppose that α = ±(e_{j} + e_{l}) for some 1 ≤ j ≠ l ≤ n, so that (α, α) = 2. In this case, σ_{α}(v) = v − (v_{j} + v_{l}) (e_{j} + e_{l}) (20.11.4)

for every v ∈ V. This interchanges the jth and lth coordinates and multiplies them by −1, while leaving the other coordinates unchanged. Thus σ_{α} maps A_{1} and A_{2} onto themselves, so that A_{1} and A_{2} are root systems in V. More precisely, A_{1} is said to be of type B_{n}, and A_{2} is said to be of type C_{n}.

Consider

(20.11.5) A_{3} = A_{1} ∪ A_{2} = {±e_{j} : 1 ≤ j ≤ n} ∪ {±2 e_{j} : 1 ≤ j ≤ n}
∪{±e_{j} ± e_{l} : 1 ≤ j ≠ l ≤ n}.

If α ∈ A_{3}, then σ_{α} maps A_{1} and A_{2} onto themselves, as in the previous paragraphs, and hence σ_{α} maps A_{3} onto itself. We also have that σ_{α}(v) − v is an integer multiple of α for every v ∈ Z^{n}, as before, and in particular for v ∈ A_{3}.

20.11. TYPES B_{N} AND C_{N} 427

This means that A_{3} is a root system in V as well, which is said to be of type BC_{n}. Note that A_{1} and A_{2} are reduced, and that A_{3} is not reduced.

If α is a nonzero element of V, then put α̂ = 2 α/(α, α), as in (19.7.9). Observe that A_{2} = Â_{1} = {α̂ : α ∈ A_{1}}, (20.11.6)

which is the inverse system of A_{1} in V with respect to (·, ·), as in Section 19.8. Of course, this means that A_{1} = Â_{2} is the inverse system of A_{2} in V with respect to (·, ·). It follows that Â_{3} = A_{3}.

The Weyl groups of A_{1}, A_{2}, and A_{3} are the same, because they are generated by the same reflections on V. This group consists of linear mappings on V that permute the coordinates of elements of V, while multiplying any collection of the coordinates by −1.

Put B_{1} = {e_{j} − e_{j+1} : 1 ≤ j ≤ n − 1} ∪ {e_{n}} (20.11.7)

which is interpreted as being {e_{1}} when n = 1. It is easy to see that this is a basis for V as a vector space over R, and one can check that it is a base for A_{1}, as a root system in V. More precisely, if 1 ≤ j < n, then

e_{j} = ∑_{r=j}^{n−1}(e_{r} − e_{r+1}) + e_{n} (20.11.8)

and

e_{j} + e_{n} = ∑_{r=j}^{n−1}(e_{r} − e_{r+1}) + 2 e_{n}. (20.11.9)

If 1 ≤ j < l ≤ n, then

e_{j} − e_{l} = ∑_{r=j}^{l−1}(e_{r} − e_{r+1}). (20.11.10)

If 1 ≤ j < l < n, then

e_{j} + e_{l} = ∑_{r=j}^{l−1}(e_{r} − e_{r+1}) + ∑_{r=l}^{n−1} 2 (e_{r} − e_{r+1}) + 2 e_{n}. (20.11.11)

Similarly, put

B_{2} = {e_{j} − e_{j+1} : 1 ≤ j ≤ n − 1} ∪ {2 e_{n}}, (20.11.12)

which is interpreted as being {2 e_{1}} when n = 1. This is a basis for V as a vector space over R, and one can verify that it is a base for A_{2} as a root system in V. Indeed, if 1 ≤ j < n, then

2 e_{j} = ∑_{r=j}^{n−1} 2 (e_{r} − e_{r+1}) + 2 e_{n}, (20.11.13)

428 CHAPTER 20. ROOT SYSTEMS, 2 and e_{j} + e_{n} can be expressed as in (20.11.9). If 1 ≤ j < l ≤ n, then e_{j} − e_{l} can be expressed as in (20.11.10), and e_{j} + e_{l} can be expressed as in (20.11.11) when l < n. Note that B_{1} is a base for A_{3} as well, using (20.11.13).

If n = 1, then A_{1} = {±e_{1}} and A_{2} = {±2 e_{1}} are isomorphic to each other, and to the root system discussed in the previous section, with n = 1. If n = 2, then one can use the bases B_{1} and B_{2} to show that A_{1} is isomorphic to A_{2}.

Let α and β be elements of A_{1}, A_{2}, or A_{3}, so that n(α, β) ∈ Z can be defined as in Section 19.9. In fact, n(α, β) = 2 (α, β) (β, β)^{−1}, (20.11.14) as before, so that n(α, β) n(β, α) = 2 (α, β)^{2} (α, α)^{−1} (β, β)^{−1}. (20.11.15) If 1 ≤ j, l ≤ n − 1, then n(e_{j} − e_{j+1}, e_{l} − e_{l+1}) = (e_{j} − e_{j+1}, e_{l} − e_{l+1}) (20.11.16) is symmetric in j and l, and equal to 2 when j = l. This is equal to −1 when |j − l| = 1, and to 0 when |j − l| ≥ 2, as before. It follows that n(e_{j} − e_{j+1}, e_{l} − e_{l+1}) n(e_{l} − e_{l+1}, e_{j} − e_{j+1}) (20.11.17) is equal to 1 when |j − 1| = 1, and to 0 when |j − l| ≥ 2.

If 1 ≤ j ≤ n − 1, then n(e_{j} − e_{j+1}, e_{n}) = 2 (e_{j} − e_{j+1}, e_{n}), (20.11.18) which is 0 when j < n − 1, and −2 when j = n − 1. Similarly, n(e_{n}, e_{j} − e_{j+1}) = (e_{n}, e_{j} − e_{j+1}), (20.11.19) which is 0 when j < n − 1 and −1 when j = n − 1. Thus n(e_{j} − e_{j+1}, e_{n}) n(e_{n}, e_{j} − e_{j+1}) (20.11.20) is equal to 0 when j < n − 1, and to 2 when j = n − 1. Using this and the remarks in the preceding paragraph, one can determine the Cartan matrix and Coxeter graph of A_{1} with respect to B_{1}. These are the same as for A_{3} with respect to B_{1}.

If 1 ≤ j ≤ n − 1 again, then n(e_{j} − e_{j+1}, 2 e_{n}) = (e_{j} − e_{j+1}, e_{n}), (20.11.21) which is 0 when j < n − 1 and −1 when j = n − 1. We also have that n(2 e_{n}, e_{j} − e_{j+1}) = 2 (e_{n}, e_{j} − e_{j+1}), (20.11.22)

20.12. TYPE D_{N} 429

which is 0 when j < n − 1, and −2 when j = n − 1. This implies that

n(e_{j} − e_{j+1}, 2 e_{n}) n(2 e_{n}, e_{j} − e_{j+1}) (20.11.23)

is equal to 0 when j < n − 1, and to 2 when j = n − 1. This can be used to determine the Cartan matrix and Coxeter graph of A_{2} with respect to B_{2}, in combination with the earlier remarks about (20.11.16) and (20.11.17). Note that the Coxeter graphs of A_{1}, A_{2}, and A_{3} are connected, so that these root systems are irreducible.

If T is an automorphism of A_{1} or A_{3} and T (B_{1}) = B_{1}, then one can check that T is the identity mapping on V, using the Cartan matrix. Similarly, if T is an automorphism of A_{2} and T (B_{2}) = B_{2}, then T is the identity mapping on V.

If n ≥ 2, then the maximal root in A_{1} with respect to B_{1} is e_{1} + e_{2}, which can be expressed in terms of B_{1} as in (20.11.9) when n = 2, and (20.11.11) when n > 2. The maximal short root in A_{1} with respect to B_{1} is e_{1}, which can be expressed in terms of B_{1} as in (20.11.8). The maximal root in A_{2} with respect to B_{2} is 2 e_{1}, which can be expressed in terms of B_{2} as in (20.11.13). The maximal short root in A_{2} with respect to B_{2} is e_{1} + e_{2} when n ≥ 2, which can be expressed in terms of B_{2} as in (20.11.9) when n = 2, and (20.11.11) when n > 2.

This corresponds to some of the remarks on p64 and Tables 1 and 2 on p66 of [14], and on p39-40 of [24], as before. Root systems of type BC_{n} are mentioned in Exercise 3 on p66 of [14], and on p41 of [24].

20.12 Type D_{n}

Let n ≥ 2 be an integer, and let us take V = R^{n} again, with its standard inner product (·, ·) and basis e_{1}, …, e_{n}. Consider

A = {α ∈ Z^{n} : (α, α) = 2}, (20.12.1)

which consists of the vectors of the form ±e_{j} ± e_{l}, 1 ≤ j ≠ l ≤ n. Of course, A is a finite set of nonzero elements of V that spans V.

If α ∈ V and (α, α) = 2, then the reflection σ_{α} on V associated to α with respect to (·, ·) is given by σ_{α}(v) = v − (v, α) α, as before. If α ∈ Z^{n} too, then σ_{α} maps Z^{n} onto itself, and hence σ_{α} maps A onto itself, because σ_{α} is an orthogonal transformation on V. We also have that σ_{α}(v) − v = (v, α) α is an integer multiple of α when v ∈ Z^{n} in this situation. Thus A is a root system in V, which is said to be of type D_{n}. Observe that A is reduced, as a root system in V.

If α = e_{j} − e_{l} for some 1 ≤ j ≠ l ≤ n, then σ_{α}(v) interchanges the jth and lth coordinates of v ∈ V, leaving the other coordinates unchanged, as before. If α = ±(e_{j} + e_{l}) for some 1 ≤ j ≠ l ≤ n, then σ_{α}(v) interchanges the jth and lth coordinates and multiplies them by −1, leaving the other coordinates unchanged, as in the previous section. The Weyl group of A consists of linear mappings on V that permute the coordinates of v ∈ V, and multiply an even number of the coordinates by −1.

430 CHAPTER 20. ROOT SYSTEMS, 2

Let Â be the inverse system of A in V with respect to (·, ·), as in Section 19.8. Observe that Â = A in this situation.

It is easy to see that B = {e_{j} − e_{j+1} : 1 ≤ j ≤ n − 1} ∪ {e_{n−1} + e_{n}} (20.12.2) is a basis for V, as a vector space over R. One can check that B is a base for A, as a root system in V. More precisely, if 1 ≤ j < l ≤ n, then

e_{j} − e_{l} = ∑_{r=j}^{l−1}(e_{r} − e_{r+1}), (20.12.3)

as usual. If 1 ≤ j < n − 1, then

e_{j} + e_{n} = ∑_{r=j}^{n−2}(e_{r} − e_{r+1}) + (e_{n−1} + e_{n}). (20.12.4)

If 1 ≤ j < l ≤ n − 1, then

e_{j} + e_{l} = ∑_{r=j}^{l−1}(e_{r} − e_{r+1}) + ∑_{r=l}^{n−2} 2 (e_{r} − e_{r+1}) + (e_{n−1} − e_{n})

+(e_{n−1} + e_{n}), (20.12.5) where the second sum on the right is interpreted as being equal to 0 when l = n − 1.

If α, β ∈ A, then n(α, β) ∈ Z can be defined as in Section 19.9, and is equal to (α, β) in this situation. If 1 ≤ j, l ≤ n − 1, then n(e_{j} − e_{j+1}, e_{l} − e_{l+1}) = (e_{j} − e_{j+1}, e_{l} − e_{l+1}) (20.12.6) is equal to 2 when j = l, to −1 when |j − l| = 1, and to 0 when |j − l| ≥ 2, as before. Thus n(e_{j} − e_{j+1}, e_{l} − e_{l+1}) n(e_{l} − e_{l+1}, e_{j} − e_{j+1}) (20.12.7) is equal to 1 when |j − l| = 1, and to 0 when |j − l| ≥ 2. If 1 ≤ j ≤ n − 1, then n(e_{j} − e_{j+1}, e_{n−1} + e_{n}) = (e_{j} − e_{j+1}, e_{n−1} + e_{n}) (20.12.8) is equal to 0 when j ≤ n−3 and when j = n−1, and equal to −1 when j = n−2. It follows that n(e_{j} − e_{j+1}, e_{n−1} + e_{n}) n(e_{n−1} + e_{n}, e_{j} − e_{j+1}) (20.12.9) is equal to 0 when j ≤ n − 3 and j = n − 1, and to 1 when j = n − 1. This can be used to determine Cartan matrix and Coxeter graph of A with respect to B. If n ≥ 3, then the Coxeter graph is connected, so that A is irreducible in V. In this case, the maximal root in A with respect to B is e_{1} + e_{2}, which can expressed in terms of B as in (20.12.5).

If n = 2, then A is isomorphic to the sum of two root systems of rank one. If n = 3, then A is isomorphic to the corresponding root system in Section 20.10. Thus one often restricts one’s attention to n ≥ 4.

As usual, one can look for automorphisms T of A that map B onto itself, other than the identity mapping. One way to do this is to interchange e_{n−1} −e_{n} with e_{n−1} + e_{n}, while fixing the other elements of B. If n = 4, then one can permute any of the basis elements e_{1} − e_{2}, e_{3} − e_{4}, and e_{3} + e_{4}.

This corresponds to some remarks on p64 and Tables 1 and 2 on p66 of [14] again, and on p39-40 of [24].

20.13 Complex numbers

Let V be a vector space over the real numbers of positive finite dimension, and let (·, ·) be an inner product on V. If α is a nonzero element of V, then let σ_{α} be the reflection on V associated to α with respect to (·, ·). Let β be another nonzero element of V. If α and β are proportional in V, then σ_{α} = σ_{β} on V. Otherwise, if α and β are not proportional in V, then they span a twodimensional linear subspace V_{0} of V. Note that σ_{α} and σ_{β} map V_{0} onto itself, as in Section 20.6. Of course, σ_{α} and σ_{β} fix every element of the orthogonal complement of V_{0} in V.

The complex plane C may be considered as a two-dimensional vector space over the real numbers. More precisely, C can be identified with R^{2} in the usual way, using the real and imaginary parts of a complex number. If u, v ∈ C, then put (u, v) = Re(u \overline{v}), (20.13.1)

which is the real part of u times the complex conjugate of v. This is the standard inner product on C, as a vector space over R, and which corresponds to the standard inner product on R^{2}. The norm on C associated to (20.13.1) is the same as the standard absolute value function | · | on C. Note that

(a u, a v) = |a|^{2} (u, v) (20.13.2)

for every a, u, v ∈ C. As before, a real-linear mapping from C into itself is a linear mapping from C into itself, as a vector space over R. Complex conjugation on C is a real-linear mapping from C into itself that preserves (20.13.2). This is the same as the reflection on C, as a vector space over R, associated to a nonzero imaginary number with respect to (20.13.1).

If a is a nonzero complex number, then put

ρ_{a}(z) = a \overline{(z/a)} = (a/\overline{a}) \overline{z} (20.13.3)

for every z ∈ C. This defines a real-linear mapping ρ_{a} from C into itself, with

ρ_{t a} = ρ_{a} (20.13.4) for every t ∈ R \ {0}. Observe that ρ_{a} is an orthogonal transformation on C, as a vector space over R, with respect to (20.13.1). If α ∈ C \ {0}, then let σ_{α} be the reflection on C, as a vector space over R, associated to α with respect to (20.13.1). Observe that ρ_{a} = σ_{a i} (20.13.5)

for every a ∈ C \ {0}. Equivalently,

σ_{α} = ρ_{α i} = −ρ_{α} (20.13.6)

for every α ∈ C \ {0}. Thus

σ_{α}(z) = −ρ_{α}(z) = −α \overline{(z/α)} = −(α/\overline{α}) \overline{z} (20.13.7)

for every α ∈ C \ {0} and z ∈ C.

If α, β ∈ C \ {0} and z ∈ C, then

σ_{α}(σ_{β}(z)) = −(α/\overline{α}) \overline{σ_{β}(z)} = (α \overline{β}/(\overline{α} β)) z. (20.13.8)

This is related to Exercise 3 on p46 of [14]. This is also related to Weyl groups of rank two root systems, as in Exercise 4 on p46 of [14], and some remarks on p27 of [24].

Let α ∈ C \ {0} and z ∈ C be given. Observe that

σ_{α}(z) − z = −α \overline{(z/α)} − z = −α \overline{(z/α)} − α(z/α)
(20.13.9)  = −2 Re(z/α) α = −2 (z, α) |α|^{−2} α.

Of course, the right side corresponds to the usual expression for reflections with respect to an inner product.

20.14 Type A_{2}

Some very nice pictures of a root system in the plane that is isomorphic to the one in Section 20.10 with n = 2 can be found on p44 of [14] and p26 of [24]. This root system can also be described as the set A of cube roots of unity and their negatives in the complex plane. Remember that points on the unit circle in C can be expressed as exp(i t) with t ∈ R, using the complex exponential function. The elements of A can be expressed as exp(i t), where t is an integer multiple of π/3. In particular, A is a subgroup of the unit circle in C, as a commutative group with respect to multiplication.

It is well known that

exp(π i/3) = 1/2 + (\sqrt{3}/2) i (20.14.1)

and exp(2 π i/3) = −1/2 + (\sqrt{3}/2) i. (20.14.2)

Indeed,

(20.14.3)  (1/2 + (\sqrt{3}/2) i)^{2} = 1/4 − 3/4 + 2 (1/2) (\sqrt{3}/2) i
= −1/2 + (\sqrt{3}/2) i

and

(20.14.4) (1/2 + (\sqrt{3}/2) i)^{3} = (1/2 + (\sqrt{3}/2) i)^{2} (1/2 + (\sqrt{3}/2) i)
= (−1/2 + (\sqrt{3}/2) i) (1/2 + (\sqrt{3}/2) i)
= −1/4 − 3/4 = −1.

Put α = 1 and β = exp(2 π i/3), so that α + β = exp(π i/3), as indicated in the pictures on p44 of [14] and p26 of [24]. Thus A consists of ±α, ±β, and ±(α + β). Of course, A is a finite set of nonzero elements of C, whose linear span in C, as a vector space over R, is equal to C.

If γ ∈ C \ {0}, then let σ_{γ} be the reflection on C, as a vector space over R, associated to γ with respect to the standard inner product, as before. Thus one should check that σ_{γ}(A) = A for every γ ∈ A. This is clear from the pictures in [14, 24], and can be verified using (20.13.7) as well. One should also check that σ_{γ}(z) − z is an integer multiple of γ ∈ A for every z ∈ A, which means that 2 (z, γ) ∈ Z for every z ∈ A. This can be seen using the expressions for the elements of A mentioned earlier, or more geometrically.

Clearly A is reduced, as a root system in the plane. It is easy to see that B = {α, β} is a base for A, as in [14, 24]. If we take n = 2 in Section 20.10, then we have seen that {e_{1} −e_{2}, e_{2} −e_{3}} is a base for the root system discussed there. One can check that these root systems are isomorphic, using a linear mapping which sends one base to the other. Note that the inner product in Section 20.10 corresponds to twice the standard inner product on C in this way.

20.15 Dynkin diagrams

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. If α, β ∈ A, then let n(α, β) ∈ Z be as in Section 19.9. Let (·, ·) be an inner product on V that is invariant under the Weyl group of A, and let ∥ · ∥ be the corresponding norm on V. If α, β ∈ A, then

n(α, β) = 2 (α, β)/∥β∥^{2}, (20.15.1)

as before. In particular, n(α, β) = 0 if and only if (α, β) = 0, which is symmetric in α and β. Otherwise, if n(β, α) ≠ 0, then

n(α, β)/n(β, α) = ∥α∥^{2}/∥β∥^{2}. (20.15.2)

In this case, ∥α∥ is greater than, equal to, or less than ∥β∥ exactly when (20.15.2) is greater than, equal to, or less than 1, respectively.

Let B be a base for A. The corresponding Dynkin diagram adds some information to the Coxeter graph of A with respect to B, so that the Cartan matrix of A with respect to B is determined. More precisely, if α and β are distinct elements of B, and if there is more than one edge between α and β in the Coxeter graph, then n(α, β), n(β, α) ≠ 0, and (20.15.2) is not equal to 1. In this situation, one can add a marking to the edges between α and β, to indicate whether (20.15.2) is greater than or less than 1. This is equivalent to indicating whether ∥α∥ is greater than or less than ∥β∥, as on p57 of [14] and p39 of [24].

Alternatively, suppose that A is irreducible, so that the Coxeter graph is connected. As on p38 of [24], one can label each α ∈ B with a positive real number that is proportional to ∥α∥^{2}. If α and β are distinct elements of B, and if there is at least one edge between α and β in the Coxeter graph, then n(α, β), n(β, α) ≠ 0, and (20.15.2) is the same as the ratio of the labels of α and β. In this case, the number of edges between α and β in the Coxeter graph is the same as this ratio. Two sets of labels of the elements of B are considered to be equivalent when they differ by a single positive multiplicative constant.

Let B_{0} be a nonempty subset of B, and let V_{0} be the linear subspace spanned by B_{0}. Thus A_{0} = A∩V_{0} is a root system in V_{0}, as in Section 20.6, and Exercise 2 on p54 of [14]. More precisely, B_{0} is a base for A_{0}. If α ∈ A, then let σ_{α} be the symmetry on V with vector α that maps A onto itself. If α ∈ A_{0}, then σ_{α} maps V_{0} onto itself, as before. This implies that σ_{α} maps A_{0} onto itself, so that the restriction of σ_{α} to V_{0} is the symmetry on V_{0} with vector α that maps A_{0} onto itself. If α, β ∈ A_{0}, then it follows that the analogue of n(α, β) for A_{0} is the same as for A. Similarly, the restriction of (·, ·) to V_{0} is invariant under the Weyl group of A_{0}. The Cartan matrix of A_{0} with respect to B_{0} is the same as the restriction to B_{0} × B_{0} of the Cartan matrix of A with respect to B. If α and β are distinct elements of B_{0}, then the number of edges between α and β in the Coxeter graph of A_{0} with respect to B_{0} is the same as in the Coxeter graph of A with respect to B. One could also use the same additional markings or labellings for the Dynkin diagram of A_{0} with respect to B_{0} as for A with respect to B. This corresponds to Exercise 6 on p54 of [14].

Root systems, 3

21.1 More on type A_{2}

Let (·, ·) be the standard inner product on the complex plane, considered as a vector space over the real numbers, as in Section 20.13. Also let A_{2,1} be the root system in C discussed in Section 20.14, consisting of the cube roots of unity and their negatives. Equivalently, A_{2,1} consists of complex numbers of the form exp(j π i/3), with j ∈ Z. Observe that

exp(π i/3) + exp(2 π i/3) = \sqrt{3} i, (21.1.1)

by (20.14.1) and (20.14.2). If j ∈ Z, then it follows that

(21.1.2) exp(j π i/3) + exp((j + 1) π i/3) = \sqrt{3} i exp((j − 1) π i/3).

Put A_{2,2} = \sqrt{3} i A_{2,1} = \{\sqrt{3} i z : z ∈ A_{2,1}\}. (21.1.3)

This is the same as the set of complex numbers that can be expressed as in (21.1.2) for some j ∈ Z. It is clear from (21.1.3) that A_{2,2} is a root system in C, as a vector space over R, that is isomorphic to A_{2,1}. If w ∈ A_{2,2}, then it is easy to see that w/\overline{w} ∈ A_{2,1}. (21.1.4)

Of course, this also holds when w ∈ A_{2,1}.

If γ ∈ C \ {0}, then let σ_{γ} be the reflection on C, as a vector space over R, associated to γ with respect to (·, ·), as before. Thus σ_{γ}(z) = −(γ/\overline{γ}) \overline{z} for every z ∈ C, as in Section 20.13. If γ ∈ A_{2,1} or A_{2,2}, then one can check that σ_{γ} maps A_{2,1} and A_{2,2} onto themselves, using (21.1.4).

Remember that σ_{γ}(z)−z = −2 (z, γ) |γ|^{−2} γ for every γ ∈ C\{0} and z ∈ C, as in Section 20.13. Let us check that

2 (z, γ) |γ|^{−2} = 2 Re(z/γ) ∈ Z (21.1.5) for every γ, z ∈ A_{2,1} ∪ A_{2,2}. This was already mentioned in Section 20.14 when γ, z ∈ A_{2,1}. If γ, z ∈ A_{2,2}, then this follows from the previous case, because γ and z are products of elements of A_{2,1} by \sqrt{3} i. If γ ∈ A_{2,1} and z ∈ A_{2,2}, then (21.1.5) can be reduced to the case where z ∈ A_{2,1}, because elements of A_{2,2} can be expressed as sums of elements of A_{2,1}, as in (21.1.2). The case where γ ∈ A_{2,2} and z ∈ A_{2,1} is a more precise version of this, because of the additional factor of |γ|^{−2} = 1/3. If γ = \sqrt{3} i and z ∈ A_{2,1}, then (21.1.5) can be verified directly. Otherwise, if γ is the product of \sqrt{3} i and any element of A_{2,1}, then one can reduce to the case where γ = \sqrt{3} i.

21.2 Type G_{2}

Some very nice pictures can be found on p44 of [14] and p27 of [24]. Let us continue with the notation in the previous section, and put

A = A_{2,1} ∪ A_{2,2}. (21.2.1)

It is easy to see that this defines a root system in the complex plane, as a vector space over the real numbers, using the remarks in the previous section. This root system is said to be of type G_{2}, and this description of it corresponds to the one on p40 of [24]. More precisely, A is reduced as a root system in C.

Put α = 1 ∈ A_{2,1} and

β = \sqrt{3} i exp(π i/3) = \sqrt{3} i (1/2 + (\sqrt{3}/2) i) = −3/2 + (\sqrt{3}/2) i, (21.2.2)

which is an element of A_{2,2}. Note that this choice of α is the same as in Section 20.14, but this choice of β is different. Of course, B = {α, β} is a basis for C as a vector space over R, and in fact it is a base for A, as in [14, 24]. Indeed,

β + α = −1/2 + (\sqrt{3}/2) i = exp(2 π i/3) (21.2.3)

and β + 2 α = 1/2 + (\sqrt{3}/2) i = exp(π i/3). (21.2.4)

Similarly,

(21.2.5)  β + 3 α = 3/2 + (\sqrt{3}/2) i = \sqrt{3} i (1/2 − (\sqrt{3}/2) i)
= \sqrt{3} i exp(−π i/3).

We also have that 2 β + 3 α = \sqrt{3} i. (21.2.6)

These expressions for elements of A are indicated in the picture on p27 of [24], and the other elements of A can be obtained by multiplying by −1.

If z, w ∈ A, then n(z, w) = 2 (z, w) |w|^{−2} ∈ Z, as in Section 19.9. Observe that n(α, β) = 2 (α, β) |β|^{−2} = 2 Re(α \overline{β}) |β|^{−2} = −1 (21.2.7) and n(β, α) = 2 (α, β) |α|^{−2} = −3. (21.2.8)

Thus the number of edges between α and β in the Coxeter graph of A with respect to B is n(α, β) n(β, α) = 3. (21.2.9)

The corresponding Dynkin diagram may be depicted as on p38 of [24], where α has the label |α|^{2} = 1 and β has the label |β|^{2} = 3. Alternatively, the Dynkin diagram may be depicted as on p57f of [14] and p39 of [24], where the edges are marked to indicate that |α| < |β|.

Let T be a one-to-one linear mapping from C onto itself, as a vector space over R. Remember that T is an automorphism of the root system A if T (A) = A. In this case, if T maps B onto itself, then T preserves the Cartan matrix of A with respect to B, as in Section 20.2. It is easy to see that this only happens when T is the identity mapping on C in this situation. This is mentioned in Table 1 on p66 of [14], and on p39 of [24].

If z ∈ C \ {0}, then put ẑ = 2 z/|z|^{2}, as in (19.7.9). If E ⊆ C \ {0}, then let Ê be the set of ẑ, z ∈ E. Thus

Â_{2,1} = 2 A_{2,1} = {2 z : z ∈ A_{2,1}} (21.2.10)

and Â_{2,2} = (2/3) A_{2,2} = {(2/3) z : z ∈ A_{2,2}}. (21.2.11)

Observe that Â_{2,1} = (2/\sqrt{3}) i A_{2,2} (21.2.12)

and Â_{2,2} = (2/\sqrt{3}) i A_{2,1}. (21.2.13)

This implies that Â = Â_{2,1} ∪ Â_{2,2} = (2/\sqrt{3}) i A. (21.2.14)

Remember that Â is the inverse system of A in C with respect to (·, ·), as in Section 19.8. It follows that Â is isomorphic to A, as root systems in C. This corresponds to part of Exercise 5 on p63 of [14].

21.3 Another description

Let us now consider the description on p65 of [14], of a root system that is isomorphic to the one in the previous section. Let (·, ·) be the standard inner product on R^{3}, and let e_{1}, e_{2}, and e_{3} be the standard basis vectors in R^{3}. Put

V = {v ∈ R^{3} : v_{1} + v_{2} + v_{3} = 0}, (21.3.1)

which is the two-dimensional linear subspace of R^{3} orthogonal to e_{1} + e_{2} + e_{3}. Consider

(21.3.2)  A_{1} = {α ∈ V ∩ Z^{3} : (α, α) = 2},
(21.3.3)  A_{2} = {α ∈ V ∩ Z^{3} : (α, α) = 6},

and A = A_{1} ∪ A_{2}. (21.3.4)

Thus V is the same as in Section 20.10 with n = 2, and A_{1} is the root system discussed there in this case. As before, the elements of A_{1} are of the form e_{j} − e_{l}, where 1 ≤ j ≠ l ≤ 3. Observe that α ∈ Z^{3} satisfies (α, α) = 6 exactly when two of the coordinates of α are ±1, and the third coordinate is ±2. It follows that α ∈ A_{2} exactly when one of the coordinates of α is 2 and the other two coordinates are −1, or one of the coordinates is −2 and the other two coordinates are 1. In particular, A is a finite set of nonzero elements of V whose linear span is V.

If α is a nonzero element of R^{3}, then let σ_{α} be the reflection on R^{3} associated to α with respect to (·, ·), as usual. If α ∈ V, then

σ_{α}(V) = V, (21.3.5)

as in Section 20.6. Suppose for the moment that (α, α) = 2, so that σ_{α}(v) = v − (v, α) α for every v ∈ R^{3}. If α ∈ Z^{3}, then σ_{α}(Z^{3}) = Z^{3}, as in Section 20.10. If α ∈ V ∩Z^{3}, then it follows that σ_{α} maps A_{1} and A_{2} onto themselves, because σ_{α} is an orthogonal transformation on R^{3}. Of course, this implies that σ_{α} maps A onto itself. If α, v ∈ Z^{3}, then (v, α) ∈ Z, so that σ_{α}(v) − v = (v, α) α is an integer multiple of α. If α = e_{j} − e_{l}, 1 ≤ j ≠ l ≤ 3, then σ_{α}(v) interchanges the jth and lth coordinates of v, without affecting the other coordinate, as before.

Suppose now that α ∈ A_{2}, so that

σ_{α}(v) = v − (v, α) α/3 (21.3.6)

for every v ∈ R^{3}. If β ∈ A, then we would like to check that

(β, α)/3 ∈ Z. (21.3.7)

Suppose first that β ∈ A_{1}, so that β = e_{j} − e_{l} for some 1 ≤ j ≠ l ≤ n. If α_{j} = α_{l}, then (β, α) = 0. Otherwise, if α_{j} ≠ α_{l}, then one can check that

(β, α) = ±3. (21.3.8)

If β = ±α ∈ A_{2}, then (21.3.7) clearly holds. If β ∈ A_{2} and β ≠ ±α, then one can verify that (21.3.8) holds.

It follows from (21.3.7) that σ_{α} maps A into Z^{3}. This implies that σ_{α} maps A_{1} and A_{2} into themselves, because σ_{α} is an orthogonal transformation on R^{3} that maps V to itself. This means that σ_{α} maps A_{1} and A_{2} onto themselves, because σ_{α} is its own inverse, so that σ_{α} maps A onto itself as well. If β ∈ A, then σ_{α}(β) − β is an integer multiple of α, by (21.3.7). This shows that A is a root system in V, and it is easy to see that A is reduced.

As a base for A, one can take

21.4. TYPE F_{4} 439

{e_{1} − e_{2}, −2 e_{1} + e_{2} + e_{3}}, (21.3.9) as on p65 of [14]. Indeed,

(21.3.10)  (e_{1} − e_{2}) + (−2 e_{1} + e_{2} + e_{3}) = e_{3} − e_{1},
(21.3.11)  2 (e_{1} − e_{2}) + (−2 e_{1} + e_{2} + e_{3}) = e_{3} − e_{2},
(21.3.12)  3 (e_{1} − e_{2}) + (−2 e_{1} + e_{2} + e_{3}) = e_{1} − 2 e_{2} + e_{3},
(21.3.13) 3 (e_{1} − e_{2}) + 2 (−2 e_{1} + e_{2} + e_{3}) = −e_{1} − e_{2} + 2 e_{3}.

Note that (21.3.13) is the maximal root in A with respect to (21.3.9), and that (21.3.11) is the maximal short root, as in Table 2 of p66 of [14].

If z, w ∈ A, then n(z, w) = 2 (z, w) (w, w)^{−2} ∈ Z, as in Section 19.9. It is easy to see that

(21.3.14)  n(e_{1} − e_{2}, −2 e_{1} + e_{2} + e_{3}) = −1,
(21.3.15)  n(−2 e_{1} + e_{2} + e_{3}, e_{1} − e_{2}) = −3.

In particular, their product is 3, which is the number of edges between the two vertices in the Coxeter graph of A.

Consider the linear mapping from the complex plane, as a vector space over the real numbers, onto V which sends the elements α and β of the base in the previous section to the first and second elements of (21.3.9), respectively. One can check that this defines an isomorphism between the root systems discussed in this and the previous section. More precisely, the subsystems A_{2,1} and A_{2,2} in the previous section correspond to A_{1} and A_{2} in this section. This is related to Exercise 4 on p67 of [14]. Observe that the restriction to V of the standard inner product on R corresponds to twice the standard inner product on C with respect to this isomorphism.

21.4 Type F_{4}

In this section, we take V = R^{4}, with its standard inner product (·, ·) and basis e_{1}, e_{2}, e_{3}, e_{4}. Put A_{1} = {α ∈ Z^{4} : (α, α) = 1 or 2}, (21.4.1)

as in Section 20.11, with n = 4. The elements of A_{1} with (α, α) = 1 are of the form ±e_{j} for some j = 1, 2, 3, 4, and the elements of A_{1} with (α, α) = 2 are of the form ±e_{j} ± e_{l} for some 1 ≤ j ≠ l ≤ n, as before. We have seen that A_{1} is a reduced root system in V.

Let L̃_{4} be the subgroup of R^{4}, as a commutative group with respect to addition, generated by e_{1}, e_{2}, e_{3}, e_{4}, and (1/2) (e_{1} + e_{2} + e_{3} + e_{4}). Of course, L_{4} = Z^{4} is the subgroup of R^{4} generated by e_{1}, e_{2}, e_{3}, e_{4}, so that L̃_{4} is the same as the subgroup of R^{4} generated by Z^{4} and (1/2) (e_{1}+e_{2}+e_{3}+e_{4}). Equivalently,

L̃_{4} = Z^{4} ∪ (Z^{4} + (1/2, 1/2, 1/2, 1/2)). (21.4.2)

Consider A = {α ∈ L̃_{4} : (α, α) = 1 or 2}. (21.4.3)

Note that A ∩ Z^{n} = A_{1}. (21.4.4)

One can check that

A ∩ (Z^{4} + (1/2, 1/2, 1/2, 1/2)) (21.4.5)

= {α ∈ R^{4} : α_{j} = ±1/2 for every j = 1, 2, 3, 4}.

More precisely, it is easy to see that if α_{j} = ±1/2 for each j = 1, 2, 3, 4, then α ∈ L̃_{4} and (α, α) = 1, so that α ∈ A. If α ∈ Z^{4} + (1/2, 1/2, 1/2, 1/2) and α_{j} ≠ ±1/2 for some j = 1, 2, 3, 4, then one can verify that (α, α) ≥ 3.

Of course, A is a finite set of nonzero elements of V, whose linear span is V. If α ∈ V \ {0}, then let σ_{α} be the reflection on V associated to α with respect to (·, ·), as usual. Suppose first that α = ±e_{j} for some j = 1, 2, 3, 4, so that σ_{α} = σ_{e_{j}} multiplies the jth coordinate of v ∈ V by −1, and leaves the other coordinates of v unchanged. Clearly σ_{α} maps Z^{4}, L̃_{4}, A_{1}, and A onto themselves. If v ∈ L̃_{4}, then 2 v_{j} ∈ Z, and σ_{α}(v) − v = −2 v_{j} e_{j} is an integer multiple of α.

Suppose now that α = ±e_{j} ± e_{l} for some 1 ≤ j ≠ l ≤ 4, so that σ_{α}(v) = v − (v, α) α for every v ∈ V. More precisely, we have seen that σ_{α} interchanges the jth and lth coordinates of v ∈ V in this case, and may multiply them by −1, while leaving the other coordinates of v unchanged. In particular, σ_{α} maps Z^{4}, L̃_{4}, A_{1}, and A onto themselves. If v ∈ L̃_{4}, then one can check that

(v, α) ∈ Z. (21.4.6)

This implies that σ_{α}(v) − v = −(v, α) α is an integer multiple of α when v ∈ L̃_{4}, and in particular when v ∈ A.

Suppose that α is an element of (21.4.5), so that α_{j} = ±1/2 for each j = 1, 2, 3, 4. If v ∈ L̃_{4}, then one can verify that

2 (v, α) ∈ Z. (21.4.7)

More precisely, if v ∈ Z^{4}, then this follows from the fact that 2 α ∈ Z^{4}. If v = (1/2, 1/2, 1/2, 1/2), then (21.4.7) can be checked directly. This implies that (21.4.7) holds for all v ∈ L̃_{4}.

Note that σ_{α}(v) = v − 2 (v, α) α for every v ∈ V, because (α, α) = 1. It follows that σ_{α} maps L̃_{4} into itself, by (21.4.7). Thus σ_{α} maps L̃_{4} onto itself, because σ_{α} is its own inverse on V. This means that σ_{α} maps A onto itself, because σ_{α} is an orthogonal transformation on V. If v ∈ L̃_{4}, then σ_{α}(v) − v is an integer multiple of α, by (21.4.7).

This shows that A is a root system in V, which is said to be of type F_{4}. This follows the descriptions on p65 of [14], and p40 of [24]. Observe that A is reduced, as a root system in V.

Put V_{0} = {v ∈ V : v_{1} = 0} (21.4.8) and A_{0} = A ∩ V_{0} = A_{1} ∩ V_{0}. (21.4.9)

Observe that A_{0} is a reduced root system in V_{0}, which is isomorphic to the analogue of A_{1} in R^{3}.

Put B = {e_{2} − e_{3}, e_{3} − e_{4}, e_{4}, (1/2) (e_{1} − e_{2} − e_{3} − e_{4})} (21.4.10)

and B_{0} = B ∩ V_{0} = {e_{2} − e_{3}, e_{3} − e_{4}, e_{4}}. (21.4.11)

It is easy to see that B_{0} is a basis for V_{0}. In fact, B_{0} is a base for A_{0} as a root system in V_{0}, as in Section 20.11. We also have that B is a basis for V, because B_{0} is a basis for V_{0}. One can check that B is a base for A, as in [14, 24].

More precisely, note that e_{2} and e_{3} can be expressed as sums of elements of B_{0}, as in Section 20.11. Using this, one can express any element of A of the form (1/2) (e_{1} ± e_{2} ± e_{3} ± e_{4}) as (1/2) (e_{1} − e_{2} − e_{3} − e_{4}) plus possibly some elements of B_{0}. If l = 2, 3, 4, then e_{1} − e_{l} can be expressed as the sum of

2 ((1/2) (e_{1} − e_{2} − e_{3} − e_{4})) = e_{1} − e_{2} − e_{3} − e_{4} (21.4.12)

and elements of B_{0}. Similarly, one can use this to express e_{1} and e_{1} + e_{l}, l = 2, 3, 4, as sums of (21.4.12) and elements of B_{0}.

The maximal root in A with respect to B is

2 (e_{2} − e_{3}) + 3 (e_{3} − e_{4}) + 4 e_{4} + 2 ((1/2) (e_{1} − e_{2} − e_{3} − e_{4})) (21.4.13)

= e_{1} + e_{2},

as in Table 2 on p66 of [14]. The maximal short root is

(e_{2} − e_{3}) + 2 (e_{3} − e_{4}) + 3 e_{4} + 2 ((1/2) (e_{1} − e_{2} − e_{3} − e_{4})) (21.4.14)

= e_{1}.

If α, β ∈ A, then n(α, β) = 2 (α, β) (β, β)^{−1} ∈ Z, as in Section 19.9. If α, β ∈ B and α ≠ β, then the number of edges between α and β in the Coxeter graph of A with respect to B is n(α, β) n(β, α). In particular, if (α, β) = 0, then n(α, β) = n(β, α) = 0, and there are no edges between α and β in the Coxeter graph of A with respect to B. Clearly

(21.4.15) (e_{2} − e_{3}, e_{4}) = (e_{2} − e_{3}, (1/2) (e_{1} − e_{2} − e_{3} − e_{4}))
= (e_{3} − e_{4}, (1/2) (e_{1} − e_{2} − e_{3} − e_{4})) = 0.

Thus there are no edges between e_{2} − e_{3} and e_{4}, or between either e_{2} − e_{3} or e_{3} − e_{4} and (1/2) (e_{1} − e_{2} − e_{3} − e_{4}) in the Coxeter graph of A with respect to B.

It is easy to see that

n(e_{2} − e_{3}, e_{3} − e_{4}) = n(e_{3} − e_{4}, e_{2} − e_{3}) = −1 (21.4.16) and

n(e_{4}, (1/2) (e_{1} − e_{2} − e_{3} − e_{4})) (21.4.17)

= n((1/2) (e_{1} − e_{2} − e_{3} − e_{4}), e_{4}) = −1.

This implies that there is one edge between e_{2} − e_{3} and e_{3} − e_{4}, and one edge between e_{4} and (1/2) (e_{1} − e_{2} − e_{3} − e_{4}), in the Coxeter graph of A with respect to B.

We also have that n(e_{3} − e_{4}, e_{4}) = −2 (21.4.18)

and n(e_{4}, e_{3} − e_{4}) = −1. (21.4.19)

This means that there are two edges between e_{3} − e_{4} and e_{4} in the Coxeter graph of A with respect to B. The corresponding Dynkin diagram may be depicted as on p38 of [24], where e_{2} − e_{3} and e_{3} − e_{4} are labelled with 2, and e_{4} and (1/2) (e_{1} − e_{2} − e_{3} − e_{4}) are labelled with 1. The Dynkin diagram may be depicted as on p57f of [14] and p39 of [24] as well, where the edges between e_{3} − e_{4} and e_{4} are marked to indicate that (e_{3} − e_{4}, e_{3} − e_{4}) = 2 > 1 = (e_{4}, e_{4}).

Let T be an automorphism of A as a root system, so that T is a one-to-one linear mapping from V onto itself that maps A onto itself. If T (B) = B, then T preserves the Cartan matrix of A with respect to B, as in Section 20.2. It it easy to see that this can only happen when T is the identity mapping on V, as indicated in Table 1 on p66 of [14], and on p39 of [24].

21.5 Inverse systems

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. If α, β ∈ A, then we let n(α, β) ∈ Z be as in Section 19.9, as usual. Let (·, ·) be an inner product on V that is invariant under the Weyl group of A, so that n(α, β) = 2 (α, β) (β, β)^{−1} for every α, β ∈ A. If v ∈ V \ {0}, then put v̂ = 2 v (v, v)^{−1}, as in (19.7.9). Note that

(v̂, v̂) = 4/(v, v). (21.5.1)

Remember that Â = {α̂ : α ∈ A} is the inverse system of A in V with respect to (·, ·), as in Section 19.8. If α, β ∈ A, then α̂, β̂ ∈ Â, and n(α̂, β̂) can be defined as before. Observe that

n(α̂, β̂) = 2 (α̂, β̂) (β̂, β̂)^{−1} = 2 (α, β) (α, α)^{−1} = n(β, α). (21.5.2)

It follows that n(α̂, β̂) n(β̂, α̂) = n(α, β) n(β, α) (21.5.3)

for every α, β ∈ A. Of course, if (α, α) = (β, β), then

n(α̂, β̂) = n(α, β). (21.5.4)

21.6. TYPE E_{8} 443

Let B be a base for A. Suppose for the moment that A is reduced, so that B̂ = {α̂ : α ∈ B} is a base for Â, as in Section 19.13. In this case, the Coxeter graph of A with respect to B corresponds to the Coxeter graph of Â with respect to B̂ under the mapping α ↦ α̂, by (21.5.3). If A is not reduced, then one can get a base for Â from B as in Section 19.13. One can check that the Coxeter graphs for A and Â with respect to these bases correspond to each other in this situation as well.

If the elements of A have the same norm with respect to (·, ·), then there is a dilation on V that sends A onto Â. This dilation sends α ∈ A to α̂ ∈ Â.

If A is as in Section 21.2, then the elements of A do not all have the same norm, but we saw that Â is isomorphic to A. Alternatively, let B = {α, β} be a base for A, so that B̂ = {α̂, β̂} is a base for Â. Consider the mapping ϕ from B onto B̂ defined by ϕ(α) = β̂ and ϕ(β) = α̂. Using ϕ, the Cartan matrix of A with respect to B corresponds to the Cartan matrix of Â with respect to B̂, because of (21.5.2). Thus ϕ extends to an isomorphism between A and Â, as in Section 20.2.

Let us now return to the situation considered in the previous section. Note that v̂ = 2 v when v ∈ V satisfies (v, v) = 1, and v̂ = v when (v, v) = 2. If A_{1} is as in (21.4.1), then its inverse system Â_{1} in V = R^{4} with respect to the standard inner product consists of elements of the form ±2 e_{j} for j = 1, 2, 3, 4 and ±e_{j} ± e_{l} for 1 ≤ j ≠ l ≤ 4, as in Section 20.11. If A is as in (21.4.3), then the inverse system Â of A in V with respect to the standard inner product consists of the elements of Â_{1}, and elements of the form ±e_{1} ± e_{2} ± e_{3} ± e_{4}.

Let B be the base for A in (21.4.10). The corresponding base B̂ for Â is given by B̂ = {e_{2} − e_{3}, e_{3} − e_{4}, 2 e_{4}, e_{1} − e_{2} − e_{3} − e_{4}}. (21.5.5)

Let ϕ be the one-to-one mapping from B onto B̂ that interchanges the ordering in these lists. One can verify that the Cartan matrix of A with respect to B corresponds to the Cartan matrix of Â with respect to B̂, using ϕ. This means that ϕ extends to an isomorphism between A and Â, as in Section 20.2 again.

This corresponds to part of Exercise 5 on p63 of [14]. Of course, the Dynkin diagram of the inverse system can be obtained by changing the labels on the vertices, or the markings of the multiple edges. In the previous two cases, the resulting Dynkin diagram is easily seen to be isomorphic to the initial one.

21.6 Type E_{8}

Let us take V = R^{8}, with its standard inner product (·, ·) and basis e_{1}, …, e_{8}. Put A_{1} = {α ∈ Z^{8} : (α, α) = 2}, (21.6.1)

as in Section 20.12, with n = 8. The elements of A_{1} are of the form ±e_{j} ± e_{l} with 1 ≤ j ≠ l ≤ 8, as before. Remember that A is a reduced root system in V.

Of course, L_{8} = Z^{8} is the subgroup of R^{8}, as a commutative group with respect to addition, generated by e_{1}, …, e_{8}. Let L̃_{8} be the subgroup of R^{8} generated by e_{1}, …, e_{8} and (1/2) (e_{1} + ⋯ + e_{8}). Thus

L̃_{8} = Z^{8} ∪ (Z^{8} + (1/2, …, 1/2)). (21.6.2)

If α ∈ R^{8} satisfies α_{j} = ±1/2 for every j = 1, …, 8, then α is an element of Z^{8} + (1/2, …, 1/2) and (α, α) = 2. If α ∈ Z^{8} + (1/2, …, 1/2) and α_{j} ≠ ±1/2 for some j = 1, …, 8, then it is easy to see that (α, α) ≥ 4.

Let L_{8,e} be the set of v ∈ Z^{8} such that

∑_{j=1}^{8} v_{j} ∈ 2 Z, (21.6.3)

and let L̃_{8,e} be the set of v ∈ L̃_{8} such that (21.6.3) holds. These are subgroups of Z^{8} and L̃_{8}, respectively. Note that

(1/2) (e_{1} + ⋯ + e_{8}) ∈ L̃_{8,e}, (21.6.4)

because the sum of its coordinates is 4. It follows that

L̃_{8,e} = L_{8,e} ∪ (L_{8,e} + (1/2, …, 1/2)). (21.6.5)

We also have that A_{1} ⊆ L_{8,e}.

Suppose that α ∈ R^{8} satisfies α_{j} = ±1/2 for each j = 1, …, 8, and let r be the number of j such that α_{j} = 1/2. In this case,

∑_{j=1}^{8} α_{j} = (1/2) (r − (8 − r)) = r − 4. (21.6.6)

This means that α ∈ L̃_{8,e} exactly when r is even.

Consider A = {α ∈ L̃_{8,e} : (α, α) = 2}. (21.6.7)

Observe that A ∩ L_{8,e} = A_{1}. (21.6.8)

Using the previous remarks, we get that α ∈ A ∩ (L_{8,e} + (1/2, …, 1/2)) if and only if α_{j} = ±1/2 for every j = 1, …, 8, and α_{j} = 1/2 for an even number of j. Clearly A is a finite set of nonzero elements of V whose linear span is V.

Let us check that (v, w) ∈ Z (21.6.9)

for every v, w ∈ L̃_{8,e}. Of course, (21.6.9) holds when v, w ∈ Z^{8}. If w = (1/2, …, 1/2), then (21.6.9) holds for every v ∈ L_{8,e}, by (21.6.3). If v = (1/2, …, 1/2) as well, then (v, w) = 2. It follows that (21.6.9) holds for every v, w ∈ L̃_{8,e}, by (21.6.5).

If α ∈ V satisfies (α, α) = 2, then the reflection σ_{α} on V associated to α with respect to (·, ·) is given by σ_{α}(v) = v − (v, α) α, as before. If α ∈ A, then σ_{α} maps L̃_{8,e} into itself, because of (21.6.9). This implies that σ_{α} maps L̃_{8,e} onto itself, because σ_{α} is its own inverse. It follows that σ_{α} maps A onto itself, because σ_{α} is an orthogonal transformation on V. If v ∈ L̃_{8,e}, then σ_{α}(v) − v is an integer multiple of α, by (21.6.9).

Thus A is a root system in V, which is said to be of type E_{8}. This follows the descriptions on p65 of [14], and p41 of [24]. Note that A is reduced, as a root system in V.

Put V_{0} = {v ∈ V : v_{8} = 0} (21.6.10)

and A_{0} = A ∩ V_{0}. (21.6.11)

Equivalently, A_{0} = A_{1} ∩ V_{0}, (21.6.12)

because L̃_{8,e} ∩ V_{0} = L_{8,e} ∩ V_{0}. This is a reduced root system in V_{0}, which is isomorphic to the one in Section 20.12 with n = 7.

Put

(21.6.13) B = {(1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})), e_{1} + e_{2}, e_{2} − e_{1},
e_{3} − e_{2}, e_{4} − e_{3}, e_{5} − e_{4}, e_{6} − e_{5}, e_{7} − e_{6}}

and

(21.6.14) B_{0} = B ∩ V_{0} = {e_{1} + e_{2}, e_{2} − e_{1}, e_{3} − e_{2}, e_{4} − e_{3},
e_{5} − e_{4}, e_{6} − e_{5}, e_{7} − e_{6}}.

Observe that B_{0} is a basis for V_{0}. More precisely, B_{0} is a base for A_{0}, as a root system in V_{0}. This is similar to the base discussed in Section 20.12, with n = 7, ordering the standard basis vectors in V_{0} the other way. If 1 ≤ l < j ≤ 7, then e_{j} − e_{l} and e_{j} + e_{l} can be expressed as sums of elements of B_{0}, as before.

It is easy to see that B is a basis for V, because B_{0} is a basis for V_{0}. In fact, B is a base for A, as on p65 of [14]. More precisely, suppose first that α ∈ A satisfies α_{j} = ±1/2 for j = 1, …, 7, and α_{8} = 1/2. One can express α as (1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})) plus possibly some elements elements of B_{0}.

One can express e_{8} − e_{7} as the sum of

2 ((1/2) (e_{1} + e_{8} − (e_{2} + ⋯ e_{7}))) = e_{8} + e_{1} − (e_{2} + ⋯ + e_{7}) (21.6.15)

and elements of B_{0}. Using this, one can express e_{8} − e_{l} as the sum of (21.6.15) and elements of B_{0} for every l = 1, …, 7. One can use this to express e_{8} + e_{l} as the sum of (21.6.15) and elements of B_{0} for every l = 1, …, 7.

The maximal root in A with respect to B is

2 ((1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7}))) + 3 (e_{1} + e_{2}) + 4 (e_{2} − e_{1}) (21.6.16)

+6 (e_{3} − e_{2}) + 5 (e_{4} − e_{3}) + 4 (e_{5} − e_{4}) + 3 (e_{6} − e_{5}) + 2 (e_{7} − e_{6})
= e_{8} + e_{7},

as in Table 2 on p66 of [14].

If α, β ∈ A, then let n(α, β) ∈ Z be as in Section 19.9, so that n(α, β) = 2 (α, β) (β, β)^{−1} = (α, β). If α, β ∈ B and α ≠ β, then n(α, β) n(β, α) = (α, β)^{2} is the number of edges between α and β in the Coxeter graph of A with respect to B. Observe that

((1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})), e_{2} − e_{1}) = −1. (21.6.17)

However, (1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})) is orthogonal to the elements of B other than e_{2} − e_{1} and itself.

The inner products between distinct elements of B_{0} can be determined as in Section 20.12. More precisely,

(e_{1} + e_{2}, e_{3} − e_{2}) = −1, (21.6.18)

and e_{1} + e_{2} is orthogonal to the elements of B_{0} other than e_{3} − e_{2} and itself. Similarly, (e_{j} − e_{j−1}, e_{l} − e_{l−1}) (21.6.19)

is equal to −1 when |j − l| = 1, and to 0 when |j − l| ≥ 2. The Coxeter graph or Dynkin diagram of A can be depicted as on p58 of [14], or p37, 39 of [24].

Let T be an automorphism of A as a root system in V, and suppose that T (B) = B. This implies that T preserves the Cartan matrix of A with respect to B, as in Section 20.2, so that T preserves the Coxeter graph of A with respect to B too. One can check that this can only happen when T is the identity mapping on V, as indicated in Table 1 on p66 of [14], and on p39 of [24].

21.7 Type E_{7}

Let us continue with the same notation and hypotheses as in the previous section. Observe that

(1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})) ∈ Z^{8} + (1/2, …, 1/2) ⊆ L̃_{8}, (21.7.1)

so that Z^{8} + (1/2, −1/2, …, −1/2, 1/2) = Z^{8} + (1/2, …, 1/2). (21.7.2)

Thus L̃_{8} = Z^{8} ∪ (Z^{8} + (1/2, −1/2, …, −1/2, 1/2)), (21.7.3)

which is the same as the subgroup of R^{8}, as a commutative group with respect to addition, generated by e_{1}, …, e_{8} and (1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})). More precisely, (1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})) ∈ L̃_{8,e}, (21.7.4)

with the sum of its coordinates equal to −2, and in fact

(1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})) ∈ L_{8,e} + (1/2, …, 1/2). (21.7.5)

21.7. TYPE E_{7} 447 This implies that L_{8,e} + (1/2, −1/2, …, −1/2, 1/2) = L_{8,e} + (1/2, …, 1/2), (21.7.6) so that L̃_{8,e} = L_{8,e} ∪ (L_{8,e} + (1/2, −1/2, …, −1/2, 1/2)). (21.7.7)

Remember that V = R^{8}, and put V_{7} = {v ∈ V : v_{7} = −v_{8}}. (21.7.8) Note that (1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})) ∈ V_{7}. The elements of A_{1} ∩ V_{7} = {α ∈ Z^{8} ∩ V_{7} : (α, α) = 2} (21.7.9) are of the form ±e_{j} ± e_{l} with 1 ≤ j ≠ l ≤ 6, or ±(e_{7} − e_{8}). It is easy to see that A_{1} ∩ V_{7} spans V_{7}. In fact, A_{1} ∩ V_{7} is a reduced root system in V_{7}, as in Section 20.6.

Consider A_{7} = A ∩ V_{7} = {α ∈ L̃_{8,e} ∩ V_{7} : (α, α) = 2}. (21.7.10) Clearly A_{7} ∩ L_{8,e} = (A ∩ L_{8,e}) ∩ V_{7} = A_{1} ∩ V_{7}. (21.7.11) Observe that A_{7} ∩ (L_{8,e} + (1/2, −1/2, …, −1/2, 1/2)) (21.7.12)

= (A ∩ (L_{8,e} + (1/2, …, 1/2))) ∩ V_{7}

consists of α ∈ R^{8} such that α_{j} = ±1/2 for every j = 1, …, 8, α_{j} = 1/2 for an even number of j, and α_{7} = −α_{8}. Of course, V_{7} is spanned by A_{7}, because V_{7} is spanned by A_{1} ∩ V_{7}. Thus A_{7} is a root system in V_{7}, as in Section 20.6. This root system is said to be of type E_{7}. Note that A_{7} is reduced, as a root system in V_{7}.

Put V_{7,0} = V_{7} ∩ V_{0} = {v ∈ V : v_{7} = v_{8} = 0} (21.7.13) and A_{7,0} = A_{7} ∩ V_{7,0} = A ∩ V_{7,0}. (21.7.14) Equivalently, A_{7,0} = A_{1} ∩ V_{7,0}, (21.7.15) because L̃_{8,e} ∩ V_{7,0} = L_{8,e} ∩ V_{7,0}. This is a reduced root system in V_{7,0}, which is isomorphic to the one in Section 20.12 with n = 6.

Put

(21.7.16)  B_{7} = {(1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})), e_{1} + e_{2},
e_{2} − e_{1}, e_{3} − e_{2}, e_{4} − e_{3}, e_{5} − e_{4}, e_{6} − e_{5}},

and B_{7,0} = {e_{1} + e_{2}, e_{2} − e_{1}, e_{3} − e_{2}, e_{4} − e_{3}, e_{5} − e_{4}, e_{6} − e_{5}}. (21.7.17)

Thus B, B_{0} in the previous section can be expressed as

B = B_{7} ∪ {e_{7} − e_{6}}, B_{0} = B_{7,0} ∪ {e_{7} − e_{6}}. (21.7.18)

It is easy to see that B_{7,0} is a basis for V_{7,0}, and in fact B_{7,0} is a base for A_{7,0}. This is similar to the base discussed in Section 20.12, ordering the standard basis vectors in V_{7,0} the other way, as before.

Observe that B_{7} is a basis for V_{7}, because B_{7,0} is a basis for V_{7,0}. More precisely, B_{7} is a base for A_{7}, because B is a base for A, as in Section 20.15. This is how the root system of type E_{7} is described on p65 of [14], and p41 of [24].

The maximal root in A_{7} with respect to B_{7} is

2 ((1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7}))) + 2 (e_{1} + e_{2}) + 3 (e_{2} − e_{1}) (21.7.19)

+4 (e_{3} − e_{2}) + 3 (e_{4} − e_{3}) + 2 (e_{5} − e_{4}) + (e_{6} − e_{5})
= e_{8} − e_{7},

as in Table 2 on p66 of [14].

The Coxeter graph of A_{7} with respect to B_{7} corresponds to a subgraph of the Coxeter graph of A with respect to B, and similarly for the Dynkin diagram, as in Section 20.15. These can be depicted as on p58 of [14], or p37, 39 of [24].

If T is an automorphism of A_{7} as a root system in V_{7}, and if T (B_{7}) = B_{7}, then T preserves the Cartan matrix of A_{7} with respect to B_{7}, as in Section 20.2. This implies that T preserves the Coxeter graph of A with respect to B too. One can verify that this can only happen when T is the identity mapping on V_{7}, as in Table 1 on p66 of [14], and on p39 of [24].

21.8 Type E_{6}

We continue with the same notation and hypotheses as in the previous two sections. Remember that V = R^{8}, and put

V_{6} = {v ∈ V : v_{6} = v_{7} = −v_{8}}, (21.8.1)

which is a linear subspace of V_{7}. Note that (1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})) ∈ V_{6}. Put V_{6,0} = V_{6} ∩ V_{0} = {v ∈ V : v_{6} = v_{7} = v_{8} = 0}, (21.8.2)

which is a linear subspace of V_{7,0}.

The elements of

A_{1} ∩ V_{6} = {α ∈ Z^{8} ∩ V_{6} : (α, α) = 2} (21.8.3)

are of the form ±e_{j} ± e_{l} with 1 ≤ j ≠ l ≤ 5. Thus

A_{1} ∩ V_{6} = A_{1} ∩ V_{6,0}, (21.8.4) and the linear span of (21.8.4) is V_{6,0}. More precisely, (21.8.4) is a reduced root system in V_{6,0}, which is isomorphic to the one in Section 20.12 with n = 5.

Consider A_{6} = A ∩ V_{6} = {α ∈ L̃_{8,e} ∩ V_{6} : (α, α) = 2}. (21.8.5)

Note that A_{6} ⊆ A_{7}, and that

A_{6} ∩ L_{8,e} = (A ∩ L_{8,e}) ∩ V_{6} = A_{1} ∩ V_{6}. (21.8.6)

Of course, A_{6} is the union of (21.8.6) and

A_{6} ∩ (L_{8,e} + (1/2, −1/2, …, −1/2, 1/2)) (21.8.7)

= (A ∩ (L_{8,e} + (1/2, …, 1/2))) ∩ V_{6}.

An element α of R^{8} is in (21.8.7) if and only if α_{j} = ±1/2 for every j = 1, …, 8, α_{j} = 1/2 for an even number of j, and α_{6} = α_{7} = −α_{8}. In particular, (1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})) is an element of (21.8.7). It follows that V_{6} is spanned by A_{6}, because V_{6,0} is spanned by (21.8.6), as in the preceding paragraph. This implies that A_{6} is a root system in V_{6}, as in Section 20.6. This root system is reduced, and said to be of type E_{6}.

Put A_{6,0} = A_{6} ∩ V_{6,0} = A ∩ V_{6,0}. (21.8.8)

Equivalently, A_{6,0} = A_{1} ∩ V_{6,0}, (21.8.9)

because L̃_{8,e} ∩ V_{6,0} = L_{8,e} ∩ V_{6,0}.

Put

(21.8.10)  B_{6} = {(1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})), e_{1} + e_{2},
e_{2} − e_{1}, e_{3} − e_{2}, e_{4} − e_{3}, e_{5} − e_{4}}

and B_{6,0} = {e_{1} + e_{2}, e_{2} − e_{1}, e_{3} − e_{2}, e_{4} − e_{3}, e_{5} − e_{4}}. (21.8.11)

This means that

B_{7} = B_{6} ∪ {e_{6} − e_{5}}, B_{7,0} = B_{6,0} ∪ {e_{6} − e_{5}}. (21.8.12)

Observe that B_{6,0} is a basis for V_{6,0}, and in fact a base for A_{6,0}. This is similar to the base discussed in Section 20.12, as before.

It is easy to see that B_{6} is a basis for V_{6}, because B_{6,0} is a basis for V_{6,0}. In fact, B_{6} is a base for A_{6}, because B_{7} is a base for A_{7}, as in Section 20.15. Of course, one could also use the fact that B is a base for A here. The root system of type E_{6} is described in this way on p65 of [14], and p41 of [24].

The maximal root in A_{6} with respect to B_{6} is

(1/2) (e_{1} + e_{8} − (e_{2} + ⋯ + e_{7})) + 2 (e_{1} + e_{2}) (21.8.13)

+2 (e_{2} − e_{1}) + 3 (e_{3} − e_{2}) + 2 (e_{4} − e_{3}) + (e_{5} − e_{4})
= (1/2) (e_{1} + e_{2} + e_{3} + e_{4} + e_{5} − e_{6} − e_{7} + e_{8}),

as in Table 2 on p66 of [14].

The Coxeter graph of A_{6} with respect to B_{6} corresponds to a subgraph of the Coxeter graph of A_{7} with respect to B_{7}, and similarly for the Dynkin diagram, as in Section 20.15. One can also look at these in terms of their analogues for A and B. The Coxeter graph and Dynkin diagram of A_{6} with respect to B_{6} may be depicted as on p58 of [14], or p37, 39 of [24].

If T is an automorphism of A_{6} as a root system in V_{6}, and T (B_{6}) = B_{6}, then T preserves the Cartan matrix of A_{6} with respect to B_{6}, and hence the Coxeter graph. There is exactly one automorphism T with this property, other than the identity mapping, which interchanges some elements of B_{6}, and satisfies T^{2} = I. This is mentioned in Table 1 on p66 of [14], and on p39 of [24].

21.9 Irreducible root systems

Let V be a vector space over the real numbers of positive finite dimension, and let A be an irreducible reduced root system in V. It is well known that A is isomorphic to one of the root systems that has been described previously. More precisely, A is isomorphic to one of the following:

(21.9.1) a root system of type A_{n}, with n ≥ 1, as in Section 20.10; (21.9.2) a root system of type B_{n}, with n ≥ 2, as in Section 20.11; (21.9.3) a root system of type C_{n}, with n ≥ 3, as in Section 20.11; (21.9.4) a root system of type D_{n}, with n ≥ 4, as in Section 20.12; (21.9.5) a root system of type E_{6}, E_{7}, or E_{8}, as in Sections 21.6 – 21.8; (21.9.6) a root system of type F_{4}, as in Section 21.4; (21.9.7) a root system of type G_{2}, as in Sections 21.2 and 21.3.

Equivalently, if B is a base for A, then the corresponding Dynkin diagram is isomorphic to one of the Dynkin diagrams in this list. This is the theorem on p57f of [14], which corresponds to Theorem 4 on p38 of [24]. The restrictions on n in the first four types ensures that there are no repetitions. If A is irreducible but not reduced, then it is well known that A is isomorphic to a root system of type BC_{n}, with n ≥ 1, as in Section 20.11. This corresponds to Exercise 3 on p66 of [14], and is also mentioned on p41 of [24].

Suppose that A is an irreducible reduced root system again, and let B be a base for A. If T is an automorphism of A that maps B onto itself, then T preserves the Cartan matrix of A with respect to B, as in Section 20.2. Conversely, any one-to-one mapping from B onto itself that preserves the Cartan matrix corresponds to a unique automorphism T of A that maps B onto itself, as before. Equivalently, automorphisms of A that map B onto itself correspond to automorphisms of the Dynkin diagram of A with respect to B.

Remember that there are pairs of vertices in the Coxeter graph of A with respect to B with multiple edges in the cases (21.9.2), (21.9.3), (21.9.6), and (21.9.7). In these cases, the only automorphism of the Dynkin diagram is the identity mapping, as before. This implies that the automorphism group of A is the same as the Weyl group of A.

In the cases (21.9.1), (21.9.4), and (21.9.5), there is at most one edge between any pair of vertices in the Coxeter graph. This means that the Coxeter graph determines the Cartan matrix of A with respect to B, or equivalently the Dynkin diagram. In these cases, automorphisms of A that map B onto itself correspond exactly to automorphisms of the Coxeter graph.

If A is of type A_{1}, E_{7}, or E_{8}, then the only automorphism of the Coxeter graph is the identity mapping. Thus the automorphism group of A is the same as the Weyl group of A in these cases.

21.10 Connected Coxeter graphs

Let V be a vector space over the real numbers of positive finite dimension, let A be a root system in V, and let B be a base for A. Also let (·, ·) be an inner product on V that is invariant under the Weyl group of A, and let ∥ · ∥ be the corresponding norm on V. If α ∈ B, then put

e_{α} = α/∥α∥. (21.10.1)

If α, β ∈ B and α ≠ β, then

(e_{α}, e_{β}) = (α, β) ∥α∥^{−1} ∥β∥^{−1} ≤ 0, (21.10.2)

where the second step is as in Section 19.11. In this case,

4 (e_{α}, e_{β})^{2} = 4 (α, β)^{2} ∥α∥^{−2} ∥β∥^{−2} (21.10.3)

is the same as the number of edges between α and β in the Coxeter graph of A with respect to B, as in Section 20.3.

Now let B be a nonempty finite set, and let Γ be a Coxeter graph, with B as the set of vertices in Γ. Thus every pair of distinct elements of B is connected by 0, 1, 2, or 3 edges in Γ. Let V be a vector space over the real numbers, with dimension equal to the number of elements of B. Suppose that for each α ∈ B, e_{α} is a nonzero element of V, and that the collection of e_{α}, α ∈ B, is a basis for V. Under these conditions, there is a unique symmetric bilinear form (v, w) on V with the following properties. First,

(e_{α}, e_{α}) = 1 (21.10.4)

for every α ∈ B. Second, if α, β ∈ B and α ≠ β, then

(e_{α}, e_{β}) ≤ 0 (21.10.5)

and (21.10.6) 4 (e_{α}, e_{β})^{2} = the number of edges between α and β in Γ.

We may also be interested in situations where

(21.10.7) (v, w) is positive definite on V.

If Γ is the Coxeter graph of a root system, then we can take e_{α} to be as in (21.10.1). Thus (21.10.7) holds in this situation, as on p37 of [24]. The properties of the e_{α}’s in the preceding paragraph, including (21.10.7), correspond to the definition of an admissible set of vectors in a finite-dimensional inner product space on p60 of [14].

If Γ is a connected Coxeter graph for which (21.10.7) holds, then it is well known that Γ isomorphic to the Coxeter graph of one of the root systems (21.9.1)

– (21.9.7). More precisely, the Coxeter graphs associated to root systems of type B_{n} and C_{n} are the same, and otherwise these Coxeter graphs are distinct. In particular, Coxeter graphs of irreducible root systems are of this form, as in Theorem 3 on p37 of [24]. This is shown in the proof of the theorem on p57f of [14]. Using this, it is not difficult to obtain the analogous statement for Dynkin diagrams, as on p62f of [14], and p39 of [24].

21.11 Lengths in the Weyl group

Let V be a vector space over the real numbers of positive finite dimension, let A be a reduced root system in V, and let B be a base for A. If α ∈ A, then σ_{α} denotes the symmetry on V with vector α that maps A onto itself, as usual. Remember that the Weyl group W of A is generated by the symmetries σ_{β} with β ∈ B, as in Section 19.14.

Suppose that σ ∈ W can be expressed as

σ = σ_{β_{1}} ∘ ⋯ ∘ σ_{β_{r}} (21.11.1)

for some β_{1}, …, β_{r} ∈ B, where r ≥ 0 is as small as possible. In this case, the expression on the right side of (21.11.1) is said to be reduced, and the length of σ with respect to B is defined by

length(σ) = r, (21.11.2)

as on p51 of [14]. More precisely, the right side of (21.11.1) is interpreted as being the identity mapping on V when r = 0, so that length(σ) = 0 exactly when σ is the identity mapping.

Let A^{+} = A^{B,+} be the set of α ∈ A that can be expressed as a linear combination of elements of B whose coefficients are nonnegative integers, so that A = A^{+} ∪ (−A^{+}). If σ ∈ W, then put

n(σ) = #{α ∈ A^{+} : σ(α) ∈ −A^{+}}, (21.11.3)

which is to say the number of elements of the set on the right, as on p51 of [14]. Lemma A at the top of p52 of [14] states that

length(σ) = n(σ) (21.11.4)

21.11. LENGTHS IN THE WEYL GROUP 453 for every σ ∈ W. Of course, if length(σ) = 0, then σ is the identity mapping on V, and n(σ) = 0 too.

Suppose now that length(σ) > 0, and that the statement holds for elements of W with smaller length. Let (21.11.1) be a reduced expression for σ, so that (21.11.2) holds. Under these conditions, we have that σ(β_{r}) ∈ −A^{+}, (21.11.5) as mentioned just after (20.1.8). Equivalently, (σ_{β_{1}} ∘ ⋯ ∘ σ_{β_{r−1}})(β_{r}) ∈ A^{+}, (21.11.6) because σ_{β_{r}}(β_{r}) = −β_{r}.

Remember that σ_{β_{r}} maps A^{+} \ {β_{r}} onto itself, as in (19.12.8). Using this, we get that n(σ_{β_{1}} ∘ ⋯ ∘ σ_{β_{r−1}}) = n(σ) − 1. (21.11.7) More precisely, the number of α ∈ A^{+} \ {β_{r}} that are mapped into −A^{+} by σ_{β_{1}} ∘ ⋯ ∘ σ_{β_{r−1}} is the same as for σ, because σ_{β_{r}} maps A^{+} \ {β_{r}} onto itself. The case where α = β_{r} corresponds to the remarks in the preceding paragraph.

The length of σ_{β_{1}} ∘ ⋯ ∘ σ_{β_{r−1}} with respect to B is clearly less than or equal to r − 1. In fact, (21.11.8) length(σ_{β_{1}} ∘ ⋯ ∘ σ_{β_{r−1}}) = r − 1 = length(σ) − 1, because otherwise σ would have length less than r with respect to B. Arguing by induction, we have that length(σ_{β_{1}} ∘ ⋯ ∘ σ_{β_{r−1}}) = n(σ_{β_{1}} ∘ ⋯ ∘ σ_{β_{r−1}}). (21.11.9) Thus (21.11.4) follows from (21.11.7), (21.11.8), and (21.11.9), as desired.

If σ ∈ W and β ∈ B, then n(σ ∘ σ_{β}) = n(σ) ± 1. (21.11.10) Indeed, the number of α ∈ A^{+} \ {β} that are mapped into −A^{+} by σ is the same as for σ ∘ σ_{β}, because σ_{β} maps A \ {β} onto itself, as before. Of course, (σ ∘ σ_{β})(β) = σ(σ_{β}(β)) = −σ(β), (21.11.11) so that exactly one of σ(β) and (σ ∘ σ_{β})(β) is in A^{+}. This implies (21.11.10).

Exercise 5 on p54 of [14] states that if σ ∈ W can be expressed as the composition of r symmetries associated to elements of B, then r has the same parity as length(σ). More precisely, it is easy to see that r has the same parity as n(σ), by (21.11.11). Thus r has the same parity as length(σ), by (21.11.4).

Exercise 6 on p54 of [14] says that σ ↦ (−1)^{length(σ)} (21.11.12) defines a group homomorphism from W into the multiplicative group {±1}. This follows from the fact that if σ ∈ W can be expressed as the composition of r symmetries associated to elements of β, then

(−1)^{r} = (−1)^{length(σ)}, (21.11.13)

as in the preceding paragraph. Alternatively, the determinant of any symmetry on V is equal to −1, so that (21.11.12) is the same as the restriction of the determinant to W.

21.12 Nonnegative linear functionals

Let us continue with the same notation and hypotheses as in the previous section. Note that (21.12.1) length(σ^{−1}) = length(σ)

for every σ ∈ W, because the inverse of a symmetry on V is itself. Similarly,

n(σ^{−1}) = n(σ) (21.12.2)

for every σ ∈ W. This can be verified directly from the definition (21.11.3) of n(σ), and it is compatible with (21.11.4) and (21.12.1) as well.

Let us say that a linear functional λ on V is nonnegative with respect to B if (21.12.3) λ(β) ≥ 0 for every β ∈ B.

Of course, this implies that

(21.12.4) λ(α) ≥ 0 for every α ∈ A^{+}.

Remember that the Weyl chamber associated to B consists of the λ ∈ V′ such that λ(β) > 0 for every β ∈ B, as in Section 19.15. The set of linear functionals on V that are nonnegative with respect to B is the same as the closure in V′ of the Weyl chamber associated to B. This uses the topology on V′ that corresponds to the standard topology on a Euclidean space of the same dimension, using any linear isomorphism.

Suppose that λ ∈ V′ is nonnegative with respect to B, σ ∈ W, and λ ∘ σ is nonnegative with respect to B too. Under these conditions, σ can be expressed as the composition of finitely many symmetries on V with vector in B that map A onto itself, in such a way that λ is invariant under each of these symmetries. In particular, λ ∘ σ = λ. (21.12.5)

This corresponds to Lemma B on p52 of [14].

If the length of σ with respect to B is 0, then σ is the identity mapping on V, and the statement in the preceding paragraph is trivial. Suppose that length(σ) > 0, which is the same as saying that length(σ^{−1}) > 0. Thus n(σ^{−1}) is positive, by (21.11.4), so that there is an α ∈ A^{+} such that σ^{−1}(α) ∈ −A^{+}, by the definition (21.11.3) of n(σ^{−1}). More precisely, there is a β ∈ B such that

σ^{−1}(β) ∈ −A^{+}, (21.12.6)

because otherwise σ^{−1} would map A^{+} into itself. Alternatively, this can be obtained as in (20.1.8).

Because λ ∘ σ is nonnegative with respect to B, we have that

λ(β) = (λ ∘ σ)(σ^{−1}(β)) ≤ 0. (21.12.7)

This implies that λ(β) = 0, (21.12.8)

because λ(β) ≥ 0, by hypothesis. It follows that

λ ∘ σ_{β} = λ (21.12.9)

on V, because σ_{β}(v) − v is a multiple of β for every v ∈ V.

Let us check that

(21.12.10) length(σ^{−1} ∘ σ_{β}) = length(σ^{−1}) − 1.

Equivalently, this means that

n(σ^{−1} ∘ σ_{β}) = n(σ^{−1}) − 1, (21.12.11)

by (21.11.4). Remember that σ_{β} maps A^{+} \{β} onto itself, as in (19.12.8). One can use this to get (21.12.11), because of (21.12.6) and the fact that σ_{β}(β) = −β. Alternatively, if β is obtained from σ^{−1} as in (20.1.8), then (21.12.10) holds by construction, as in (21.11.8).

Observe that λ ∘ (σ_{β} ∘ σ) = (λ ∘ σ_{β}) ∘ σ = λ ∘ σ, (21.12.12)

by (21.12.9). Thus λ ∘ (σ_{β} ∘ σ) is nonnegative with respect to B, by hypothesis. We also have that (21.12.13) length(σ_{β} ∘ σ) = length(σ) − 1,

by (21.12.1), (21.12.10), and the fact that σ_{β} is its own inverse. This permits us to use induction to obtain the initial statement about σ.

If λ is strictly positive on B, then σ has to be the identity mapping on V. More precisely, (21.12.7) cannot hold in this situation. This corresponds to Exercise 12 on p55 of [14].

Alternatively, if λ is strictly positive on B, then B is the same as the base B_{λ} for A associated to λ as in Section 19.11. If σ ∈ W, then σ(B) is a base for A too. If λ ∘ σ is strictly positive on B, then λ is strictly positive on σ(B). This implies that σ(B) = B_{λ}, as before, so that σ(B) = B. It follows that σ is the identity mapping on V, as in Section 20.1.

Remember that τ ∈ V′ is said to be regular with respect to A if τ(α) ≠ 0 for each α ∈ A, as in Section 19.15. In this case, there is a unique base B_{τ} for A on which τ is strictly positive, as in Section 19.11. If σ ∈ W and τ ∘ σ = τ, then τ is strictly positive on σ(B_{τ}), so that σ(B_{τ}) = B_{τ}. This implies that σ is the identity mapping on V, as before.

Let (·, ·) be an inner product on V that is invariant under W. If u ∈ V, then let μ_{u}(v) = (v, u) be the corresponding linear functional on V with respect to (·, ·). If σ is an orthogonal transformation on V with respect to (·, ·), then

μ_{u}(σ(v)) = (σ(v), u) = (v, σ^{−1}(u)) = μ_{σ^{−1}(u)}(v) (21.12.14)

for every v ∈ V, so that μ_{u} ∘ σ = μ_{σ^{−1}(u)} on V. In particular, this holds when σ ∈ W. If σ(u) = u, then σ^{−1}(u) = u, and μ_{u} ∘ σ = μ_{u}.

Note that μ_{u} is regular with respect to A exactly when u is not orthogonal to any element of A with respect to (·, ·). In this case, if σ(u) = u, then we get that σ is the identity mapping on V, because μ_{u} ∘ σ = σ, as before.

If σ ∈ W is a reflection on V with respect to (·, ·), then σ is the reflection associated to an element of A, as in Exercise 13 on p55 of [14]. To see this, it suffices to show that the hyperplane in V of vectors fixed by σ is contained in the hyperplane of vectors fixed by σ_{α} for some α ∈ A. Otherwise, for each α ∈ A, the intersection of the hyperplane in V of vectors fixed by σ with the hyperplane of vectors fixed by σ_{α} is a hyperplane in the hyperplane of vectors fixed by σ. Because A has only finitely many elements, there is an element u of the hyperplane of vectors fixed by σ that is not in the hyperplane of vectors fixed by σ_{α} for any α ∈ A, as in Section 18.12.

Equivalently, this means that u is not orthogonal to any α ∈ A with respect to (·, ·). By construction, σ(u) = u, so that σ is the identity mapping on V, as before. This contradicts the hypothesis that σ be a reflection on V.

21.13 The mapping α ↦ −α

Let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. It is easy to see that the mapping

α ↦ −α (21.13.1)

on V is an automorphism of A, as in the first part of Exercise 6 on p67 of [14]. The second part of the exercise is to try to decide for which irreducible reduced root systems (21.13.1) is an element of the Weyl group of A. Of course, this holds automatically when the automorphism group of A is the same as the Weyl group of A. Remember that this happens for root systems of type B_{n} and C_{n}, as well as types A_{1}, F_{4}, G_{2}, E_{7}, and E_{8}, as in Section 21.9.

If n ≥ 2, then (21.13.1) is not in the Weyl group of the root system of type A_{n}. This can be seen using the description of the Weyl group in Section 20.10.

The Weyl group of root systems of type D_{n} consist of linear mappings on V = R^{n} that permute the coordinates, and multiply an even number of coordinates by −1, as in Section 20.12. Thus (21.13.1) is an element of the Weyl group exactly when n is even.

21.14. ANOTHER MAPPING ON V 457

To deal with type E_{6}, let us take V = R^{8}, equipped with the standard inner product (·, ·), and put

(21.13.2)  V_{6} = {v ∈ V : v_{6} = v_{7} = −v_{8}},
(21.13.3)  V_{7} = {v ∈ V : v_{7} = −v_{8}},

as in Sections 21.7 and 21.6. Consider the line

Z = {v ∈ V : v_{1} = ⋯ = v_{5} = 0, v_{7} = −v_{6}/2 = −v_{8}} (21.13.4)

in V_{7}. If v ∈ V_{6} and z ∈ Z, then

(v, z) = ∑_{j=1}^{n} v_{j} z_{j} = v_{6} z_{6} + v_{7} z_{7} + v_{8} z_{8}
(21.13.5)  = v_{7} (−2 z_{7}) + v_{7} z_{7} + (−v_{7}) (−z_{7}) = 0.

Thus Z is the orthogonal complement of V_{6} in V_{7}, with respect to the standard inner product on V.

Let A_{6} and A_{7} be the root systems in V_{6} and V_{7} of types E_{6} and E_{7}, respectively, as in Sections 21.8 and 21.7. If α ∈ A_{6}, then the reflection on V_{6} associated to α is the same as the restriction to V_{6} of the reflection on V_{7} associated to α. The restriction to Z of the reflection on V_{7} associated to α ∈ V_{6} is the identity mapping on Z, because Z is orthogonal to α.

Remember that A_{6} = A_{7} ∩ V_{6} ⊆ A_{7}, by construction. If T is a one-to-one linear mapping from V_{6} onto itself, then let T̃ be the extension of T to a oneto-one linear mapping from V_{7} onto itself whose restriction to Z is the identity mapping on Z. If T is in the Weyl group of A_{6}, then it follows that T̃ is in the Weyl group of A_{7}.

Consider the mapping T from V_{6} into itself defined by T (v) = −v for every v ∈ V_{6}. If T is in the Weyl group of A_{6}, then T̃ is in the Weyl group of A_{7}, as before. Remember that the mapping on V_{7} that sends v ∈ V_{7} to −v is in the Weyl group of A_{7}, as mentioned earlier. It follows that the composition of this mapping with T̃ is in the Weyl group of A_{7} as well.

The composed mapping on V_{7} just mentioned sends every element of V_{6} to itself, and sends z ∈ Z to −z. Note that this is a reflection on V_{7}. If this composed mapping is an element of the Weyl group of A_{7}, then it is the reflection associated to an element of A_{7}, as in the previous section. This is not possible, because there are no elements of A_{7} in Z. Thus the mapping v ↦ −v on V_{6} is not an element of the Weyl group of A_{6}.

21.14 Another mapping on V

Let V be a vector space over the real numbers of positive finite dimension again, and let A be a reduced root system in V. Also let B be a base for A, and observe that −B is a base for A as well. It follows that there is an element σ of the Weyl group W of A such that

σ(B) = −B, (21.14.1)

as in Section 19.14. In fact, σ is unique, as in Section 20.1.

Let A^{+} = A^{B,+} be the set of α ∈ A that can be expressed as linear combinations of elements of B whose coefficients are nonnegative integers, as before. Of course, (21.14.1) implies that

σ(A^{+}) = −A^{+}. (21.14.2)

If n(σ) is as in (21.11.3), then (21.14.2) is the same as saying that

n(σ) = #A^{+}. (21.14.3)

Note that this is the largest possible value of n(σ). Remember that this is equal to the length of σ with respect to B, as in (21.11.4).

If σ is any element of W that satisfies (21.14.2), then σ satisfies (21.14.1). Indeed, (21.14.2) implies that

σ(B) ⊆ −A^{+} = A^{−B,+}. (21.14.4)

This implies that (21.14.1) holds, because σ(B) is a base for A, as in (19.15.7).

Let β_{0} ∈ B be given, and let V_{0} be the hyperplane in V spanned by β ∈ B with β ≠ β_{0}. If v ∈ V, then v can be expressed in a unique way as a linear combination of elements of B, and we let τ_{0}(v) ∈ R be the coefficient of β_{0} in this expression for v. Thus V_{0} is the same as the kernel of τ_{0}. If β ∈ B \ {β_{0}} and v ∈ V, then σ_{β}(v) − v ∈ V_{0}, so that

τ_{0} ∘ σ_{β} = τ_{0} (21.14.5)

on V. If σ is an element of the subgroup of W generated by σ_{β} with β ∈ B\{β_{0}}, then it follows that τ_{0} ∘ σ = τ_{0} (21.14.6)

on V too. In particular, this means that

τ_{0}(σ(β_{0})) = τ_{0}(β_{0}) = 1. (21.14.7)

This implies that σ does not satisfy (21.14.2).

This corresponds to Exercise 9 on p54 of [14].

21.15 Invertibility of Cartan matrices

Let k be a field, and let V be a vector space over k of positive finite dimension r. Also let k^{r} be the space of r-tuples of elements of k, as usual, which is an r-dimensional vector space over k with respect to coordinatewise addition and scalar multiplication. If v_{1}, …, v_{r} is a basis for V, then we get a one-to-one linear mapping from k^{r} onto V, which sends t = (t_{1}, …, t_{r}) ∈ k^{r} to ∑_{j=1}^{r} t_{j} v_{j}. Similarly, if λ_{1}, …, λ_{r} are linear functionals on V that form a basis for the dual space V′, then we get a one-to-one mapping from V onto k^{r}, which sends v ∈ V to (λ_{1}(v), …, λ_{r}(v)) ∈ k^{r}. Thus the composition of these two mappings is a one-to-one linear mapping from k^{r} onto itself. The composition of these two mappings corresponds to the r × r matrix with entries λ_{l}(v_{j}), j, l = 1, …, r, with respect to the standard basis in k^{r}. Thus this matrix is invertible, as an r × r matrix with entries in k, under these conditions.

Now let V be a vector space over the real numbers of positive finite dimension, and let A be a root system in V. As before, let σ_{α} be the symmetry on V with vector α ∈ A that maps A onto itself, and let λ_{α} be the corresponding linear functional on V. Remember that n(α, β) = λ_{β}(α) is an integer for every α, β ∈ A, by the definition of a root system. Let (·, ·) be an inner product on V that is invariant under the Weyl group of A, so that σ_{α} is the reflection on V with respect to (·, ·) associated to α ∈ A, and λ_{α}(v) = 2 (v, α) (α, α)^{−1} for every v ∈ V. Thus n(α, β) = 2 (α, β) (β, β)^{−1} for every α, β ∈ A.

Let B be a base for A, so that B is a basis for V in particular. Using the inner product, we can identify V with its dual space, and it follows that λ_{β}, β ∈ B, is a basis for V′. This implies that the Cartan matrix n(α, β), α, β ∈ B, is invertible, as before, and as on p55 of [14]. Equivalently, the determinant of the Cartan matrix is nonzero. Of course, the determinant is an integer.

More precisely, the Cartan matrix may be considered as the product of the symmetric matrix (α, β) and the diagonal matrix with diagonal entries 2 (β, β)^{−1}. The symmetric matrix (α, β) is positive definite, because of the positivity of the inner product, and the fact that B is a basis for V. Thus its determinant is a positive real number. The determinant of the diagonal matrix is postive too, because the diagonal entries of the matrix are positive. This means that the determinant of the Cartan matrix is positive as well.

Roots and Lie algebras

22.1 A basic situation

Let k be a field of characteristic 0, and let (A, [·, ·]_{A}) be a Lie algebra over k with positive finite dimension as a vector space over k. Also let B be a Lie subalgebra of A such that every element of B is ad-diagonalizable as an element of A. This implies that B is commutative as a Lie algebra, as in Section 17.1. Let B′ be the dual space of linear functionals on B, as a vector space over k, as usual. If α ∈ B′, then put

(22.1.1) A_{α} = {x ∈ A : ad_{w}(x) = [w, x]_{A} = α(w) x for every w ∈ B},

as before, which is a linear subspace of A. This is the same as the centralizer C_{A}(B) of B in A when α = 0. In particular, B ⊆ A_{0}, because B is commutative as a Lie algebra. Let us suppose for the rest of the section that

A_{0} = C_{A}(B) = B (22.1.2)

and Z(A) = {0}, (22.1.3)

where Z(A) is the center of A as a Lie algebra.

It follows that B ≠ {0}, because A ≠ {0}, by hypothesis. Put

Φ_{B} = {α ∈ B′ : α ≠ 0 and A_{α} ≠ {0}}, (22.1.4)

so that Φ_{B} ∪ {0} is the same as the set of α ∈ B′ such that A_{α} ≠ {0}, as before. Remember that A corresponds to the direct sum of the subspaces A_{α} with α ∈ Φ_{B} ∪ {0}, as a vector space over k, and in particular that Φ_{B} has only finitely many elements, as in Section 17.2. If x ∈ A_{α} and y ∈ A_{β} for some α, β ∈ B′, then [x, y]_{A} ∈ A_{α+β}, (22.1.5)

as before. We also have that the linear span of Φ_{B} in B′ is equal to B′, because of (22.1.3), as in Section 17.5. Let b(·, ·) be a nondegenerate bilinear form on A, and suppose that b(·, ·) is associative on A, or equivalently invariant under the adjoint representation on A, as in Sections 6.10 and 7.7. This implies that the restriction of b(·, ·) to (22.1.2) is nondegenerate, as in Section 17.3.

Of course, there is a natural embedding of Q into k, because k has characteristic 0, so that B′ may also be considered as a vector space over Q. Let E_{Q} be the linear subspace of B′, as a vector space over Q, spanned by Φ_{B}, as in Section 17.12. If α_{1}, …, α_{n} are elements of Φ_{B} that form a basis for B′, as a vector space over k, then α_{1}, …, α_{n} form a basis for E_{Q}, as a vector space over Q, as in Section 17.10. Using this, we can get a vector space E_{R} over R of the same dimension n that contains E_{Q}, as in Section 17.13. In particular, Φ_{B} is contained in E_{R}, and the linear span of Φ_{B} in E_{R}, as a vector space over R, is equal to E_{R}.

If α ∈ B′, then there is a unique t_{b,α} ∈ B such that α(w) = b(w, t_{b,w}) for every w ∈ B, because b(·, ·) is nondegenerate on B. Remember that α(t_{b,α}) = b(t_{b,α}, t_{b,α}) ≠ 0 when α ∈ Φ_{B}, as in Section 17.6. In this case, we put

h_{α} = 2 α(t_{b,α})^{−1} t_{b,α}, (22.1.6)

as before, so that α(h_{α}) = 2. If β ∈ Φ_{B}, then β − β(h_{α}) α ∈ Φ_{B}, as in Section 17.8. We have also seen that β(h_{α}) corresponds to an integer in this situation, with respect to the standard embedding of Q into k.

Of course, β ↦ β(h_{α}) (22.1.7)

defines a linear functional on B′, as a vector space over k. If β ∈ E_{Q}, then β(h_{α}) is in the image of Q in k, by the remarks in the previous paragraphs. Let λ_{α} be the corresponding linear functional on E_{Q}, as a vector space over Q. This extends to a linear functional on E_{R}, as a vector space over R, in a natural way, that we shall also denote by λ_{α}. As before, we have that

β − λ_{α}(β) α ∈ Φ_{B} (22.1.8)

and λ_{α}(β) ∈ Z for every β ∈ Φ_{B}.

It follows that Φ_{B} is a root system in E_{R}, as in Section 19.3. More precisely, if α ∈ Φ_{B}, then the corresponding symmetry on E_{R} is defined by

σ_{α}(β) = β − λ_{α}(β) α (22.1.9)

for every β ∈ E_{R}. Note that the definition of a root system is extended to finite-dimensional vector spaces over the complex numbers in Definition 7 on p41 of [24]. If k = C, then one can consider Φ_{B} as a root system in B′ as a vector space over C.

This root system is reduced, in the sense that for each α ∈ Φ_{B}, the only multiples of α in Φ_{B} are ±α. This was discussed in Sections 17.7 and 17.8, for multiples of α by elements of k, and by elements of Q in particular. One should also consider multiples of α by real numbers in E_{R}, but it is easy to reduce to considering multiples by elements of Q, because Φ_{B} is contained in E_{Q}. Of course, in any root system, one only has to consider multiples by ±2 or ±1/2.

Let (·, ·) be any inner product on E_{R} that is invariant under the Weyl group of Φ_{B}. If α ∈ Φ_{B}, then it follows that (22.1.9) is the reflection on E_{R} associated to α with respect to (·, ·). This means that

λ_{α}(β) = 2 (β, α) (α, α)^{−1} (22.1.10)

for every β ∈ E_{R}. In particular, this corresponds to β(h_{α}) when β ∈ Φ_{B} too.

Suppose for the moment that A is semisimple as a Lie algebra over k, so that we can take b(·, ·) to be the Killing form on A, as in Section 17.11. This leads to an inner product (·, ·)_{E_{R}} on E_{R}, as in Section 17.13. More precisely, a bilinear form (·, ·)_{E_{Q}} was defined on E_{Q} in Section 17.12, which was extended to E_{R}. If α, β ∈ Φ_{B}, then

λ_{α}(β) = 2 (α, β)_{E_{Q}} (α, α)_{E_{Q}}^{−1}, (22.1.11)

by (17.10.8) and the definition of (·, ·)_{E_{Q}}. Of course, this implies that

λ_{α}(β) = 2 (α, β)_{E_{R}} (α, α)_{E_{R}}^{−1}. (22.1.12)

In fact, this holds for every α ∈ Φ_{B} and β ∈ E_{R}, because E_{R} is spanned by Φ_{B}, and both sides are linear in β. This means that (22.1.9) is the same as the reflection on E_{R} associated to α ∈ Φ_{B} with respect to (·, ·)_{E_{R}} in this situation. This corresponds to the theorem on p40 of [14].

22.2 Inverse roots and B

Let us continue with the same notation and hypotheses as in the previous section, except for the last paragraph. Let α_{1}, …, α_{n} be elements of Φ_{B} that form a basis for B′ as a vector space over k, so that α_{1}, …, α_{n} form a basis for E_{Q} as a vector space over Q, as before. Thus α_{1}, …, α_{n} is a basis for E_{R} too, as a vector space over R, by construction.

Let E′_{Q} and E′_{R} be the dual spaces of linear functionals on E_{Q} and E_{R}, as vector spaces over Q and R, respectively. Of course, any linear functional on E_{Q} or E_{R} is uniquely determined by its values on α_{1}, …, α_{n}. Linear functionals on E_{Q} may take arbitrary values in Q on α_{1}, …, α_{n}, and linear functionals on E_{R} may take arbitrary values in R on α_{1}, …, α_{n}. Every linear functional on E_{Q} has a natural unique extension to a linear functional on E_{R}. If μ is a linear functional on E_{R}, and μ maps E_{Q} ⊆ E_{R} into Q, then the restriction of μ to E_{Q} is a linear functional on E_{Q}. Thus we may identify E′_{Q} with

{μ ∈ E′_{R} : μ(E_{Q}) ⊆ Q}. (22.2.1)

This is a linear subspace of E′_{R}, as a vector space over Q.

If w ∈ B, then put

T (w) = (α_{1}(w), …, α_{n}(w)), (22.2.2)

22.2. INVERSE ROOTS AND B 463

which is an element of the space k^{n} of n-tuples of elements of k. This defines a linear mapping from B into k^{n}, as a vector space over k with respect to coordinatewise addition and scalar multiplication, because α_{j} is a linear functional on B as a vector space over k for each j = 1, …, n. More precisely, T is a one-to-one mapping from B onto k^{n}, because α_{1}, …, α_{n} is a basis for B′, as a vector space over k.

Remember that k has characteristic 0, so that there is a standard embedding of Q into k. Let us identify Q with its image in k under the standard embedding, so that Q^{n} may be identified with a subset of k^{n}. Of course, Q^{n} is a linear subspace of k^{n}, as a vector space over Q. Put

C_{Q} = T^{−1}(Q^{n}), (22.2.3)

which is a linear subspace of B, as a vector space over Q. Equivalently, C_{Q} consists of the w ∈ B such that α_{j}(w) ∈ Q for every j = 1, …, n. This is the same as (22.2.4) C_{Q} = {w ∈ B : β(w) ∈ Q for every β ∈ E_{Q}},

by the definition of E_{Q}. Note that the dimension of C_{Q}, as a vector space over Q, is equal to n, and that the linear span of C_{Q} in B, as a vector space over k, is equal to B. The standard basis for Q^{n} corresponds to a basis for C_{Q}, as a vector space over Q, which is also a basis for B, as a vector space over k.

If β ∈ E_{Q} ⊆ B′, then β is a linear functional on B, as a vector space over k, that maps C_{Q} into Q. This implies that the restriction of β to C_{Q} is a linear functional on C_{Q}, as a vector space over Q. Of course, β is uniquely determined, as an element of B′, by its restriction to C_{Q}, because the linear span of C_{Q} in B, as a vector space over k, is equal to B. It is easy to see that every linear functional on C_{Q}, as a vector space over Q, corresponds to some β ∈ E_{Q} in this way, using T. Thus the dual space C′_{Q} of linear functionals on C_{Q}, as a vector space over Q, can be identified with E_{Q}.

If w ∈ B, then β ↦ β(w) (22.2.5)

defines a linear functional on the dual B′ of B, as a vector space over k. This defines a one-to-one linear mapping from B onto the dual of B′, as vector spaces over k. Similarly, if w ∈ C_{Q}, then (22.2.5) defines a linear functional on E_{Q}, as a vector space over Q. This defines a one-to-one linear mapping from C_{Q} onto E′_{Q}, as vector spaces over Q.

Let α ∈ Φ_{B} be given, and let h_{α} ∈ B be as in the previous section. Remember that β(h_{α}) ∈ Z for every β ∈ Φ_{B}, so that

h_{α} ∈ C_{Q} (22.2.6)

in particular. The linear functional λ_{α} on E_{Q} that corresponds to h_{α} as in the preceding paragraph is the same as in the previous section. Note that λ_{α}, considered as a linear functional on E_{R}, is the inverse root associated to α, as in Section 19.8.

Put Φ′_{B} = {λ_{α} : α ∈ Φ_{B}}, (22.2.7)

which is a subset of E′_{Q} ⊆ E′_{R}. As a subset of E′_{R}, this is the inverse system of Φ_{B}, as in Section 19.8.

In [24], k = C, and the notion of root systems is extended to complex vector spaces, so that h_{α} is identified directly with the inverse root of α, as in part (b) of Theorem 2 on p43f.

In [14], the Killing form on a semisimple Lie algebra A is used to get an inner product on E_{R}, as in the previous section. Using this, the inverse root of α ∈ Φ_{B} is identified with an element of E_{R}. More precisely, the inverse root of α corresponds to an element of E_{Q}, and thus an element of B′. This corresponds to h_{α} ∈ B using the Killing form, as on p43 of [14].

22.3 Semigroups and submodules

Let Ξ be a commutative semigroup, with the semigroup operation expressed additively. Also let Ψ be a nonempty subset of Ξ. If α ∈ Ξ and n is a positive integer, then n · α is the sum of n α’s in Ξ, as usual.

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Suppose that for each α ∈ Ψ, A_{α} is a submodule of A, as a module over k. We ask that A correspond to the direct sum of the A_{α}’s, α ∈ Ψ, as a module over k, so that every element of A can be expressed in a unique way as a sum of elements of the A_{α}’s. It is convenient to put A_{α} = {0} when α ∈ Ξ \ Ψ, (22.3.1)

so that A_{α} is defined as a submodule of A, as a module over k, for every α ∈ Ξ. If x ∈ A_{α} and y ∈ A_{β} for some α, β ∈ Ξ, then we ask that

[x, y]_{A} ∈ A_{α+β}. (22.3.2)

Of course, this holds trivially when α or β is in Ξ \ Ψ, by (22.3.1). Observe that

[x, y]_{A} = 0 when α + β ∉ Ψ, (22.3.3)

by (22.3.1) and (22.3.2). If Ξ has an additive identity element 0, then (22.3.2) implies that A_{0} is a Lie subalgebra of A.

If x ∈ A_{α} for some α ∈ Ψ, then

ad_{x}(A_{γ}) ⊆ A_{α+γ} (22.3.4)

for every γ ∈ Ξ. If n ∈ Z_{+}, then we get that

(ad_{x})^{n}(A_{γ}) ⊆ A_{n·α+γ}, (22.3.5)

where (ad_{x})^{n} is the nth power of ad_{x} on A, with respect to composition of mappings. Suppose that γ ∈ Ψ and n(γ) ∈ Z_{+} have the property that

n(γ) · α + γ ∉ Ψ. (22.3.6)

This implies that (ad_{x})^{n(γ)} = 0 on A_{γ}, and hence that (ad_{x})^{n} = 0 on A_{γ} for every integer n ≥ n(γ). If Ψ has only finitely many elements, and for each γ ∈ Ψ there is an n(γ) ∈ Z_{+} such that (22.3.6) holds, then it follows that (ad_{x})^{n} = 0 on A when n ≥ max_{γ∈Ψ} n(γ), so that x is ad-nilpotent as an element of A.

In some situations, we may have that

n ↦ n · α + γ (22.3.7)

is injective as a mapping from Z_{+} into Ξ. In particular, this holds for every γ ∈ Ξ when Ξ is a commutative group without torsion, and α ≠ 0. Suppose that Ψ has only finitely many elements, so that the injectivity of (22.3.7) implies that n · α + γ ∈ Ψ (22.3.8)

for only finitely many n ∈ Z_{+}. This means that (ad_{x})^{n} = 0 on A_{γ} for all but finitely many n ∈ Z_{+}, as before. If this holds for every γ ∈ Ψ, then it follows that (ad_{x})^{n} = 0 on A for all but finitely many n ∈ Z_{+}.

As usual, one may say that Ξ is a semigroup with cancellation if for every α, β, γ ∈ Ξ with α + γ = β + γ, (22.3.9)

we have that α = β. Note that this holds when Ξ is a subsemigroup of a group.

Suppose now that Ξ has an additive identity element 0. If α ∈ Ξ satisfies n · α = 0 for some n ∈ Z_{+}, then α is said to be a torsion element of Ξ. If 0 is the only torsion element of Ξ, then one may say that Ξ is torsion-free, or that Ξ has no nontrivial torsion.

Suppose that Ξ is a semigroup with cancellation. If α, γ ∈ Ξ and α is not a torsion element, then it is easy to see that (22.3.8) is injective as a mapping from Z_{+} into Ξ. If x ∈ A_{α} and Ψ has only finitely many elements, then it follows that x is ad-nilpotent as an element of A, as before.

22.4 Semigroups and algebras

Let Ξ be a commutative semigroup, where the semigroup operation is expressed additively, and let Ψ be a nonempty subset of Ξ. Also let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, where multiplication of x, y ∈ A is expressed as x y. Suppose that for each α ∈ Ψ, A_{α} is a submodule of A, as a module over k, and that A corresponds to the direct sum of the A_{α}’s, α ∈ Ψ, as a module over k. As before, it is convenient to put A_{α} = {0} when α ∈ Ξ \ Ψ.

Suppose that x y ∈ A_{α+β} (22.4.1)

when x ∈ A_{α} and y ∈ A_{β} for some α, β ∈ Ξ. Thus

x y = 0 (22.4.2) when α, β, or α+β is not in Ψ. If Ξ has an additive identity element 0, then A_{0} is a subalgebra of A, by (22.4.1). Of course, the remarks in the previous section correspond to the case where A is a Lie algebra over k.

Note that x y − y x ∈ A_{α+β} (22.4.3)

when x ∈ A_{α} and y ∈ A_{β} for some α, β ∈ Ξ, by (22.4.1). If A is an associative algebra over k, then A may be considered as a Lie algebra with respect to the corresponding commutator bracket, as usual.

Suppose that A is an associative algebra over k, and that x ∈ A_{α} for some α ∈ Ψ. If n ∈ Z_{+}, then x^{n} ∈ A_{n·α}, (22.4.4)

by (22.4.1). In particular, if n · α ∉ Ψ, then x^{n} = 0. If n ↦ n · α is injective as a mapping from Z_{+} into Ξ, and Ψ has only finitely many elements, then n · α ∈ Ψ for only finitely many n ∈ Z_{+}, so that x is nilpotent in A. If Ξ is a semigroup with cancellation and an additive identity element 0, and if α is not a torsion element of Ξ, then n ↦ n · α is an injective mapping from Z_{+} into Ξ, as before.

Let A be an algebra over k in the strict sense again. If Ψ_{0} is a subset of Ξ, then let A(Ψ_{0}) be the subset of A consisting of elements that can be expressed as a sum of elements of A_{α}, for finitely many α ∈ Ψ_{0}. This is a submodule of A, as a module over k, which may be interpreted as being {0} when Ψ_{0} = ∅, and which corresponds to the direct sum of the A_{α}’s, α ∈ Ψ_{0}, as a module over k. Of course, A(Ψ_{0}) = A(Ψ_{0} ∩ Ψ), (22.4.5)

and one may wish to restrict one’s attention to Ψ_{0} ⊆ Ψ. However, it is sometimes convenient for A(Ψ_{0}) to be defined for any Ψ_{0} ⊆ Ξ.

Suppose that Ψ_{0} ⊆ Ξ has the property that

(22.4.6) if α, β ∈ Ψ_{0} and α + β ∈ Ψ, then α + β ∈ Ψ_{0}.

In particular, if Ξ_{0} is a subsemigroup of Ξ, then Ξ_{0} ∩Ψ has this property. Under these conditions, A(Ψ_{0}) is a subalgebra of A, as an algebra over k in the strict sense.

Suppose that Ψ_{1} ⊆ Ξ has the property that

(22.4.7) if α ∈ Ψ_{1} and β, α + β ∈ Ψ, then α + β ∈ Ψ_{1}.

In this case, A(Ψ_{1}) is a two-sided ideal in A. If Ξ_{1} ⊆ Ξ has the property that

(22.4.8) if α ∈ Ξ_{1} and β ∈ Ξ, then α + β ∈ Ξ_{1},

then Ξ_{1} ∩ Ψ satisfies (22.4.7).

Similarly, suppose that Ψ_{2} ⊆ Ξ satisfies (22.4.6), and that Ψ_{1} ⊆ Ψ_{2} satisfies

(22.4.9) if α ∈ Ψ_{1} and β, α + β ∈ Ψ_{2}, then α + β ∈ Ψ_{1}.

In this situation, A(Ψ_{1}) is a two-sided ideal in A(Ψ_{2}), as an algebra over k in the strict sense.

22.5. SOME NILPOTENCY PROPERTIES 467

22.5 Some nilpotency properties

Let us continue with the same notation and hypotheses as in the previous section. In particular, Ξ is a commutative semigroup, and A is an algebra over k in the strict sense. If Ψ_{1}, …, Ψ_{r} ⊆ Ξ for some r ∈ Z_{+}, then put

(22.5.1) Ψ_{1} + ⋯ + Ψ_{r} = { ∑_{j=1}^{r} α_{j} : α_{j} ∈ Ψ_{j} for each j = 1, …, r},

as usual. This subset of Ξ may be denoted ∑_{j=1}^{r} Ψ_{j} as well. Note that A(Ψ_{1}) · A(Ψ_{2}) ⊆ A(Ψ_{1} + Ψ_{2}) (22.5.2) for every Ψ_{1}, Ψ_{2} ⊆ Ξ, by (22.4.1), where the left side is the set of finite sums of products of elements of A(Ψ_{1}) and A(Ψ_{2}), as in Section 9.2.

If Ψ_{0} ⊆ Ξ, then (22.4.6) may be expressed as (Ψ_{0} + Ψ_{0}) ∩ Ψ ⊆ Ψ_{0}. (22.5.3) Of course, Ξ_{0} ⊆ Ξ is a subsemigroup of Ξ when Ξ_{0} + Ξ_{0} ⊆ Ξ_{0}. (22.5.4) Similarly, if Ψ_{1} ⊆ Ξ, then (22.4.7) may be expressed as (Ψ_{1} + Ψ) ∩ Ψ ⊆ Ψ_{1}. (22.5.5) If Ξ_{1} ⊆ Ξ, then (22.4.8) may be expressed as Ξ_{1} + Ξ ⊆ Ξ_{1}. (22.5.6) If Ψ_{1} ⊆ Ψ_{2} ⊆ Ξ, then (22.4.9) may be expressed as (Ψ_{1} + Ψ_{2}) ∩ Ψ_{2} ⊆ Ψ_{1}. (22.5.7)

If r ∈ Z_{+}, and Ψ_{j} ⊆ Ξ, x_{j} ∈ A(Ψ_{j}) for each j = 1, …, r, then the product of x_{1}, …, x_{r} in A, with any ordering or grouping, is an element of A( ∑_{j=1}^{r} Ψ_{j}). In particular, if

( ∑_{j=1}^{r} Ψ_{j}) ∩ Ψ = ∅, (22.5.8)

then any such product of x_{1}, …, x_{n} in A is equal to 0.

Let Ψ_{0} be a subset of Ξ, and put

Ψ_{0}(r) = ∑_{j=1}^{r} Ψ_{0} (22.5.9)

468 CHAPTER 22. ROOTS AND LIE ALGEBRAS for each r ∈ Z_{+}, which is the subset of Ξ consisting of sums of r elements of Ψ_{0}. If x_{1}, …, x_{r} are elements of A(Ψ_{0}), then the product of x_{1}, …, x_{r} in A, with any grouping, is an element of A(Ψ_{0}(r)), as in the preceding paragraph. If Ψ_{0}(r) ∩ Ψ = ∅, (22.5.10) then such a product of x_{1}, …, x_{r} in A is equal to 0.

If y ∈ A = A(Ψ), then the product of x_{1}, …, x_{r}, y in A, with any ordering or grouping, is an element of A(Ψ_{0}(r) + Ψ), as before. If (Ψ_{0}(r) + Ψ) ∩ Ψ = ∅, (22.5.11) then such a product of x_{1}, …, x_{r}, y in A is equal to 0.

Suppose now that A is a Lie algebra over k, so that (22.5.2) can be expressed as [A(Ψ_{1}), A(Ψ_{2})] ⊆ A(Ψ_{1} + Ψ_{2}) (22.5.12) for every Ψ_{1}, Ψ_{2} ⊆ Ξ. Equivalently, if x ∈ Ψ_{1}, then ad_{x}(A(Ψ_{2})) ⊆ A(Ψ_{1} + Ψ_{2}). (22.5.13) If Ψ_{0} ⊆ Ξ, x ∈ A(Ψ_{0}), and r ∈ Z_{+}, then ad_{x}(A(Ψ_{0}(r) + Ψ)) ⊆ A(Ψ_{0}(r + 1) + Ψ). (22.5.14) Thus (ad_{x})^{r}(A) ⊆ A(Ψ_{0}(r) + Ψ) (22.5.15) for every r ≥ 1. In particular, if x ∈ A(Ψ_{0}) and r ∈ Z_{+} satisfies (22.5.11), then we get that (ad_{x})^{r} = 0 on A.

Suppose that Ψ_{0} satisfies (22.5.3), so that A(Ψ_{0}) is a Lie subalgebra of A. Let (A(Ψ_{0}))^{j}, j ≥ 0, be the lower central series in A(Ψ_{0}), as in Section 9.5. If r ∈ Z_{+}, then (A(Ψ_{0}))^{r−1} ⊆ A(Ψ_{0}(r)), (22.5.16) as before. If (22.5.10) holds, then (A(Ψ_{0}))^{r−1} = {0}, (22.5.17) so that A(Ψ_{0}) is nilpotent as a Lie algebra.

Suppose that Ξ has an additive identity element 0, so that A_{0} is a Lie subslagebra of A. If Ψ_{1}, Ψ_{2} ⊆ Ξ, then (Ψ_{1} ∪ {0}) + (Ψ_{2} ∪ {0}) = (Ψ_{1} + Ψ_{2}) ∪ Ψ_{1} ∪ Ψ_{2} ∪ {0}. (22.5.18) If A_{0} is commutative as a Lie algebra, then [A(Ψ_{1} ∪ {0}), A(Ψ_{2} ∪ {0})] ⊆ A((Ψ_{1} + Ψ_{2}) ∪ Ψ_{1} ∪ Ψ_{2}). (22.5.19)

Suppose that Ψ_{0} ⊆ Ξ satisfies (22.5.3) again. This implies that Ψ_{0} ∪ {0} satisfies the analogous condition,

((Ψ_{0} ∪ {0}) + (Ψ_{0} ∪ {0})) ∩ Ψ ⊆ Ψ_{0} ∪ {0}. (22.5.20)

Thus A(Ψ_{0} ∪ {0}) is a Lie subalgebra of A too, and it is easy to see that A(Ψ_{0}) is an ideal in A(Ψ_{0} ∪ {0}). If A_{0} is commutative as a Lie algebra, then

[A(Ψ_{0} ∪ {0}), A(Ψ_{0} ∪ {0})] ⊆ A(Ψ_{0}). (22.5.21)

It follows that A(Ψ_{0} ∪ {0}) is solvable as a Lie algebra over k when (22.5.10) holds for some r, so that A(Ψ_{0}) is nilpotent as a Lie algebra.

If [A_{0}, A_{α}] = A_{α} (22.5.22)

for every α ∈ Ψ_{0}, then [A_{0}, A(Ψ_{0})] = A(Ψ_{0}). (22.5.23)

If A_{0} is commutative as a Lie algebra, then we get that

[A(Ψ_{0} ∪ {0}), A(Ψ_{0} ∪ {0})] = A(Ψ_{0}). (22.5.24)

22.6 Roots and submodules

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a root system in V. Also let Θ be the subgroup of V, as a commutative group with respect to addition, generated by Φ. We would like to consider situations like those in Section 22.3, with

Ξ = Θ, Ψ = Φ ∪ {0}. (22.6.1)

Note that Θ has no nontrivial torsion, because V has none.

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. If α ∈ Φ ∪ {0}, then we ask that A_{α} be a submodule of A, as a module over k, and that A correspond to the direct sum of the A_{α}’s, α ∈ Φ ∪ {0}, as a module over k. As before, we put A_{α} = {0} when α ∈ Θ \ (Φ ∪ {0}), and we ask that

[A_{α}, A_{β}] ⊆ A_{α+β} (22.6.2)

for every α, β ∈ Θ, as in (22.3.2). Thus A_{0} is a Lie subalgebra of A, as before. If x ∈ A_{α} for some nonzero α ∈ Θ, then x is ad-nilpotent in A, as in Section 22.3.

If Ψ_{0} ⊆ Θ, then let A(Ψ_{0}) be the subset of A consisting of finite sums of elements of A_{α}, α ∈ Ψ_{0}, as in Section 22.4. Let Δ ⊆ Φ be a base for Φ, as a root system in V, and let Φ^{+} = Φ^{Δ,+} be the set of positive roots in Φ with respect to Δ. Thus Φ^{+} consists of the α ∈ Φ that can be expressed as a linear combination of elements of Δ whose coefficients are nonnegative integers, and Φ = Φ^{+} ∪ (−Φ^{+}). Note that

(Φ^{+} + Φ^{+}) ∩ (Φ ∪ {0}) ⊆ Φ^{+}, (22.6.3)

so that A(Φ^{+}) is a Lie subalgebra of A, as before. Similarly,

((−Φ^{+}) + (−Φ^{+})) ∩ (Φ ∪ {0}) ⊆ −Φ^{+}, (22.6.4)

and A(−Φ^{+}) is a Lie subalgebra of A too.

It is easy to see that

( ∑_{j=1}^{r} Φ^{+}) ∩ (Φ ∪ {0}) = ∅ (22.6.5)

when r is sufficiently large, because Φ has only finitely many elements. In fact,

(( ∑_{j=1}^{r} Φ^{+}) + (Φ ∪ {0})) ∩ (Φ ∪ {0}) = ∅ (22.6.6)

when r is sufficiently large. This implies that A(Φ^{+}) is nilpotent as a Lie algebra over k, and that the elements of A(Φ^{+}) are ad-nilpotent in A, as in the previous section. Similarly, A(−Φ^{+}) is nilpotent as a Lie algebra over k, and the elements of A(−Φ^{+}) are ad-nilpotent in A. This corresponds to some remarks near the top of p84 of [14], and part (b) of Theorem 4 on p47 of [24].

Observe that

((Φ^{+} ∪ {0}) + (Φ^{+} ∪ {0})) ∩ (Φ ∪ {0}) ⊆ Φ^{+} ∪ {0} (22.6.7)

and (((−Φ^{+}) ∪ {0}) + ((−Φ^{+}) ∪ {0})) ∩ (Φ ∪ {0}) ⊆ (−Φ^{+}) ∪ {0}, (22.6.8)

so that A(Φ^{+} ∪ {0}) and A((−Φ^{+}) ∪ {0}) are Lie subalgebras of A. As in the previous section, A(Φ^{+}) is an ideal in A(Φ^{+} ∪ {0}), and A(−Φ^{+}) is an ideal in A((−Φ^{+}) ∪ {0}).

If A_{0} is commutative as a Lie algebra over k, then

[A(Φ^{+} ∪ {0}), A(Φ^{+} ∪ {0})] ⊆ A(Φ^{+}) (22.6.9)

and [A((−Φ^{+}) ∪ {0}), A((−Φ^{+}) ∪ {0})] ⊆ A(−Φ^{+}), (22.6.10)

as in (22.5.21). This implies that A(Φ^{+}∪{0}) and A((−Φ^{+})∪{0}) are solvable as Lie algebras over k, because A(Φ^{+}) and A(−Φ^{+}) are nilpotent as Lie algebras over k. This corresponds to some of the remarks near the top of p84 of [14] again, and part of part (c) of Theorem 4 on p47 of [24].

If [A_{0}, A_{α}] = A_{α} (22.6.11) for every α ∈ Φ, then [A_{0}, A(Φ^{+})] = A(Φ^{+}) (22.6.12)

and [A_{0}, A(−Φ^{+})] = A(−Φ^{+}). (22.6.13)

If A_{0} is commutative as a Lie algebra over k, then it follows that

[A(Φ^{+} ∪ {0}), A(Φ^{+} ∪ {0})] = A(Φ^{+}) (22.6.14)

and [A((−Φ^{+}) ∪ {0}), A((−Φ^{+}) ∪ {0})] = A(−Φ^{+}), (22.6.15)

as in (22.5.24). This corresponds to the second part of part (c) of Theorem 4 on p47 of [24].

22.7 Borel subalgebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. A Lie subalgebra B of A is said to be a Borel subalgebra if B is solvable as a Lie algebra over k, and B is maximal with respect to inclusion among Lie subalgebras of A that are solvable as Lie algebras. If solvable subalgebras in A satisfy an ascending chain condition, then every solvable subalgebra of A is contained in a Borel subalgebra. In particular, this holds for finite-dimensional Lie algebras over fields.

Let A be an algebra over k in the strict sense for the moment, where multiplication of a, b ∈ A is expressed as a b, and let A_{1}, A_{2} be subalgebras of A. In particular, A_{1} and A_{2} are submodules of A, as a module over k, so that A_{1} +A_{2} is a submodule of A. Let a_{1}, b_{1} ∈ A_{1} and a_{2}, b_{2} ∈ A_{2} be given, so that

(a_{1} + a_{2}) (b_{1} + b_{2}) = a_{1} b_{1} + a_{1} b_{2} + a_{2} b_{1} + a_{2} b_{2}. (22.7.1)

Of course, a_{1} b_{1} ∈ A_{1} and a_{2} b_{2} ∈ A_{2}, by hypothesis. If A_{1} is a right ideal in A, then a_{1} b_{2} ∈ A_{1}, and if A_{2} is a left ideal in A, then a_{1} b_{2} ∈ A_{2}. Similarly, if A_{1} is a left ideal in A, then a_{2} b_{1} ∈ A_{1}, and if A_{2} is a right ideal in A, then a_{2} b_{1} ∈ A_{2}. It follows that (22.7.1) is an element of A_{1} + A_{2} when either A_{1} is a right ideal in A or A_{2} is a left ideal in A, and either A_{1} is a left ideal in A, or A_{2} is a right ideal in A. In particular, this holds when A_{1} or A_{2} is a two-sided ideal in A. This means that A_{1} + A_{2} is a subalgebra of A under these conditions.

Let A be a Lie algebra over k again, let A_{1} be a Lie subalgebra of A, and let A_{2} be an ideal in A. Thus A_{1} + A_{2} is a Lie subalgebra of A, as in the preceding paragraph. If A_{1} and A_{2} are both solvable as Lie algebras over k, then A_{1} + A_{2} is solvable as well. This was mentioned in Section 9.4 when A_{1} and A_{2} are both ideals in A, and essentially the same argument works in this situation. More precisely, note that A_{2} is an ideal in A_{1} + A_{2}. It suffices to verify that (A_{1} + A_{2})/A_{2} is solvable as a Lie algebra, because A_{2} is solvable, as in Section 9.3. Let q_{2} be the canonical quotient mapping from A_{1} + A_{2} onto (A_{1} + A_{2})/A_{2}, which is essentially the same as the restriction of the canonical quotient mapping from A onto A/A_{2} to A_{1} + A_{2}.

It is easy to see that A_{1} ∩ A_{2} is an ideal in A_{1}, because A_{2} is an ideal in A. In fact, A_{1} ∩ A_{2} is the same as the kernel of the restriction of q_{2} to A_{1}, as a subalgebra of A_{1} + A_{2}. Note that q_{2}(A_{1}) = q_{2}(A_{1} + A_{2}). The restriction of q_{2} to A_{1} can be identified with the canonical quotient mapping from A_{1} onto A_{1}/(A_{1} ∩ A_{2}), as a Lie algebra homomorphism. This leads to a natural Lie algebra isomorphism between A_{1}/(A_{1} ∩ A_{2}) and (A_{1} + A_{2})/A_{2}. By hypothesis, A_{1} is solvable, which implies that A_{1}/(A_{1} ∩ A_{2}) is solvable, as in Section 9.3. This means that (A_{1} + A_{2})/A_{2} is solvable, as desired.

If A_{1} is a Borel subalgebra of A, then we get that A_{1} = A_{1} + A_{2}, which means that A_{2} ⊆ A_{1}. (22.7.2)

If A has a solvable radical Rad A, as in Section 9.4, then it follows that

Rad A ⊆ A_{1}. (22.7.3)

This corresponds to part of the proof of Lemma B on p83 of [14].

Let A_{0} be a solvable ideal in A, and let q_{0} be the canonical quotient mapping from A onto A/A_{0}. If B is a Lie subalgebra of A that is solvable as a Lie algebra over k, then q_{0}(B) is solvable too, as in Section 9.3. If B_{0} is a solvable Lie subalgebra of A/A_{0}, then q_{0}^{−1}(B_{0}) is a solvable Lie subalgebra of A, as before. If B is a Borel subalgebra of A, then A_{0} ⊆ B, as in the preceding paragraph. This leads to a natural one-to-one correspondence between Borel subalgebras of A and A/A_{0}, as in Lemma B on p83 of [14].

Let B be a Lie subalgebra of A, and let x be an element of the normalizer N_{A}(B) of B in A. Under these conditions, it is easy to see that

B(x) = {t x + y : t ∈ k, y ∈ B} (22.7.4)

is a Lie subalgebra of A, with

[B(x), B(x)] ⊆ B. (22.7.5)

If B is solvable, then it follows that B(x) is solvable as well. If B is a Borel subalgebra of A, then we get that B = B(x), so that x ∈ B. This means that N_{A}(B) = B when B is a Borel subalgebra of A, as in Lemma A on p83 of [14].

22.8 Some diagonalizability conditions

Let Ξ be a commutative semigroup, with the semigroup operation expressed additively, and with an additive identity element 0. Also let Φ be a set of nonzero elements of Ξ. We would like to consider the same type of situation as in Section 22.3, with Ψ = Φ ∪ {0}.

Let k be a commutative ring with a multiplicative identity element again, and let (A, [·, ·]_{A}) be a Lie algebra over k. As before, suppose that A_{α} is a submodule of A, as a module over k, for each α ∈ Φ ∪ {0}, and that A corresponds to the direct sum of the A_{α}’s, as a module over k. We put A_{α} = {0} when α ∈ Ξ \ (Φ ∪ {0}), and ask that

[A_{α}, A_{β}] ⊆ A_{α+β} (22.8.1)

for every α, β ∈ Ξ, as in (22.3.2). In particular, this means that A_{0} is a Lie subalgebra of A.

Let Hom_{k}(A_{0}, k) be the space of homomorphisms from A_{0} into k, as modules over k. Remember that this is a module over k, with respect to pointwise addition and scalar multiplication. Suppose that

(22.8.2) ϕ is a semigroup homomorphism from Ξ into Hom_{k}(A_{0}, k),

as a commutative semigroup with respect to addition. Note that

ϕ(0) = 0 (22.8.3)

in Hom_{k}(A_{0}, k), because the additive identity element in a group is the only element of the group whose sum with itself is equal to itself.

If α ∈ Ξ, then it is convenient to let

ϕ_{α} = ϕ(α) (22.8.4)

be the corresponding element of Hom_{k}(A_{0}, k). Thus, if w ∈ A_{0}, then ϕ_{α}(w) is an element of k.

If α ∈ Φ ∪ {0}, w ∈ A_{0}, and x ∈ A_{α}, then we ask that

ad_{w}(x) = [w, x]_{A} = ϕ_{α}(w) x. (22.8.5)

In particular, this is equal to 0 when α = 0. This means that A_{0} is commutative as a Lie algebra. If α ∈ Ξ \ (Φ ∪ {0}) and x ∈ A_{α}, then x = 0, so that (22.8.5) holds trivially.

Note that (22.8.1) follows from (22.8.5) when α or β is 0. If w ∈ A_{0}, and x ∈ A_{α}, y ∈ A_{β} for some α, β ∈ Ξ, then

[w, [x, y]_{A}]_{A} = [[w, x]_{A}, y]_{A} + [x, [w, y]_{A}]_{A}
(22.8.6)  = ϕ_{α}(w) [x, y]_{A} + ϕ_{β}(w) [x, y]_{A} = ϕ_{α+β}(w) [x, y]_{A}.

Of course, this is compatible with (22.8.1) and (22.8.5).

If w ∈ A_{0} satisfies ϕ_{α}(w) = 0 (22.8.7)

for every α ∈ Φ, then w is in the center Z(A) of A as a Lie algebra. In particular, if Z(A) = {0}, then w = 0.

If α ∈ Ξ, then [A_{0}, A_{α}] ⊆ A_{α}, as in (22.8.1). If there is a w ∈ A_{0} such that ϕ_{α}(w) has a multiplicative inverse in k, then we get that

[A_{0}, A_{α}] = A_{α}. (22.8.8)

22.9 Diagonalizability and subspaces

Let k be a field, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. Suppose that B is a Lie subalgebra of A such that every element of B is addiagonalizable as an element of A. This implies that B is commutative as a Lie algebra, as in Section 17.1. Let B′ be the dual of B, as a vector space over k, and for each α ∈ B′, let A_{α} be the set of x ∈ A such that ad_{w}(x) = α(w) x for every w ∈ B. Consider the set Φ_{B} of α ∈ B′ such that α ≠ 0 and A_{α} ≠ {0}. Under these conditions, Φ_{B} has only finitely many elements, and A corresponds to the direct sum of the subspaces A_{α} with α ∈ Φ_{B} ∪ {0}, as a vector space over k, as in Section 17.2. Remember that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ B′, as before. In this situation, we can take Ξ_{B} to be a subsemigroup of B′, as a commutative group with respect to addition, with

Φ_{B} ∪ {0} ⊆ Ξ_{B}. (22.9.1)

Note that [B, A_{α}] = A_{α} (22.9.2)

for every α ∈ Φ_{B}. It follows that [A_{0}, A_{α}] = A_{α} for every α ∈ Φ_{B}, because B ⊆ A_{0}, by the commutativity of B as a Lie algebra.

As before, A_{0} is the centralizer C_{A}(B) of B in A. Suppose that C_{A}(B) = B, so that A_{0} = B, and Hom_{k}(A_{0}, k) = A′_{0} = B′. (22.9.3)

In this case, we can take ϕ to be the obvious inclusion mapping of Ξ_{B} into (22.9.3), so that (22.8.5) follows from the definition of A_{α}.

Let us return for the moment to the situation before the preceding paragraph, so that we do not necessarily ask the centralizer of B in A to be B. Let C be a linear subspace of A such that

[B, C] ⊆ C. (22.9.4)

In particular, this holds when C is a Lie subalgebra of A that contains B. If w ∈ B, then ad_{w}(C) ⊆ C, (22.9.5)

and ad_{w} is diagonalizable on A, by hypothesis. This implies that the restriction of ad_{w} to C is diagonalizable on C, as in Section 10.6.

If u, v ∈ B, then [u, v]_{A} = 0, which implies that ad_{u} and ad_{v} commute on A, as in Section 2.4. Thus the restrictions of ad_{u} and ad_{v} to C commute as well. This means that the restrictions of ad_{u}, u ∈ B, to C are simultaneously diagonalizable on C, by standard arguments. Put

Ψ_{C} = {α ∈ B′ : A_{α} ∩ C ≠ {0}}, (22.9.6)

which is contained in Φ_{B} ∪ {0}. It follows that C corresponds to the direct sum of A_{α} ∩ C with α ∈ Ψ_{C}, as a vector space over k. If C is a Lie subalgebra of A, then [A_{α} ∩ C, A_{β} ∩ C] ⊆ A_{α+β} ∩ C (22.9.7) for every α, β ∈ B′. Of course,

ad_{w}(A_{α} ∩ C) = A_{α} ∩ C (22.9.8)

when w ∈ B, α ∈ B′, and α(w) ≠ 0.

22.10 Injectivity on Ψ

Let us consider some situations as in Section 22.8, with k a field. Thus we let Ξ be a commutative semigroup, with the semigroup operation expressed additively, and with an additive identity element 0. We also let Φ be a finite set of nonzero elements in Ξ, and start with situations as in Section 22.3, with Ψ = Φ ∪ {0}.

Let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k, and let A_{α} be a linear subspace of A, as a vector space over k, for each α ∈ Φ∪{0}. Suppose that A corresponds to the direct sum of the A_{α}’s, α ∈ Φ ∪ {0}, and put A_{α} = {0} when α ∈ Ξ \ (Φ ∪ {0}). As before, we ask that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ Ξ, which implies that A_{0} is a Lie subalgebra of A.

Suppose that

(22.10.1) ϕ is a semigroup homomorphism from Ξ into A′_{0},

where A′_{0} is the dual of A_{0} as a vector space over k, considered as a commutative semigroup with respect to addition in (22.10.1). As before, ϕ(0) = 0 in A′_{0}, because A′_{0} is a group with respect to addition. If α ∈ Ξ, then it is convenient to put ϕ_{α} = ϕ(α) again. We ask that ad_{w}(x) = [w, x]_{A} = ϕ_{α}(w) x for every α ∈ Φ ∪ {0}, w ∈ A_{0}, and x ∈ A_{α}, which holds trivially when α ∈ Ξ \ (Φ ∪ {0}). This implies that A_{0} is commutative as a Lie subalgebra of A.

Note that [A_{0}, A_{α}] = A_{α} for every α ∈ Φ such that ϕ_{α} ≠ 0, as in (22.8.8). More precisely, if ϕ_{α} ≠ 0, then there is a w ∈ A_{0} such that ϕ_{α}(w) ≠ 0, and ad_{w}(A_{α}) = A_{α}.

Let us take B = A_{0}, so that every element of B is ad-diagonalizable, by hypothesis. If ρ is an element of the dual B′ = A′_{0} of B, then put

(22.10.2) Ã_{ρ} = {x ∈ A : for every w ∈ B, ad_{w}(x) = [w, x]_{A} = ρ(w) x},

which is a linear subspace of A. Thus

A_{α} ⊆ Ã_{ϕ_{α}} (22.10.3)

for every α ∈ Ξ. If ρ is any element of B′, then it follows that Ã_{ρ} corresponds to the direct sum of A_{α} with α ∈ Φ ∪ {0} and ϕ_{α} = ρ. This means that Ã_{ρ} = {0} when ρ ≠ ϕ_{α} for every α ∈ Φ ∪ {0}.

Suppose that

(22.10.4) for every α ∈ Φ, we have that ϕ_{α} ≠ 0.

This implies that A_{0} = Ã_{0}, (22.10.5)

as in the preceding paragraph. Note that Ã_{0} is the same as the centralizer of B in A, as before. Thus the centralizer C_{A}(A_{0}) of A_{0} in A is equal to A_{0} in this case. We also get that [A_{0}, A_{α}] = A_{α} for every α ∈ Φ when (22.10.4) holds.

Suppose now that (22.10.6) ϕ is injective on Φ ∪ {0},

so that ϕ_{α} ≠ ϕ_{β}, as elements of A′_{0}, for every α, β ∈ Φ ∪ {0} with α ≠ β. Of course, (22.10.6) is the same as saying that (22.10.4) holds, and that ϕ is injective on Φ. It follows that A_{α} = Ã_{ϕ_{α}} (22.10.7)

for every α ∈ Φ ∪ {0}, as before.

Let C be a linear subspace of A such that

[A_{0}, C] ⊆ C, (22.10.8)

and put Ψ̃_{C} = {ρ ∈ B′ : Ã_{ρ} ∩ C ≠ {0}}, (22.10.9)

which corresponds to (22.9.6) with the notation in this section. As in the previous section, C corresponds to the direct sum of Ã_{ρ} ∩ C with ρ ∈ Ψ̃_{C}, as a vector space over k. Note that this does not use (22.10.4) or (22.10.6). Put

Ψ_{C} = {α ∈ Φ ∪ {0} : A_{α} ∩ C ≠ {0}}. (22.10.10)

If (22.10.6) holds, then C corresponds to the direct sum of A_{α} ∩C with α ∈ Ψ_{C}, as a vector space over k.

22.11 Some additional conditions

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a root system in V. Thus, for every α ∈ Φ, there is a unique symmetry σ_{α} on V with vector α that maps Φ onto itself. Let λ_{α} be the linear functional on V that is equal to 0 on the hyperplane fixed by σ_{α} and satisfies λ_{α}(α) = 2, as before. Equivalently, σ_{α}(v) = v − λ_{α}(v) α for every v ∈ V. Remember that λ_{α}(β) ∈ Z for every α, β ∈ Φ, by definition of a root system.

Let Θ be the subgroup of V, as a commutative group with respect to addition, generated by Φ. This is the same as the subsemigroup of V generated by Φ. We would like to consider situations like those discussed in Section 22.8, with Ξ = Θ. Note that the elements of Θ can be expressed as linear combinations of the elements of any base for Φ with coefficients in Z.

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. If α ∈ Φ ∪ {0}, then we suppose that A_{α} is a submodule of A, as a module over k, and that A corresponds to the direct sum of the A_{α}’s, as a module over k. As usual, it is convenient to put A_{α} = {0} when α ∈ Θ \ (Φ ∪ {0}), and we ask that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ Θ. We also ask that ϕ be a group homomorphism from Θ into Hom_{k}(A_{0}, k), as a commutative group with respect to addition, such that ad_{w}(x) = ϕ_{α}(w) x for every w ∈ A_{0} and x ∈ A_{α}, α ∈ Φ ∪ {0}, where ϕ_{α} = ϕ(α). This implies that A_{0} is a Lie subalgebra of A that is commutative as a Lie algebra over k, as before.

If α ∈ Φ, then −α ∈ Φ, and [A_{α}, A_{−α}] ⊆ A_{0}. We may be interested in situations where there is an h_{α} ∈ A_{0} such that

[A_{α}, A_{−α}] = {t h_{α} : t ∈ k}. (22.11.1)

We may also ask that ϕ_{α}(h_{α}) = 2(= 1 + 1) (22.11.2)

in k. In particular, (22.11.1) implies that there are x_{α} ∈ A_{α} and y_{α} ∈ A_{−α} such that [x_{α}, y_{α}]_{A} = h_{α}. (22.11.3)

More precisely, we may ask that

(22.11.4) A_{α} and A_{−α} are generated by x_{α} and y_{α}, respectively,

as modules over k, so that every element of A_{α} or A_{−α} can be expressed as a multiple of x_{α} or y_{α} by an element of k, as appropriate. Of course, (22.11.3) and (22.11.4) imply (22.11.1). If β ∈ Φ, then we may ask that

ϕ_{β}(h_{α}) = λ_{α}(β) · 1 (22.11.5)

in k, which reduces to (22.11.2) when α = β.

In the situation considered in Section 22.1, we take V = E_{R} and Φ = Φ_{B}. More precisely, k is a field of characteristic 0 in this case, and Φ may be considered as a subset of the dual of A_{0} = B, as a vector space over k. Remember that E_{Q} is the linear subspace of the dual of A_{0}, as a vector space over Q, spanned by Φ. By construction, E_{Q} is also a linear subspace of V = E_{R}, as a vector space over Q. In particular,

Θ ⊆ E_{Q}. (22.11.6)

Thus ϕ may be taken to be the obvious inclusion mapping of Θ into the dual of A_{0}. The one-dimensionality of [A_{α}, A_{−α}] for α ∈ Φ was discussed in Section 17.5, and h_{α} was defined in Section 17.6. The one-dimensionality of A_{α}, A_{−α} was discussed in Section 17.7, and (22.11.5) follows from the definition of λ_{α} in Section 22.1.

We also have that

(22.11.7) A_{0} is spanned by the h_{α}’s, α ∈ Φ,

as a vector space over k, in the situation considered in Section 22.1. More precisely, A_{0} = B is spanned by t_{b,α}, α ∈ Φ, because the dual of B is spanned by Φ.

Similarly, if Δ ⊆ Φ is a base for Φ, then

(22.11.8) A_{0} is spanned by the h_{α}’s, α ∈ Δ,

as a vector space over k, in the situation considered in Section 22.1. To see this, it suffices to check that A_{0} is spanned by t_{b,α}, α ∈ Δ, because h_{α} was defined to be the product of t_{b,α} by a nonzero element of k. This reduces to the previous case, because every element of Φ can be expressed as a linear combination of elements of Δ with integer coefficients.

In fact, (22.11.9) h_{α}, α ∈ Δ, form a basis for A_{0}

as a vector space over k, in the situation considered in Section 22.1. To see this, remember that the number of elements of Δ is the same as the dimension of V = E_{R} as a vector space over R. This is the same as the dimension of E_{Q}, as a vector space over Q. This is also the same as the dimension of the dual of A_{0} = B, as a vector space over k. Of course, this is the same as the dimension of A_{0}, as a vector space over k.

If α, β ∈ Φ, then ad_{x_{α}} maps A_{β} into A_{α+β}, as before. In the situation considered in Section 22.1, if we also have that α + β ∈ Φ, then

(22.11.10) the restriction of ad_{x_{α}} to A_{β} is a one-to-one mapping onto A_{α+β},

as in Section 17.9.

22.12 Standard Borel subalgebras

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a root system in V. Also let Θ be the subgroup of V, as a commutative group with respect to addition, generated by Φ. We would like to consider situations with some of the conditions mentioned in the previous section, with k a field. Let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k, and let A_{α} be a linear subspace of A, as a vector space over k, for each α ∈ Φ∪{0}. As before, we suppose that A corresponds to the direct sum of the A_{α}’s, α ∈ Φ ∪ {0}, as a vector space over k, and we put A_{α} = {0} when α ∈ Θ \ (Φ ∪ {0}), for convenience. We suppose that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ Θ, and that there is a group homomorphism ϕ from Θ into the dual A′_{0} of A_{0} as a vector space over k such that ad_{w}(x) = ϕ_{α}(w) x for every w ∈ A_{0} and x ∈ A_{α}, α ∈ Φ ∪ {0}. Here ϕ_{α} = ϕ(α) for every α ∈ Θ, as usual.

Suppose that for each α ∈ Φ there are x_{α} ∈ A_{α} and y_{α} ∈ A_{−α} such that A_{α}, A_{−α} are spanned in A by x_{α}, y_{α}, respectively. Put h_{α} = [x_{α}, y_{α}]_{A} for every α ∈ Φ, so that h_{α} ∈ A_{0}. We ask that

ϕ_{α}(h_{α}) = 2 · 1 (22.12.1)

in k for every α ∈ Φ, as before. Suppose in addition that

(22.12.2) ϕ is injective on Φ ∪ {0}, so that ϕ_{α} ≠ ϕ_{β} when α and β are distinct elements of Φ ∪ {0}. Note that this holds in the situation considered in Section 22.1, as in the previous section.

In particular, we are in the type of situation considered in Section 22.6. Let Δ ⊆ Φ be a base for Φ, as a root system in V, and let Φ^{+} be the set of positive roots in Φ with respect to Δ, as before. If Ψ_{0} ⊆ Θ, then we let A(Ψ_{0}) be the linear subspace of A spanned by A_{α} with α ∈ Ψ_{0}, as in Section 22.4. Remember that A_{0} is commutative as a Lie subalgebra of A, as in Section 22.8. It follows that A(Φ^{+} ∪ {0}) is a solvable Lie subalgebra of A, as in Section 22.6.

Let C be a Lie subalgebra of A such that

A(Φ^{+} ∪ {0}) ⊆ C (22.12.3)

In particular, this means that A_{0} ⊆ C, so that [A_{0}, C] ⊆ C. If we put

Ψ_{C} = {α ∈ Φ ∪ {0} : A_{α} ∩ C ≠ {0}}, (22.12.4)

then C corresponds to the direct sum of A_{α} ∩ C with α ∈ Ψ_{C}, as in Section 22.10. Note that this uses (22.12.2).

Suppose that A(Φ^{+} ∪ {0}) is a proper subset of C. This implies that Ψ_{C} is not contained in Φ^{+} ∪ {0}, as in the preceding paragraph. This means that there is an α ∈ Ψ_{C} ∩ (−Φ^{+}), because Φ = Φ^{+} ∪ (−Φ^{+}). It follows that x_{α} ∈ C, because A_{α} is spanned by x_{α}.

Under these conditions, −α ∈ Φ^{+}, and y_{α}, h_{α} ∈ A(Φ^{+} ∪ {0}), so that y_{α}, h_{α} are elements of C too. Suppose that the characteristic of k is not 2, which implies that ϕ_{α}(h_{α}) ≠ 0. (22.12.5)

In particular, x_{α}, y_{α}, h_{α} ≠ 0, and their linear span in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k), as a Lie algebra over k. Remember that this Lie subalgebra is not solvable in this case. It follows that C is not solvable as a Lie algebra over k.

Equivalently, if the characteristic of k is not 2, and if C is solvable as a Lie algebra, then A(Φ^{+} ∪ {0}) = C. This means that A(Φ^{+} ∪ {0}) is a Borel subalgebra of A when the characteristic of k is not 2. This may be called the standard Borel subalgebra of A with respect to Φ, Δ, and the A_{α}’s. This corresponds to some remarks near the top of p84 in [14], and at the bottom of p47 of [24].

22.13 Subalgebras and derivations

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, where multiplication of x, y ∈ A is expressed as x y. If B and C are submodules of A, as a module over k, then remember that their product B · C is defined to be the subset of A consisting of finite sums of elements of A of the form x y, with x ∈ B and y ∈ C. This is a submodule of A too, as a module over k, as in Section 9.2.

Let B be a submodule of A again, as a module over k, and put B_{1} = B. If B_{1}, …, B_{l} ⊆ A have been defined for some positive integer l, then let B_{l+1} ⊆ A be the set of finite sums of elements of B_{j} · B_{l+1−j}, j = 1, …, l. Continuing in this way, we can define B_{l} as a submodule of A, as a module over k, for every l ∈ Z_{+}. Equivalently, B_{l} consists of finite sums of elements of A that can be expressed as the product of l elements of B, with some grouping of the factors.

Let B_{∞} ⊆ A be the set of finite sums of elements of B_{l}, l ∈ Z_{+}. It is easy to see that B_{∞} is a submodule of A as a module over k, and in fact a subalgebra of A. Of course, B = B_{1} ⊆ B_{∞}, by construction. More precisely, B_{∞} is the smallest subalgebra of A that contains B. We may call B_{∞} the subalgebra of A generated by B.

Let δ be a derivation on A such that

δ(B) ⊆ B. (22.13.1)

One can check that δ(B_{l}) ⊆ B_{l} (22.13.2)

for every l ∈ Z_{+}, by induction. This implies that

δ(B_{∞}) ⊆ B_{∞}. (22.13.3)

22.14 Reducibility and semigroups

Let us return to the type of situation considered in Section 22.3. Thus we let Ξ be a commutative semigroup, with the semigroup operation expressed additively, and we let Ψ be a nonempty subset of Ξ. Also let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Suppose that A_{α} is a submodule of A, as a module over k, for each α ∈ Ψ, and that A corresponds to the direct sum of the A_{α}’s, α ∈ Ψ, as a module over k. It is convenient to put A_{α} = {0} when α ∈ Ξ \ Ψ, as before. We ask that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ Ψ, which holds trivially when α or β is in Ξ \ Ψ. If Ξ has an additive identity element 0, then it follows that A_{0} is a Lie subalgebra of A.

If Ψ_{0} ⊆ Ξ, then let A(Ψ_{0}) be the subset of A consisting of finite sums of elements of A_{α}, α ∈ Ψ_{0}, as in Section 22.4. If (Ψ_{0} + Ψ_{0}) ∩ Ψ ⊆ Ψ_{0}, then A(Ψ_{0}) is a Lie subalgebra of A, as before.

Let Ψ_{1}, Ψ_{2} be nonempty subsets of Ψ. If

(Ψ_{j} + Ψ_{j}) ∩ Ψ ⊆ Ψ_{j} (22.14.1)

for j = 1, 2, then A(Ψ_{1}), A(Ψ_{2}) are Lie subalgebras of A, as in the preceding paragraph. If Ψ_{1} ∪ Ψ_{2} = Ψ (22.14.2)

and Ψ_{1} ∩ Ψ_{2} = ∅, (22.14.3) then A corresponds to the direct sum of A(Ψ_{1}) and A(Ψ_{2}), as a module over k. If (Ψ_{1} + Ψ_{2}) ∩ Ψ = ∅, (22.14.4)

then it follows that [A(Ψ_{1}), A(Ψ_{2})] = {0}. (22.14.5)

If all four of these conditions hold, then A corresponds to the direct sum of A(Ψ_{1}) and A(Ψ_{2}), as a Lie algebra over k.

Suppose from now on in this section that Ξ has an additive identity element 0, and that 0 ∈ Ψ. In this case, (22.14.2) implies that 0 is an element of Ψ_{1} or Ψ_{2}. This means that (22.14.4) does not hold, because Ψ_{1}, Ψ_{2} ≠ ∅.

Let Φ_{1}, Φ_{2} be nonempty subsets of

Φ = Ψ \ {0}. (22.14.6)

If j ∈ {1, 2} and (Φ_{j} + Φ_{j}) ∩ Ψ ⊆ Φ_{j} ∪ {0}, (22.14.7)

then ((Φ_{j} ∪ {0}) + (Φ_{j} ∪ {0})) ∩ Ψ ⊆ Φ_{j} ∪ {0}, (22.14.8)

and A(Φ_{j} ∪ {0}) is a Lie subalgebra of A, as before. Let

(22.14.9) A^{j} be the Lie subalgebra of A generated by A(Φ_{j}),

so that A^{j} ⊆ A(Φ_{j} ∪ {0}) (22.14.10)

in this situation.

If (Φ_{1} + Φ_{2}) ∩ Ψ = ∅, (22.14.11)

then [A(Φ_{1}), A(Φ_{2})] = {0}, (22.14.12)

as before. Equivalently, this means that

A(Φ_{1}) ⊆ C_{A}(A(Φ_{2})), A(Φ_{2}) ⊆ C_{A}(A(Φ_{1})). (22.14.13)

This implies that

A^{1} ⊆ C_{A}(A(Φ_{2})), A^{2} ⊆ C_{A}(A(Φ_{1})), (22.14.14)

because the centralizer C_{A}(E) of any E ⊆ A in A is a Lie subalgebra of A, as in Section 7.6. This is the same as saying that

A(Φ_{2}) ⊆ C_{A}(A^{1}), A(Φ_{1}) ⊆ C_{A}(A^{2}). (22.14.15)

It follows that A^{2} ⊆ C_{A}(A^{1}), A^{1} ⊆ C_{A}(A^{2}), (22.14.16) as before, which are of course equivalent.

If w ∈ A_{0}, then ad_{w}(A(Φ_{j})) ⊆ A(Φ_{j}) (22.14.17)

for j = 1, 2, because ad_{w}(A_{α}) ⊆ A_{α} for every α ∈ Ψ, by hypothesis. This implies that ad_{w}(A^{j}) ⊆ A^{j} (22.14.18)

for j = 1, 2, as in the previous section, because ad_{w} is a derivation on A. Let us check that

(22.14.19) A^{j} is an ideal in A for j = 1, 2

when (22.14.11) holds. Of course, this is the same as saying that the normalizer N_{A}(A^{j}) of A^{j} in A is equal to A. By construction, A(Φ_{j}) ⊆ A^{j} ⊆ N_{A}(A^{j}) for j = 1, 2. Using (22.14.14), we get that A(Φ_{2}) ⊆ N_{A}(A^{1}) and A(Φ_{1}) ⊆ N_{A}(A^{2}). We also have that A_{0} ⊆ N_{A}(A^{j}) for j = 1, 2, by (22.14.18), so that N_{A}(A^{j}) = A, as desired.

If

(22.14.20) for each j = 1, 2, there is an α_{j} ∈ Φ_{j} such that A_{α_{j}} ≠ {0},

then A^{j} ≠ {0} for j = 1, 2. (22.14.21)

If (22.14.7) holds for j = 1, 2, (22.14.20) holds, and

Φ_{1} ∩ Φ_{2} = ∅, (22.14.22)

then A^{j} ≠ A for j = 1, 2, (22.14.23)

by (22.14.10). If (22.14.11) holds too, and k is a field, then it follows that A is not simple as a Lie algebra, because of (22.14.19). This is related to the proposition on p73 of [14], and to part of the corollary to Theorem 9 on p50 of [24].

Note that A^{1} ∩ A^{2} ⊆ A_{0} (22.14.24)

when (22.14.7) holds for j = 1, 2 and (22.14.22) holds, because of (22.14.10). If (22.14.11) holds as well, Φ_{1} ∪ Φ_{2} = Φ, (22.14.25)

and (22.14.26) A_{0} is commutative as a Lie algebra over k,

then A^{1} ∩ A^{2} ⊆ Z(A), (22.14.27)

where Z(A) is the center of A as a Lie algebra, as usual. More precisely, if x ∈ A^{1} ∩ A^{2}, then [x, y]_{A} = 0 for every y ∈ A^{1} or A^{2}, by (22.14.16). This holds when y ∈ A_{0} too, by (22.14.24) and (22.14.26). It follows that this holds for every y ∈ A, because of (22.14.25), as desired.

If Z(A) = {0}, then (22.14.27) implies that

A^{1} ∩ A^{2} = {0}. (22.14.28)

If we also have that

(22.14.29) every element of A can be expressed as the sum of elements of A^{1} and A^{2},

then A corresponds to the direct sum of A^{1} and A^{2}, as a module over k. If (22.14.11) holds too, then A corresponds to the direct sum of A^{1} and A^{2} as a Lie algebra over k, because of (22.14.16). If (22.14.25) holds, and every element of A_{0} can be expressed as a sum of elements of A^{1} and A^{2}, then (22.14.29) holds.

If j ∈ {1, 2}, α, β ∈ Φ_{j}, and α + β = 0, then

[A_{α}, A_{β}] ⊆ A^{j} ∩ A_{0}. (22.14.30)

If (22.14.7) and (22.14.26) hold, then one can check that set of elements of A that can be expressed as finite sums of elements of A(Φ_{j}) and [A_{α}, A_{β}] with α, β ∈ Φ_{j} and α + β = 0 is a Lie subalgebra of A. It follows that this is the same as A^{j} in this case.

22.15 Reducibility and roots

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a root system in V. Also let Θ be the subgroup of V, as a group with respect to addition, generated by Φ. We would like to consider situations like those in the previous section, with Ξ = Θ and Ψ = Φ ∪ {0}. Thus we let k, A, and A_{α} be as before.

Suppose that Φ is reducible as a root system in V, as in Section 20.4. Thus V is the direct sum of two nontrivial linear subspaces V_{1} and V_{2}, as a vector space over R, and Φ ⊆ V_{1} ∪ V_{2}. (22.15.1)

Put Φ_{j} = Φ ∩ V_{j} for j = 1, 2, so that Φ = Φ_{1} ∪ Φ_{2}. If α ∈ Φ_{1} and β ∈ Φ_{2}, then α ≠ β, because V_{1} ∩ V_{2} = {0}, and

α + β ∉ Φ ∪ {0}, (22.15.2)

because α + β ∉ V_{1} ∪ V_{2}. This means that (22.14.7), (22.14.11), (22.14.22), and (22.14.25) hold in this situation.

If j ∈ {1, 2} and α ∈ Φ_{j}, then −α ∈ Φ_{j}, and

[A_{α}, A_{−α}] ⊆ A^{j} ∩ A_{0}, (22.15.3)

as in (22.14.30). Suppose from now on in this section that A_{0} is commutative as a Lie algebra over k, as in (22.14.26). This implies that the set of elements of A that can be expressed as finite sums of elements of A(Φ_{j}) and [A_{α}, A_{−α}] with α ∈ Φ_{j} is a Lie subalgebra of A, as in the previous section. This Lie subalgebra is the same as A^{j} under these conditions, as before.

We can describe A^{j}, j = 1, 2, more precisely with some additional conditions, as in Section 22.11. Suppose that for each α ∈ Φ, there is an h_{α} ∈ A_{0} such that [A_{α}, A_{−α}] is the set of multiples of h_{α} by elements of k, as before. This means that A^{j} consists of sums of elements of A(Φ_{j}) and linear combinations of the h_{α}’s, α ∈ Φ_{j}, with coefficients in k, for j = 1, 2.

Let Δ_{1} and Δ_{2} be bases for Φ_{1} and Φ_{2} in V_{1} and V_{2}, respectively. This implies that Δ_{1} ∪ Δ_{2} is a base for Φ, as in Section 20.4. In the situation considered in Section 22.1, k is a field, and h_{α}, α ∈ Δ_{1} ∪ Δ_{2}, form a basis for A_{0} as a vector space over k, as in Section 22.11. If α ∈ Φ_{j} for j = 1, 2, then h_{α} can be expressed as a linear combination of h_{β}, β ∈ Δ_{j}, in A_{0} in this case. This follows from the fact that α can be expressed as a linear combination of the elements of Δ_{j} with integer coefficients, so that t_{b,α} can be expressed as a linear combination of t_{b,β}, β ∈ Δ_{j}, where t_{b,γ} is as in Section 22.1.

Under these conditions, A^{j} consists of sums of elements of A(Φ_{j}) and linear combinations of the h_{α}’s, now with α ∈ Δ_{j}, for j = 1, 2. It follows that every element of A can be expressed in a unique way as a sum of elements of A^{1} and A^{2}, so that A corresponds to the direct sum of A^{1} and A^{2}, as a Lie algebra over k.

As before, this is related to the proposition on p73 of [14], and to part of the corollary to Theorem 9 on p50 of [24].

Roots and Lie algebras, 2

23.1 Inverse roots and homomorphisms

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a root system in V. If α ∈ Φ, then there is a unique symmetry σ_{α} on V with vector α that maps Φ onto itself, as usual. Let λ_{α} be the linear functional on V that is equal to 0 on the hyperplane fixed by σ_{α} and satisfies λ_{α}(α) = 2, so that σ_{α}(v) = v − λ_{α}(v) α for every v ∈ V. Remember that

Φ′ = {λ_{α} : α ∈ Φ} (23.1.1)

is a root system in the dual V′ of V, as in Section 19.8.

Note that α ↦ λ_{α} (23.1.2)

is a one-to-one mapping from Φ onto Φ′. More precisely, if (·, ·) is an inner product on V that is invariant under the Weyl group of Φ and α ∈ Φ, then λ_{α} corresponds to α̂ = 2 α (α, α)^{−1} with respect to (·, ·), and ˆ α̂ = α. Alternatively, the symmetry on V′ with vector λ_{α} that maps Φ′ onto itself corresponds to the dual of σ_{α}, and the associated linear functional on V′ corresponds to α in a natural way.

Let Θ = Θ_{Φ} be the subgroup of V, as a commutative group with respect to addition, generated by Φ. If Δ is a base for Φ, then Θ consists of the linear combinations of elements of Δ with coefficients in Z.

Let Hom(Θ, Z) be the collection of homomorphisms from Θ in Z, as commutative groups with respect to addition. Of course, commutative groups may be considered as modules over Z, and group homomorphisms between them may be considered as module homomorphisms. Thus we may also use Hom_{Z}(Θ, Z) for space of homomorphisms from Θ into Z, as commutative groups or modules over Z. This is a commutative group with respect to pointwise addition of functions into Z, or equivalently a module over Z.

Elements of Hom(Θ, Z) are uniquely determined by their restrictions to Δ. Every Z-valued function on Δ can be extended to a homomorphism from Θ into Z. Similarly, linear functionals on V are uniquely determined by their restrictions to Δ, and every real-valued function on Δ can be extended to a linear functional on V. If λ is a linear functional on V that maps Θ into Z, then the restriction of λ to Θ defines an element of Hom(Θ, Z). Every element of Hom(Θ, Z) is the restriction to Θ of a unique linear functional on V that maps Θ into Z, which can be obtained from its restriction to Δ, as before.

Thus Hom(Θ, Z) can be identified with the set of linear functionals on V that map Θ into Z. This is a subgroup of the dual space V′, as a commutative group with respect to addition. If α ∈ Φ, then λ_{α} maps Φ into Z, by the definition of a root system. This means that λ_{α} maps Θ into Z, so that the restriction of λ_{α} to Θ is an element of Hom(Θ, Z).

Let Θ_{Φ′} be the subgroup of V′, as a commutative group with respect to addition, generated by Φ′. Note that every element of Θ_{Φ′} maps Θ into Z, so that its restriction to Θ is an element of Hom(Θ, Z).

As before, Θ_{Φ′} consists of linear combinations of elements of a base for Φ′ with coefficients in Z. If Φ is reduced as a root system in V, then

Δ′ = {λ_{α} : α ∈ Δ} (23.1.3)

is a base for Φ′, as in Section 19.13.

In the types of situations considered in Section 22.11, one can ask for a homomorphism from Θ_{Φ′} into A_{0}, as commutative groups with respect to addition, that sends λ_{α} to h_{α} for every α ∈ Φ. In particular, this happens in the type of situation considered in Section 22.1, as in Section 22.2.

23.2 Bases and submodules

Let us return to the type of situation considered in Section 22.6, and look at some additional properties related to a base for the root system. Thus we let V be a vector space over the real numbers of positive finite dimension, and let Φ be a root system in V. Also let Δ be a base for Φ, and let Φ^{+} = Φ^{Δ,+} be the set of positive roots in Φ with respect to Δ. Let Θ be the subgroup of V, as a commutative group with respect to addition, generated by Φ, as before. Of course, this is the same as the subgroup of V generated by Δ.

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Suppose that A_{α} is a submodule of A, as a module over k, for each α ∈ Φ ∪ {0}, and that A corresponds to the direct sum of the A_{α}’s, as a module over k. Put A_{α} = {0} when α ∈ Θ \ (Φ ∪ {0}), as before, and suppose that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ Θ. Of course, this implies that [A_{α}, A_{−α}] ⊆ A_{0} (23.2.1)

for every α ∈ Θ. In particular, A_{0} is a Lie subalgebra of A, as before.

If α, β ∈ Δ, then α − β ∉ Φ, by the definition of a base for a root system. If α ≠ β too, then α − β ∉ Φ ∪ {0}, so that

[A_{α}, A_{−β}] = {0}. (23.2.2)

23.3. SOME GENERATING SUBMODULES 487 This corresponds to part of the proposition on p96 of [14], and part of part (b) of Theorem 6 on p48 of [24].

If α ∈ Φ, then let σ_{α} be the symmetry on V with vector α that maps Φ onto itself, and let λ_{α} be the corresponding linear functional on V, so that σ_{α}(v) = v − λ_{α}(v) α for every v ∈ V. Let α, β ∈ Δ be given, with α ≠ β. Remember that λ_{α}(β) ∈ Z, by definition of a root system, and that λ_{α}(β) ≤ 0 in this case, as in Section 20.2. Observe that β − λ_{α}(β) α + α = σ_{α}(β − α), (23.2.3) because σ_{α}(α) = −α. This implies that β − λ_{α}(β) α + α ∈ Θ \ (Φ ∪ {0}), (23.2.4) because λ_{α}(β) ∈ Z and α − β ∉ Φ ∪ {0}, as before. It follows that A_{β−λ_{α}(β) α+α} = {0} (23.2.5) under these conditions. Similarly, −β − λ_{α}(β) (−α) − α = −(β − λ_{α}(β) α + α) ∈ Θ \ (Φ ∪ {0}), (23.2.6) so that A_{−β−λ_{α}(β) (−α)−α} = {0}. (23.2.7) This corresponds to another part of the proposition on p96 of [14], and to part (c) of Theorem 6 on p48 of [24].

Let α, β ∈ Θ be given, so that [[A_{α}, A_{β}], A_{−α−β}] ⊆ [A_{α+β}, A_{−α−β}] ⊆ A_{0}. (23.2.8) Using the Jacobi identity, we get that [[A_{α}, A_{β}], A_{−α−β}] ⊆ [[A_{β}, A_{−α−β}], A_{α}] + [[A_{−α−β}, A_{α}], A_{β}]. (23.2.9) This implies that [[A_{α}, A_{β}], A_{−α−β}] ⊆ [A_{α}, A_{−α}] + [A_{β}, A_{−β}]. (23.2.10)

23.3 Some generating submodules

Let us continue with the same notation and hypotheses as in the previous section. If Ψ_{0} ⊆ Θ, then let A(Ψ_{0}) be the subset of A consisting of finite sums of elements of A_{α}, α ∈ Ψ_{0}, as before. This is a submodule of A, as a module over k.

Suppose that for each α ∈ Δ and β ∈ Φ^{+} with α + β ∈ Φ we have that [A_{α}, A_{β}] = A_{α+β}. (23.3.1)

Of course, α + β ∈ Φ^{+} in this case. This condition holds in the situation considered in Section 22.1, as in Section 17.9. Note that α + β ≠ 0 when α, β are elements of Φ^{+}, because of the nonnegativity of the coefficients of α and β when expressed as linear combinations of elements of Δ. If α + β ∉ Φ, then A_{α+β} = {0}, by construction, and (23.3.1) holds automatically, by hypothesis.

Let β ∈ Φ^{+} be given, and remember that β can be expressed as ∑_{j=1}^{r} α_{j} for some α_{1}, …, α_{r} ∈ Δ, where

β_{l} = ∑_{j=1}^{l} α_{j} ∈ Φ (23.3.2)

for each l = 1, …, r, as in Section 19.12. If l < r, then we get that

A_{β_{l+1}} = [A_{α_{l+1}}, A_{β_{l}}], (23.3.3)

as in (23.3.1), because β_{l} ∈ Φ^{+}.

Remember that A(Φ^{+}) is a Lie subalgebra of A, as in Section 22.6. If (23.3.1) holds for every α ∈ Δ and β ∈ Φ^{+} with α + β ∈ Φ, then

(23.3.4) A(Φ^{+}) is generated by the submodules A_{α}, α ∈ Δ, as a Lie algebra over k.

More precisely, if β ∈ Φ^{+}, then one can use (23.3.3) to get that A_{β} is contained in the Lie subalgebra of A generated by A_{α}, α ∈ Δ. This corresponds to part of the proof of the proposition on p74 of [14], and is related to part of part (a) of Theorem 6 on p48 of [24].

Similarly, suppose that

[A_{−α}, A_{−β}] = A_{−α−β} (23.3.5)

for every α ∈ Δ and β ∈ Φ^{+} with α + β ∈ Φ. This also holds in the situation considered in Section 22.1, as in Section 17.9. If α +β ∉ Φ, then −α −β ∉ Φ, so that A_{−α−β} = {0}, and (23.3.5) holds automatically, by hypothesis. If β ∈ Φ^{+}, and α_{1}, …, α_{r} ∈ Δ, β_{l} ∈ Φ^{+} are as in (23.3.2), then we get that

A_{−β_{l+1}} = [A_{−α_{l+1}}, A_{−β_{l}}] (23.3.6)

when l < r.

We also have that A(−Φ^{+}) is a Lie subalgebra of A, as in Section 22.6 again. If (23.3.5) holds for every α ∈ Δ and β ∈ Φ^{+} with α + β ∈ Φ, then

(23.3.7) A(−Φ^{+}) is generated by the submodules A_{−α}, α ∈ Δ, as a Lie algebra over k.

This can be obtained from (23.3.6), as before. This corresponds to another part of the proof of the proposition on p74 of [14], and is related to another part of part (a) of Theorem 6 on p48 of [24].

23.4. SOME GENERATORS 489

If α ∈ Δ, β ∈ Φ^{+}, and (23.3.1) holds, then [A_{α+β}, A_{−α−β}] ⊆ [A_{α}, A_{−α}] + [A_{β}, A_{−β}], (23.3.8) by (23.2.10). Suppose that (23.3.1) holds for every α ∈ Δ and β ∈ Φ^{+} with α + β ∈ Φ again. If β ∈ Φ^{+}, then (23.3.9) [A_{β}, A_{−β}] is contained in the submodule of A_{0}

generated by [A_{α}, A_{−α}], α ∈ Δ, as a module over k.

To see this, one can express β as in (23.3.2) with l = r, as before. If l < r, then [A_{β_{l+1}}, A_{−β_{l+1}}] ⊆ [A_{α_{l+1}}, A_{−α_{l+1}}] + [A_{β_{l}}, A_{−β_{l}}], (23.3.10) by (23.3.3) and (23.3.8). One can use this repeatedly to get (23.3.9). Of course, one can use an analogous argument when (23.3.5) holds for every α ∈ Δ and β ∈ Φ^{+} with α + β ∈ Φ, instead of (23.3.1).

In the situation considered in Section 22.1, (23.3.11) A_{0} is generated by [A_{α}, A_{−α}], α ∈ Φ^{+}, as a module over k. as in (22.11.7). More precisely, (23.3.12) A_{0} is generated by [A_{α}, A_{−α}], α ∈ Δ, as a module over k. in this case, as in (22.11.8). If (23.3.9) holds for every β ∈ Φ^{+}, then (23.3.11) implies (23.3.12).

Suppose that (23.3.1) and (23.3.5) hold for every α ∈ Δ and β ∈ Φ^{+} with α + β ∈ Φ. If (23.3.11) holds too, then we get that (23.3.13) A is generated by the submodules

A_{α} and A_{−α}, α ∈ Δ, as a Lie algebra over k.

This corresponds to the proposition on p74 of [14], which is related to another part of part (a) of Theorem 6 on p48 of [24].

23.4 Some generators

Let us continue with the same notation and basic hypotheses as in the previous two sections. Suppose now that for each α ∈ Δ, there is an x_{α} ∈ A_{α} such that (23.4.1) A_{α} is generated by x_{α} as a module over k, so that every element of A_{α} can be expressed as a multiple of x_{α} by an element of k. This holds in the situation considered in Section 22.1, as in Section 17.7. If α ∈ Δ and β ∈ Φ^{+} satisfy α + β ∈ Φ, then (23.3.1) is the same as saying that ad_{x_{α}}(A_{β}) = A_{α+β}. (23.4.2)

Suppose that this holds for all such α and β, let β ∈ Φ^{+} be given, and let α_{1}, …, α_{r} ∈ Δ be as before, so that (23.3.2) holds for every l = 1, …, r, and β_{r} = β. If l < r, then A_{β_{l+1}} = ad_{x_{α_{l+1}}} (A_{β_{l}}), (23.4.3)

by (23.4.2). This implies that A_{β_{l}} is generated by a single element as a module over k for each l = 1, …, r. In particular, A_{β} is generated by a single element, as a module over k, because β = β_{r}.

In this situation, (23.3.4) is the same as saying that

(23.4.4) A(Φ^{+}) is generated by x_{α}, α ∈ Δ, as a Lie algebra over k.

This corresponds to part of part (a) of Theorem 6 on p48 of [24].

Similarly, suppose that for each α ∈ Δ there is a y_{α} ∈ A_{−α} such that

(23.4.5) A_{−α} is generated by y_{α} as a module over k.

This holds in the situation considered in Section 22.1, as in Section 17.7 again. If α ∈ Δ, β ∈ Φ^{+}, and α + β ∈ Φ, then (23.3.5) is the same as saying that

ad_{y_{α}}(A_{−β}) = A_{−α−β}. (23.4.6)

If this holds for all such α and β, then we get that A_{−β} is generated by a single element as a module over k for every β ∈ Φ^{+}, as before.

In this case, (23.3.7) is the same as saying that

(23.4.7) A(−Φ^{+}) is generated by y_{α}, α ∈ Δ, as a Lie algebra over k.

This corresponds to another part of part (a) of Theorem 6 on p48 of [24]. Suppose that (23.4.1) and (23.4.5) hold for every α ∈ Δ, so that

(23.4.8) [A_{α}, A_{−α}] is generated by [x_{α}, y_{α}]_{A} as a module over k

for every α ∈ Δ. In this situation, (23.3.12) is the same as saying that

(23.4.9) A_{0} is generated by [x_{α}, y_{α}]_{A}, α ∈ Δ, as a module over k.

If (23.4.2) and (23.4.6) hold for every α ∈ Δ and β ∈ Φ^{+} with α + β ∈ Φ, and if (23.4.9) holds, then we get that

(23.4.10) A is generated by x_{α} and y_{α}, α ∈ Δ, as a Lie algebra over k.

This corresponds to the second formulation of the proposition on p74 of [14], and to another part of part (a) of Theorem 6 on p48 of [24].

23.5 Some relations

Let us continue with the same notation and basic hypotheses as in the previous three sections. If α ∈ Φ, then let σ_{α} be the symmetry on V with vector α that maps Φ onto itself, and let λ_{α} be the corresponding linear functional on V. Thus λ_{α} is equal to 0 on the hyperplane fixed by σ_{α} and satisfies λ_{α}(α) = 2, so that σ_{α}(v) = v − λ_{α}(v) α for every v ∈ V. Suppose that ϕ is a group homomorphism from Θ into Hom_{k}(A_{0}, k), as a commutative group with respect to addition, such that ad_{w}(x) = ϕ_{α}(w) x for every w ∈ A_{0} and x ∈ A_{α}, α ∈ Φ ∪ {0}, where ϕ_{α} = ϕ(α), as before. In particular, this means that A_{0} is commutative as a Lie subalgebra of A.

Suppose that for each α ∈ Δ, x_{α} ∈ A_{α} and y_{α} ∈ A_{−α}, and put

h_{α} = [x_{α}, y_{α}]_{A} (23.5.1)

which is an element of A_{0}. We also suppose that

ϕ_{β}(h_{α}) = λ_{α}(β) · 1 (23.5.2)

in k for every α ∈ Δ and β ∈ Φ. Remember that λ_{α}(β) ∈ Z for every α, β ∈ Φ. In the situation considered in Section 22.1, we can choose x_{α}, y_{α} so that (23.5.1) satisfies (23.5.2), as in Section 22.11. These choices of x_{α} and y_{α} satisfy the other conditions mentioned in the previous section in that situation too, as before.

If α, β ∈ Δ, then [h_{α}, h_{β}]_{A} = 0, (23.5.3)

because A_{0} is commutative as a Lie algebra over k. If α ≠ β, then

[x_{α}, y_{β}]_{A} = 0, (23.5.4)

as in (23.2.2). We also have that

[h_{α}, x_{β}]_{A} = ϕ_{β}(h_{α}) x_{β} = λ_{α}(β) · x_{β} (23.5.5)

for every α, β ∈ Δ, using (23.5.2) and the fact that x_{β} ∈ A_{β}, by construction. Similarly, [h_{α}, y_{β}]_{A} = ϕ_{−β}(h_{α}) y_{β} = −λ_{α}(β) · y_{β} (23.5.6)

for every α, β ∈ Δ, because y_{β} ∈ A_{−β}. These relations, including (23.5.1), are called the Weyl relations, as in part (b) of Theorem 6 on p48 of [24].

Let α, β ∈ Δ be given, with α ≠ β. Remember that λ_{α}(β) ≤ 0, as in Section 20.2. Observe that ad_{x_{α}}^{−λ_{α}(β)+1}(x_{β}) = 0, (23.5.7)

by (23.2.5). Similarly, ad_{y_{α}}^{−λ_{α}(β)+1}(y_{β}) = 0, (23.5.8)

by (23.2.7). This corresponds to part (c) of Theorem 6 on p48 of [24], and the Weyl relations together with these additional relations are in the proposition on p96 of [14].

23.6 Isomorphisms and diagonalizability

Let k be a field, and let (A_{1}, [·, ·]_{A_{1}}) and (A_{2}, [·, ·]_{A_{2}}) be finite-dimensional Lie algebras over k. Also let T be a Lie algebra isomorphism from A_{1} onto A_{2}. Suppose that B_{1} is a Lie subalgebra of A_{1} such that every element of B_{1} is ad-diagonalizable as an element of A_{1}. This implies that B_{1} is commutative as a Lie algebra over k, as in Section 17.1. It follows that

B_{2} = T (B_{1}) (23.6.1)

has the analogous properties in A_{2}.

Let B′_{1}, B′_{2} be the duals of B_{1}, B_{2}, respectively, as vector spaces over k. The restriction T_{B} of T to B_{1} is a one-to-one linear mapping from B_{1} onto B_{2}. If α_{2} ∈ B′_{2}, then T′_{B}(α_{2}) = α_{2} ∘ T_{B} ∈ B′_{1}, (23.6.2)

as usual. This defines the dual linear mapping T′_{B} from B′_{2} into B′_{1}. More precisely, T′_{B} is a one-to-one mapping from B′_{2} onto B′_{1} in this situation.

If α_{j} ∈ B′_{j}, j = 1, 2, then let A_{j,α_{j}} be the set of x_{j} ∈ A_{j} such that

ad_{A_{j},w_{j}}(x_{j}) = [w_{j}, x_{j}]_{A_{j}} = α_{j}(w_{j}) x_{j} (23.6.3)

for every w_{j} ∈ B_{j}, as before. Thus A_{j,α_{j}} is a linear subspace of A_{j}, as a vector space over k. Observe that x_{1} ∈ A_{1}, w_{1} ∈ B_{1}, and α_{1} ∈ B′_{1} satisfy

[w_{1}, x_{1}]_{A_{1}} = α_{1}(w_{1}) x_{1} (23.6.4)

exactly when [T (w_{1}), T (x_{1})]_{A_{2}} = α_{1}(w_{1}) T (x_{1}). (23.6.5)

Let α_{2} ∈ B′_{2} be given, and let us apply this to α_{1} = T′_{B}(α_{2}). This implies that x_{1} ∈ A_{1} and w_{1} ∈ B_{1} satisfy

[w_{1}, x_{1}]_{A_{1}} = (T′_{B}(α_{2}))(w_{1}) x_{1} (23.6.6)

exactly when [T (w_{1}), T (x_{1})]_{A_{2}} = α_{2}(T (w_{1})) T (x_{1}), (23.6.7)

because (T′_{B}(α_{2}))(w_{1}) = α_{2}(T_{B}(w_{1})) = α_{2}(T (w_{1})).

Using this, it is easy to see that

T (A_{1,T′_{B}(α_{2})}) = A_{2,α_{2}} (23.6.8)

for every α_{2} ∈ B′_{2}. Put

Φ_{j,B_{j}} = {α_{j} ∈ B′_{j} : α_{j} ≠ 0 and A_{j,α_{j}} ≠ {0}} (23.6.9)

for j = 1, 2, as before. It follows that

T′_{B}(Φ_{2,B_{2}}) = Φ_{1,B_{1}}, (23.6.10)

23.7. AUTOMORPHISMS AND SL_{2}(K) 493

by (23.6.8). Remember that Φ_{j,B_{j}} has finitely many elements, as in Section 17.2.

Suppose now that k has characteristic 0. Let E_{j,Q} be the linear subspace of B′_{j}, as a vector space over Q, spanned by Φ_{j,B_{j}} for j = 1, 2, as before. Using (23.6.10), we get that T′_{B}(E_{2,Q}) = E_{1,Q}. (23.6.11)

Thus the restriction of T′_{B} to E_{2,Q} is a one-to-one mapping onto E_{1,Q} that is linear over Q.

Let E_{j,R} be the vector space over R obtained from E_{j,Q} as in Section 17.13, for j = 1, 2. More precisely, E_{j,Q} corresponds to a linear subspace of E_{j,R}, as a vector space over Q, and any basis for E_{j,Q}, as a vector space over Q, is also a basis for E_{j,R}, as a vector space over R. The restriction of T′_{B} to E_{2,Q} leads to a one-to-one mapping from E_{2,R} onto E_{1,R} that is linear over R. This mapping agrees with T′_{B} on E_{2,Q}, and in particular maps Φ_{2,B_{2}} onto Φ_{1,B_{1}}.

If Φ_{1,B_{1}} is a root system in E_{1,R}, then Φ_{2,B_{2}} is a root system in E_{2,R}. In this case, the linear mapping from E_{2,R} onto E_{1,R} obtained from T′_{B} as in the preceding paragraph is an isomorphism between these root systems. Remember that Φ_{j,B_{j}} is a root system in E_{j,R} when A_{j}, B_{j} satisfy the conditions in Section 22.1.

23.7 Automorphisms and sl_{2}(k)

Let k be a field of characteristic 0, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. If w ∈ A is ad-nilpotent, then the exponential of ad_{w} defines a Lie algebra automorphism of A, as in Section 14.11. If u ∈ A and [w, u]_{A} = 0, then it is easy to see that (exp ad_{w})(u) = u. (23.7.1)

Let x, y be elements of A, and put

h = [x, y]_{A}. (23.7.2)

Suppose that [h, x]_{A} = 2 · x, [h, y]_{A} = −2 · y. (23.7.3)

Let C be the linear span of x, y, and h in A, which is a Lie subalgebra of A that is isomorphic to sl_{2}(k), as a Lie algebra over k. Using (23.7.3), we get that x and y are ad-nilpotent as elements of A, as in Section 14.2. More precisely, this uses the adjoint representation of A.

Of course, −y is ad-nilpotent as an element of A too. Thus the exponentials of ad_{x} and − ad_{y} define Lie algebra automorphisms on A, as before. It follows that (23.7.4) θ = (exp ad_{x}) ∘ (exp −ad_{y}) ∘ (exp ad_{x})

is a Lie algebra automorphism on A too. If u ∈ A satisfies

[x, u]_{A} = [y, u]_{A} = 0, (23.7.5) then (23.7.6) (exp ad_{x})(u) = (exp −ad_{y})(u) = u,

as in (23.7.1). This implies that

θ(u) = u. (23.7.7)

Clearly ad_{x} and ad_{y} map C into itself, because C is a Lie subalgebra of A that contains x and y. The restrictions of ad_{x} = ad_{A,x} and ad_{y} = ad_{A,y} to C are the same as ad_{C,x} and ad_{C,y}, respectively, corresponding to the adjoint representation on C. It is easy to see that the restrictions of exp ad_{x} = exp ad_{A,x} and exp(− ad_{y}) = exp(− ad_{A,y}) to C are the same as exp ad_{C,x} and exp(− ad_{C,y}). This means that the restriction of θ to C is the same as

(23.7.8) (exp ad_{C,x}) ∘ (exp −ad_{C,y}) ∘ (exp ad_{C,x}).

Using this, we get that

θ(x) = −y, θ(y) = −x, θ(h) = −h. (23.7.9)

More precisely, we can take θ to be as in (23.7.8) here, as in the preceding paragraph. This permits one to obtain (23.7.9) as in Section 15.7, because C is isomorphic to sl_{2}(k) as a Lie algebra over k, with its usual basis. Note that (23.7.4) is an inner automorphism of A, as in Section 14.11.

23.8 Automorphisms and roots

Let k be a field of characteristic 0 again, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. Also let B be a Lie subalgebra of A, all of whose elements are ad-diagonalizable as elements of A. If α is an element of the dual B′ of B, as a vector space over k, then we let A_{α} be the set of x ∈ A such that [w, x]_{A} = α(w) x for every w ∈ B, as before. The set of α ∈ B′ such that α ≠ 0 and A_{α} ≠ {0} is denoted Φ_{B}, as usual.

Let α ∈ Φ_{B} be given, and suppose that x_{α} ∈ A_{α} and y_{α} ∈ A_{−α} have the property that h_{α} = [x_{α}, y_{α}]_{A} satisfies α(h_{α}) = 2. Thus [h_{α}, x_{α}]_{A} = α(h_{α}) x_{α} = 2 x_{α}, and [h_{α}, y_{α}]_{A} = −α(h_{α}) y_{α} = −2 y_{α}. In particular, x_{α} and y_{α} are adnilpotent as elements of A, as in Sections 14.2 and 17.2. This implies that −y_{α} is ad-nilpotent as an element of A, and that the exponentials of ad_{x_{α}} and − ad_{y_{α}} define Lie algebra automorphisms of A, as in Section 14.11. Note that there are such x_{α}, y_{α} in the situation considered in Section 22.1, as in Section 17.6.

Put (23.8.1) θ_{α} = (exp ad_{x_{α}}) ∘ (exp −ad_{y_{α}}) ∘ (exp ad_{x_{α}}),

which defines a Lie algebra automorphism on A. If u ∈ A satisfies [x_{α}, u]_{A} = [y_{α}, u]_{A} = 0, then θ_{α}(u) = u, (23.8.2) as in the previous section. In particular, this holds for every u ∈ B with α(u) = 0, because x_{α} ∈ A_{α} and y_{α} ∈ A_{−α}. We also have that

θ_{α}(h_{α}) = −h_{α}, (23.8.3)

as before. This implies that θ_{α}(B) = B, (23.8.4)

because B is spanned by the kernel of α and h_{α}.

Let θ_{α,B} be the restriction of θ_{α} to B, which is a one-to-one linear mapping from B onto itself. If β ∈ B′, then

θ′_{α,B}(β) = β ∘ θ_{α,B} ∈ B′, (23.8.5)

and θ′_{α,B} is a one-to-one linear mapping from B′ onto itself. As in Section 23.6,

θ_{α}(A_{θ′_{α,B}(β)}) = A_{β} (23.8.6)

for every β ∈ B′. This implies that

θ′_{α,B}(Φ_{B}) = Φ_{B}, (23.8.7)

as before.

It is easy to see that θ_{α}(u) = u − α(u) h_{α} (23.8.8)

for every u ∈ B, using (23.8.2), (23.8.3), and the fact that α(h_{α}) = 2. If β ∈ B′, then we get that θ′_{α,B}(β) = β − β(h_{α}) α. (23.8.9)

Let E_{Q} be the linear subspace of B′, as a vector space over Q, spanned by Φ_{B}, as before. Thus θ′_{α,B}(E_{Q}) = E_{Q}, (23.8.10)

as in Section 23.6, by (23.8.7).

Suppose that for each β ∈ Φ_{B}, β(h_{α}) corresponds to an element of Q with respect to the standard embedding into k, which holds in the situation considered in Section 22.1. This implies that β(h_{α}) is in the image of Q in k when β ∈ E_{Q}, by definition of E_{Q}. Let λ_{α} be the linear functional on E_{Q}, as a vector space over Q, corresponding to β ↦ β(h_{α}). If β ∈ E_{Q}, then (23.8.9) can be expressed as θ′_{α,B}(β) = β − λ_{α}(β) α. (23.8.11)

Let E_{R} be the vector space over R obtained from E_{Q} as in Sections 17.13 and 23.6, so that E_{Q} corresponds to a linear subspace of E_{R}, as a vector space over Q, and any basis for E_{Q}, as a vector space over Q, is a basis for E_{R} too, as a vector space over R. The restriction of θ′_{α,B} to E_{Q} leads to a one-to-one mapping from E_{R} onto itself that is linear over R, as in Section 23.6.

There is a natural extension of λ_{α} to a linear functional on E_{R}, that we shall denote as λ_{α} as well. The mapping on E_{R} corresponding to the restriction of θ′_{α,B} to E_{Q} can be given by the same expression as in (23.8.11), using this extension of λ_{α} to E_{R}. This defines a symmetry on E_{R} with vector α that maps Φ_{B} onto itself.

In the situation considered in Section 22.1, Φ_{B} is a root system in E_{R}, and every α ∈ Φ_{B} can be handled in this way. It follows that every element of the Weyl group of Φ_{B} corresponds to a Lie algebra automorphism of A that maps B onto itself as in Section 23.6 in this case. This corresponds to the first part of the remark on p47 of [24]. This also corresponds to some remarks on p77 of [14]. More precisely, every element of the Weyl group of Φ_{B} corresponds to an inner automorphism of A that maps B into itself in this way, because (23.8.1) is an inner automorphism of A, as in the previous section.

23.9 Automorphisms and bases for Φ_{B}

We would like to consider some conditions combining those in Section 22.12 and the previous section, and which hold in the situation described in Section 22.1, as usual. We begin as in the previous section, so that (A, [·, ·]_{A}) is a finitedimensional Lie algebra over a field k of characteristic 0. Let B, A_{α} for α ∈ B′, and Φ_{B} be as before. Remember that B is commutative as a Lie subalgebra of A, as in Section 17.1, and that A corresponds to the direct sum of the linear subspaces A_{α}, α ∈ Φ_{B} ∪ {0}, as a vector space over k, as in Section 17.2. We also have that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ B′, as before.

Of course, B ⊆ A_{0}, because B is commutative as a Lie subalgebra of A. We ask that B = A_{0}, which will be compatible with Section 22.12. This is the same as saying that B is equal to its centralizer in A, as usual.

Suppose that for each α ∈ Φ_{B}, we have elements x_{α} of A_{α} and y_{α} of A_{−α} such that h_{α} = [x_{α}, y_{α}]_{A} satisfies α(h_{α}) = 2, as in the previous section. Suppose also that A_{α}, A_{−α} are spanned by x_{α}, y_{α}, respectively, as in Section 22.12.

Let E_{Q} be the linear subapce of B′, as a vector space over Q, spanned by Φ_{B}, and let E_{R} be the vector space over R obtained from E_{Q} in the usual way, as in the previous section. Suppose that for each α, β ∈ Φ_{B}, β(h_{α}) corresponds to an element of Q with respect to the standard embedding into k, as in the previous section. If α ∈ Φ_{B}, then we get a linear functional λ_{α} on E_{Q}, as a vector space over Q, corresponding to β ↦ β(h_{α}), as before. This has a natural extension to a linear functional on E_{R}, which is denoted λ_{α} too, as before. This can be used to get a symmetry on E_{R} with vector α that maps Φ_{B} onto itself, as in the prevous section.

We would like Φ_{B} to be a root system in E_{R}. This amounts to asking that for every α, β ∈ Φ_{B}, β(h_{α}) corresponds to an integer with respect to the standard embedding into k, so that λ_{α}(β) ∈ Z. Under these conditions, we can take V = E_{R} and Φ = Φ_{B} in Section 22.12.

If Θ is the subgroup of E_{R}, as a commutative group with respect to addition, generated by Φ_{B}, then Θ is in fact a subgroup of E_{Q}, and thus of B′. Equivalently, Θ is a subgroup of the dual A′_{0} of A_{0}, as a vector space over k. The group homomorphism ϕ from Θ into A′_{0} mentioned in Section 22.12 may 23.9. AUTOMORPHISMS AND BASES FOR Φ_{B} 497

be taken to be the obvious inclusion mapping of Θ into A′_{0} here.

If Ψ_{0} ⊆ Φ_{B} ∪{0}, then we take A(Ψ_{0}) to be the linear subspace of A spanned by A_{α}, α ∈ Ψ_{0}, as in Section 22.12. Let Δ be a base for Φ_{B}, as a root system in E_{R}. This leads to the set Φ_{B}^{Δ,+} of positive roots in Φ_{B} with respect to Δ, which can be expressed as linear combinations of elements of Δ with nonnegative coefficients. Remember that A(Φ_{B}^{Δ,+} ∪ {0}) is a solvable Lie subalgebra of A, and in fact a Borel subalgebra of A, as in Section 22.12. This may be called the standard Borel subalgebra of A associated to B and Δ, as before.

If σ is an element of the Weyl group of Φ_{B}, then there is an inner automorphism θ of A that corresponds to σ as in the previous section. More precisely, θ(B) = B, and we let θ_{B} be the restriction of θ to B, which is a one-to-one linear mapping from B onto itself. If β ∈ B′, then θ′_{B}(β) = β ∘ θ_{B} ∈ B′, and θ′_{B} is a one-to-one linear mapping from B′ onto itself. As before, θ maps A_{θ′_{B}(β)} onto A_{β} for every β ∈ B′, which implies that θ′_{B} maps Φ_{B} onto itself. It follows that θ′_{B} maps E_{Q} onto itself, as in the previous section.

The restriction of θ′_{B} to E_{Q} leads to a one-to-one linear mapping from E_{R} onto itself, as before. We can choose θ so that this linear mapping on E_{R} is σ. Indeed, if α ∈ Φ_{B}, then the automorphism θ_{α} in (23.8.1) corresponds in this way to the symmetry on E_{R} with vector α that maps Φ_{B} onto itself. If σ is any element of the Weyl group of Φ, then σ can be expressed as the composition of finitely many such symmetries on E_{R}. One can take θ to be the composition of the corresponding θ_{α}’s in the other order, because the dual of a composition of linear mappings is the composition of the dual linear mappings in the other order.

If Ψ_{0} ⊆ Φ_{B} ∪ {0}, then θ′_{B}(Ψ_{0}) = σ(Ψ_{0}) ⊆ Φ_{B} ∪ {0}, and

θ(A(σ(Ψ_{0}))) = θ(A(θ′_{B}(Ψ_{0}))) = A(Ψ_{0}). (23.9.1)

Let Δ be a base for Φ_{B} again, so that σ(Δ) is a base for Φ_{B} too. Thus Φ_{B}^{σ(Δ),+} can be defined in the same way as before, and in fact

σ(Φ_{B}^{Δ,+}) = Φ_{B}^{σ(Δ),+}. (23.9.2)

This implies that θ(A(Φ_{B}^{σ(Δ),+})) = A(Φ_{B}^{Δ,+}), (23.9.3)

by (23.9.1).

Suppose that Φ_{B} is reduced as a root system in E_{R}. This implies that every base for Φ_{B} can be expressed as σ(Δ) for some σ in the Weyl group of Φ_{B}, as in Section 19.14. It follows that the standard Borel subalgebra of A associated to any base of Φ_{B} is related to A(Φ_{B}^{Δ,+} ∪ {0}) by an inner automorphism of A, as in (23.9.3). This basically corresponds to the second part of Lemma C on p84 of [14].

More precisely, that result also states that θ can be taken to be an element of the subgroup E(A) of the group Int(A) of all inner automorphisms of A defined in Section 24.3. To see this, it is enough to check that if α ∈ Φ_{B}, then the automorphism θ_{α} in (23.8.1) is an element of E(A). To get this, it is enough to verify that exp ad_{x_{α}} and exp(− ad_{y_{α}}) are elements of E(A). This reduces to showing that x_{α} and y_{α} are strongly ad-nilpotent in A, in the sense described in Section 24.3, by definition of E(A). This is easy to do, using the properties of x_{α} and y_{α} mentioned in the previous section.

23.10 Sums and diagonalizability

Let k be a field, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. Also let B be a Lie subalgebra of A such that every element of B is ad-diagonalizable as an element of A, which implies that B is commutative as a Lie algebra, as in Section 17.1. As usual, B′ denotes the dual of B, as a vector space over k, and if α ∈ B′, then A_{α} is the set of x ∈ A such that [w, x]_{A} = α(w) x for every w ∈ B. Let Φ_{B} be the set of α ∈ B′ such that α ≠ 0 and A_{α} ≠ {0}, as before. Remember that Φ_{B} has only finitely many elements, and that A corresponds to the direct sum of the linear subspaces A_{α}, α ∈ Φ_{B} ∪ {0}, as a vector space over k, as in Section 17.2.

If C is an ideal in A, then ad_{w}(C) ⊆ C for every w ∈ B, and the restrictions of ad_{w}, w ∈ B, to C are simultaneously diagonalizable, as in Section 22.9. This means that C corresponds to the direct sum of the linear subspaces A_{α} ∩ C, as a vector space over k, where α ∈ Φ_{B} ∪ {0} and A_{α} ∩ C ≠ {0}, as before.

Let C_{1}, …, C_{n} be finitely many nontrivial ideals in A, and suppose that A corresponds to their direct sum, as a vector space over k, and thus as a Lie algebra. Each C_{j} corresponds to a direct sum of subspaces of the form A_{α} ∩ C_{j}, as a vector space over k, as in the preceding paragraph. It follows that for each α ∈ Φ_{B} ∪ {0}, A_{α} corresponds to the direct sum of linear subspaces of the form A_{α} ∩ C_{j}, j = 1, …, n, as a vector space over k.

Let us suppose from now on in this section that B is equal to its centralizer in A, which means that B = A_{0}. This implies that B corresponds to the direct sum of the linear subspaces

B_{j} = B ∩ C_{j} = A_{0} ∩ C_{j}, (23.10.1)

j = 1, …, n, as in the previous paragraph. It is easy to see that every element of the centralizer of B_{j} in C_{j} is contained in the centralizer of B in A, for each j = 1, …, n. Thus

(23.10.2) the centralizer of B_{j} in C_{j} is equal to B_{j}

for every j = 1, …, n.

If j ∈ {1, …, n}, then put

(23.10.3) B̃′_{j} = {α ∈ B′ : α(w_{l}) = 0 for every w_{l} ∈ B_{l}, 1 ≤ l ≠ j ≤ n}.

This is a linear subspace of B′, and B′ corresponds to the direct sum of B̃′_{j}, 1 ≤ j ≤ n, as a vector space over k. The mapping from α ∈ B̃′_{j} to the restriction of α to B_{j} is a one-to-one linear mapping from B̃′_{j} onto the dual B′_{j} of B_{j} for each j = 1, …, n, because B corresponds to the direct sum of the B_{j}’s, 1 ≤ j ≤ n, as a vector space over k.

Let α ∈ B′ and j ∈ {1, …, n} be given, and suppose that

(23.10.4) there is a w_{j} ∈ B_{j} such that α(w_{j}) ≠ 0.

If x ∈ A, then [w_{j}, x]_{A} ∈ C_{j}, because B_{j} ⊆ C_{j}, and C_{j} is an ideal in A. If x ∈ A_{α}, then we get that α(w_{l}) x ∈ C_{j}, so that x ∈ C_{j}. Thus

A_{α} ⊆ C_{j} (23.10.5)

when (23.10.4) holds.

Suppose that there are also an l ∈ {1, …, n} and a w_{l} ∈ B_{l} such that j ≠ l and α(w_{l}) ≠ 0. This implies that A_{α} ⊆ C_{l}, as before, so that A_{α} = {0}, because C_{j} ∩ C_{l} = {0}, by hypothesis.

If α ∈ Φ_{B}, then (23.10.4) holds for some j ∈ {1, …, n}, because α ≠ 0 and B corresponds to the direct sum of the B_{j}’s. If 1 ≤ l ≤ n and j ≠ l, then it follows from the remarks in the preceding paragraph that α(w_{l}) = 0 for every w_{l} ∈ B_{l}, because A_{α} ≠ {0}. This means that α ∈ B̃′_{j}, so that

Φ_{B} ⊆ ⋃_{j=1}^{n} B̃′_{j}. (23.10.6)

Let j ∈ {1, …, n} be given, and put

Φ_{B,j} = Φ_{B} ∩ B̃′_{j}. (23.10.7)

If α ∈ Φ_{B,j}, then (23.10.4) holds, because α ≠ 0 and α ∈ B̃′_{j}. This implies that (23.10.5) holds, as before. Of course,

Φ_{B} = ⋃_{j=1}^{n} Φ_{B,j}, (23.10.8)

by (23.10.6). Note that Φ_{B,j} ∩ Φ_{B,l} = ∅ (23.10.9)

when 1 ≤ j ≠ l ≤ n, because B̃′_{j} ∩ B̃′_{l} = {0}, by construction.

Let α ∈ B̃′_{j} be given for some j ∈ {1, …, n}, and suppose that α ≠ 0, so that (23.10.4) holds. Under these conditions,

(23.10.10) A_{α} = {x_{j} ∈ C_{j} : [u_{j}, x_{j}]_{A} = α(u_{j}) x_{j} for every u_{j} ∈ B_{j}}.

More precisely, A_{α} is contained in the right side, by (23.10.5). If x_{j} ∈ C_{j} and w_{l} ∈ B_{l} for some 1 ≤ l ≠ j ≤ n, then [w_{l}, x_{j}]_{A} = 0 and α(w_{l}) = 0. This implies that the right side of (23.10.10) is contained in A_{α}, as desired.

Suppose now that k has characteristic 0, and that Φ_{B} ≠ ∅. Let E_{Q} be the linear subspace of B′, as a vector space over Q, spanned by Φ_{B}, as usual.

Similarly, if Φ_{B,j} ≠ ∅ for some j ∈ {1, …, n}, then let E_{Q,j} be the linear subspace of B̃′_{j}, as a vector space over Q, spanned by Φ_{B,j}. Thus

E_{Q,j} ⊆ E_{Q}, (23.10.11)

and it is easy to see that E_{Q} corresponds to the direct sum of the E_{Q,j}’s for these j’s, as a vector space over Q.

Let E_{R} be the vector space over R obtained from E_{Q} in the usual way, so that E_{Q} corresponds to a linear subspace of E_{R}, as a vector space over Q, and any basis for E_{Q}, as a vector space over Q, is a basis for E_{R}, as a vector space over R. Similarly, we can get a vector space E_{R,j} over R from E_{Q,j} for each j ∈ {1, …, n} such that Φ_{B,j} ≠ ∅. More precisely, E_{R,j} is a linear subspace of E_{R}, and E_{R} corresponds to the direct sum of the E_{R,j}’s for these j’s.

Suppose that Φ_{B} is a root system in E_{R}. This implies that Φ_{B,j} is a root system in E_{R,j} when Φ_{B,j} ≠ ∅. If this happens for more that one j ∈ {1, …, n}, then it follows that Φ_{B} is reducible as a root system in E_{R}. This is related to the corollary on p74 of [14], and to part of the corollary to Theorem 9 on p50 of [24].

23.11 Reducibility and diagonalizability

Let us return to the same type of situation as at the beginning of the previous section. Thus we let k be a field, and (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. We also let B be a Lie subalgebra of A such that every element of B is ad-diagonalizable as an element of A, which implies that B is commutative as a Lie algebra, as before. If α is an element of the dual B′ of B, as a vector space over k, then A_{α} is the set of x ∈ A such that [w, x]_{A} = α(w) x for every w ∈ B. We let Φ_{B} be the set of α ∈ B′ such that α ≠ 0 and A_{α} ≠ {0}, which is a finite set with the property that A corresponds to the direct sum of A_{α}, α ∈ Φ_{B} ∪ {0}, as a vector space over k.

Remember that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ B′, as in Section 17.2. Let Ξ_{B} be a subsemigroup of B′, as a commutative group with respect to addition, that contains Φ_{B} ∪ {0}. In particular, we are in the same type of situation as in Section 22.3, with Ξ = Ξ_{B} and Ψ = Φ_{B} ∪ {0}. If Ψ_{0} ⊆ Ξ_{B}, then let A(Ψ_{0}) be the subset of A_{0} consisting of finite sums of elements of A_{α}, α ∈ Ψ_{0}, as in Section 22.4. If (Ψ_{0} + Ψ_{0}) ∩ (Φ_{B} ∪ {0}) ⊆ Ψ_{0}, (23.11.1)

then A(Ψ_{0}) is a Lie subalgebra of A, as before.

Let B̂′_{1}, B̂′_{2} be complementary linear subspaces of B′, so that

B̂′_{1} ∩ B̂′_{2} = {0} (23.11.2)

and B̂′_{1} + B̂′_{2} = B′. (23.11.3)

23.11. REDUCIBILITY AND DIAGONALIZABILITY 501

Suppose that Φ_{B} ⊆ B̂′_{1} ∪ B̂′_{2}, (23.11.4) and put Φ_{B,j} = Φ_{B} ∩ B̂′_{j} (23.11.5) for j = 1, 2. Note that Φ_{B} = Φ_{B,1} ∪ Φ_{B,2} and Φ_{B,1} ∩ Φ_{B,2} = ∅, (23.11.6) by (23.11.2). If α ∈ Φ_{B,1} and β ∈ Φ_{B,2}, then α + β ∉ Φ_{B} ∪ {0}, (23.11.7) because α + β ∉ B̂′_{1} ∪ B̂′_{2}. It follows that [A_{α}, A_{β}] = {0} (23.11.8) in this case.

We are now in a situation like one considered in Section 22.14, with Φ = Φ_{B} and Φ_{j} = Φ_{B,j} for j = 1, 2. More precisely, (22.14.7), (22.14.11), (22.14.22), and (22.14.25) hold under these conditions, as in the preceding paragraph. In particular, A(Φ_{B,j} ∪ {0}) is a Lie subalgebra of A for j = 1, 2, as before. Let A^{j} be the Lie subalgebra of A generated by A(Φ_{B,j}) for j = 1, 2, as before, which is contained in A(Φ_{B,j} ∪ {0}). Remember that [A^{1}, A^{2}] = {0}, (23.11.9) and that A^{j} is an ideal in A for j = 1, 2, as a Lie algebra over k.

Put

(23.11.10)  B_{1} = {w ∈ B : β(w) = 0 for every β ∈ B̂′_{2}},
(23.11.11)  B_{2} = {w ∈ B : β(w) = 0 for every β ∈ B̂′_{1}}.

These are complementary linear subspaces of B, because B̂′_{1}, B̂′_{2} are complementary linear subspaces of B′, by hypothesis. Observe that [B_{1}, A_{β}] = {0} (23.11.12) for every β ∈ Φ_{B,2}, and similarly that [B_{2}, A_{β}] = {0} (23.11.13) for every β ∈ Φ_{B,1}, by the definition of A_{β}. Equivalently, this means that [B_{1}, A(Φ_{B,2})] = [B_{2}, A(Φ_{B,1})] = {0}, (23.11.14) which is the same as saying that A(Φ_{B,1}) ⊆ C_{A}(B_{2}), A(Φ_{B,2}) ⊆ C_{A}(B_{1}). (23.11.15)

It follows that A^{1} ⊆ C_{A}(B_{2}), A^{2} ⊆ C_{A}(B_{1}), (23.11.16)

because the centralizer C_{A}(E) of E ⊆ A is a Lie subalgebra of A.

Suppose from now on in this section that B is its own centralizer in A, so that B = A_{0}. In particular, this means that A_{0} is commutative as a Lie algebra over k. This implies that A^{1} ∩ A^{2} is contained in the center Z(A) of A as a Lie algebra, as in Section 22.14.

If j ∈ {1, 2} and α ∈ Φ_{B,j}, then

[A_{α}, A_{−α}] ⊆ A^{j} ∩ A_{0} = A^{j} ∩ B. (23.11.17)

Note that A_{−α} = {0} unless −α ∈ Φ_{B,j} too. As in Section 22.14, A^{j} is the same as the set of elements of A that can be expressed as finite sums of elements of A(Φ_{B,j}) and [A_{α}, A_{−α}] with α ∈ Φ_{B,j}, because A_{0} is commutative as a Lie algebra over k.

Suppose for the moment that α ∈ Φ_{B,1}, β ∈ Φ_{B,2}, x_{α} ∈ A_{α}, y_{α} ∈ A_{−α}, and z_{β} ∈ A_{β}. This implies that

[x_{α}, z_{β}]_{A} = [y_{α}, z_{β}]_{A} = 0, (23.11.18)

because ±α + β ∉ B̂′_{1} ∪ B̂′_{2}. It follows that

[[x_{α}, y_{α}]_{A}, z_{β}]_{A} = 0, (23.11.19)

by the Jacobi identity. We also have that [x_{α}, y_{α}]_{A} ∈ A_{0} = B, so that

[[x_{α}, y_{α}]_{A}, z_{β}]_{A} = β([x_{α}, y_{α}]_{A}) z_{β}, (23.11.20)

by definition of A_{β}. If z_{β} ≠ 0, then we get that

β([x_{α}, y_{α}]_{A}) = 0. (23.11.21)

Of course, we can always choose z_{β} to be a nonzero element of A_{β}, because β ∈ Φ_{B,2} ⊆ Φ_{B}. This shows that (23.11.21) holds for every x_{α} ∈ A_{α} and y_{α} ∈ A_{−α} when β ∈ Φ_{B,2}. This argument works as well when α ∈ Φ_{B,2} and β ∈ Φ_{B,1}.

Let us suppose from now on in this section that Z(A) = {0}. This implies that the linear span of Φ_{B} in B′ is equal to B′, as in Section 17.5. It follows that the linear span of Φ_{B,j} in B̂′_{j} is equal to B̂′_{j} for j = 1, 2 in this situation. This means that (23.11.21) holds for every x_{α} ∈ A_{α} and y_{α} ∈ A_{−α} when α ∈ Φ_{B,j} and β ∈ B̂′_{2}, and when α ∈ Φ_{B,2} and β ∈ B̂′_{1}. This is the same as saying that

[A_{α}, A_{−α}] ⊆ B_{j} (23.11.22)

for j = 1, 2 and α ∈ Φ_{B,j}.

Using (23.11.22), we get that

A^{j} ⊆ A(Φ_{B,j}) + B_{j} (23.11.23) for j = 1, 2. One can check that A(Φ_{B,j})+B_{j} is a Lie subalgebra of A for j = 1, 2, because B is commutative as a Lie subalgebra of A, and [B, A(Φ_{B,j})] ⊆ A(Φ_{B,j}) for j = 1, 2. We also have that

[A(Φ_{B,1}) + B_{1}, A(Φ_{B,2}) + B_{2}] = {0}, (23.11.24)

because of (23.11.8), (23.11.14), and the fact that B is commutative as a Lie algebra. It is easy to see that

A(Φ_{B,1}) + B_{1}, A(Φ_{B,2}) + B_{2} (23.11.25)

are complementary linear subspaces of A, because B_{1}, B_{2} are complementary linear subspaces of B = A_{0}. It follows that these are ideals in A, and that A corresponds to their direct sum, as a Lie algebra over k.

This is related to the proposition on p73 of [14], and to part of the corollary to Theorem 9 on p50 of [24], as before.

Strong ad-nilpotence and automorphisms

24.1 Invariant subspaces and nilpotent vectors

Let k be a field, let V be a vector space over k, and let T be a linear mapping from V into itself. Remember that (24.1.1) V_{0} = {v ∈ V : T^{l}(v) = 0 for some l ∈ Z_{+}} is a linear subspace of V, as in Section 10.7. Clearly T (V_{0}) ⊆ V_{0}. (24.1.2) In fact, T^{−1}(V_{0}) = V_{0}. (24.1.3) More precisely, if v ∈ V and T (v) ∈ V_{0}, then it is easy to see that v ∈ V_{0}.

Let W be a linear subspace of V, and suppose that T (W) ⊆ W. (24.1.4) Put (24.1.5) W_{0} = W ∩ V_{0} = {w ∈ W : T^{l}(w) = 0 for some l ∈ Z_{+}}, which is a linear subspace of W. Of course, T (W_{0}) ⊆ W_{0}, (24.1.6) as in (24.1.2). Similarly, W ∩ T^{−1}(W_{0}) = W_{0}, (24.1.7) as in (24.1.3).

Let q be the canonical quotient mapping from V onto the quotient vector space V/W. As usual, there is a unique linear mapping T_{V/W} from V/W into itself such that T_{V/W} ∘ q = q ∘ T (24.1.8)

on V. This implies that (T_{V/W})^{l} ∘ q = q ∘ T^{l} (24.1.9)

on V for every positive integer l.

Put

(24.1.10) (V/W)_{0} = {z ∈ V/W : (T_{V/W})^{l}(z) = 0 for some l ∈ Z_{+}},

which is a linear subspace of V/W, as before. If v ∈ V, then

(T_{V/W})^{l}(q(v)) = q(T^{l}(v)) (24.1.11)

for every positive integer l, as in (24.1.9). Thus (24.1.10) consists of the q(v) with v ∈ V such that (24.1.11) is equal to 0 for some l ≥ 1. Equivalently, this means that T^{l}(v) ∈ W (24.1.12)

for some l ≥ 1. In particular,

q(V_{0}) ⊆ (V/W)_{0}. (24.1.13)

Let q_{0} be the canonical quotient mapping from W onto the quotient vector space W/W_{0}. As before, there is a unique linear mapping T_{W/W_{0}} from W/W_{0} into itself such that T_{W/W_{0}} ∘ q_{0} = q_{0} ∘ T (24.1.14)

on W. Thus (T_{W/W_{0}})^{l} ∘ q_{0} = q_{0} ∘ T^{l} (24.1.15)

on W for every positive integer l, as before. Observe that the kernel of T_{W/W_{0}} is trivial in W/W_{0}, because of (24.1.7).

24.2 Finite-dimensional spaces

Let us continue with the same notation and hypotheses as in the previous section. Suppose in addition that the dimension of W is finite, as a vector space over k. Of course, this implies that the dimension of W/W_{0} is finite. Remember that the kernel of T_{W/W_{0}} is trivial in W/W_{0}. It follows that T_{W/W_{0}} maps W/W_{0} onto itself.

If w ∈ W and l ∈ Z_{+}, then we get that there is a u_{l} ∈ W such that

(T_{W/W_{0}})^{l}(q_{0}(u_{l})) = q_{0}(w). (24.2.1)

Equivalently, this means that

q_{0}(T^{l}(u_{l})) = q_{0}(w). (24.2.2)

506 CHAPTER 24. STRONG AD-NILPOTENCE AND AUTOMORPHISMS

This is the same as saying that

T^{l}(u_{l}) − w ∈ W_{0}. (24.2.3)

Let v ∈ V be given, and suppose that q(v) ∈ (V/W)_{0}. This implies that T^{l}(v) ∈ W for some positive integer l, as in (24.1.12). It follows that there is a u_{l} ∈ W such that T^{l}(u_{l}) − T^{l}(v) ∈ W_{0}, (24.2.4)

as in the preceding paragraph. Equivalently, this means that

T^{l}(v − u_{l}) ∈ W_{0}. (24.2.5)

Hence there is a positive integer r such that

T^{r}(T^{l}(v − u_{l})) = 0, (24.2.6)

by the definition (24.1.5) of W_{0}.

Thus T^{l+r}(v − u_{l}) = 0, so that

v − u_{l} ∈ V_{0}. (24.2.7)

Of course, q(v) = q(v − u_{l}), because u_{l} ∈ W. It follows that

q(v) = q(v − u_{l}) ∈ q(V_{0}). (24.2.8)

Combining this with (24.1.13), we get that

q(V_{0}) = (V/W)_{0} (24.2.9)

under these conditions.

Alternatively, suppose that V has finite dimension, as a vector space over k. It is well known that the characteristic polynomial of T on V is equal to the product of the characteristic polynomial of the restriction of T to W and the characteristic polynomial of T_{V/W} on V/W. In particular, the order of vanishing of the characteristic polynomial of T on V at 0 is the same as the sum of the orders of vanishing of the characteristic polynomials of the restriction of T to W and T_{V/W} at 0. This means that the dimension of V_{0} is equal to the sum of the dimensions of W_{0} and (V/W)_{0}, as vector spaces over k. Note that the kernel of the restriction of q to V_{0} is equal to W_{0}, because W_{0} = W ∩ V_{0}, as in (24.1.5). This implies that the dimension of V_{0} is equal to the sum of the dimensions of W_{0} and q(V_{0}) too. Hence the dimension of q(V_{0}) is equal to the dimension of (V/W)_{0}. This permits one to obtain (24.2.9) from (24.1.13).

24.3 Strong ad-nilpotence

Let k be a field of characteristic 0, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. An element x of A is said to be strongly ad-nilpotent if there are a y ∈ A, α ∈ k, and l ∈ Z_{+} such that α ≠ 0 and

(ad_{y} − α I)^{l}(x) = 0, (24.3.1) where I is the identity mapping on A. This definition is given on p82 of [14], where k is also taken to be algebraically closed. In that case, if x is strongly ad-nilpotent, then x is ad-nilpotent in A, by the remarks in Section 18.1, as in [14].

Otherwise, let k_{1} be an algebraically closed field that contains k. We may as well identify A with k^{n} for some positive integer n as a vector space over k, using a basis for A. Let A_{1} be k_{1}^{n}, as a vector space over k_{1}. We can extend the Lie bracket on A to A_{1}, so that A_{1} becomes a Lie algebra over k_{1}. If x ∈ A is strongly ad-nilpotent as an element of A, then x is strongly ad-nilpotent as an element of A_{1} too. This implies that x is ad-nilpotent as an element of A_{1}, as before. It follows that x is ad-nilpotent as an element of A as well. Another approach to this will be given in Section 24.5.

If A is semisimple, and x ∈ A is ad-nilpotent, then x is strongly ad-nilpotent, as in Sections 14.1 and 14.2.

Let N (A) be the set of x ∈ A such that x is strongly ad-nilpotent in A. If B is a Lie subalgebra of A, then it is easy to see that

N (B) ⊆ N (A), (24.3.2)

as on p82 of [14]. Note that N (A) is invariant under Lie algebra automorphisms of A.

If x ∈ A is ad-nilpotent in A, then the exponential exp ad_{x} of ad_{x} on A defines a Lie algebra automorphism of A, as in Section 14.11. Remember that Int A denotes the subgroup of the automorphism group of A generated by these automorphisms. Let E(A) be the subgroup of Int A generated by exponentials of strongly ad-nilpotent elements of A. One can check that E(A) is a normal subgroup of the group of all Lie algebra automorphisms of A, because N (A) is invariant under Lie algebra automorphisms of A, as on p82 of [14]. If A is semisimple, then E(A) = Int A, because N (A) contains all ad-nilpotent elements of A, as before.

Let B be a Lie subalgebra of A, and let x be an element of B. Thus ad_{B,x} is the same as the restriction of ad_{A,x} to B. Suppose that x ∈ N (B), so that x ∈ N (A), and ad_{A,x} is nilpotent on A. Note that exp ad_{B,x} is the same as the restriction of exp ad_{A,x} to B.

Let E(A, B) be the subgroup of E(A) generated by elements of the form exp ad_{A,x} with x ∈ N (B). The restriction of every element of E(A, B) to B is an element of E(B). It is easy to see that every element of E(B) can be obtained from an element of E(A, B) in this way. This corresponds to some remarks on p82 of [14].

24.4 Strong ad-nilpotence and homomorphisms

Let k be a field of characteristic 0, and let (A, [·, ·]_{A}), (C, [·, ·]_{C}) be finitedimensional Lie algebras over k. Also let ψ be a Lie algebra homomorphism from A into C. If y ∈ A, then

ψ ∘ ad_{A,y} = ad_{C,ψ(y)} ∘ ψ, (24.4.1) as in Section 14.8. This implies that ψ ∘ (ad_{A,y} − α I_{A}) = (ad_{C,ψ(y)} − α I_{C}) ∘ ψ (24.4.2) for every α ∈ k, where I_{A}, I_{C} are the identity mappings on A, C, respectively. It follows that ψ ∘ (ad_{A,y} − α I_{A})^{l} = (ad_{C,ψ(y)} − α I_{C})^{l} ∘ ψ (24.4.3) for every l ∈ Z_{+}.

Suppose that x ∈ A is strongly ad-nilpotent in A, so that there are y ∈ A, α ∈ k, and l ∈ Z_{+} such that α ≠ 0 and (ad_{A,y} − α I_{A})^{l}(x) = 0, (24.4.4) as in (24.3.1). This implies that (ad_{C,ψ(y)} − α I_{C})^{l}(ψ(x)) = 0, (24.4.5) by (24.4.3). This means that ψ(x) is strongly ad-nilpotent in C, so that ψ(N (A)) ⊆ N (C). (24.4.6) In particular, x and ψ(x) are ad-nilpotent in A and C, respectively, as before. We also get that (24.4.7) ψ ∘ (exp ad_{A,x}) = (exp ad_{C,ψ(x)}) ∘ ψ, as in Section 14.12.

Let y ∈ A and α ∈ k be given again, and observe that (24.4.8) {x ∈ A : (ad_{A,y} − α I_{A})^{l}(x) = 0 for some l ∈ Z_{+}} is a linear subspace of A. This corresponds to (24.1.1), with V = A and T = ad_{A,y} −α I_{A}. Similarly, (24.4.9) {z ∈ C : (ad_{C,ψ(y)} − α I_{C})^{l}(z) = 0 for some l ∈ Z_{+}} is a linear subspace of C. Clearly ψ maps (24.4.8) into (24.4.9), because of (24.4.3). If, in the notation of Section 24.1, we take W to be the kernel of ψ, then (24.1.4) holds, because the kernel of ψ is an ideal in A.

Suppose now that ψ maps A onto C, so that V/W corresponds to C, and the quotient mapping q from V onto V/W corresponds to ψ. Note that the linear mapping T_{V/W} induced on V/W by T as in Section 24.1 corresponds to ad_{C,ψ(y)} −α I_{C}, and that (24.1.10) corresponds to (24.4.9). In this case, ψ maps (24.4.8) onto (24.4.9), as in (24.2.9). It follows that ψ(N (A)) = ψ(N (C)) (24.4.10) under these conditions. This corresponds to some remarks on p82 of [14].

If σ_{C} ∈ E(C), then there is a σ_{A} ∈ E(A) such that ψ ∘ σ_{A} = σ_{C} ∘ ψ. (24.4.11) This can be obtained from (24.4.7) and (24.4.10). This corresponds to the lemma on p82 of [14].

24.5 Another criterion for nilpotence

Let k be a field of characteristic 0, let A be an associative algebra over k, and let δ be a derivation on A. If λ ∈ k, then put

(24.5.1) E_{λ}(δ) = {a ∈ A : (δ − λ I)^{l}(a) = 0 for some l ∈ Z_{+}},

as in Sections 10.7 and 10.9, where I is the identity mapping on A. If b ∈ E_{λ}(δ) and c ∈ E_{μ}(δ) for some μ ∈ k, then b c ∈ E_{λ+μ}(δ), as in Section 10.9. It follows that b^{j} ∈ E_{j·λ}(δ) (24.5.2)

for every positive integer j.

Suppose that λ ≠ 0, so that the elements of k of the form j · λ, j ∈ Z_{+}, are distinct, because k has characteristic 0. If, for each j ∈ Z_{+}, we have that b^{j} ≠ 0, then the b^{j}’s are linearly independent in A, as in Section 10.7. If A has finite dimension as a vector space over k, then it follows that b^{j} = 0 for some j ≥ 1. This extends the remarks at the beginning of Section 14.2.

If a ∈ A, then δ_{a}(x) = [a, x] = a x − x a defines a derivation on A. Thus if b ∈ E_{λ}(δ_{a}) for some nonzero λ ∈ k and A has finite dimension as a vector space over k, then b is nilpotent in A.

Remember that A may be considered as a Lie algebra over k, with respect to the commutator bracket corresponding to multiplication in A. If b ∈ A is strongly ad-nilpotent in A as a Lie algebra, then b is nilpotent in A, as in the preceding paragraph.

Let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. If x ∈ A is strongly ad-nilpotent, then ad_{x} is strongly ad-nilpotent as an element of the Lie algebra of linear mappings from A into itself. This follows from (24.4.6), using the adjoint representation of A. This implies that ad_{x} is nilpotent as a linear mapping on A, as in the previous paragraph.

24.6 Nilpotence and automorphisms

Let k be a field, and let f(t) = ∑_{j=0}^{n} f_{j} t^{j} be a polynomial function of t ∈ k with coefficients in a vector space over k. If f_{j} ≠ 0 for some j, then f(t) = 0 for at most n t ∈ k, by standard arguments.

Suppose from now on in this section that k has characteristic 0. Let A be an associative algebra over k with a multiplicative identity element e, and let a be a nilpotent element of A. Thus exp(t a) defines a polynomial function of t ∈ k with values in A. If a ≠ 0, then exp(t a) = e for only finitely many t ∈ k, as in the previous paragraph.

Let (A, [·, ·]_{A}) be a Lie algebra over k. If x ∈ A is ad-nilpotent, then exp(t ad_{x}) defines a polynomial function of t ∈ k with values in the algebra of linear mappings from A into itself. If ad_{x} ≠ 0 as a linear mapping from A into itself, then exp(t ad_{x}) is equal to the identity mapping on A for only finitely many t ∈ k, as before.

Let y ∈ A be given, and for each α ∈ k, let A_{y,α} be the set of x ∈ A such that (ad_{y} −α I)^{l}(x) = 0 for some positive integer l, as in Section 18.1. Remember that A_{y,α} ≠ {0} exactly when α is an eigenvalue of ad_{y} on A. In particular, if y is ad-nilpotent as an element of A, then A_{y,α} = {0} when α ≠ 0.

If A is nilpotent as a Lie algebra over k, then every y ∈ A is ad-nilpotent, and it follows that 0 is the only element of A that is strongly ad-nilpotent. This implies that E(A) contains only the identity mapping on A, as in Exercise 1 on p87 of [14].

Suppose now that k is algebraically closed, and that A has finite dimension, as a vector space over k. If y ∈ A, then A corresponds to the direct sum of the linear subspaces A_{y,α}, where α ∈ k is an eigenvalue of ad_{y} on A, as a vector space over k, as in Section 18.1.

If ad_{y} is not nilpotent on A, then it follows that ad_{y} has a nonzero eigenvalue α ∈ k. This means that there is an x ∈ A such that x ≠ 0 and ad_{y}(x) = α x, so that x is strongly ad-nilpotent in particular. Note that ad_{x}(y) = − ad_{y}(x) = −α x ≠ 0, so that ad_{x} ≠ 0. Of course, t ad_{x} is strongly ad-nilpotent for every t ∈ k too, so that exp(t ad_{x}) ∈ E(A) (24.6.1)

for every t ∈ k.

If A is not nilpotent as a Lie algebra, then there is a y ∈ A that is not ad-nilpotent, by Engel’s theorem. Under these conditions, we get that E(A) is nontrivial, as in Exercise 1 on p87 of [14].

24.7 Conjugacy and solvability

Let k be an algebraically closed field of characteristic 0, and let (A, [·, ·]_{A}) be a finite-dimensional solvable Lie algebra over k. If C_{1}, C_{2} are Cartan subalgebras of A, then there is an element of the group E(A) defined in Section 24.3 that maps C_{1} onto C_{2}. This is the theorem on p82 of [14], which is related to Theorem 2 on p12 of [24].

Of course, if A is nilpotent as a Lie algebra, then A may be considered as a Cartan subalgebra of itself. More precisely, A is the only Cartan subalgebra of itself in this case, because Cartan subalgebras are maximal among nilpotent Lie subalgebras, as mentioned at the beginning of Section 18.11.

To prove the theorem, one can use induction on the dimension of A. We may also suppose that A is not nilpotent as a Lie algebra over k, as in the preceding paragraph. In particular, this means that the dimension of A is at least two.

Because A ≠ {0} is solvable as a Lie algebra, A has nontrivial ideals that are commutative as Lie algebras. Let A_{0} be a nontrivial ideal in A that is commutative as a Lie algebra, and with minimal dimension among such ideals.

Let q_{0} be the natural quotient mapping from A onto A/A_{0}, and note that A/A_{0} is a solvable Lie algebra over k, with dimension strictly less than the dimension of A. Under these conditions, q_{0}(C_{1}) and q_{0}(C_{2}) are Cartan subalgebras of A/A_{0}, as in Section 18.11.

24.8. THE CASE WHERE A = B_{2} 511

Our induction hypothesis implies that there is an element σ_{0} of E(A/A_{0}) such that σ_{0}(q_{0}(C_{1})) = q_{0}(C_{2}). (24.7.1) This leads to an element σ of E(A) such that q_{0} ∘ σ = σ_{0} ∘ q_{0}, (24.7.2) as in Section 24.4.

Put B_{1} = q_{0}^{−1}(q_{0}(C_{1})), B_{2} = q_{0}^{−1}(q_{0}(C_{2})). (24.7.3) These are Lie subalgebras of A, with C_{1} ⊆ B_{1}, C_{2} ⊆ B_{2}. (24.7.4) It is easy to see that σ(B_{1}) = B_{2}, (24.7.5) using (24.7.1) and (24.7.2).

Note that C_{1}, C_{2} may be considered as Cartan subalgebras of B_{1}, B_{2}, respectively, as in Section 18.11. It follows that σ(C_{1}) may be considered as a Cartan subalgebra of B_{2}, by (24.7.5).

Suppose for the moment that B_{2} ≠ A. (24.7.6) In this case, our induction hypothesis implies that there is an element τ_{2} of E(B_{2}) such that τ_{2}(σ(C_{1})) = C_{2}. (24.7.7)

As in Section 24.3, there is an element τ of E(A, B_{2}) ⊆ E(A) whose restriction to B_{2} is equal to τ_{2}. In particular, this implies that τ(σ(C_{1})) = C_{2}. (24.7.8) This completes the proof of the theorem in this case, because τ ∘ σ ∈ E(A).

24.8 The case where A = B_{2}

Let us continue with the same notation and hypotheses as in the previous section, except that we suppose now that A = B_{2}. In this case, σ(B_{1}) = B_{2} = A, so that B_{1} = A as well. Equivalently, this means that C_{1} + A_{0} = C_{2} + A_{0} = A. (24.8.1)

If x ∈ A and α ∈ k, then let A_{x,α} be the set of y ∈ A such that (ad_{x} − α I)^{l}(y) = 0 (24.8.2) for some positive integer l, as in Section 18.1, and where I is the identity mapping on A. This is a linear subspace of A, and A corresponds to the direct sum of A_{x,α}, as a vector space over k, where α ∈ k is an eigenvalue of ad_{x} on A, as before, because k is algebraically closed, by hypothesis.

Remember that A_{0} is an ideal in A, so that ad_{x}(A_{0}) ⊆ A_{0}. If α ∈ k, then A_{0} ∩ A_{x,α} consists of y ∈ A_{0} such that (24.8.2) holds for some l ∈ Z_{+}, which is the analogue of A_{x,α} for the restriction of ad_{x} to A_{0}. As before, A_{0} corresponds to the direct sum of A_{0} ∩ A_{x,α}, where α ∈ k is an eigenvalue of ad_{x} on A_{0}, as a vector space over k.

Remember that [A_{x,α}, A_{x,β}] ⊆ A_{x,α+β} for every α, β ∈ k, as in Section 18.1. This implies that [A_{x,0}, A_{0} ∩ A_{x,α}] ⊆ A_{0} ∩ A_{x,α} (24.8.3)

for every α ∈ k, because A_{0} is an ideal in A.

Because C_{2} is a Cartan subalgebra of A, there is an x ∈ A such that

C_{2} = A_{x,0}, (24.8.4)

as in Section 18.8. In this case, one can check that

(24.8.5) A_{0} ∩ A_{x,α} is an ideal in A

for every α ∈ k. More precisely, this uses (24.8.1), (24.8.3), and the fact that A_{0} is commutative as a Lie subalgebra of A.

If A_{0} ∩ A_{x,α} ≠ {0} (24.8.6)

for some α ∈ k, then A_{0} ∩ A_{x,α} = A_{0}, (24.8.7)

because A_{0} is supposed to be minimal among nontrivial ideals that are commutative as Lie subalgebras of A. Note that (24.8.6) holds for some α ∈ k, because A_{0} ≠ {0}, and A_{0} corresponds to the direct sum of A_{0} ∩ A_{x,α} over some α ∈ k, as a vector space over k.

If (24.8.7) holds with α = 0, then A_{0} ⊆ A_{x,0} = C_{2}, which means that A = C_{2}, by (24.8.1). This would imply that A is nilpotent as a Lie algebra, by definition of a Cartan subalgebra. However, A is not supposed to be nilpotent, as in the previous section, and so we get that A_{0} ∩ A_{x,0} = {0}.

It follows that (24.8.7) holds for some nonzero α ∈ k, so that A_{0} ⊆ A_{x,α}. This implies that A_{0} = A_{x,α}, (24.8.8)

because of (24.8.1), (24.8.4), and the fact that A corresponds to the direct sum of A_{x,β} over some β ∈ k, as a vector space over k.

Using (24.8.1), we get that x can be expressed as

x = y + z, (24.8.9)

with y ∈ C_{1} and z ∈ A_{0}. More precisely, z ∈ A_{x,α}, by (24.8.8).

24.8. THE CASE WHERE A = B_{2} 513

It is easy to see that ad_{x}(A_{x,α}) ⊆ A_{x,α}, by the definition of A_{x,α}. Because z ∈ A_{x,α} and α ≠ 0, one can express z as

z = ad_{x}(w), (24.8.10)

where w is a linear combination of (ad_{x})^{j}(z), for finitely many positive integers j. In particular, this implies that w ∈ A_{x,α}.

Equivalently, w ∈ A_{0}, so that ad_{w}(A) ⊆ A_{0}, because A_{0} is an ideal in A. This implies that

ad_{w} ∘ ad_{w} = 0 (24.8.11)

on A, because A_{0} is commutative as a Lie subalgebra of A.

Thus

exp(ad_{w}) = I + ad_{w}, (24.8.12)

so that

(24.8.13) (exp(ad_{w}))(x) = x + ad_{w}(x) = x − ad_{x}(w) = x − z = y.

Put

C = (exp(ad_{w}))(C_{2}), (24.8.14)

which is a Cartan subalgebra of A, because C_{2} is a Cartan subalgebra of A, and exp(ad_{w}) is an automorphism of A. One can check that

C = A_{y,0}, (24.8.15)

using (24.8.4) and (24.8.13).

Remember that y ∈ C_{1}, by construction, so that ad_{y} maps C_{1} into itself. More precisely, ad_{y} is nilpotent on C_{1}, because C_{1} is a Cartan subalgebra of A, and thus nilpotent as a Lie algebra. It follows that

C_{1} ⊆ C, (24.8.16)

by (24.8.15).

This implies that

C_{1} = C, (24.8.17)

by the characterization of Cartan subalgebras as minimal Engel subalgebras, as in Section 18.8. Alternatively, one can use the fact that Cartan subalgebras are maximal among nilpotent Lie subalgebras of A, as in Section 18.11.

Remember that w ∈ A_{x,α}, so that w is strongly ad-nilpotent in A, and exp(ad_{w}) ∈ E(A). It follows that there is an element of E(A) that maps C_{1} onto C_{2}, by (24.8.14) and (24.8.17). This completes the proof of the theorem stated at the beginning of the previous section.

24.9 Conjugacy in arbitrary Lie algebras

Let k be an algebraically closed field of characteristic 0 again, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. If B_{1}, B_{2} are Borel subalgebras of A, then the theorem on p84 of [14] states that

(24.9.1) there is an element of E(A) that maps B_{1} onto B_{2}.

This is closely related to Theorem 5 on p48 of [24].

If C_{1}, C_{2} are Cartan subalgebras of A, then the corollary on p84 states that

(24.9.2) there is an element of E(A) that maps C_{1} onto C_{2}.

This is closely related to Theorem 2 on p12 of [24].

To obtain the corollary from the theorem, remember that Cartan subalgebras are nilpotent, and solvable in particular. This implies that there are Borel subalgebras B_{1}, B_{2} of A that contain C_{1}, C_{2}, respectively. Using the theorem, we get σ ∈ E(A) such that σ(B_{1}) = B_{2}. It follows that

σ(C_{1}) ⊆ B_{2}. (24.9.3)

Of course, σ(C_{1}) is a Cartan subalgebra of A as well. As in Section 18.11, σ(C_{1}) and C_{2} may also be considered as Cartan subalgebras of B_{2}. Because B_{2} is solvable as a Lie algebra over k, there is an element of E(B_{2}) that maps σ(C_{1}) onto C_{2}, as in Section 24.7.

Every element of E(B_{2}) can be obtained as the restriction of an element of E(A, B_{2}) ⊆ E(A) to B_{2}, as in Section 24.3. This implies that there is an element of E(A) that maps σ(C_{1}) onto C_{2}. It follows that there is an element of E(A) that maps C_{1} onto C_{2}, as desired.

24.9.1 The beginning of the proof of the theorem

Let us now begin the proof of the theorem. Note that the theorem holds trivially when A is solvable as a Lie algebra, in which case A is the only Borel subalgebra of itself.

To prove the theorem, one uses induction on the dimension of A as a vector space over k. If the dimension of A is one, then the theorem is trivial, as in the preceding paragraph.

Remember that Rad A denotes the solvable radical of A, as in Section 9.4. If B_{1}, B_{2} are Borel subalgebras of A, then

Rad A ⊆ B_{1}, B_{2}, (24.9.4)

as in Section 22.7.

Remember that Rad A is an ideal in A, and let q be the natural quotient mapping from A onto A/ Rad(A). Note that q(B_{1}), q(B_{2}) are Borel subalgebras of A/ Rad A, as in Section 22.7. We also have that

B_{j} = q^{−1}(q(B_{j})) for j = 1, 2, (24.9.5) by (24.9.4).

If Rad A ≠ {0}, then the dimension of A/ Rad A is less than the dimension of A, and our induction hypothesis implies that there is an element σ of E(A/ Rad A) such that σ(q(B_{1})) = q(B_{2}). (24.9.6)

This leads to an element σ_{A} of E(A) such that

q ∘ σ_{A} = σ ∘ q, (24.9.7)

as in Section 24.4.

It is easy to see that σ_{A}(B_{1}) = B_{2}, (24.9.8)

using (24.9.5), (24.9.6), and (24.9.7). This completes the proof when Rad A ≠ {0}.

Thus we may suppose from now on that Rad A = {0}, so that A is semisimple as a Lie algebra over k. Of course, we may suppose that A ≠ {0} too.

24.9.2 Standard Borel subalgebras of A

Let A_{0} be a toral subalgebra of A, as in Section 17.1, so that every element of A_{0} is ad-diagonalizable as an element of A. Remember that A_{0} is commutative as a Lie algebra, as before. More precisely, let A_{0} be a toral subalgebra of A that is maximal with respect to inclusion. Note that A_{0} ≠ {0}, as in Section 17.1, because A ≠ {0}, and using our hypotheses on k. We also have that the centralizer of A_{0} in A is equal to A_{0}, as in Section 17.4.

Let A′_{0} be the dual of A_{0}, as a vector space over k, as usual. If α ∈ A′_{0}, then let A_{α} be the set of x ∈ A such that ad_{w}(x) = α(w) x for every w ∈ A_{0}. This is the same as A_{0} when α = 0, because A_{0} is its own centralizer in A, as in the preceding paragraph.

Let Φ = Φ_{A_{0}} be the set of α ∈ A′_{0} such that α ≠ 0 and A_{α} ≠ {0}. Remember that A corresponds to the direct sum of the subspaces A_{α}, α ∈ Φ ∪ {0}, as a vector space over k, as in Section 17.2. If α, β ∈ A′_{0}, then [A_{α}, A_{β}] ⊆ A_{α+β}, as before.

If α ∈ Φ, then A_{α} has dimension one as a vector space over k, as in Section 17.7. We may choose x_{α} ∈ A_{α} and y_{α} ∈ A_{−α} so that h_{α} = [x_{α}, y_{α}]_{A} satisfies α(h_{α}) = 2, as in Section 17.6.

Let E_{Q} be the linear subspace of A′_{0}, as a vector space over Q, spanned by Φ, as in Section 17.12. Let E_{R} be the corresponding vector space over R, as in Section 17.13. Remember that Φ is a reduced root system in E_{R}, as in Section 22.1.

If Ψ_{0} ⊆ Φ ∪ {0}, then let A(Ψ_{0}) be the linear subspace of A spanned by A_{α}, α ∈ Ψ_{0}, as in Section 22.4. Let Δ be a base for Φ, as a root system in E_{R}, and let Φ^{+} = Φ^{Δ,+} be the corresponding set of positive roots, which are the elements of Φ that can be expressed as linear combinations of elements of Δ with nonnegative coefficients. Remember that

B_{Δ} = A(Φ^{+} ∪ {0}) (24.9.9) is a Borel subalgebra of A, as in Section 22.12, which is the standard Borel subalgebra associated to A_{0} and Δ.

Let B be any other Borel subalgebra of A. It suffices to show that there is an element of E(A) that maps B_{Δ} onto B.

If B_{Δ} ⊆ B, then B_{Δ} = B, because B_{Δ} is maximal among solvable Lie subalgebras of A. Of course, the problem is trivial in this case.

24.9.3 The second induction hypothesis

To continue the proof, we also use induction on the dimension of B_{Δ} ∩ B, as a vector space over k, as on p84 of [14]. If the dimension is equal to the dimension of B_{Δ}, then B_{Δ} ∩ B = B_{Δ}, so that B_{Δ} ⊆ B, and thus B_{Δ} = B, as before. If B̃ is a Borel subalgebra of A such that

(24.9.10) dim(B_{Δ} ∩ B̃) > dim B_{Δ} ∩ B,

then we suppose from now on that there is an element of E(A) that maps B_{Δ} onto B̃.

Of course, B_{Δ} depends on the choice of the maximal toral subalgebra A_{0} of A, as well as the base Δ for the corresponding root system Φ, as in the previous subsection. In Section 24.12, we shall consider another maximal toral subalgebra of A. Thus A_{0} should be considered as an arbitrary maximal toral subalgebra of A here, and Δ as an arbitrary base for the corresponding root system Φ.

As in [14], we consider separately the cases where B_{Δ} ∩ B is nontrivial or equal to {0}. In the first case, we consider the subcases where B_{Δ} ∩ B does or does not have nonzero elements that are ad-nilpotent in A. These two subcases are discussed in the next two sections.

24.10 The first subcase

Let us continue with the same notation and hypotheses as in the previous section. In this section, we suppose that

B_{Δ} ∩ B ≠ {0}, (24.10.1)

and that

(24.10.2) B_{Δ} ∩ B has a nonzero element that is ad-nilpotent in A.

This corresponds to Case (i) of (1) on p85 of [14].

We may as well suppose too that

B_{Δ} ∩ B ≠ B_{Δ}, B. (24.10.3)

Otherwise, we would have B_{Δ} ⊆ B or B ⊆ B_{Δ}, which would imply that B_{Δ} = B.

Put N_{Δ} = A(Φ^{+}), (24.10.4)

so that N_{Δ} ⊆ B_{Δ}. Remember that N_{Δ} is a Lie subalgebra of A that is nilpotent as a Lie algebra over k, and that the elements of N_{Δ} are ad-nilpotent in A, as in Section 22.6.

If α ∈ A′_{0} and α ≠ 0, then it is easy to see that [A_{0}, A_{α}] = A_{α}. This implies that [B_{Δ}, B_{Δ}] = N_{Δ}, (24.10.5)

as in Section 22.6.

We would like to check that

(24.10.6) N_{Δ} = {v ∈ B_{Δ} : ad_{v} is nilpotent on A}.

We have seen that N_{Δ} is contained in the right side, and so we only have to verify that the right side is contained in N_{Δ}. Note that B_{Δ} corresponds to the direct sum of A_{0} and N_{Δ}, as a vector space over k.

Let v ∈ B_{Δ} be given, so that v can be expressed as

v = w + u, (24.10.7)

with w ∈ A_{0} and u ∈ N_{Δ}. Suppose that v ∉ N_{Δ}, so that w ≠ 0. We would like to check that ad_{v} is not nilpotent on A.

Remember that the linear span of Φ in A′_{0}, as a vector space over k, is equal to A′_{0}, as in Section 17.5. This implies that the linear span of Δ in A′_{0} is equal to A′_{0}, because every element of Φ can be expressed as a linear combination of elements of Δ with coefficients in Z. It follows that α(w) ≠ 0 for some α ∈ Δ, because w is a nonzero element of A_{0}.

Observe that ad_{v} maps N_{Δ} into itself, and that N_{Δ} corresponds to the direct sum of A_{β}, β ∈ Φ^{+}, as a vector space over k. If z ∈ N_{Δ}, then the component of ad_{u}(z) in A_{α} is equal to 0. This implies that the component of ad_{v}(z) in A_{α} is the same as the component of ad_{w}(z) in A_{α}, which is α(w) times the component of z in A_{α}. One can use this to get that ad_{v} is not nilpotent on N_{Δ}, because A_{α} ≠ {0}.

24.10.1 Nilpotent elements of B_{Δ} ∩ B

Let N be the set of elements of B_{Δ} ∩ B that are ad-nilpotent in A, so that N ≠ {0}, by hypothesis. Observe that

N = N_{Δ} ∩ B, (24.10.8)

by (24.10.6).

Thus N is a linear subspace of A, and in fact a Lie subalgebra of A. We also have that [B_{Δ} ∩ B, B_{Δ} ∩ B] ⊆ [B_{Δ}, B_{Δ}] ∩ B = N_{Δ} ∩ B = N. (24.10.9)

In particular, N is an ideal in B_{Δ} ∩ B, as a Lie algebra over k.

Of course, x ↦ ad_{x} = ad_{A,x} defines a representation of B on A, as a vector space over k. If x ∈ [B, B], then ad_{x} is nilpotent on A, because B is solvable, as in Section 14.14. This implies that

B_{Δ} ∩ [B, B] ⊆ N. (24.10.10)

Note that N is nilpotent as a Lie algebra over k, because N_{Δ} is nilpotent. This means that N is solvable as a Lie algebra over k in particular. It follows that N is not an ideal in A, because A is semisimple, and N ≠ {0}.

Let M be the normalizer of N in A, so that M is a proper Lie subalgebra of A. Clearly B_{Δ} ∩ B ⊆ M, (24.10.11)

because N is an ideal in B_{Δ} ∩ B.

Let us check that B_{Δ} ∩ B ≠ B_{Δ} ∩ M. (24.10.12)

Of course, B_{Δ} ∩ B is a linear subspace of B_{Δ}, so that the quotient

B_{Δ}/(B_{Δ} ∩ B) (24.10.13)

may be considered as a vector space over k. If x ∈ N, then ad_{x} maps B_{Δ} and B_{Δ} ∩ B into themselves, because x ∈ B_{Δ} ∩ B. This leads to an induced linear mapping from (24.10.13) into iself. More precisely, this induced linear mapping is nilpotent, because ad_{x} is nilpotent.

The collection of linear mappings from (24.10.13) into itself induced by ad_{x}, x ∈ N, is a Lie subalgebra of the algebra of all linear mappings from (24.10.13) into itself, because N is a Lie algebra over k. Note that (24.10.13) is nontrivial, by (24.10.3). It follows that there is a nonzero element of (24.10.13) that is sent to 0 by the mapping induced by ad_{x} for every x ∈ N, as in Section 9.9.

Equivalently, this means that there is a y ∈ B_{Δ} such that y ∉ B and ad_{x}(y) ∈ B_{Δ} ∩ B for every x ∈ N. We also have that ad_{x}(y) ∈ N_{Δ} for every x ∈ N, because N_{Δ} is an ideal in B_{Δ}. This implies that

[y, x] = −ad_{x}(y) ∈ N (24.10.14)

for every x ∈ N, so that y ∈ M. It follows that (24.10.12) holds, because y ∈ B_{Δ} and y ∉ B.

Similarly, let us verify that

B_{Δ} ∩ B ≠ B ∩ M. (24.10.15)

The quotient B/(B_{Δ} ∩ B) is a nontrivial vector space over k, on which ad_{x} induces a linear mapping for every x ∈ N. Using the same type of argument as before, we get that there is a z ∈ B such that z ∉ B_{Δ} and ad_{x}(z) ∈ B_{Δ} ∩ B for every x ∈ N.

If x ∈ N, then [x, z] ∈ [B, B], because N ⊆ B and z ∈ B. This means that

[x, z] ∈ B_{Δ} ∩ [B, B] ⊆ N, (24.10.16)

using (24.10.10) in the second step. It follows that z ∈ M. Thus z is an element of the right side of (24.10.15), and not the left.

24.10.2 Some Borel subalgebras of M and A

Note that B_{Δ} ∩ M, B ∩ M are solvable as Lie algebras over k, because B_{Δ}, B are solvable. Let C_{Δ}, C be Borel subalgebras of M such that

B_{Δ} ∩ M ⊆ C_{Δ}, B ∩ M ⊆ C. (24.10.17)

Remember that M is a proper Lie subalgebra of A, so that the dimension of M is less than the dimension of A. Thus our induction hypothesis implies that there is an element of E(M) that maps C onto C_{Δ}.

Every element of E(M) can be obtained by restricting an element of E(A, M) to M. It follows that there is a σ ∈ E(A, M) ⊆ E(A) such that

σ(C) = C_{Δ}. (24.10.18)

Let B_{1} be a Borel subalgebra of A that contains C_{Δ}. Note that

B_{Δ} ∩ B ⊆ B_{Δ} ∩ M ⊆ B_{Δ} ∩ C_{Δ} ⊆ B_{Δ} ∩ B_{1}, (24.10.19)

by (24.10.11) and (24.10.17). We also have that

B_{Δ} ∩ B ≠ B_{Δ} ∩ B_{1}, (24.10.20)

by (24.10.12). This implies that the dimension of B_{Δ} ∩B is strictly less than the dimension of B_{Δ} ∩ B_{1}. This permits us to use our second induction hypothesis, to get an element τ of E(A) such that

τ(B_{1}) = B_{Δ}. (24.10.21)

Combining (24.10.18) and (24.10.21), we obtain that

τ(σ(C)) = τ(C_{Δ}) ⊆ τ(B_{1}) = B_{Δ}. (24.10.22)

Observe that

(24.10.23) τ(σ(B_{Δ} ∩ B)) ⊆ τ(σ(B ∩ M))
⊆ τ(σ(C)) ∩ τ(σ(B)) ⊆ B_{Δ} ∩ τ(σ(B)),

by (24.10.11), (24.10.17), and (24.10.22). We also have that the first inclusion is strict, by (24.10.15).

This shows that the dimension of B_{Δ} ∩ B is strictly less than the dimension of B_{Δ} ∩ τ(σ(B)). Of course, τ(σ(B)) is a Borel subalgebra of A too. Thus we may use our second induction hypothesis again, to get that there is an element of E(A) that maps B_{Δ} onto τ(σ(B)). This means that there is an element of E(A) that maps B_{Δ} onto B, as desired.

520 CHAPTER 24. STRONG AD-NILPOTENCE AND AUTOMORPHISMS

24.11 The second subcase

We continue with the same notation and hypotheses as in Section 24.9. In this section, we continue to suppose that B_{Δ} ∩ B ≠ {0} as well, and now also that (24.11.1) B_{Δ} ∩ B does not have any nonzero elements

that are ad-nilpotent in A.

This corresponds to Case (ii) of (1) on p85 of [14].

Let B_{0} be a Borel subalgebra of A, and let x ∈ B_{0} be given. Remember that there are unique x_{1}, x_{2} ∈ A such that x = x_{1} + x_{2}, [x_{1}, x_{2}]_{A} = 0, and ad_{x_{1}}, ad_{x_{2}} are the diagonalizable and nilpotent parts of ad_{x}, as a linear mapping from A into itself, as in Section 14.3. We also have that ad_{x_{1}}(B_{0}), ad_{x_{2}}(B_{0}) ⊆ B_{0}, (24.11.2) because ad_{x}(B_{0}) ⊆ B_{0}, as in Section 10.8. This means that x_{1}, x_{2} are elements of the normalizer of B_{0} in A. It follows that x_{1}, x_{2} ∈ B_{0}, (24.11.3) because the normalizer of a Borel subalgebra of A is itself, as in Section 22.7.

If x ∈ B_{Δ} ∩ B, then the ad-diagonalizable and ad-nilpotent parts of x are elements of B_{Δ} ∩ B too, by the remarks in the preceding paragraph. This implies that the ad-nilpotent part of x is 0, by hypothesis. This means that every element of B_{Δ} ∩ B is ad-diagonalizable in A under these conditions.

Of course, T = B_{Δ} ∩ B (24.11.4) is a Lie subalgebra of A, and in fact a toral subalgebra of A, as in Section 17.1. In particular, T is commutative as a Lie algebra over k, as before. It follows that ad_{x}, x ∈ T, can be simultaneously diagonalized on A.

Let N_{Δ} be as in (24.10.4), and note that N_{Δ} ∩ T = {0}, (24.11.5) by hypothesis. Thus [B_{Δ}, B_{Δ}] ∩ T = {0}, (24.11.6) by (24.10.5). Of course, this implies that [B_{Δ}, T] ∩ T = {0}, (24.11.7) because T ⊆ B_{Δ}. It follows that the normalizer of T in B_{Δ} is the same as the centralizer of T in B_{Δ}, which is to say that N_{B_{Δ}}(T) = C_{B_{Δ}}(T). (24.11.8)

24.11. THE SECOND SUBCASE 521

24.11.1 Reducing to T ⊆ A_{0}

Let C be a Cartan subalgebra of C_{B_{Δ}}(T), as a Lie algebra over k. Observe that T is contained in the normalizer N_{C_{B_{Δ}}} (T)(C) of C in C_{B_{Δ}}(T), by definition of C_{B_{Δ}}(T). This implies that T ⊆ C, (24.11.9) because N_{C_{B}(T)}(C) = C, by the definition of a Cartan subalgebra.

Of course, C_{B_{Δ}}(T) ⊆ B_{Δ}, by definition. Let u be an element of the normalizer N_{B_{Δ}}(C) of C in B_{Δ}, and let t ∈ T be given. Clearly ad_{t}(u) = [t, u]_{A} ∈ C, (24.11.10) because t ∈ C. This implies that ad_{t}(ad_{t}(u)) = 0, (24.11.11) because C ⊆ C_{B_{Δ}}(T). It follows that ad_{t}(u) = 0, (24.11.12) because ad_{t} is diagonalizable on A, as before.

This shows that u ∈ C_{B_{Δ}}(T), so that N_{B_{Δ}}(C) ⊆ C_{B_{Δ}}(T). (24.11.13) Using this, it is easy to see that N_{B_{Δ}}(C) = N_{C_{B_{Δ}} (T)}(C), (24.11.14) by definition of the normalizer. This means that N_{B_{Δ}}(C) = C, (24.11.15) because C is a Cartan subalgebra of C_{B_{Δ}}(T). Of course, C is nilpotent as a Lie algebra over k, because it is a Cartan subalgebra of C_{B_{Δ}}(T). It follows that C is a Cartan subalgebra of B_{Δ} as well.

Remember that A_{0} is a maximal toral subalgebra of A, as in Section 24.9. This implies that A_{0} is a Cartan subalgebra of A, as in Section 18.10. We also have that A_{0} ⊆ B_{Δ}, by the definition (24.9.9) of B_{Δ}. It follows that A_{0} is a Cartan subalgebra of B_{Δ}, as in Section 18.11.

Because B_{Δ} is solvable as a Lie algebra over k, and A_{0}, C are Cartan subalgebras of B_{Δ}, there is an element of E(B_{Δ}) that maps C onto A_{0}, as in Section 24.7. This implies that there is an element of E(A, B_{Δ}) ⊆ E(A) that maps C onto A_{0}, as in Section 24.3. In particular, this element of E(A, B_{Δ}) maps T into A_{0}. Using this, we can reduce to the case where T ⊆ A_{0}. (24.11.16)

24.11.2 The case where T = A_{0}

Suppose for the moment that T = A_{0}. Remember that A_{0} ≠ {0}, as in Section 24.9. We also have A ≠ A_{0}, because A is semisimple, and A_{0} is commutative as a Lie algebra. This implies that Φ ≠ ∅, so that B_{Δ} ≠ A_{0}. Thus A_{0} is not a Borel subalgebra of A, and A_{0} ≠ B.

If T = A_{0}, then A_{0} ⊆ B. If x ∈ A_{0}, then it follows that ad_{x} maps B into itself. This implies that the restrictions of the ad_{x}’s, x ∈ A_{0}, to B are simultaneously diagonalizable on B. Remember that for each α ∈ Φ, A_{α} has dimension one as a vector space over k. This means that B corresponds, as a vector space over k, to the direct sum of A_{0} and some collection of A_{α}’s. In particular, there is at least one α_{0} ∈ Φ such that

A_{α_{0}} ⊆ B, (24.11.17)

because B ≠ A_{0}. More precisely, α_{0} ∉ Φ^{+}, because B_{Δ} ∩ B = T ⊆ A_{0}. Thus α_{0} ∈ −Φ^{+}.

Let θ_{α_{0}} be the Lie algebra automorphism of A corresponding to α_{0} as in (23.8.1). Note that the Lie subalgebra of A denoted B in Section 23.8 corresponds to A_{0} here. It is easy to see that θ_{α_{0}} ∈ E(A), because x_{α}, y_{α} are strongly ad-nilpotent in A, as mentioned in Section 23.9. We also have that

θ_{α_{0}}(A_{0}) = A_{0} (24.11.18)

and θ_{α_{0}}(A_{α_{0}}) = A_{−α_{0}}. (24.11.19)

More precisely, (24.11.18) corresponds to (23.8.4), and can also be obtained from (23.8.6), while (24.11.19) can be obtained from (23.7.9) or (23.8.6).

It follows that θ_{α_{0}}(B) is a Borel subalgebra of A with

A_{0} + A_{−α_{0}} ⊆ θ_{α_{0}}(B). (24.11.20)

Of course, A_{0} + A_{−α_{0}} ⊆ B_{Δ}, because −α_{0} ∈ Φ^{+}. Thus

A_{0} + A_{−α_{0}} ⊆ B_{Δ} ∩ θ_{α_{0}}(B). (24.11.21)

In particular, the dimension of B_{Δ} ∩θ_{α_{0}}(B) is strictly larger than the dimension of B_{Δ} ∩B = T = A_{0}, so that our second induction hypothesis implies that there is an element of E(A) that maps B_{Δ} onto θ_{α_{0}}(B). This implies that there is an element of E(A) that maps B_{Δ} onto B, as desired.

24.11.3 The case where T ≠ A_{0}, B ⊆ C_{A}(T)

Thus we suppose from now on in this section that

T ≠ A_{0}. (24.11.22)

Suppose for the moment that

B ⊆ C_{A}(T). (24.11.23)

Remember that T = B_{Δ} ∩ B ≠ {0}, by hypothesis. This implies that

C_{A}(T) ≠ A, (24.11.24)

because otherwise T would be contained in the center of A, which would contradict the semisimplicity of A.

Observe that A_{0} ⊆ C_{A}(T), (24.11.25)

because T ⊆ A_{0} and A_{0} is commutative as a Lie algebra. Let B_{1} be a Borel subalgebra of C_{A}(T), as a Lie algebra over k, that contains A_{0}. Of course, B may be considered as a Borel subalgebra of C_{A}(T) too, as in Section 18.11. Thus our first induction hypothesis implies that there is an element of E(C_{A}(T)) that maps B onto B_{1}. It follows that there is an element σ of E(A, C_{A}(T)) ⊆ E(A) such that σ(B) = B_{1}, (24.11.26)

as in Section 24.3.

Using (24.11.26) we get that B_{1} is a Borel subalgebra of A. Note that

A_{0} ⊆ B_{Δ} ∩ B_{1}, (24.11.27)

while B_{Δ} ∩ B = T is a proper linear subspace of A_{0}. This means that the dimension of B_{Δ} ∩ B is strictly less than the dimension of B_{Δ} ∩ B_{1}, so that our second induction hypothesis implies that there is an element of E(A) that maps B_{Δ} onto B_{1}. It follows that there is an element of E(A) that maps B_{Δ} onto B, by (24.11.26).

24.11.4 The case where T ≠ A_{0}, B ⊈ C_{A}(T)

Suppose for the rest of the section that (24.11.23) does not hold. This means that there is a t_{1} ∈ T such that ad_{t_{1}} is not identically zero on B. Note that ad_{t_{1}} maps B into itself, because T ⊆ B and B is a Lie subalgebra of A. We also have that ad_{t_{1}} is diagonalizable on B, because it is diagonalizable on A. It follows that there is an a_{1} ∈ k and x_{1} ∈ B such that a_{1} ≠ 0, x_{1} ≠ 0, and

ad_{t_{1}}(x_{1}) = a_{1} x_{1}. (24.11.28)

Remember that if Ψ_{0} ⊆ Φ ∪ {0}, then A(Ψ_{0}) is the linear subspace of A spanned by A_{α}, α ∈ Ψ_{0}, as in Subsection 24.9.2. Put

Ψ_{1} = {α ∈ Φ : α(t_{1}) = a_{1}} (24.11.29)

and (24.11.30) Ψ_{2} = {α ∈ Φ : α(t_{1}) = n · a_{1} for some n ∈ Z_{+}},

so that Ψ_{1} ⊆ Ψ_{2}. One can use (24.11.28) to check that

x_{1} ∈ A(Ψ_{1}). (24.11.31)

One can also verify that A(Ψ_{2}), A(Ψ_{2}∪{0}) are Lie subalgebras of A, with A(Ψ_{2}) nilpotent, and A(Ψ_{2} ∪ {0}) solvable. More precisely, this can be obtained as in Section 22.5.

Let B_{2} be a Borel subalgebra of A that contains A(Ψ_{2} ∪ {0}). Thus

A_{0} ⊆ B_{Δ} ∩ B_{2}, (24.11.32)

which implies that the dimension of B_{Δ} ∩ B = T is strictly less than the dimension of B_{Δ} ∩ B_{2}, because T is a proper linear subspace of A_{0}. Using our second induction hypothesis, we get that there is an element σ_{2} of E(A) such that

σ_{2}(B_{Δ}) = B_{2}. (24.11.33)

Observe that x_{1} ∈ B_{2} ∩ B, (24.11.34)

using (24.11.31). Clearly T ⊆ A_{0} ⊆ B_{2}, so that

T ⊆ B_{2} ∩ B. (24.11.35)

Of course, x_{1} is not in T, by (24.11.28). This implies that B_{Δ} ∩ B = T is a proper linear subspace of B_{2} ∩ B. In particular, the dimension of B_{Δ} ∩ B is strictly less than the dimension of B_{2} ∩ B.

It follows that the dimension of B_{Δ} ∩ B is strictly less than the dimension of σ_{2}^{−1}(B_{2} ∩ B) = B_{Δ} ∩ σ_{2}^{−1}(B). (24.11.36)

Note that σ_{2}^{−1}(B) is a Borel subalgebra of A, so that our second induction hypothesis implies that there is an element of E(A) that maps B_{Δ} onto σ_{2}^{−1}(B). This implies that there is an element of E(A) that maps B_{Δ} onto B, as desired.

24.12 The second case

We continue with the same notation and hypotheses as in Section 24.9 again. The arguments in the previous two sections show that if B_{Δ} ∩ B ≠ {0}, then there is an element of E(A) that maps B_{Δ} onto B. More precisely, this works for any choice of maximal toral subalgebra A_{0} of A, and base Δ for the corresponding root system Φ, as in Subsection 24.9.3.

In this section, we suppose that

B_{Δ} ∩ B = {0}. (24.12.1)

This corresponds to (2) on p86 of [14].

Remember that A corresponds to the direct sum of A_{α}, α ∈ Φ ∪ {0}, as a vector space over k, as in Subsection 24.9.2. In particular, the dimension of A, as a vector space over k, is given by

dim A = dim A_{0} + #Φ, (24.12.2) where #Φ is the number of elements of Φ, because A_{α} has dimension one as a vector space over k when α ∈ Φ. Similarly,

(24.12.3) dim B_{Δ} = dim A(Φ^{+} ∪ {0}) = dim A_{0} + #Φ^{+}.

Note that #Φ^{+} = #Φ/2, (24.12.4)

because Φ is the union of Φ^{+} and −Φ^{+}, which are disjoint. Thus

(24.12.5) dim B_{Δ} = dim A_{0} + #Φ/2.

It follows that (dim A)/2 < dim B_{Δ}, (24.12.6)

because A_{0} ≠ {0}. We also have that

(24.12.7) dim B_{Δ} + dim B ≤ dim A,

by (24.12.1). This means that

dim B < (dim A)/2. (24.12.8)

Suppose for the moment that every element of B is ad-nilpotent in A. Of course, this implies that every element of B is ad-nilpotent in B. It follows that B is nilpotent as a Lie algebra over k, as in Section 9.10.

Remember that the normalizer of B in A is equal to B, because B is a Borel subalgebra of A, as in Section 22.7. This means that B is a Cartan subalgebra of A.

Under these conditions, we get that B is a toral subalgebra of A, as in Section 18.10. Thus the elements of B are ad-diagonalizble in A. This implies that B = {0}, because the elements of B are also supposed to be ad-nilpotent in A. This contradicts the hypothesis that B be a Borel subalgebra of A, since A ≠ {0}.

Thus we may suppose now that B has an element that is not ad-nilpotent in A. Remember that B contains the ad-diagonalizable and ad-nipotent parts of its elements, as in (24.11.3). It follows that B has a nonzero element x_{0} that is ad-diagonalizable in A.

Let T_{0} be a maximal toral subalgebra of A that contains x_{0}. Using any base for the root system corresponding to T_{0}, we get a standard Borel subalgebra B_{0} of A with respect to T_{0}, as in Subsection 24.9.2. Note that T_{0} ⊆ B_{0}, so that

B_{0} ∩ B ≠ {0}. (24.12.9)

This implies that there is an element of E(A) that maps B_{0} onto B, as mentioned at the beginning of the section. In particular,

dim B = dim B_{0}. (24.12.10)

However, dim B_{0} > (dim A)/2, as in (24.12.6). This contradicts (24.12.8).

This shows that (24.12.1) is not possible. It follows that there is an element of E(A) that maps B_{Δ} onto B. This completes the proof of the theorem stated at the beginning of Section 24.9.

24.13 Some consequences

Let k be an algebraically closed field of characteristic 0 again, and let (A, [·, ·]_{A}) be a finite-dimensional Lie algebra over k. The rank of A as a Lie algebra over k may be defined as the dimension of any Cartan subalgebra of A, as on p86 of [14]. The corollary stated at the beginning of Section 24.9 implies that

(24.13.1) the rank of A does not depend on the choice of the Cartan subalgebra.

The rank of a finite-dimensional complex Lie algebra is defined another way in Definition 2 on p11 of [24], and Corollary 1 on p13 of [24] states that these two definitions are equivalent.

Suppose now that A ≠ {0} is semisimple, as a Lie algebra over k. Let A_{0} be a maximal toral subalgebra of A, and let Φ be the corresponding root system, as in Subsection 24.9.2. Also let Δ be a base for Φ, and B_{Δ} be the standard Borel subalgebra of A associated to A_{0} and Δ, as in Subsection 24.9.2.

Suppose that N is a Lie subalgebra of A such that every element of N is adnilpotent as an element of A. In particular, every element of N is ad-nilpotent as an element of N, so that N is nilpotent as a Lie algebra over k, as in Section 9.10. Of course, this implies that N is solvable as a Lie algebra over k, so that there is a Borel subalgebra B of A that contains N.

The theorem stated at the beginning of Section 24.9 implies that there is an element σ of E(A) such that σ(B) = B_{Δ}. In particular, σ(N) ⊆ B_{Δ}. In fact,

σ(N) ⊆ N_{Δ}, (24.13.2)

where N_{Δ} is as in (24.10.4). This follows from (24.10.6), because the elements of σ(N) are ad-nilpotent in A. This corresponds to Exercise 2 on p87 of [14], and the corollary to Theorem 5 on p48 of [24].

Note that the standard Borel subalgebras of A associated to A_{0} and any base for Φ contain A_{0}, by construction. Conversely, let B be any Borel subalgebra of A such that A_{0} ⊆ B. (24.13.3)

Under these conditions,

(24.13.4) B is a standard Borel subalgebra of A with respect to A_{0} and some base for Φ,

as mentioned on p86 of [14].

To see this, let Δ be a base for Φ again, and let B_{Δ} be the corresponding standard Borel subalgebra of A, as before. Remember that there is an element σ of E(A) such that σ(B_{Δ}) = B, by the theorem stated at the beginning of Section 24.9. In particular,

σ(A_{0}) ⊆ σ(B_{Δ}) ⊆ B. (24.13.5)

24.13. SOME CONSEQUENCES 527

Remember that A_{0} is a Cartan subalgebra of A, as in Section 18.10. Of course, this implies that σ(A_{0}) is a Cartan subalgebra of A too. It follows that A_{0} and σ(A_{0}) may be considered as Cartan subalgebras of B, as in Section 18.11.

Thus there is an element of E(B) that maps σ(A_{0}) onto A_{0}, as in Section 24.7. This implies that there is an element τ of E(A, B) ⊆ E(A) such that τ(σ(A_{0})) = A_{0}, (24.13.6) as in Section 24.3. Note that τ(σ(B_{Δ})) = τ(B) = B. (24.13.7)

Put σ_{1} = τ ∘ σ ∈ E(A), so that σ_{1}(A_{0}) = A_{0} and σ_{1}(B_{Δ}) = B. Let σ_{0} be the restriction of σ_{1} to A_{0}, and let σ′_{0} be the corresponding dual linear mapping from the dual A′_{0} of A_{0} into itself. If α ∈ A′_{0} and A_{α} is as in Subsection 24.9.2, then σ_{1}(A_{σ′_{0}(α)}) = A_{α} (24.13.8) for every α ∈ A′_{0}, as in Section 23.6. In particular, σ′_{0}(Φ) = Φ, (24.13.9) as before.

Let E_{Q} be the linear subspace of A′_{0}, as a vector space over Q, spanned by Φ, as in Subsection 24.9.2. Thus σ′_{0}(E_{Q}) = E_{Q}, (24.13.10) by (24.13.9), as in Section 23.6. This leads to a one-to-one linear mapping from the corresponding real vector space E_{R} onto itself, which is an automorphism of Φ as a root system in E_{R}, as before. Using this, one can check that Δ̃ = (σ′_{0})^{−1}(Δ) (24.13.11) is a base for Φ as a root system in E_{R} too.

Let B_{Δ} be the standard Borel subalgebra of A associated to A_{0} and Δ̃ as in Subsection 24.9.2. One can verify that

σ_{1}(B_{Δ}) = B_{Δ}, (24.13.12)

using (24.13.8). This means that

B = B_{Δ}, (24.13.13)

as desired, because σ_{1}(B_{Δ}) = B, by construction.

24.14 The case of sl_{2}(k)

Exercise 4 on p87 of [14] asks one to consider simplifications of the proof of the theorem stated at the beginning of Section 24.9, in the case where the Lie algebra is sl_{2}(k). Let us begin with a preliminary remark.

Let k be an algebraically closed field, and let (A, [·, ·]_{A}) be a Lie algebra over k of finite dimension at least 2. Suppose that x is a nonzero element of A, so that the quotient A/{t x : t ∈ k} (24.14.1)

may be considered as a vector space over k of positive finite dimension. Note that ad_{x} induces a linear mapping from (24.14.1) into itself. This mapping has a nonzero eigenvector in (24.14.1), because k is algebraically closed.

Equivalently, there is an element y of A such that x and y are linearly independent in A, and ad_{x}(y) can be expressed as a linear combination of x and y. This means that the linear span of x and y is a 2-dimensional Lie subalgebra of A, which is solvable as a Lie algebra over k. It follows that

(24.14.2) every Borel subalgebra of A has dimension at least two

under these conditions.

Now let k be an algebraically closed field of characteristic 0, as in Section 24.9. Note that the Borel subalgebras of sl_{2}(k) are proper subalgebras of sl_{2}(k), because sl_{2}(k) is not solvable. This implies that every Borel subalgebra of sl_{2}(k) has dimension two as a vector space over k, by (24.14.2). Conversely, every twodimensional Lie subalgebra of sl_{2}(k) is solvable, and thus a Borel subalgebra.

Let x = \binom{0 1}{0 0}, y = \binom{0 0}{1 0}, and h = \binom{1 0}{0 −1} be the usual basis for sl_{2}(k), as a vector space over k. Thus [x, y] = h, [h, x] = 2 · x, and [h, y] = −2 · y, as before. The linear span of h is a maximal toral subalgebra of sl_{2}(k), for which Φ corresponds to the linear functionals defined by multiplication by 2 and −2. The linear functional defined by multiplication by 2 may be considered as a base for Φ as a root system, for which the corresponding standard Borel subalgebra B_{Δ} of sl_{2}(k) is spanned by h and x.

Let B be any Borel subalgebra of sl_{2}(k), so that B has dimension two as a vector space over k, as before. In particular, B_{Δ} ∩ B ≠ {0} in this case. We would like to show that there is an element of E(sl_{2}(k)) that maps B_{Δ} onto B. Of course, this is trivial when B = B_{Δ}, and so we may suppose that B ≠ B_{Δ}.

Suppose for the moment that x ∈ B. If B ⊆ B_{Δ}, then B = B_{Δ}, and so we may suppose that B ⊈ B_{Δ}. Thus

a x + b y + c h ∈ B (24.14.3)

for some a, b, c ∈ k, with b ≠ 0. This implies that

[x, a x + b y + c h] = b h − 2 c x ∈ B, (24.14.4)

because x ∈ B and B is a Lie subalgebra of sl_{2}(k). It follows that h ∈ B, because x ∈ B. This means that y ∈ B, by (24.14.3), which is to say that B = sl_{2}(k). This is a contradiction, because sl_{2}(k) is not solvable.

Thus we may suppose that x ∉ B. It follows that

a x + h ∈ B (24.14.5)

for some a ∈ k, because B_{Δ} ∩ B ≠ {0}. One can use the exponential of a multiple of ad_{x} to reduce to the case where h ∈ B. Using this, one can check that either B = B_{Δ}, or B is spanned by y and h. If B is spanned by y and h, then B = σ(B_{Δ}), where σ ∈ E(sl_{2}(k)) is as in Section 15.7.

24.15 Automorphisms and semisimplicity

Let k be an algebraically closed field of characteristic 0, and let (A, [·, ·]_{A}) be a semisimple Lie algebra over k of positive finite dimension. Also let A_{0} be a maximal toral subalgebra of A, and let Φ be the corresponding root system, as in Subsection 24.9.2. Choose a base Δ for Φ, and let B_{Δ} be the standard Borel subalgebra of A associated to A_{0} and Δ, as before.

Suppose that τ is an automorphism of A, as a Lie algebra over k. Thus τ(B) is another Borel subalgebra of A. The theorem stated at the beginning of Section 24.9 implies that there is an element σ_{1} of E(A) such that

σ_{1}(τ(B_{Δ})) = B_{Δ}. (24.15.1)

Remember that A_{0} is a Cartan subalgebra of A, as in Section 18.10. This implies that σ_{1}(τ(A_{0})) is a Cartan subalgebra of A as well. It follows that A_{0} and σ_{1}(τ(A_{0})) are Cartan subalgebras of B_{Δ} too, as a Lie algebra over k, as in Section 18.11. Thus there is an element of E(B_{Δ}) that maps σ_{1}(τ(A_{0})) onto A_{0}, as in Section 24.7. This means that there is an element σ_{2} of E(A, B_{Δ}) ⊆ E(A) with σ_{2}(σ_{1}(τ(A_{0}))) = A_{0}, (24.15.2)

as in Section 24.3.

Put τ_{2} = σ_{2} ∘ σ_{1} ∘ τ, (24.15.3)

which is an automorphism of A that maps A_{0} onto itself. Let τ_{2,0} be the restriction of τ_{2} to A_{0}, and let τ′_{2,0} be the corresponding dual linear mapping from A′_{0} into itself. Thus τ_{2}(A_{τ′_{2,0}(α)}) = A_{α} (24.15.4)

for every α ∈ A′_{0}, as in Section 23.6, where A_{α} is as in Subsection 24.9.2. This implies that τ′_{2,0}(Φ) = Φ, (24.15.5)

as before.

It follows that τ′_{2,0}(E_{Q}) = E_{Q}, (24.15.6)

as in Section 23.6, where E_{Q} is the linear subspace of A′_{0}, as a vector space over Q, spanned by Φ. As before, this leads to a one-to-one linear mapping from the corresponding real vector space E_{R} onto itself, which is an automorphism of Φ as a root system in E_{R}.

Note that τ_{2}(B_{Δ}) = B_{Δ}, (24.15.7)

by (24.15.1), and because σ_{2} ∈ E(A, B_{Δ}). Let Φ^{+} be the set of positive roots with respect to Δ, as in Subsection 24.9.2. One can check that

τ′_{2,0}(Φ^{+}) = Φ^{+}, (24.15.8)

using (24.15.4) and (24.15.7).

Using (24.15.8), one can verify that

τ′_{2,0}(Δ) = Δ. (24.15.9)

More precisely, Δ consists of exactly the elements of Φ^{+} that cannot be expressed as the sum of at least two other elements of Φ^{+}.

We shall see in Section 28.6 that every automorphism of Φ as a root system that sends Δ onto itself corresponds to an automorphism of A that maps A_{0} onto itself. This also uses the remarks in Section 28.7.

The automorphisms of A that are equal to the identity mapping on A_{0} are called diagonal automorphisms, and they will be discussed further in Sections 28.6 and 29.7.

This corresponds to some remarks on p87 of [14].

Part IV

Some related constructions Universal enveloping algebras

25.1 Polynomial algebras

Let I be a nonempty set, and let M_{I} be the set of functions α on I with values in the set Z_{+} ∪{0} of nonnegative integers such that α(j) = 0 for all but finitely many j ∈ I. The elements of M_{I} are considered as multi-indices with respect to I, which reduces to the usual notion when I = {1, …, n} for some positive integer n. If α ∈ M_{I}, then we put

|α| = ∑_{j∈I} α(j), (25.1.1)

which reduces to a finite sum. If β ∈ M_{I} too, then α + β ∈ M_{I}, and

|α + β| = |α| + |β|. (25.1.2)

Let X_{j}, j ∈ I, be a commuting family of indeterminates. If α ∈ M_{I}, then X^{α} is considered as a formal monomial in the X_{j}’s of degree |α|. This may be identified with the formal product

X_{j_{1}}^{α(j_{1})} ⋯ X_{j_{n}}^{α(j_{n})} (25.1.3)

when j_{1}, …, j_{n} are finitely many distinct elements of I, and α(j) = 0 for every j ∈ I \{j_{1}, …, j_{n}}. If β ∈ M_{I} as well, then the corresponding product of formal monomials can be defined as usual by

X^{α} · X^{β} = X^{α+β}. (25.1.4)

Let k be a commutative ring with a multiplicative identity element. A formal polynomial in the X_{j}’s, j ∈ I, with coefficients in k can be expressed as

f(X) = ∑_{α∈M_{I}} f_{α} X^{α}, (25.1.5)

where f_{α} ∈ k for every α ∈ M_{I}, and f_{α} = 0 for all but finitely many α ∈ M_{I}. The space k[{X_{j} : j ∈ I}] of these formal polynomials can be defined as the space c_{00}(M_{I}, k) of all k-valued functions on M_{I} with finite support, which is to say that are equal to 0 at all but finitely many elements of M_{I}. This is a submodule of the space c(M_{I}, k) of all k-valued functions on M_{I}, as a module over k with respect to pointwise addition and scalar multiplication of functions. Of course, this corresponds to termwise addition and scalar multiplication of formal polynomials as in (25.1.5).

Let

g(X) = ∑_{β∈M_{I}} g_{β} X^{β} (25.1.6)

be another formal polynomial in the X_{j}’s, j ∈ I, with coefficients in k. If γ ∈ M_{I}, then put

h_{γ} = ∑_{α+β=γ} f_{α} g_{β}, (25.1.7)

where more precisely the sum is taken over all α, β ∈ M_{I} such that α + β = γ. It is easy to see that there are only finitely many such α, β ∈ M_{I}, so that the right side of (25.1.7) is a finite sum in k. We also have that (25.1.7) is equal to 0 for all but finitely many γ ∈ M_{I}, because of the analogous conditions for f_{α} and g_{β}. Thus

h(X) = ∑_{γ∈M_{I}} h_{γ} X^{γ} (25.1.8)

is a formal polynomial in the X_{j}’s, j ∈ I, with coefficients in k, and we put f(X) g(X) = h(X). One can check that k[X_{j} : j ∈ I}] is a commutative associative algebra over k with respect to this definition of multiplication. Let us identify elements of k with the formal polynomials whose coefficients of X^{α} are equal to 0 when α is a nonzero element of M_{I}, and are equal to the given element of k when α = 0. This is a subalgebra of k[{X_{j} : j ∈ I}], and 1 ∈ k corresponds to the multiplicative identity element in k[{X_{j} : j ∈ I}].

Let B be an associative algebra over k with a multiplicative identity element e_{B}, and let b_{j} be an element of B for each j ∈ I. The Cartesian power B^{I} is the same as the Cartesian product of the family of copies of B indexed by I, and we let b be the element of B^{I} whose jth coordinate is equal to b_{j} for each j ∈ I. Suppose that the b_{j}’s commute in B, which is to say that

b_{j} b_{l} = b_{l} b_{j} (25.1.9)

for every j, l ∈ I. If α ∈ M_{I}, and j_{1}, …, j_{n} are finitely many distinct elements of I such that α(j) = 0 when j ∈ I \ {j_{1}, …, j_{n}}, then put

b^{α} = b_{j_{1}}^{α(j_{1})} ⋯ b_{j_{n}}^{α(j_{n})}. (25.1.10)

As usual, b_{j}^{α(j)} is interpreted as being e_{B} when α(j) = 0, so that the right side of (25.1.10) does not depend on the choice of j_{1}, …, j_{n}. If β ∈ M_{I} too, then

b^{α+β} = b^{α} b^{β}, (25.1.11) because of (25.1.9). If f(X) is as in (25.1.5), then put

f(b) = ∑_{α∈M_{I}} f_{α} b^{α}, (25.1.12)

which reduces to a finite sum in B. It is easy to see that f(X) ↦ f(b) defines an algebra homomorphism from k[{X_{j} : j ∈ I}] into B.

25.2 Tensor products of associative algebras

Let k be a commutative ring with a multiplicative identity element, and let B_{1}, B_{2} be associative algebras over k. Thus B_{1}, B_{2} are modules over k in particular, and we take B = B_{1} ⨂ B_{2}, as a module over k. If b_{1}, b′_{1} ∈ B_{1} and b_{2}, b′_{2} ∈ B_{2}, then (b_{1} b′_{1}) ⊗ (b_{2} b′_{2}) (25.2.1)

is an element of B, and this defines a multilinear mapping from B_{1}×B_{2}×B_{1}×B_{2} into B. This multilinear mapping corresponds to a unique linear mapping from B_{1} ⨂ B_{2} ⨂ B_{1} ⨂ B_{2} into B, as usual. Remember that B_{1} ⨂ B_{2} ⨂ B_{1} ⨂ B_{2} can be identified with B ⨂ B, as in Section 7.15. This leads to a bilinear mapping from B × B into B, by composing the standard bilinear mapping from B × B into B ⨂ B with the previous linear mapping from B ⨂ B into B. This bilinear mapping is used to define multiplication on B. By construction,

(b_{1} ⊗ b_{2}) (b′_{1} ⊗ b′_{2}) = (b_{1} b′_{1}) ⊗ (b_{2} b′_{2}) (25.2.2)

in B for every b_{1}, b′_{1} ∈ B_{1} and b_{2}, b′_{2} ∈ B_{2}. One can check that multiplication on B is associative, because multiplication is associative on B_{1} and B_{2}, by hypothesis. If multiplication is commutative on B_{1} and B_{2}, then multiplication on B is commutative as well.

Suppose for the moment that B_{1} and B_{2} have multiplicative identity elements e_{1} and e_{2}, respectively. In this case, e_{1} ⊗ e_{2} is the multiplicative identity element in B. Observe that b_{1} ↦ b_{1} ⊗ e_{2} (25.2.3)

and b_{2} ↦ e_{1} ⊗ b_{2} (25.2.4)

are algebra homomorphisms from B_{1} and B_{2} into B. We also have that

(b_{1} ⊗ e_{2}) (e_{1} ⊗ b_{2}) = (e_{1} ⊗ b_{2}) (b_{1} ⊗ e_{2}) = b_{1} ⊗ b_{2} (25.2.5)

for every b_{1} ∈ B_{1} and b_{2} ∈ B_{2}.

Let C be another associative algebra over k, and let ϕ_{1}, ϕ_{2} be algebra homomorphisms from B_{1}, B_{2} into C, respectively. Suppose that

ϕ_{1}(b_{1}) ϕ_{2}(b_{2}) = ϕ_{2}(b_{2}) ϕ_{1}(b_{1}) (25.2.6) for every b_{1} ∈ B_{1} and b_{2} ∈ B_{2}. Of course, (b_{1}, b_{2}) ↦ ϕ_{1}(b_{1}) ϕ_{2}(b_{2}) is a bilinear mapping from B_{1} × B_{2} into C. This leads to a unique linear mapping ϕ from B into C, with ϕ(b_{1} ⊗ b_{2}) = ϕ_{1}(b_{1}) ϕ_{2}(b_{2}) (25.2.7)

for every b_{1} ∈ B_{1} and b_{2} ∈ B_{2}. One can verify that ϕ is an algebra homomorphism from B into C under these conditions.

25.3 Tensor algebras

Let k be a commutative ring with a multiplicative identity element, and let V be a module over k. If n is a positive integer, then let V^{n} be the space of n-tuples of elements of V, which is the same as the Cartesian product of n copies of V. Remember that the nth tensor power T^{n}V of V is defined as a module over k as in Section 7.15, as the tensor product of n copies of V. This is interpreted as being equal to V when n = 1, as before. In particular, the mapping from (v_{1}, …, v_{n}) ∈ V^{n} to v_{1} ⊗ ⋯ ⊗ v_{n} ∈ T^{n}V is multilinear over k.

Let n_{1} and n_{2} be positive integers, and remember that there is a natural isomorphism between (T^{n_{1}}V) ⨂(T^{n_{2}}V) and T^{n_{1}+n_{2}}V, as modules over k, as in Section 7.15. There is also a natural bilinear mapping from T^{n_{1}}V × T^{n_{2}}V into (T^{n_{1}}V) ⨂(T^{n_{2}}V), by the definition of the tensor product. This leads to a natural bilinear mapping from (T^{n_{1}}V) × (T^{n_{2}}V) into T^{n_{1}+n_{2}}V. If a_{1} ∈ T^{n_{1}}V and a_{2} ∈ T^{n_{2}}V, then let a_{1} a_{2} be the image of (a_{1}, a_{2}) in T^{n_{1}+n_{2}}V under this mapping. If a_{1} = v_{1} ⊗ ⋯ ⊗ v_{n_{1}} and a_{2} = w_{1} ⊗ ⋯ ⊗ w_{n_{2}} for some v_{1}, …, v_{n_{1}} and w_{1}, …, w_{n_{2}} in V, then

(v_{1} ⊗ ⋯ ⊗ v_{n_{1}}) (w_{1} ⊗ ⋯ ⊗ w_{n_{2}}) = v_{1} ⊗ ⋯ ⊗ v_{n_{1}} ⊗ w_{1} ⊗ ⋯ ⊗ w_{n_{2}} (25.3.1)

in T^{n_{1}+n_{2}}V, by construction.

It is customary to define T^{n}V when n = 0 to be k, as a module over itself. If W is any module over k, then scalar multiplication on W can be used to define bilinear mappings from k × W and from W × k into W. This leads to standard isomorphisms from k ⨂ W and from W ⨂ k onto W, as modules over k. In particular, we have a natural bilinear mapping from T^{n_{1}}V × T^{n_{2}}V into T^{n_{1}+n_{2}}V when n_{1} = 0 or n_{2} = 0 as well. If a_{1} ∈ T^{n_{1}}V, a_{2} ∈ T^{n_{2}}V, and a_{3} ∈ T^{n_{3}}V for some nonnegative integers n_{1}, n_{2}, n_{3}, then one can check that

(a_{1}a_{2}) a_{3} = a_{1} (a_{2} a_{3}) (25.3.2)

in T^{n_{1}+n_{2}+n_{3}}V.

The tensor algebra on V is denoted T V, and defined initially as a module over k to be the direct sum of T^{n}V over all nonnegative integers n,

T V = ⨁_{n=0}^{∞} T^{n}V. (25.3.3)

There is a natural bilinear mapping from (T V)×(T V) into T V, which is defined using the natural bilinear mapping from (T^{n_{1}}V) × (T^{n_{2}}V) into T^{n_{1}+n_{2}}V for n_{1}, n_{2} ≥ 0 mentioned in the previous two paragraphs. If a, b ∈ T V, then let a b be the image of (a, b) in T V under this mapping. It is easy to see this definition of multiplication on T V is associative, using (25.3.2). We may consider k = T^{0}V as a subalgebra of T V, and the multiplicative identity element 1 in k is the multiplicative identity element in T V too.

Let A be an associative algebra over k with a multiplicative identity element e, and let ϕ be a homomorphism from V into A, as modules over k. If n is a positive integer, then there is a natural multilinear mapping from V^{n} into A, which sends (v_{1}, …, v_{n}) ∈ V^{n} to ϕ(v_{1}) ⋯ ϕ(v_{n}). This leads to a unique linear mapping ψ_{n} from T^{n}V into A, with

ψ_{n}(v_{1} ⊗ ⋯ ⊗ v_{n}) = ϕ(v_{1}) ⋯ ϕ(v_{n}) (25.3.4)

for every (v_{1}, …, v_{n}) ∈ V^{n}. Put ψ_{0}(t) = t e for every t ∈ k = T^{0}V. If a_{1} ∈ T^{n_{1}}V and a_{2} ∈ T^{n_{2}}V for some n_{1}, n_{2} ≥ 0, then one can check that

ψ_{n_{1}+n_{2}}(a_{1} a_{2}) = ψ_{n_{1}}(a_{1}) ψ_{n_{2}}(a_{2}). (25.3.5)

Let ψ be the mapping from T V into A which corresponds to ψ_{n} on T^{n}V for every n ≥ 0. This defines an algebra homomorphism from T V into A, because of (25.3.5). More precisely, this is the unique algebra homomorphism from T V into A that is equal to ϕ on V = T^{1}V and sends 1 ∈ k = T^{0}V to e. Note that every algebra homomorphism ψ from T V into A with ψ(1) = e occurs in this way, where ϕ corresponds to ψ on V = T^{1}V. This corresponds to some of the remarks on p11 of [25], and on p89 of [14].

25.4 Defining universal enveloping algebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. A universal enveloping algebra of A is an associative algebra UA over k with a multiplicative identity element e = e_{UA} and a mapping i = i_{UA} from A into UA with the following two properties. First, i is a Lie algebra homomorphism from A into UA, considered as a Lie algebra over k with respect to the corresponding commutator bracket. This means that i is a homomorphism from A into UA as modules over k such that

i([x, y]_{A}) = i(x) i(y) − i(y) i(x) (25.4.1)

for every x, y ∈ A.

Second, let B be an associative algebra over k with a multiplicative identity element e_{B}, and let ϕ be a Lie algebra homomorphism from A into B, as a Lie algebra over k with respect to the corresponding commutator bracket. As before, this means that ϕ is a homomorphism from A into B, as modules over k, such that ϕ([x, y]_{A}) = ϕ(x) ϕ(y) − ϕ(y) ϕ(x) (25.4.2) for every x, y ∈ B. Under these conditions, there should be a unique algebra homomorphism ψ from UA into B such that ψ(e_{UA}) = e_{B} and

ψ ∘ i = ϕ (25.4.3)

on A. This corresponds to Definition 1.1 on p11 of [25], and to the definition on p90 of [14].

In particular, we can apply the second condition to B = UA and ϕ = i. This implies that there is a unique algebra homomorphism ψ from UA into itself such that ψ(e_{UA}) = e_{UA} and ψ ∘ i = i. The identity mapping on UA satisfies these properties, which means that the identity mapping on UA is the only mapping on UA with these properties.

It is easy to see that the pair (UA, i) is unique when it exists, up to isomorphic equivalence. Indeed, let ŨA be another associative algebra over k with a multiplicative identity element e_{_{U}A} and a Lie algebra homomorphism i_{_{U}A} from A into ŨA that is a universal enveloping algebra over A. Because i_{UA} and i_{_{U}A} are Lie algebra homomorphisms, there are unique algebra homomorphisms ψ from UA into ŨA and ψ̃ from ŨA into UA such that ψ(e_{UA}) = e_{_{U}A},

ψ̃(e_{_{U}A}) = e_{UA}, ψ ∘ i_{UA} = i_{_{U}A}, and ψ̃ ∘ i_{_{U}A} = i_{UA}. Observe that

(ψ̃ ∘ ψ) ∘ i_{UA} = ψ̃ ∘ (ψ ∘ i_{UA}) = ψ̃ ∘ i_{UA} = i_{UA}, (25.4.4)

and similarly (ψ ∘ ψ̃) ∘ i_{_{U}A} = i_{_{U}A}. Of course,

(ψ̃ ∘ ψ)(e_{UA}) = ψ̃(ψ(e_{UA})) = ψ̃(e_{_{U}A}) = e_{UA}, (25.4.5)

and similarly (ψ ∘ ψ̃)(e_{_{U}A}) = e_{_{U}A}. Clearly ψ̃ ∘ ψ is an algebra homomorphism from UA into itself, and ψ ∘ ψ̃ is an algebra homomorphism from ŨA into itself. It follows that ψ̃ ∘ ψ is the identity mapping on UA, and that ψ ∘ ψ̃ is the identity mapping on ŨA, as in the preceding paragraph. This corresponds to some remarks on p11 of [25], and on p90f of [14].

25.5 Constructing universal enveloping algebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. In particular, A is a module over k, so that the corresponding tensor algebra T A can be defined as in Section 25.3. Let I_{A} be the two-sided ideal in T A generated by elements of the form

[x, y]_{A} − x ⊗ y − y ⊗ x, (25.5.1)

where x, y ∈ A. Of course, this uses the obvious embeddings of A = T^{1}A and T^{2}A in T A. Thus I_{A} consists of elements of T A that can be obtained by multiplying elements of the form (25.5.1) by other elements of T A on the left and right, and taking finite sums of such products.

The corresponding quotient

UA = (T A)/I_{A} (25.5.2)

is an associative algebra over k. Let q_{A} be the canonical quotient mapping from T A onto UA. Remember that 1 ∈ k = T^{0}A is the multiplicative identity element in T A, so that e_{UA} = q_{A}(1) is the multiplicative identity element in UA. Let i = i_{UA} be the composition of the natural inclusion mapping of A = T^{1}A into T A with q_{A}. Let us check that (UA, i) is a universal enveloping algebra of A, as in Theorem 1.2 on p11 of [25], and as mentioned on p91 of [14].

It is easy to see that i is a Lie algebra homomorphism from A into UA, by construction. Let B be an associative algebra over k with a multiplicative identity element e_{B}, and let ϕ be a Lie algebra homomorphism from A into B. In particular, ϕ is a homomorphism from A into B as modules over k, and so there is a unique algebra homomorphism ρ from T A into B that is equal to ϕ on A = T^{1}A and sends 1 ∈ k = T^{0}A to e_{B}, as in Section 25.3. If x, y ∈ A, then one can check that (25.5.1) is in the kernel of ρ, because ϕ is a Lie algebra homomorphism from A into B. This implies that I_{A} is contained in the kernel of ρ, so that ρ can be expressed as

ρ = ψ ∘ q_{A} (25.5.3)

on T A for some algebra homomorphism ψ from UA into B. Note that ψ(e_{UA}) = ψ(q_{A}(1)) = ρ(1) = e_{B}. We can compose both sides of (25.5.3) with the natural inclusion of A = T^{1}A into T A to get that

ϕ = ψ ∘ i (25.5.4)

on A.

Suppose that ψ̃ is another algebra homomorphism from UA into B such that ψ̃(e_{UA}) = e_{B} and ϕ = ψ̃ ∘ i. This implies that ρ̃ = ψ̃ ∘ q_{A} is an algebra homomorphism from T A into B such that ρ̃(1) = ψ̃(q_{A}(1)) = ψ̃(e_{UA}) = e_{B} and ρ̃ = ϕ on A = T^{1}A. It follows that ρ̃ = ρ on T A, as in Section 25.3. This means that ψ̃ = ψ on UA, so that ψ is unique. This could also be obtained from the fact that UA is generated by e_{UA} and i(A), as an algebra over k, because T A is generated by the images of k = T^{0}A and A = T^{1}A in T A, as an algebra over k.

25.6 Some properties of UA

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. Also let q_{0} be the mapping from T A into k that sends each element of T A to its component in T^{0}A = k. This is an algebra homomorphism from T A onto k. The kernel of q_{0} is the two-sided ideal I_{0} in T A consisting of elements of T A whose component in T^{0}A is equal to 0, which corresponds to the direct sum of T^{n}A over n ≥ 1 in T A.

Suppose now that (A, [·, ·]_{A}) is a Lie algebra over k. Let I_{A} be the two-sided ideal in T A associated to A as in the previous section, and observe that

I_{A} ⊆ I_{0}. (25.6.1)

Remember that q_{A} is the canonical quotient mapping from T A onto UA = (T A)/I_{A}, as before. Using (25.6.1), we get that the restriction of q_{A} to T^{0}A = k is injective, as mentioned on p91 of [14]. We can also express q_{0} as the composition of q_{A} with an algebra homomorphism from UA onto k. Alternatively, the map from A into k that sends every element of A to 0 is a Lie algebra homomorphism. This leads to an algebra homomorphism from UA into k that sends e to 1.

Let V be a module over k, and remember that the space Hom_{k}(V, V) of module homomorphisms from V into itself is an associative algebra over k with respect to composition of mappings, and with the identity mapping on V as the multiplicative identity element. A Lie algebra homomorphism from A into Hom_{k}(V, V) is the same as a representation of A, as a Lie algebra over k, on V. This leads to an algebra homomorphism from UA into Hom_{k}(V, V) that sends e_{UA} to the identity mapping on V, and whose composition with i_{UA} is the given representation of A on V. Thus, if V is a module over A as a Lie algebra, then V may be considered as a left module over UA as an associative algebra, as in the remark on p11f of [25].

There is an exercise on p12 of [25], which is attributed to Bergman, and which asks one to show that UA = k if and only if A = {0}. More precisely, UA = k means that UA = q_{A}(k), which happens exactly when I_{A} = I_{0}. Of course, if A = {0}, then I_{0} = {0}, and I_{A} = I_{0}. Conversely, let us check that A = {0} when I_{A} = I_{0}. The exercise comes with the hint that one should use the adjoint representation. Remember that A may be considered as a module over itself, as a Lie algebra over k, using the adjoint representation. Using this, we may consider A as a left module over UA, as an associative algebra over k, as in the preceding paragraph. If I_{A} = I_{0}, then it follows that the action of any element of A on A by the adjoint representation is equal to 0, so that A is commutative as a Lie algebra. In this case, the condition that I_{A} = I_{0} implies that A = T^{1}A = {0}, as desired.

Let A_{0} be a Lie subalgebra of A, so that A_{0} also has a universal enveloping algebra UA_{0}, with multiplicative identity element e_{0} = e_{UA_{0}} and corresponding Lie algebra homomorphism i_{0} = i_{UA_{0}} from A_{0} into UA_{0}. Consider the restriction ϕ_{0} of i_{UA} to A_{0}, which is a Lie algebra homomorphism from A_{0} into UA. This leads to a unique algebra homomorphism ψ_{0} from UA_{0} into UA such that ψ_{0}(e_{0}) = e_{UA} and ψ_{0} ∘ i_{0} = ϕ_{0}.

25.7 Commuting Lie subalgebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA} and corresponding Lie algebra homomorphism i = i_{UA} from A into UA. Suppose that A_{1} and A_{2} are Lie subalgebras of A, and let UA_{1}, UA_{2} be universal enveloping algebras of A_{1}, A_{2}, respectively, with multiplicative identity elements e_{j} = e_{UA_{j}} and corresponding Lie algebra homomorphisms i_{j} = i_{UA_{j}} from A_{j} into UA_{j} for j = 1, 2. As in the previous section, the restriction ϕ_{j} of i to A_{j} is a Lie algebra homomorphism from A_{j} into UA for j = 1, 2, which leads to a unique algebra homomorphism ψ_{j} from UA_{j} into UA such that ψ_{j}(e_{j}) = e and ψ_{j} ∘ i_{j} = ϕ_{j}.

Suppose that A_{1} and A_{2} commute in A, in the sense that

[a_{1}, a_{2}]_{A} = 0 (25.7.1)

for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}. This implies that

i(a_{1}) i(a_{2}) = i(a_{2}) i(a_{1}) (25.7.2)

in UA for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}, because i is a Lie algebra homomorphism from A into UA. Using this, we get that

ψ_{1}(x_{1}) ψ_{2}(x_{2}) = ψ_{2}(x_{2}) ψ_{1}(x_{1}) (25.7.3)

in UA for every x_{1} ∈ UA_{1} and x_{2} ∈ UA_{2}. More precisely, this follows from (25.7.2) when x_{j} ∈ i_{j}(A_{j}) for j = 1, 2. Of course, (25.7.3) holds automatically when x_{1} = e_{1} or x_{2} = e_{2}. It follows that (25.7.3) holds for every x_{1} ∈ UA_{1} and x_{2} ∈ UA_{2}, because UA_{j} is generated by e_{j} and i_{j}(A_{j}) as an algebra over k for j = 1, 2. Let ψ be the unique linear mapping from (UA_{1}) ⨂(UA_{2}) into UA such that ψ(x_{1} ⊗ x_{2}) = ψ_{1}(x_{1}) ψ_{2}(x_{2}) (25.7.4)

for every x_{1} ∈ UA_{1} and x_{2} ∈ UA_{2}. This is an algebra homomorphism, as in Section 25.2, with ψ(e_{1} ⊗ e_{2}) = ψ_{1}(e_{1}) ψ_{2}(e_{2}) = e.

Suppose now that A corresponds to the direct sum of A_{1} and A_{2}, so that every element of A can be expressed in a unique way as a sum of elements of A_{1} and A_{2}. Observe that a_{1} ↦ i_{1}(a_{1}) ⊗ e_{2} (25.7.5)

is a Lie algebra homomorphism from A_{1} into (UA_{1}) ⨂(UA_{2}), and similarly that

a_{2} ↦ e_{1} ⊗ i_{2}(a_{2}) (25.7.6)

is a Lie algebra homomorphism from A_{2} into (UA_{1}) ⨂(UA_{2}). If a_{1} ∈ A_{1} and a_{2} ∈ A_{2}, then put

ρ(a_{1} + a_{2}) = i_{1}(a_{1}) ⊗ e_{2} + e_{1} ⊗ i_{2}(a_{2}), (25.7.7)

which defines a mapping from A into (UA_{1}) ⨂(UA_{2}). It is easy to see that ρ is a Lie algebra homomorphism from A into (UA_{1}) ⨂(UA_{2}), using (25.7.1). This implies that there is a unique algebra homomorphism θ from UA into (UA_{1}) ⨂(UA_{2}) such that θ(e) = e_{1} ⊗ e_{2} and ρ = θ ∘ i.

Let us check that ψ∘θ is the identity mapping on UA. Of course, (ψ∘θ)(e) = ψ(θ(e)) = ψ(e_{1} ⊗ e_{2}) = e. If a_{1} ∈ A_{1} and a_{2} ∈ A_{2}, then

(25.7.8) (ψ ∘ θ)(i(a_{1} + a_{2})) = ψ(ρ(a_{1} + a_{2}))
= ψ(i_{1}(a_{1}) ⊗ e_{2}) + ψ(e_{1} ⊗ i_{2}(a_{2}))
= ψ_{1}(i_{1}(a_{1})) ψ_{2}(e_{2}) + ψ_{1}(e_{1}) ψ_{2}(i_{2}(a_{2}))
= ϕ_{1}(a_{1}) + ϕ_{2}(a_{2}) = i(a_{1} + a_{2}).

This implies that ψ ∘ θ is the identity mapping on UA, because UA is generated as an algebra over k by e and i(A).

Similarly, we would like to verify that θ ∘ ψ is the identity mapping on (UA_{1}) ⨂(UA_{2}). It suffices to show that

(θ ∘ ψ)(x_{1} ⊗ x_{2}) = x_{1} ⊗ x_{2} (25.7.9)

for every x_{1} ∈ UA_{1} and x_{2} ∈ UA_{2}. Observe that

(θ ∘ ψ)(x_{1} ⊗ x_{2}) = θ(ψ(x_{1} ⊗ x_{2}))
(25.7.10)  = θ(ψ_{1}(x_{1}) ψ_{2}(x_{2})) = θ(ψ_{1}(x_{1})) θ(ψ_{2}(x_{2}))

for every x_{1} ∈ UA_{1} and x_{2} ∈ UA_{2}.

Let us check that θ(ψ_{1}(x_{1})) = x_{1} ⊗ e_{2} (25.7.11)

for every x_{1} ∈ UA_{1}. Of course, θ(ψ_{1}(e_{1})) = θ(e) = e_{1} ⊗ e_{2}, so that (25.7.11) holds when x_{1} = e_{1}. If a_{1} ∈ A_{1}, then

θ(ψ_{1}(i_{1}(a_{1}))) = θ(i(a_{1})) = ρ(a_{1}) = i_{1}(a_{1}) ⊗ e_{2}. (25.7.12)

This shows that (25.7.11) holds when x_{1} ∈ i_{1}(A_{1}). It follows that (25.7.11) holds for every x_{1} ∈ UA_{1}, because UA_{1} is generated by e_{1} and i_{1}(A_{1}), as an algebra over k.

If x_{2} ∈ UA_{2}, then θ(ψ_{2}(x_{2})) = e_{1} ⊗ x_{2}, (25.7.13)

for the same reasons. If x_{1} ∈ UA_{1} as well, then we get that

(25.7.14)  (θ ∘ ψ)(x_{1} ⊗ x_{2}) = θ(ψ_{1}(x_{1})) θ(ψ_{2}(x_{2}))
= (x_{1} ⊗ e_{2}) (e_{1} ⊗ x_{2}) = x_{1} ⊗ x_{2},

as desired. Thus UA is isomorphic to (UA_{1}) ⨂(UA_{2}) in a natural way, as on p12 of [25].

25.8 Symmetric algebras

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. We may consider A as a commutative Lie algebra over k, by putting [x, y]_{A} = 0 for every x, y ∈ A. In this case, a universal enveloping algebra of A as a Lie algebra over k is called a symmetric algebra of A as a module over k, and is denoted SA, as on p12 of [25].

More precisely, a symmetric algebra SA of A comes with a multiplicative identity element e = e_{SA}, and a Lie algebra homomorphism i = i_{SA} from A into SA. In this situation, the condition that i be a Lie algebra homomorphism means that i is a homomorphism from A into SA, as modules over k, and that

i(x) i(y) = i(y) i(x) (25.8.1)

for every x, y ∈ A. This implies that SA is commutative as an algebra over k, because it is generated by i(A) and e, as before.

The two-sided ideal I_{A} in the tensor algebra T A defined in Section 25.5 is generated by elements of the form

x ⊗ y − y ⊗ x (25.8.2)

with x, y ∈ A in this situation. This means that I_{A} consists of elements of T A that can be obtained by multiplying elements of the form (25.8.2) by other elements of T A on the left and right, and taking finite sums of such elements, as before. Thus a symmetric algebra of A can be obtained as the quotient

SA = (T A)/I_{A}, (25.8.3)

as in Section 25.5. This corresponds to the definition of the symmetric algebra on p89 of [14], and the characterization as a universal enveloping algebra of a commutative Lie algebra is mentioned in the example on p91 of [14].

Let us identify T^{n}A with a submodule of T A, as a module over k, in the obvious way. It is easy to see that I_{A} corresponds to the direct sum of I_{A}∩(T^{n}A) over n ≥ 2 in T V. More precisely, I_{A} ∩(T^{n}A) consists of finite sums of elements that can be expressed as the product of the form (25.8.2) by an element of T^{j}A on the left and T^{l}A on the right, where j, l as nonnegative integers such that j + l + 2 = n. In particular, I_{A} ∩ (T^{n}A) = {0} when n = 0 or 1. If n is any nonnegative integer, then I_{A} ∩ (T^{n}A) is a submodule of T^{n}, as a module over k, so that the quotient

S^{n}A = (T^{n}A)/(I_{A} ∩ (T^{n}A)) (25.8.4)

can be defined as a module over k. Thus (25.8.3) may be considered as the direct sum of S^{n}A over n ≥ 0, as a module over k. Note that S^{0}A = T^{0}A = k, and S^{1}A = T^{1}A = A.

Let SA be a symmetric algebra of A again, with e = e_{SA} and i = i_{SA} as before. The second property in the definition of a universal enveloping algebra says that if B is an associative algebra over k with a multiplicative identity element e_{B}, and ϕ is a Lie algebra homomorphism from A into B, then there is a unique algebra homomorphism ψ from SA into B such that ψ(e) = e_{B} and ψ ∘ i = ϕ. In this situation, a mapping ϕ from A into B is a Lie algebra homomorphism if ϕ is a homomorphism from A into B, as modules over k, and

ϕ(x) ϕ(y) = ϕ(y) ϕ(x) (25.8.5)

25.9. A FILTRATION ON UA 543

for every x, y ∈ A. Of course, (25.8.5) holds automatically when B is commutative, in which case this corresponds to a remark on p90 of [14]. Remember that SA is unique up to isomorphism, as in Section 25.4.

Suppose now that A is free as a module over k, so that A corresponds to the direct sum of a nonempty family of copies of k. Equivalently, there is a nonempty set I and an element e_{j} of A for each j ∈ I so that every element of A can be expressed in a unique way as a finite linear combinations of e_{j}’s, with coefficients in k. Let X_{j}, j ∈ I, be a commuting family of indeterminates, so that the corresponding polynomial algebra k[{X_{j} : j ∈ I}] can be defined as in Section 25.1. There is a unique homomorphism i from A into k[{X_{j} : j ∈ I}], as modules over k, such that i(e_{j}) = X_{j} (25.8.6)

for every j ∈ I. Let us check that this satisfies the requirements of a symmetric algebra of A.

Note that i is a Lie algebra homomorphism from A into k[{X_{j} : j ∈ I}], because k[{X_{j} : j ∈ I}] is a commutative algebra over k. Let B be an associative algebra over k with a multiplicative identity element e_{B}, and let ϕ be a Lie algebra homomorphism from A into B, where A is considered as a commutative Lie algebra over k. This means that ϕ is a homomorphism from A into B, as modules over k, that satisfies (25.8.5), as before. Put

b_{j} = ϕ(e_{j}) (25.8.7)

for each j ∈ I, so that b_{j} b_{l} = b_{l} b_{j} for every j, l ∈ I. Let b be the element of the Cartesian power B^{I} whose jth coordinate is equal to b_{j} for each j ∈ I. If f(X) ∈ k[{X_{j} : j ∈ I}], then put

ψ(f(X)) = f(b), (25.8.8)

where the right side is as in Section 25.1. Thus ψ is an algebra homomorphism from k[{X_{j} : j ∈ I}] into B, as before, and it is easy to see that ψ ∘ i = ϕ on A. Of course, ψ(1) = e_{B}, by construction, and ψ is uniquely determined by these properties. This corresponds to some remarks on p12 of [25], and is mentioned on p90 of [14] as well.

25.9 A filtration on UA

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let UA be a universal enveloping algebra of A, with multiplicative identity element e_{UA} and Lie algebra homomorphism i_{UA} from A into UA. If n is a nonnegative integer, then we let U_{n}A be the submodule of UA, as a module over k, generated by products of the form

i_{UA}(x_{1}) ⋯ i_{UA}(x_{m}), (25.9.1)

where x_{1}, …, x_{m} ∈ A and m ≤ n. We interpret e_{UA} as being such a product with m = 0, so that multiples of e_{UA} by elements of k are in U_{n}A for every n ≥ 0. More precisely, U_{0}A consists exactly of multiples of e_{UA} by elements of k, and U_{n}A ⊆ U_{n+1}A (25.9.2)

for every n ≥ 0. Note that ⋃_{n=0}^{∞} U_{n}A = UA. This is the way that U_{n}A is defined on p13 of [25].

Alternatively, we can take UA to be obtained from the tensor algebra T A as in Section 25.5. If n is a nonnegative integer, then let T_{n}A be the submodule of T A that corresponds to the direct sum of T^{m}A over m = 0, …, n. Thus T_{0}A corresponds to T^{0}A, as a submodule of T A, T_{n}A ⊆ T_{n+1}A for every n ≥ 0, and T A = ⋃_{n=0}^{∞} T_{n}A. If UA is the quotient of T A discussed before, then U_{n}A corresponds to the image of T_{n}A under the quotient mapping from T A onto UA. This is how U_{n}A is defined on p91 of [14].

If n is a positive integer, then put

gr_{n}UA = (U_{n}A)/(U_{n−1}A), (25.9.3)

where the quotient is defined as a module over k. Also put gr_{0} UA = U_{0}A, and

gr UA = ⨁_{n=0}^{∞} gr_{n}UA, (25.9.4)

considered as a module over k for the moment. If n, r are nonnegative integers, then multiplication on UA defines a bilinear mapping from (U_{n}A) × (U_{r}A) into U_{n+r}A. It is easy to see that this determines a bilinear mapping from (gr_{n} UA) × (gr_{r} UA) into gr_{n+r} UA in a natural way. This defines a bilinear mapping of multiplication from (gr UA) × (gr UA) into gr UA. One can check that multiplication on gr UA is associative, and that the element of gr UA corresponding to e_{UA} ∈ U_{0}A = gr_{0} UA is the multiplicative identity element. This is called the graded algebra associated to UA, as on p13 of [25].

Note that i_{UA} maps A into U_{1}A, which is mapped into gr_{1} UA by the quotient mapping. This gives a natural homomorphism from A into gr UA, as modules over k. One can check that gr UA is generated, as an algebra over k, by the image of A under this mapping and the multiplicative identity element, as in Proposition 4.1 on p13 of [25].

Theorem 4.2 on p13 of [25] states that gr UA is commutative as an algebra over k. To see this, let x, y ∈ A be given, and remember that

i_{UA}(x) i_{UA}(y) − i_{UA}(y) i_{UA}(x) = i_{UA}([x, y]_{A}) (25.9.5)

in UA, because i_{UA} is a Lie algebra homomorphism from A into UA. Of course, i_{UA}(x) i_{UA}(y) and i_{UA}(y) i_{UA}(x) are elements of U_{2}A, which are mapped to the same element of gr_{2} UA, by (25.9.5). This means that the images of x, y in gr UA commute. It follows that gr UA is commutative, because gr UA is generated as an algebra by the image of A and the multiplicative identity element, as in the preceding paragraph.

Let SA be a symmetric algebra of A, as a module over k, with multiplicative identity element e_{SA}, and homomorphism i_{SA} from A into SA, as modules over k. Because gr UA is commutative, there is a unique algebra homomorphism ρ from SA into gr UA such that ρ(e_{SA}) is the multiplicative identity element in gr UA, and ρ ∘ i_{SA} is the natural mapping from A into gr UA. More precisely, ρ maps SA onto gr UA, because gr UA is generated as an algebra by the image of A and the multiplicative identity element, as mentioned on p13 of [25].

This corresponds to the lemma on p91 of [14], which is formulated with UA given by the quotient of T A as in Section 25.5. If n is a nonnegative integer, then there is a natural module homomorphism from T^{n}A into U_{n}A, using the quotient mapping from T A onto UA. More precisely, this mapping can be obtained by first mapping T^{n}A into T_{n}A, and mapping T_{n}A onto U_{n}A using the quotient mapping from T A onto UA. This leads to a module homomorphism from T^{n}A into gr_{n} UA, by composition with the quotient mapping from U_{n}A onto gr_{n} UA. In fact, this module homomorphism maps T^{n}A onto gr_{n} UA. We can combine these module homomorphisms to get a module homomorphism from T A onto gr UA. This maps the multiplicative identity element in T A to the multiplicative identity element in gr UA. One can check that this is an algebra homomorphism from T A onto gr UA.

Remember that a symmetric algebra SA of A can be obtained as a quotient of T A as well. One can verify that the ideal in T A used to get SA is contained in the kernel of the homomorphism from T A onto gr UA mentioned in the preceding paragraph, using (25.9.5). This leads to an algebra homomorphism from SA onto gr UA, as before.

25.10 The Poincaré–Birkhoff–Witt theorem

Let us continue with the same notation and hyotheses as in the previous section, and suppose also that A is a free module over k. The Poincaré–Birkhoff–Witt theorem states that the homomorphism ρ from SA onto gr UA defined in the previous section is an isomorphism, as in Theorem 4.3 on p14 of [25]. Of course, if k is a field, so that A is a vector space over k, then A is automatically free as a module over k. This corresponds to the formulation of the theorem on p92 of [14].

To say that A is a free module over k means that there is a nonempty set I and a family {x_{j}}_{j∈I} of elements of A indexed by I such that every element of A can be expressed in a unique way as a finite linear combination of the x_{j}’s, with coefficients in k. We may refer to {x_{j}}_{j∈I} as a basis for A, as a free module over k. Let ⪯ be a linear ordering on I.

Let n be a nonnegative integer, and let j_{1}, …, j_{m} be m elements of I for some m ≤ n, with j_{1} ⪯ ⋯ ⪯ j_{m}. (25.10.1)

Thus the product i_{UA}(x_{j_{1}}) ⋯ i_{UA}(x_{j_{m}}) (25.10.2)

is an element of U_{n}A. As before, we interpret (25.10.2) as being equal to e_{UA} when m = 0, in which case (25.10.1) is considered to hold vacuously. Lemma 4.4 on p14 of [25] states that U_{n}A is generated, as a module over k, by products of the form (25.10.2).

The proof is by induction on n, with the base case n = 0 being trivial. Suppose that n ≥ 1, and remember that U_{n}A is generated as a module over k by products of the form (25.10.2) without the condition (25.10.1), by definition. Using (25.9.5), we can express an element of U_{n}A as a linear combination of terms of the form (25.10.2), with coefficients in k, m = n, and the condition (25.10.1), plus an element of U_{n−1}A. This implies that U_{n}A is generated as a module over k in the desired way, because of the corresponding property for U_{n−1}A, by the induction hypothesis.

Consider the following statement:

the family of products of the form (25.10.2) with (25.10.1)

(25.10.3)

and m ≥ 0 is a basis for UA as a module over k.

Lemma 4.5 on p14 of [25] asserts that this is equivalent to the Poincaré–Birkhoff– Witt theorem. Lemma C on p92 of [14] corresponds to the fact that the theorem implies (25.10.3).

If m ∈ Z_{+}, then let M_{m}(I) be the set of m-tuples M = (j_{1}, …, j_{m}) of elements of I that satisfy (25.10.1). We consider M = ∅ to be the unique element of M_{0}(I), and put

M(I) = ⋃_{m=0}^{∞} M_{m}(I). (25.10.4)

If M ∈ M_{m}(I) for some m ≥ 0, then m is called the length of M, and may be denoted length(M).

Remember that M_{I} is the set of all multi-indices with respect to I, as in Section 25.1. If α ∈ M_{I}, then let M_{α} ∈ M(I) be the m-tuple with m = |α| obtained by taking the j ∈ I with α(j) ≠ 0 in order and with multiplicity α(j), so that M_{0} = ∅. Similarly, if M ∈ M_{m}(I) for some m ≥ 0, then we can define α_{M} ∈ M_{I} by taking α_{M}(j) to be the number of coordinates of M equal to j for each j ∈ I. This defines a one-to-one correspondence between M(I) and M_{I}.

Let X_{j}, j ∈ I, be a commuting family of indeterminates, with the corresponding polynomial algebra k[{X_{j} : j ∈ I}], as in Section 25.1. There is a unique homomorphism from A into k[{X_{j} : j ∈ I}], as modules over k, that sends x_{j} ∈ A to X_{j} for every j ∈ I. Using this module homomorphism, k[{X_{j} : j ∈ I}] satisfies the requirements of a symmetric algebra of A, as in Section 25.8. Of course, the collection of monomials X^{α}, α ∈ M_{I}, is a basis for k[{X_{j} : j ∈ I}] as a module over k.

If M = (j_{1}, …, j_{n}) ∈ M_{n}(I) for some n ≥ 0, then put

x_{M} = i_{UA}(x_{j_{1}}) ⋯ i_{UA}(x_{j_{n}}), (25.10.5)

which is interpreted as being e_{UA} when M = ∅, as before. This is an element of U_{n}A, and we let \overline{x_{M}} be its image in gr_{n} UA under the corresponding quotient mapping. Using k[{X_{j} : j ∈ I}] as the symmetric algebra of A, the mapping ρ from SA onto gr UA defined in the previous section corresponds exactly to the module homomorphism from k[{X_{j} : j ∈ I}] into gr UA with

X^{α} ↦ \overline{x_{M_{α}}} (25.10.6)

for every α ∈ M_{I}.

The Poincaré–Birkhoff–Witt theorem is equivalent to the statement that ρ is injective, because we already know that it is a surjective algebra homomorphism. The injectivity of ρ is equivalent to saying that if n ≥ 0, c_{M} ∈ k for every M ∈ M_{n}(I), and

∑_{M∈M_{n}(I)} c_{M} \overline{x_{M}} = 0 (25.10.7)

in gr_{n} UA, then c_{M} = 0 for every M ∈ M_{n}(I). This holds trivially when n = 0, and so we may as well take n ≥ 1. Note that (25.10.7) is the same as saying that

∑_{M∈M_{n}(I)} c_{M} x_{M} ≡ 0 modulo U_{n−1}A (25.10.8)

in U_{n}A when n ≥ 1. This means that there are c_{M} ∈ k for M ∈ M(I) with length(M) ≤ n − 1 such that

∑_{M∈M_{n}(I)} c_{M} x_{M} = ∑_{{M∈M(I) \atop length(M)<n}} c_{M}x_{M} (25.10.9)

in U_{n}A, because U_{n−1}A is generated as a module over k by the x_{M} with M in M(I) and length(M) ≤ n − 1, as before.

Of course, (25.10.3) is the same as saying that the family of x_{M}, M ∈ M(I), is a basis for UA as a module over k. If this holds, then (25.10.9) implies that c_{M} = 0 for every M ∈ M(I) with length(M) ≤ n, and in particular when length(M) = n, as desired. Conversely, if ρ is injective, then (25.10.9) implies that c_{M} = 0 for every M ∈ M(I) with length(M) = n, as in the preceding paragraph. In this case, one can repeat the process to get that c_{M} = 0 for every M ∈ M(I) with length(M) ≤ n. This implies that the family of x_{M}, M ∈ M(I) is a basis for UA as a module over k, because UA is generated by the x_{M}’s, M ∈ M(I), as a module over k.

25.11 The rest of the proof

Let us continue with the same notation and hypotheses as in the previous two sections. In order to prove the Poincaré–Birkhoff–Witt theorem, we take ⪯ to be a well-ordering on I. Let V be a free module over k with a basis consisting of elements Z_{M} for each M ∈ M(I), as on p14 of [25]. As a module over k, V is isomorphic to k[{X_{j} : j ∈ I}], which is a symmetric algebra of A, and the argument on p93 of [14] uses this.

If j ∈ I and M = (j_{1}, …, j_{n}) ∈ M(I), then we put j ⪯ M when j ⪯ j_{1}. In this case, we put j M = (j, j_{1}, …, j_{n}), (25.11.1)

which is an element of M(I) too. If M = ∅, then j ⪯ M is interpreted as holding automatically, and j M has only the one coordinate j. The main lemma on p14 of [25] states that V can be made a module over A, as a Lie algebra over k, in such a way that x_{j} · Z_{M} = Z_{j M} (25.11.2)

for every j ∈ I and M ∈ M(I) with j ⪯ M. This corresponds to Lemma B on p94 of [14], part (a) in particular.

We first want to define (x, v) ↦ x · v as a bilinear mapping from A × V into V. To do this, it suffices to define x_{j} · Z_{M} as an element of V for every j ∈ I and M ∈ M(I). More precisely, we want to do this in such a way that

(25.11.3) x_{j} · Z_{M} is a finite linear combination of Z_{L}’s with coefficients in k, L ∈ M(I), and length(L) ≤ length(M) + 1.

Note that (25.11.2) has this property.

Let j ∈ I and M ∈ M(I) be given. Suppose that x_{j_{0}} · Z_{N} ∈ V has already been defined and satisfies (25.11.3) for j_{0} ∈ I and N ∈ M(I) such that either

(25.11.4) length(N) < length(M),

or (25.11.5) length(N) = length(M), j_{0} ⪯ j, and j_{0} ≠ j.

If j ⪯ M, then x_{j} · Z_{M} can be defined as in (25.11.2). Otherwise, there are j_{1} ∈ I and N ∈ M(I) such that

j_{1} ⪯ N, M = j_{1} N, j_{1} ⪯ j, and j_{1} ≠ j. (25.11.6)

In this case, we would like to define x_{j} · Z_{M} ∈ V by

x_{j} · Z_{M} = x_{j_{1}} · (x_{j} · Z_{N}) + ([x_{j}, x_{j_{1}}]_{A}) · Z_{N}. (25.11.7)

It is easy to see that the right side is defined as an element of V under these conditions, and that it satisfies (25.11.3) as well. One can use this to define x_{j} · Z_{M} for every j ∈ I and M ∈ M(I), because I is well ordered by ⪯.

In order to show that V is a module over A as a Lie algebra over k, we need to check that x · (y · v) − y · (x · v) = ([x, y]_{A}) · v (25.11.8)

for every x, y ∈ A and v ∈ V. To do this, it suffices to verify that

x_{j} · (x_{l} · Z_{N}) − x_{l} · (x_{j} · Z_{N}) = ([x_{j}, x_{l}]_{A}) · Z_{N} (25.11.9)

for every j, l ∈ I and N ∈ M(I). Arguing by induction on length(N) and min(j, l) as on p15 of [25], we may suppose that

x_{j_{0}} · (x_{l_{0}} · Z_{N_{0}}) − x_{l_{0}} · (x_{j_{0}} · Z_{N_{0}}) = ([x_{j_{0}}, x_{l_{0}}]_{A}) · Z_{N_{0}} (25.11.10)

25.11. THE REST OF THE PROOF 549 holds for every j_{0}, l_{0} ∈ I and N_{0} ∈ M(I) such that either (25.11.11) length N_{0} < length N or (25.11.12) length(N_{0}) = length(N), min(j_{0}, l_{0}) ⪯ min(j, l),

and min(j_{0}, l_{0}) ≠ min(j, l).

We may as well suppose that l ⪯ j and l ≠ j, (25.11.13) because both sides of (25.11.9) are antisymmetric in j and l, and are equal to 0 when j = l. Suppose first that l ⪯ N, so that x_{l} · Z_{N} = Z_{l N}. Thus (25.11.7) holds with M = l N and j_{1} = l. This implies (25.11.9), as desired.

Otherwise, there are r ∈ I and L ∈ M(I) such that r ⪯ L, N = r L, r ⪯ l, and r ≠ l. (25.11.14) This means that Z_{N} = Z_{r L} = x_{r} · Z_{l}, as in (25.11.2). In this case, (25.11.9) is the same as saying that x_{j} · (x_{l} · (x_{r} · Z_{L})) − x_{l} · (x_{j} · (x_{r} · Z_{L})) = ([x_{j}, x_{l}]_{A}) · (x_{r} · Z_{L}). (25.11.15)

Consider the analogous equations x_{l} · (x_{r} · (x_{j} · Z_{L})) − x_{r} · (x_{l} · (x_{j} · Z_{L})) = ([x_{l}, x_{r}]_{A}) · (x_{j} · Z_{L}) (25.11.16) and x_{r} · (x_{j} · (x_{l} · Z_{L})) − x_{j} · (x_{r} · (x_{l} Z_{L})) = ([x_{r}, x_{j}]_{A}) · (x_{l} Z_{L}), (25.11.17) where the indices j, l, and r have been cyclically permuted. These two equations hold by induction, because min(l, r) = min(r, j) = r is strictly less than min(j, l) = l. This also uses the fact that x_{j} · Z_{L} and x_{l} · Z_{L} can be expressed as finite linear combinations of Z_{N_{0}}’s with N_{0} ∈ M(I) and (25.11.18) length(N_{0}) ≤ length(L) + 1 = length(N), as in (25.11.3).

If x, y ∈ A, then x · (y · Z_{L}) − y · (x · Z_{L}) = ([x, y]_{A}) · Z_{L}, (25.11.19) by induction, because (25.11.20) length L = length(N) − 1 < length(N).

This implies that

([x_{j}, x_{l}]_{A}) · (x_{r} · Z_{L}) = x_{r} · ([x_{j}, x_{l}]_{A} · Z_{L}) + ([[x_{j}, x_{l}]_{A}, x_{r}]_{A}) · Z_{L}
(25.11.21)  = x_{r} · (x_{j} · (x_{l} · Z_{L})) − x_{r} · (x_{l} · (x_{j} · Z_{L}))
+([[x_{j}, x_{l}]_{A}, x_{r}]_{A}) · Z_{L},

using (25.11.19) in both steps. The left side of (25.11.21) is the same as the right side of (25.11.15), so that (25.11.15) is equivalent to

x_{j} · (x_{l} · (x_{r} · Z_{L})) − x_{l} · (x_{j} · (x_{r} · Z_{L})) (25.11.22)

= x_{r} · (x_{j} · (x_{l} · Z_{L})) − x_{r} · (x_{l} · (x_{j} · Z_{L}))
+([[x_{j}, x_{l}]_{A}, x_{r}]_{A}) · Z_{L}.

Similarly, (25.11.16) and (25.11.17) imply that the equations analogous to (25.11.22) but with the indices j, l, and r permuted cyclically hold. Consider the equation obtained by adding the left and right sides of the three equations corresponding to (25.11.22) and the two analogous equations for (25.11.16) and (25.11.17). The terms on the left side of the new equation all cancel with terms on the right side of the new equation. There are also three terms on the right side of the new equation that correspond to the last term on the right side of (25.11.22), and the two analogous terms with the indices permuted cyclically. The sum of these three terms is equal to 0, by the Jacobi identity.

This means that the new equation holds automatically. Because the two equations analogous to (25.11.22) with the indices permuted cyclically hold, we get that (25.11.22) holds as well. It follows that (25.11.15) holds, so that (25.11.9) holds, as desired. This shows that V is a module over A as a Lie algebra over k.

We may consider V as a left module over UA too, as an associative algebra over k, as in Section 25.6. If M ∈ M(I), then x_{M} ∈ UA can be defined as in (25.10.5). One can check that

x_{M} · Z_{∅} = Z_{M}, (25.11.23)

using (25.11.2) and induction on the length of M. In particular, if M = ∅, then x_{M} is interpreted as being e_{UA}, which acts on V by the identity mapping.

Suppose that c_{M} is an element of k for every M ∈ M(I), with c_{M} = 0 for all but finitely many M, and

∑_{M∈M(I)} c_{M} x_{M} = 0 (25.11.24)

in UA. This implies that

∑_{M∈M(I)} c_{M} Z_{M} = ∑_{M∈M(I)} c_{M} x_{M} · Z_{∅} = 0 (25.11.25)

in V. It follows that c_{M} = 0 for every M ∈ M(I), because the Z_{M}’s form a basis for V as a free module over k, by construction. This means that the family of x_{M}, M ∈ M(I), is a basis for UA as a module over k, because UA is generated by the x_{M}’s as a module over k, as in the previous section. This completes the proof of the Poincaré–Birkhoff–Witt theorem, as before.

25.12 Some corollaries

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA} and Lie algebra homomorphism i = i_{UA} from A into UA. Remember that the submodule U_{n}A of UA, as a module over k, is defined for nonnegative integers n as in Section 25.9. In particular, U_{0}A consists of multiples of e by elements of k, and U_{1}A consists of sums of elements of U_{0}A and i(A).

More precisely, U_{1}A corresponds to the direct sum of U_{0}A and i(A), as a module over k, as indicated on p13 of [25]. This can be seen using the natural algebra homomorphism from UA onto k mentioned in Section 25.6. This implies that the restriction of the quotient mapping from U_{1}A onto gr_{1} UA = (U_{1}A)/(U_{0}A) to i(A) is a one-to-one mapping from i(A) onto gr_{1} UA.

If A is free as a module over k, then the Poincaré–Birkhoff–Witt theorem implies that i is an injective mapping from A into UA. This is Corollary 1 on p16 of [25], which corresponds to Corollary B on p92 of [14].

Let A_{0} be a Lie subalgebra of A, and let UA_{0} be a universal enveloping algebra of A_{0}, with multiplicative identity element e_{0} = e_{UA_{0}} and Lie algebra homomorphism i_{0} = i_{UA_{0}} from A_{0} into UA_{0}. As in Section 25.6, the restriction ϕ_{0} of i to A_{0} is a Lie algebra homomorphism from A_{0} into UA, which leads to a unique algebra homomorphism ϕ_{0} from UA_{0} into UA such that ψ_{0}(e_{0}) = e and ψ_{0} ∘ i_{0} = ϕ_{0}. Suppose that A_{0} is free as a module over k, with basis {x_{j}}_{j∈I_{0}} for some nonempty set I_{0}. Suppose also that A is free as a module over k, and that {x_{j}}_{j∈I_{0}} can be extended to a basis {x_{j}}_{j∈I} of A, with I_{0} ⊆ I. Let ⪯ be a linear ordering on I, with j_{0} ⪯ l for every j_{0} ∈ I_{0} and l ∈ I \I_{0}. Thus M(I) and x_{M} ∈ UA, M ∈ M(I), can be defined as in Section 25.10, and the family of x_{M}, M ∈ I, is a basis for UA, as a module over k, by the Poincaré–Birkhoff–Witt theorem. This implies that ψ_{0} is an injective mapping from UA_{0} into UA, and more precisely that UA may be considered as a free module over UA_{0}, using x_{M} with M ∈ M(I \ I_{0}). This corresponds to Corollary D on p92 of [14].

Let A_{1} and A_{2} be Lie subalgebras of A, and let UA_{1}, UA_{2} be universal enveloping algebras of A_{1}, A_{2}, respectively, with multiplicative identity elements e_{j} = e_{UA_{j}} and Lie algebra homomorphisms i_{j} = i_{UA_{j}} from A_{j} into UA_{j} for j = 1, 2. As before, the restriction ϕ_{j} of i to A_{j} is a Lie algebra homomorphism from A_{j} into UA for j = 1, 2, which leads to a unique algebra homomorphism ψ_{j} from UA_{j} into UA such that ψ_{j}(e_{j}) = e and ψ_{j} ∘ i_{j} = ϕ_{j}. Observe that

(w_{1}, w_{2}) ↦ ψ_{1}(w_{1}) ψ_{2}(w_{2}) (25.12.1)

is a bilinear mapping from (UA_{1}) × (UA_{2}) into UA. Using this, we get a unique linear mapping from (UA_{1}) ⨂(UA_{2}) into UA with

w_{1} ⊗ w_{2} ↦ ψ_{1}(w_{1}) ψ_{2}(w_{2}) (25.12.2)

for every w_{1} ∈ UA_{1} and w_{2} ∈ UA_{2}.

Suppose that A corresponds to the direct sum of A_{1} and A_{2}, as a module over k, and that A_{1} and A_{2} are free as modules over k. Let {x_{j}}_{j∈I_{1}} and {x_{j}}_{j∈I_{2}} be bases for A_{1} and A_{2}, respectively, where I_{1} ∩ I_{2} = ∅. Of course, A is a free module over k as well, with basis {x_{j}}_{j∈I}, where I = I_{1} ∪ I_{2}. Let ⪯ be a linear ordering on I, with j_{1} ⪯ j_{2} for every j_{1} ∈ I_{1} and j_{2} ∈ I_{2}. Note that the restrictions of ⪯ to I_{1} and I_{2} define linear orderings on those sets.

Let M(I) and x_{M} ∈ UA, M ∈ M(I), be as in Section 25.10, so that the family of x_{M}, M ∈ M(I), is a basis for UA, as a module over k, by the Poincaré– Birkhoff–Witt theorem. Of course, we get analogous bases for UA_{1} and UA_{2}, as modules over k, indexed by M(I_{1}) and M(I_{2}), respectively. This leads to a basis for (UA_{1}) ⨂(UA_{2}), as a module over k, indexed by M(I_{1})×M(I_{2}). Elements of M(I_{1}) and M(I_{2}) can be combined to get elements of M(I), and every element of M(I) can be obtained by combining unique elements of M(I_{1}) and M(I_{2}) in this way. This defines a one-to-one correspondence between M(I_{1}) × M(I_{2}) and M(I).

The linear mapping from (UA_{1}) ⨂(UA_{2}) into UA determined by (25.12.2) defines a one-to-one correspondence between the bases for (UA_{1}) ⨂(UA_{2}) and UA, as modules over k, defined in the preceding paragraph. In particular, this linear mapping is an isomorphism between (UA_{1}) ⨂(UA_{2}) and UA, as modules over k, as in Corollary 2 on p16 of [25].

In this case, we also have an induced isomorphism from (gr UA_{1}) ⨂(gr UA_{2}) onto gr UA, as on p16 of [25]. More precisely, remember that gr UA is isomorphic to SA, as in the Poincaré–Birkhoff–Witt theorem, and similarly gr UA_{1}, gr UA_{2} are isomorphic to SA_{1}, SA_{2}, respectively. The corresponding isomorphism from (SA_{1}) ⨂(SA_{2}) onto SA can be seen in terms of bases here, or as in Section 25.7.

25.13 Some related filtrations

Let us look a bit more at the filtrations associated to the type of situation mentioned at the end of the previous section. Let k be a commutative ring with a multiplicative identity element again, let (A, [·, ·]_{A}) be a Lie algebra over k, and let A_{1}, A_{2} be Lie subalgebras of A. Also let UA, UA_{1}, and UA_{2} be universal enveloping algebras of A, A_{1}, and A_{2}, respectively, with multiplicative identity elements e = e_{UA}, e_{1} = e_{UA_{1}} and e_{2} = e_{UA_{2}}, and Lie algebra homomorphisms i = i_{UA}, i_{1} = i_{UA_{1}}, and i_{2} = i_{UA_{2}} from A, A_{1}, and A_{2} into UA, UA_{1}, and UA_{2}, respectively. If n is a nonnegative integer, then the corresponding submodules U_{n}A, U_{n}A_{1}, and U_{n}A_{2} of UA, UA_{1}, and UA_{2}, respectively, can be defined as in Section 25.9.

The restriction ϕ_{j} of i to A_{j} is a Lie algebra homomorphism from A_{j} into UA for j = 1, 2, which leads to a unique algebra homomorphism ψ_{j} from UA_{j} into UA such that ψ_{j}(e_{j}) = e and ψ_{j} ∘ i = ϕ_{j}, as before. Using this, we get a bilinear mapping from (UA_{1}) × (UA_{2}) into UA as in (25.12.1), which leads to a unique linear mapping from (UA_{1}) ⨂(UA_{2}) into UA that satisfies (25.12.2). If w_{1} ∈ U_{n_{1}}A_{1} and w_{2} ∈ U_{n_{2}}A_{2} for some n_{1}, n_{2} ≥ 0, then it is easy to see that

ψ_{1}(w_{1}) ψ_{2}(w_{2}) ∈ U_{n_{1}+n_{2}}A. (25.13.1)

Let n be a nonnegative integer, and let ((UA_{1}) ⨂(UA_{2}))_{n} be the submodule of (UA_{1}) ⨂(UA_{2}), as a module over k, generated by elements of the form w_{1} ⊗w_{2}, with w_{1} ∈ U_{n_{1}}A_{1}, w_{2} ∈ U_{n_{2}}A_{2}, and n_{1} + n_{2} ≤ n. Thus the linear mapping from (UA_{1}) ⨂(UA_{2}) into UA mentioned before maps ((UA_{1}) ⨂(UA_{2}))_{n} into U_{n}A for every n ≥ 0, by (25.13.1).

Observe that

((UA_{1}) ⨂(UA_{2}))_{n} ⊆ ((UA_{1}) ⨂(UA_{2}))_{n+1} (25.13.2)

for every n ≥ 0, and that ⋃_{n=0}^{∞}((UA_{1}) ⨂(UA_{2}))_{n} = (UA_{1}) ⨂(UA_{2}). Remember that gr_{n}UA is defined as in Section 25.9 for each nonnegative integer n, and similarly for gr_{n} UA_{1}, gr_{n} UA_{2}. Put

gr_{n}((UA_{1}) ⨂(UA_{2})) = ((UA_{1}) ⨂(UA_{2}))_{n}/((UA_{1}) ⨂(UA_{2}))_{n−1} (25.13.3)

for each positive integer n, where the quotient is defined as a module over k, and gr_{0}((UA_{1}) ⨂(UA_{2}))_{0} = ((UA_{1}) ⨂(UA_{2}))_{0}. Using the linear mapping from (UA_{1}) ⨂(UA_{2}) into UA mentioned in the preceding paragraph, we get an induced linear mapping from gr_{n}((UA_{1}) ⨂(UA_{2}))_{n} into gr_{n} UA for every n ≥ 0. If we put

gr((UA_{1}) ⨂(UA_{2})) = ⨁_{n=0}^{∞} gr_{n}((UA_{1}) ⨂(UA_{2})), (25.13.4)

then we get a homomorphism from gr((UA_{1}) ⨂(UA_{2})) into gr UA, as modules over k.

Of course, (w_{1}, w_{2}) ↦ w_{1} ⊗ w_{2} (25.13.5)

defines a bilinear mapping from (UA_{1})×(UA_{2}) into ((UA_{1}) ⨂(UA_{2})). If n_{1}, n_{2} are nonnegative integers, then this bilinear mapping sends (U_{n_{1}}A_{1}) × (U_{n_{2}}A_{2}) into ((UA_{1}) ⨂(UA_{2}))_{n_{1}+n_{2}}. This leads to a bilinear mapping

(25.13.6) from (gr_{n_{1}}UA_{1}) × (gr_{n_{2}}UA_{2}) into gr_{n_{1}+n_{2}}((UA_{1}) ⨂(UA_{2})).

Using this, we get a bilinear mapping

(25.13.7) from (gr UA_{1}) × (gr UA_{2}) into gr((UA_{1}) ⨂(UA_{2})).

This leads to a linear mapping

(25.13.8) from (gr UA_{1}) ⨂(gr UA_{2}) into gr((UA_{1}) ⨂(UA_{2})).

More precisely, if n is a nonnegative integer, then then gr_{n}((UA_{1}) ⨂(UA_{2})) is generated, as a module over k, by the images of (gr_{n_{1}} UA_{1}) × (gr_{n_{2}} UA_{2}), with n_{1} + n_{2} = n. This means that

(25.13.9) (gr UA_{1}) ⨂(gr UA_{2}) maps onto gr((UA_{1}) ⨂(UA_{2})),

under the mapping just mentioned.

Suppose that A_{1} and A_{2} are free as modules over k, with bases {x_{j}}_{j∈I_{1}} and {x_{j}}_{j∈I_{2}}, respectively. Suppose that I_{1} and I_{2} are also linearly ordered, so that M(I_{1}), M(I_{2}) can be defined as in Section 25.10. Using the Poincaré–Birkhoff– Witt theorem, we get bases for UA_{1} and UA_{2}, as modules over k, indexed by M(I_{1}) and M(I_{2}), respectively. In particular, the mapping indicated in (25.13.8) is an isomorphism between modules over k in this situation. Using these bases for UA_{1} and UA_{2}, we get a basis for (UA_{1}) ⨂(UA_{2}) as well, as a module over k.

Suppose that A corresponds to the direct sum of A_{1} and A_{2}, as a module over k. Thus A is free as a module over k too, with basis {x_{j}}_{j∈I}, where I = I_{1} ∪ I_{2}, and I_{1} ∩ I_{2} = ∅. Let ⪯ be a linear ordering on I, with j_{1} ⪯ j_{2} for every j_{1} ∈ I_{1} and j_{2} ∈ I_{2}, and let us take I_{1} and I_{2} to be linearly ordered by the restrictions of ⪯ to the subsets of I. Using the Poincaré–Birkhoff–Witt theorem again, we get a basis for UA, as a module over k, indexed by the corresponding set M(I). As in the previous section, the linear mapping from (UA_{1}) ⨂(UA_{2}) into UA determined by (25.12.2) is an isomorphism between these spaces, as modules over k, because it defines a one-to-one correspondence between their bases.

Similarly, the homomorphism from gr((UA_{1}) ⨂(UA_{2})) into gr UA, as modules over k, mentioned earlier is an isomorphism under these conditions. This can be composed with the isomorphism indicated in (25.13.8), to get an isomorphism from (gr UA_{1}) ⨂(gr UA_{2}) onto gr UA, as modules over k, under these conditions.

25.14 The diagonal mapping

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA} and Lie algebra homomorphism i = i_{UA} from A into UA, as before. Of course, A × A may be considered as a Lie algebra over k too, where addition, scalar multiplication, and the Lie bracket are defined coordinatewise. Note that A_{1} = A × {0} and A_{2} = {0} × A are Lie subalgebras of A × A, and that A × A corresponds to the direct sum of A_{1} and A_{2}, as a Lie algebra over k. We may consider UA to be a universal enveloping algebra of A_{1} and A_{2}, using the obvious isomorphisms a ↦ (a, 0) and a ↦ (0, a) from A onto A_{1} and A_{2}, respectively.

If a_{1}, a_{2} ∈ A, then put

ρ((a_{1}, a_{2})) = i(a_{1}) ⊗ e + e ⊗ i(a_{2}), (25.14.1) which defines a mapping from A × A into (UA) ⨂(UA). More precisely, ρ is a Lie algebra homomorphism from A × A into (UA) ⨂(UA), as in Section 25.7. This leads to an algebra homomorphism θ from a universal enveloping algebra of A × A into (UA) ⨂(UA), which is an algebra isomorphism in this situation, as before. One can use this to consider (UA) ⨂(UA) as a universal enveloping algebra of A × A.

Clearly a ↦ (a, a) is a Lie algebra homomorphism from A into A × A. We can compose this with ρ to get a Lie algebra homomorphism

a ↦ i(a) ⊗ e + e ⊗ i(a) (25.14.2)

from A into (UA) ⨂(UA). This leads to a unique algebra homomorphism Δ from UA into (UA) ⨂(UA), where Δ(e) = e ⊗ e and Δ ∘ i is the same as (25.14.2). This is called the diagonal map, as in Definition 5.1 and Proposition 5.2 on p16 of [25].

An element α of UA is said to be primitive if

Δ(α) = α ⊗ e + e ⊗ α, (25.14.3)

as in Definition 5.3 on p17 of [25]. Every element of i(A) is primitive, as in the preceding paragraph.

If t ∈ k and n ∈ Z_{+}, then n · t denotes the sum of n t’s in k, as usual. Suppose that k is torsion-free, so that n · t = 0 implies t = 0. If A is free as a module over k, then Theorem 5.4 on p17 of [25] states that the only primitive elements of UA are in i(A).

Suppose first that A is commutative as a Lie algebra over k, as in [25]. Let {x_{j}}_{j∈I} be a basis for A, as a module over k, and let X_{j}, j ∈ I, be a commuting family of indeterminates. There is a unique module homomorphism from A into the polynomial algebra k[{X_{j} : j ∈ I}] that sends x_{j} to X_{j} for every j ∈ I. Using this, k[{X_{j} : j ∈ I}] may be considered as a symmetric algebra of A, as in Section 25.8. This means that k[{X_{j} : j ∈ I}] is a universal enveloping algebra of A in this case.

Let X′_{j} and X′′_{j} be additional indeterminates for each j ∈ I, which commute with each other and with X′_{l}, X′′_{l}, l ∈ I. The tensor product of k[{X_{j} : j ∈ I}] with itself can be identified with the polynomial algebra

k[{X′_{j}, X′′_{j} : j ∈ I}], (25.14.4)

where X′_{j} corresponds to X_{j} ⊗1 and X′′_{j} corresponds to 1⊗X_{j} for every j ∈ I. In this situation, Δ corresponds to the mapping from k[{X_{j} : j ∈ I}] to (25.14.4) that sends a formal polynomial f(X) in the X_{j}’s to the formal polynomial f(X′ + X′′) in the X′_{j}’s and X′′_{j}’s, which replaces X_{j} with X′_{j} + X′′_{j} for every j ∈ I. The primitive elements correspond to f(X) ∈ k[{X_{j} : j ∈ I}] such that

f(X′ + X′′) = f(X′) + f(X′′). (25.14.5)

If f(X) is homogeneous of degree n for some nonnegative integer n, then this would imply that

2^{n} · f(X) = f(X + X) = f(X) + f(X) = 2 · f(X), (25.14.6) which is to say that (2^{n} − 2) · f(X) = 0. If n ≠ 1, then this implies that f(X) = 0, because k is torsion-free, by hypothesis. Otherwise, one can apply this argument to the homogeneous components of f(X), which satisfy the same condition.

Let us now consider the case where A is not necessarily commutative as a Lie algebra over k, as in [25] again. Remember that a ↦ (a, a) is a Lie algebra homomorphism from A into A×A. If U(A×A) is a universal enveloping algebra of A × A, then we can compose the previous Lie algebra homomorphism from A into A×A with the usual Lie algebra homomorphism from A×A into U(A×A) to get a Lie algebra homomorphism from A into U(A×A). This leads to a unique algebra homomorphism from UA into U(A×A), with the usual properties. It is easy to see that Δ is the same as the composition of this homomorphism with the algebra isomorphism θ from U(A × A) onto (UA) × (UA) mentioned earlier.

The algebra homomorphism from UA into U(A×A) mentioned in the preceding paragraph induces an algebra homomorphism from gr UA into gr U(A × A). We also have that θ induces an algebra isomorphism from gr U(A × A) onto (gr UA) ⨂(gr UA), as in the previous sections. Using this, Δ induces an algebra homomorphism gr Δ from gr UA into (gr UA) ⨂(gr UA). Remember that gr UA is isomorphic to SA, by the Poincaré–Birkhoff–Witt theorem. Thus gr Δ corresponds to an algebra homomorphism from SA into (SA) ⨂(SA).

Of course, SA is the same as a universal enveloping algebra of A as a commutative Lie algebra over k. The algebra homomorphism from SA into (SA) ⨂(SA) mentioned in the preceding paragraph corresponds exactly to the one considered in the first case. This can be seen by considering elements of gr_{1} UA that come from elements of A in the usual way.

Let x be a primitive element of UA, and suppose that x ∈ U_{n}A for some n ≥ 0. This implies the analogous condition for the image of x in gr_{n} UA with respect to gr Δ. If n > 1, then it follows that the image of x in gr_{n} UA is equal to 0, as in the first case. This means that x ∈ U_{n−1}A, and we can repeat the process to get that x ∈ U_{1}A.

Thus x = t e + i(a) for some t ∈ k and a ∈ A, which implies that

Δ(x) = t (e ⊗ e) + i(a) ⊗ e + e ⊗ i(a). (25.14.7)

Observe that

x ⊗ e + e ⊗ x = t (e ⊗ e) + i(a) ⊗ e + t (e ⊗ e) + e ⊗ i(a). (25.14.8)

The hypothesis that x be primitive means that (25.14.7) and (25.14.8) are the same, so that t = 0. This shows that x = i(a) ∈ i(A), as desired.

25.15 More on primitive elements

Let us return to the situation considered at the beginning of the previous section. Thus we let k be a commutative ring with a multiplicative identity element again, and (A, [·, ·]_{A}) be a Lie algebra over k. Also let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA} and Lie algebra homomorphism i = i_{UA} from A into UA. Remember that the diagonal map Δ is an algebra homomorphism from UA into (UA) ⨂(UA).

Let P UA be the set of primitive elements of UA. It is easy to see that P UA is a submodule of UA, as a module over k. Execise 1 on p17 of [25] asks one to show that if x, y ∈ P UA, then [x, y] = x y − y x ∈ P UA. This means that

Δ([x, y]) = ([x, y]) ⊗ e + e ⊗ ([x, y]). (25.15.1)

Of course, Δ([x, y]) = [Δ(x), Δ(y)], because Δ is an algebra homomorphism. It follows that Δ([x, y]) = [x ⊗ e + e ⊗ x, y ⊗ e + e ⊗ y], (25.15.2)

because x, y are primitive by hypothesis. It is easy to obtain (25.15.1) from (25.15.2), because of the way multiplication is defined in (UA) ⊗ (UA), as in Section 25.2.

Of course, we may as well suppose that k ≠ {0}, so that 1 ≠ 0 in k. Let us suppose from now on in this section that there is a prime number p such that p · 1 = 0 in k, as in Exercise 2 on p17 of [25]. Of course, this means that p · t = 0 for every t ∈ k.

Let n be a positive integer that is not a multiple of p, so that there is a positive integer m such that m n − 1 is a multiple of p. If t ∈ k, then

m · (n · t) = (m n) · t = t. (25.15.3)

In particular, if n · t = 0, then t = 0.

If y ∈ P UA, then part (a) of Exercise 2 on p17 of [25] asks us to show that

y^{p} ∈ P UA. (25.15.4)

Observe that Δ(y^{p}) = (Δ(y))^{p} = (y ⊗ e + e ⊗ y)^{p}, (25.15.5)

because Δ is an algebra homomorphism. The right side can be expanded into a sum, using the binomial theorem, because y ⊗ e commutes with e ⊗ y in (UA) ⨂(UA). It is well known and easy to see that the binomial coefficient \binom{p}{j} is a multiple of p when j = 1, …, p − 1. It follows that

Δ(y^{p}) = (y ⊗ e)^{p} + (e ⊗ y)^{p} = y^{p} ⊗ e + e ⊗ y^{p}, (25.15.6)

as desired.

Let us suppose for the rest of the section that A is free as a module over k, and let {x_{j}}_{j∈I} be a basis for A, as a module over k. If j ∈ I, then i(x_{j}) ∈ P UA, as in the previous section. This implies that

i(x_{j})^{p^{ν}} ∈ P UA (25.15.7)

for every positive integer ν, by (25.15.4). Part (b) of Exercise 2 on p17 of [25] asks us to show that the family of elements of P UA of the form

(25.15.8) i(x_{j})^{p^{ν}} , with j ∈ I and ν ∈ Z_{+} ∪ {0}, is a basis for P UA, as a module over k. It suffices to show that every element of P UA can be expressed as a finite linear combination of these elements, because of the Poincaré–Birkhoff–Witt theorem.

Suppose first that A is commutative as a Lie algebra over k, as in the previous section, and let X_{j}, j ∈ I, be a commuting family of indeterminates again. As before, the polynomial algebra k[{X_{j} : j ∈ I}] may be considered as a symmetric algebra of A, using the unique module homomorphism from A into this algebra that sends x_{j} to X_{j} for every j ∈ I. This is the same as a universal enveloping algebra of A in this case. Let X′_{j} and X′′_{j} be additional indeterminates for each j ∈ I again, which commute with each other and X′_{l}, X′′_{l}, l ∈ I, so that the tensor product of k[{X_{j} : j ∈ I}] with itself can be identified with

k[{X′_{j}, X′′_{j} : j ∈ I}], (25.15.9)

as before. Remember that Δ corresponds in this situation to the mapping from k[{X_{j} : j ∈ I}] to (25.15.9) that sends a formal polynomial f(X) in the X_{j}’s to the formal polynomial f(X′ + X′′) in the X′_{j} and X′′_{j}’s. The primitive elements correspond to the formal polynomials f(X) such that

f(X′ + X′′) = f(X′) + f(X′′), (25.15.10)

as before. In particular, this implies that the constant term in f(X) is equal to 0.

If f(X) satisfies (25.14.5, 2), then we would like to show that f(X) can be expressed as a finite linear combination of the monomials X_{j}^{p^{ν}}, with j ∈ I and ν ∈ Z_{+} ∪ {0}. We may as well suppose that the linear terms in f(X) are equal to 0, which would correspond to ν = 0. Under these conditions, one can check that the formal partial derivative of f(X) in the jth variable is equal to 0 for every j ∈ I. To see this, consider the sum of the terms in f(X′ + X′′) with exactly one factor of X′′_{j} and no factors of X′′_{l} when j ≠ l, and any number of factors of X′_{l} for l ∈ I. This sum is equal to the formal partial derivative of f(X′) in the jth variable times X′′_{j}. This would have to be a scalar multiple of X′′_{j}, by (25.15.10), which would be a term in f(X′′). In fact, this has to be equal to 0, because the linear terms in f(X) are supposed to be equal to 0.

If j ∈ I, then f(X) can be identified with a formal polynomial in X_{j} whose coefficients are formal polynomials in X_{l}, j ≠ l. The formal partial derivative of f(X) in the jth variable corresponds to the usual formal derivative of f(X) as a formal polynomial in this way. In this situation, the condition that the derivative be equal to 0 implies that this formal polynomial in X_{j} can be expressed as a formal polynomial in X_{j}^{p}. This means that f(X) can be expressed as a formal polynomial in X_{j}^{p}, j ∈ I, with coefficients in k. Let us express this formally as

f(X) = g(X^{p}). (25.15.11)

Using the binomial theorem again, we get that (X′_{j} + X′′_{j})^{p} = (X′_{j})^{p} + (X′′_{j})^{p} for every j ∈ I. This implies that g corresponds to a primitive element in the polynomial algebra as well. Note that the linear terms in g correspond to terms in f(X) of the form X_{j}^{p}, j ∈ I. We can subtract the linear terms from g, and repeat the previous argument. After a finite number of steps, we can express f(X) as a linear combination of monomials X_{j}^{p^{ν}}, with j ∈ I, ν ≥ 0, as desired.

If A is not necessarily commutative as a Lie algebra over k, then one can reduce to the previous case, as in the preceding section. Let x ∈ P UA be given, and suppose that x ∈ U_{n}A for some n ≥ 0. This implies that the image of x in gr_{n} UA satisfies the analogous primitivity condition with respect to gr Δ, as before. Using the previous case, we can express the image of x in gr_{n} UA in the analogous way. This means that x can be expressed as the sum of a finite linear combination of elements of P UA of the form (25.15.8) and an element of U_{n−1}A. This element of U_{n−1}A is primitive as well, because P UA is a submodule of UA, as a module over k. Thus we can repeat the process, to get that x is a finite linear combination of elements of P UA of the form (25.15.8), as desired.

If x, y ∈ i(A), then part (c) of Exercise 2 on p17 of [25] asks us to show that

(x + y)^{p} − x^{p} − y^{p} ∈ i(A). (25.15.12)

Note that x^{p}, y^{p}, (x + y)^{p} ∈ P UA, as in (25.15.7), and because i(A) ⊆ P UA. This implies that (x + y)^{p} − x^{p} − y^{p} ∈ P UA, (25.15.13)

because P AU is a submodule of UA, as a module over k.

We also have that

(x + y)^{p} − x^{p} − y^{p} ∈ U_{p−1}A. (25.15.14)

More precisely, (x + y)^{p} − x^{p} − y^{p} ∈ U_{p}A, and (25.15.14) is the same as saying that its image in gr_{p} UA is equal to 0. This can be obtained from the binomial theorem, because \binom{p}{l} is a multiple of p when l = 1, …, p − 1. It is easy to see that (25.15.12) follows from (25.15.13) and (25.15.14), using the basis (25.15.8) for P UA.

Free Lie algebras

26.1 Magmas and their homomorphisms

A magma is a set M together with a mapping from M × M into M, which may be expressed as (x, y) ↦ x y. (26.1.1)

This is Definition 1.1 on p18 of [25]. A mapping ϕ from M into another magma N is said to be a magma homomorphism if

ϕ(x y) = ϕ(x) ϕ(y) (26.1.2)

for every x, y ∈ M.

Let X be a set, and put X_{1} = X. If n ≥ 2 is an integer, then X_{n} can be defined recursively as the disjoint union of X_{j} × X_{l}, where j, l ∈ Z_{+} and j +l = n. Let M_{X} be the disjoint union of the X_{n}’s, n ∈ Z_{+}. There is a natural binary operation on M_{X}, defined using the embedding of X_{j} × X_{l} into X_{j+l} for every j, l ≥ 1, by construction. This makes M_{X} into a magma, which is the free magma on X, as on p18 of [25]. The elements of M_{X} may be described as non-associative words on X. If n ∈ Z_{+}, then the elements of X_{n} are said to have length n.

If f is any mapping from X into a magma N, then there is a unique extension of f to a magma homomorphism F from M_{X} into N. This is Theorem 1.2 on p18 of [25]. More precisely, if j, l ∈ Z_{+}, x ∈ X_{j}, and y ∈ X_{l}, then (x, y) ∈ X_{j+l}, and F should satisfy F ((x, y)) = F (x) F (y). (26.1.3)

This can be used to define F recursively on M_{X}.

Of course, M_{X} is generated by elements of X in the obvious way. Every element of M_{X} \ X can be expressed in a unique way as a product of elements of M_{x}, as mentioned on p18 of [25].

26.2 Magmas and algebras

Let k be a commutative (associative) ring with a multiplicative identity element, and let M be a nonempty magma. We can define an algebra A over k in the strict sense using M in the usual way, whose elements are formal sums

∑_{m∈M} c_{m} m (26.2.1)

with c_{m} ∈ k for every m ∈ M, and c_{m} = 0 for all but finitely many m ∈ M. More precisely, this can be defined as the space of k-valued functions on M with finite support, and it is convenient to express the elements of A as formal sums in this way. Of course, A is a module over k with respect to pointwise addition and scalar multiplication of functions on M, which corresponds to termwise addition and scalar multiplication of these formal sums. Multiplication on M extends to a bilinear mapping from A × A into A, which defines multiplication on A.

Let X be a nonempty set, and let M_{X} be the free magma on X, as in the previous section. Using M_{X}, we get an algebra A_{X}(k) = A_{X} over k in the strict sense, as in the preceding paragraph. This is called the free algebra on X over k, as in Definition 2.1 on p18 of [25].

Let B be another algebra over k in the strict sense, and let f be a mapping from X into B. Under these conditions, there is a unique extension of f to an algebra homomorphism F from A_{X} into B, as in Theorem 2.2 on p19 of [25]. Note that B may be considered as a magma with respect to multiplication. Thus f can first be extended to a magma homomorphism from M_{X} into B, as in the previous section. This can be extended to a mapping F from A_{X} into B that is linear over k. It is easy to see that F is an algebra homomorphism on A_{X}, because it is a magma homomorphism on M_{X}. The uniqueness of F follows from the fact that A_{X} is generated by X, as an algebra over k.

Remember that the length of an element of M_{X} is defined as in the previous section. Let us say that an element

a = ∑_{m∈M_{X}} c_{m} m (26.2.2)

of A_{X} is homogeneous of degree n for some positive integer n if c_{m} = 0 for every m ∈ M_{X} such that the length of m is not equal to n. The set A_{X}^{n}(k) = A_{X}^{n} of elements of A_{X} that are homogeneous of degree n is a submodule of A_{X}, as a module over k, and A_{X} corresponds to the direct sum of these submodules, as a module over k. If a_{1}, a_{2} ∈ A_{X} are homogeneous of degrees n_{1}, n_{2} ≥ 1, respectively, then a_{1} a_{2} is homogeneous of degree n_{1} + n_{2}. Thus A_{X} is graded as an algebra over k in the strict sense in this way, as mentioned on p19 of [25].

26.3 Constructing free Lie algebras

Let k be a commutative ring with a multiplicative identity element again, and let X be a nonempty set. Also let A_{X} be the free algebra on X over k, as in the previous section. Consider the two-sided ideal I_{X} of A_{X} generated by elements of the form a a, (26.3.1)

with a ∈ A_{X}, and

J(a, b, c) = (a b) c + (b c) a + (c a) b, (26.3.2)

with a, b, c ∈ A_{X}. Thus an element of I_{X} is a finite sum of elements of A_{X}, each of which can be obtained by starting with an element of the form (26.3.1) or (26.3.2), and multiplying it on the left and right by elements of A_{X}. More precisely, one can multiply elements of the form (26.3.1) or (26.3.2) by elements of A_{X} on the left or right by elements of A_{X} repeatedly, and in any order.

Thus the quotient A_{X}/I_{X} (26.3.3)

is defined as an algebra over k in the strict sense. It is easy to see that (26.3.3) is a Lie algebra over k, because elements of A_{X} of the form (26.3.1) and (26.3.2) are mapped to 0 in (26.3.3). More precisely, the Lie bracket on (26.3.3) is the binary operation that corresponds to multiplication on A_{X}. This is called the free Lie algebra on X over k, as in Definition 3.1 on p19 of [25]. This may be denoted L_{X}(k), or L_{X} when the choice of k is clear.

Of course, there is a natural mapping from X into L_{X}, obtained by restricting the quotient mapping from A_{X} onto L_{X} to X, as a subset of A_{X}. Note that L_{X} is generated as an algebra over k by the image of X in L_{X}, because A_{X} is generated as an algebra over k by X, by construction.

Let B be a Lie algebra over k, and let f be a mapping from X into B. As in the previous section, there is a unique extension of f to an algebra homomorphism F from A_{X} into B. This extension maps elements of A_{X} of the form (26.3.1) and (26.3.2) to 0 in B, because B is a Lie algebra over k. This implies that I_{X} is contained in the kernel of F. This leads to an algebra homomorphism from L_{X} into B. By construction, the composition of this algebra homomorphism with the natural mapping from X into L_{X} is f. This algebra homomorphism from L_{X} into B is uniquely determined by this property, because L_{X} is generated as an algebra by the image of X, as in the preceding paragraph.

This property is essentially used to define free Lie algebras on X on p94f of [14]. It is easy to see that the free Lie algebra on X is uniquely determined by this property, up to a unique isomorphism, as on p95 of [14]. The existence of a free Lie algebra on X is obtained another way on p95 of [14], and we shall return to this later.

Let Y be another nonempty set, and let L_{Y} be the corresponding free algebra on Y over k. If f is a mapping from X into Y, then we get a mapping from X into L_{Y}, by composing f with the natural mapping from Y into L_{Y}. This leads to a unique Lie algebra homomorphism from L_{X} into L_{Y}, whose composition with the natural mapping from X into L_{X} is the mapping from X into L_{Y} just mentioned, as before. This is one of the properties of free Lie algebras mentioned on p19 of [25].

26.4. DEGREE 2 IN L_{X} 563

Remember that A_{X} is graded as an algebra over k in the strict sense, as in the previous section. Let I_{X}^{#} be the collection of elements of A_{X} whose homogeneous components are elements of I_{X}. Thus I_{X}^{#} ⊆ I_{X} automatically, and we would like to verify that

I_{X}^{#} = I_{X}, (26.3.4)

as on p19 of [25]. It suffices to check that I_{X} is generated by homogeneous elements of A_{X}, as a two-sided ideal in A_{X}.

Let a ∈ A_{X} be given, and let a_{n} be the homogeneous component of a_{n} for each positive integer n, so that a_{n} = 0 for all but finitely many n ≥ 1, and a = ∑_{n=1}^{∞} a_{n}. Observe that

a a = ∑_{n=1}^{∞} a_{n} a_{n} + ∑_{n=2}^{∞} ∑_{l=1}^{n−1}(a_{l} a_{n} + a_{n} a_{l}). (26.3.5)

Clearly a_{n} a_{n} ∈ I_{X} for every n ≥ 1, and

a_{l} a_{n} + a_{n} a_{l} = (a_{l} + a_{n}) (a_{l} + a_{n}) − a_{l} a_{l} − a_{n} a_{n} (26.3.6)

is an element of I_{X} when l < n. This implies that a a can be expressed as a finite sum of homogeneous elements of I_{X}.

Similarly, let b, c ∈ A_{X} be given, with homogeneous components b_{n}, c_{n} for every n ≥ 1, respectively. It is easy to see that

J(a, b, c) = ∑_{j=1}^{∞} ∑_{l=1}^{∞} ∑_{n=1}^{∞} J(a_{j}, b_{l}, c_{n}). (26.3.7)

Thus J(a, b, c) can be expressed as a finite sum of homogeneous elements of I_{X} too. It follows that (26.3.4) holds, as before.

Let us say that an element of L_{X} is homogeneous of degree n ∈ Z_{+} if it is the image of an element of A_{X} that is homogeneous of degree n under the natural quotient mapping. This makes L_{X} a graded algebra over k as well, because of (26.3.4), as on p19 of [25].

In particular, 0 is the only element of I_{X} that is homogeneous of degree 1. Thus the quotient mapping from A_{X} onto L_{X} is injective on the submodule of A_{X} consisting of elements that are homogeneous of degree 1. This corresponds to part of the statement at the top of p20 of [25].

26.4 Degree 2 in L_{X}

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. Thus A ⨂ A can be defined as a module over k in the usual way. Let B be the submodule of A ⨂ A generated by elements of A ⨂ A of the form a ⊗ a, with a ∈ A. Equivalently, B consists of finite linear combinations of elements of this form, with coefficients in k. In particular, if x, y ∈ A, then

x ⊗ y + y ⊗ x = (x + y) ⊗ (x + y) − x ⊗ x − y ⊗ y (26.4.1) is an element of B. We can define ⋀^{2} A to be the quotient

(A ⨂ A)/B, (26.4.2)

as a module over k. If x, y ∈ A, then let x ∧ y be the image of x ⊗ y under the quotient mapping from A ⨂ A onto (26.4.2). By construction, a ∧ a = 0 for every a ∈ A, and x ∧ y = −y ∧ x for every x, y ∈ A.

Consider A × (⋀^{2} A), as a module over k with respect to coordinatewise addition and scalar multiplication. Of course, this is the same as the direct sum of A and ⋀^{2} A, as a module over k. If x, y ∈ A and u, v ∈ ⋀^{2} A, then put

[(x, u), (y, v)] = (0, x ∧ y). (26.4.3)

One can check that A × (⋀^{2} A) is a Lie algebra over k with respect to this definition of Lie bracket, as on p2 of [25].

Let X be a nonempty set, and let A_{X} be as in Section 26.2. Remember that A_{X} is a graded as an algebra over k in the strict sense, and let A be the submodule of A consisting of elements that are homogeneous of degree 1. The elements of A correspond to formal sums of elements of X with coefficients in k, where all but finitely many of the coefficients are equal to 0. This can be identified with the space of all k-valued functions on X with finite support, and is free as a module over k.

Similarly, the elements of A_{X} that are homogeneous of degree 2 correspond to formal sums of elements of X ×X with coefficients in k, where all but finitely many of the coefficients are equal to 0. The submodule of A_{X} consisting of elements that are homogeneous of degree 2 is free as a module over k, and can be identified with A ⨂ A. The intersection of this submodule with the ideal I_{X} in A_{X} defined in the previous section corresponds in this way exactly to the submodule B of A ⨂ A mentioned earlier. Remember that the free Lie algebra L_{X} on X over k is graded as an algebra over k, as in the previous section. The submodule of L_{X} consisting of homogeneous elements of degree 2 corresponds to ⋀^{2} A, as on p20 of [25].

Of course, we can identify A and ⋀^{2} A with submodules of A × (⋀^{2} A), as a module over k, in the obvious way. Thus the natural mapping from X into A can be identified with a mapping from X into A × (⋀^{2} A). This leads to a Lie algebra homomorphism from L_{X} into A × (⋀^{2} A), as in the previous section. This homomorphism sends homogeneous elements of L_{X} of degree 1 and 2 to their counterparts in A and ⋀^{2} A, as on p20 of [25]. Homogeneous elements of L_{X} of larger degree are sent to 0 by this homomorphism.

If a_{1}, a_{2} ∈ A, then a_{1}, a_{2} can be identified with homogeneous elements of L_{X} of degree 1, whose Lie bracket [a_{1}, a_{2}] is homogeneous of degree 2 in L_{X}. This corresponds to a_{1} ∧ a_{2} in ⋀^{2} A, as before. In particular, if x, y ∈ X, then x, y can be identified with elements of A, and [x, y] defined as a homogeneous element of L_{X} of degree 2.

Let ⪯ be a linear ordering on X. The collection of elements of L_{X} of the form [x, y], where x, y ∈ X, x ⪯ y, and x ≠ y, is a basis for the set of homogeneous elements of L_{X} of degree 2, as a module over k, as on p20 of [25].

26.5 Free associative algebras

Let k be a commutative ring with a multiplicative identity element, and let X be a nonempty set. Also let E = E_{X} be the free module over k with the elements of X as a basis, so that the elements of E can be given as formal sums

∑_{x∈X} c_{x} x, (26.5.1)

with c_{x} ∈ k for every x ∈ X, and c_{x} = 0 for all but finitely many x ∈ X. This can be defined as the space of k-valued functions on X with finite support, which can be expressed as formal sums in this way for convenience. This is a module over k with respect to termwise addition and scalar multiplication of formal sums, which corresponds to pointwise addition and scalar multiplication of functions on X.

The free associative algebra on X over k, Assoc_{X}(k) = Assoc_{X}, is defined to be the tensor algebra T E on E, as in Definition 4.1 on p20 of [25]. If f is any mapping from X into a module A over k, then f has a natural unique extension to a module homomorphism ϕ from E into A. If A is an associative algebra over k with a multiplicative identity element e, then there is a unique algebra homomorphism ψ from T E into A that is equal to ϕ on E = T^{1}E and sends 1 ∈ k = T^{0}E to e, as in Section 25.3.

Of course, Assoc_{X} may be considered as a Lie algebra over k, using the commutator bracket corresponding to multiplication on Assoc_{X}. The natural inclusion mapping from X into E = T^{1}E may be considered as a mapping from X into Assoc_{X} = T E, which leads to a unique Lie algebra homomorphism ϕ from the free Lie algebra L_{X} on X over k into Assoc_{X}, as in Section 26.3. More precisely,

(26.5.2) the composition of ϕ with natural mapping from X into L_{X} is the same as the natural mapping from X into Assoc_{X}.

Let UL_{X} be a universal enveloping algebra of L_{X}, with corresponding multiplicative identity element e_{UL_{X}} and mapping i_{UL_{X}} from L_{X} into UL_{X}. The Lie algebra homomorphism ϕ from L_{X} into Assoc_{X} mentioned in the preceding paragraph leads to a unique algebra homomorphism Φ from UL_{X} into Assoc_{X} such that Φ(e_{UL_{X}}) = 1 ∈ k ∈ T^{0}E and

Φ ∘ i_{UL_{X}} = ϕ, (26.5.3)

as in Section 25.4.

Similarly,

(26.5.4) there is a natural mapping from X into UL_{X}, which is obtained by composing the natural mapping from X into L_{X} with i_{UL_{X}} .

This leads to a unique algebra homomorphism Ψ from Assoc_{X} into UL_{X}, such that

(26.5.5) the composition of Ψ with the natural mapping from X

into Assoc_{X} is the same as the mapping in (26.5.4),

and which sends 1 ∈ k = T^{0}E to e_{UL_{X}}.

Thus Φ ∘ Ψ is an algebra homomorphism from Assoc_{X} to itself, which sends 1 ∈ k = T^{0}E to itself. By construction, the composition of Φ ∘ Ψ with the natural mapping from X into Assoc_{X} is that mapping. More precisely, the composition of Φ ∘ Ψ with the natural mapping from X into Assoc_{X} is the same as the composition of (26.5.3) with the natural mapping from X into L_{X}. This is the same as the natural mapping from X into Assoc_{X}, as in (26.5.2). It follows that Φ ∘ Ψ is the identity mapping on Assoc_{X}.

We also have that Ψ ∘ Φ is an algebra homomorphism from UL_{X} to itself that sends e_{UL_{X}} to itself. In order to show that Ψ ∘ Φ is the identity mapping on UL_{X}, it suffices to verify that

(Ψ ∘ Φ) ∘ i_{UL_{X}} = i_{UL_{X}} . (26.5.6)

Observe that (Ψ ∘ Φ) ∘ i_{UL_{X}} = Ψ ∘ (Φ ∘ i_{UL_{X}}) = Ψ ∘ ϕ, (26.5.7)

by (26.5.3). The composition of this mapping with the natural mapping from X into L_{X} is the same as the composition of Ψ with the natural mapping from X into Assoc_{X}, by (26.5.2). This is the same as (26.5.4), as in (26.5.5). This means that the compositions of both sides of (26.5.6) with the natural mapping from X into L_{X} are the same. Of course, both sides of (26.5.6) are Lie algebra homomorphisms from L_{X} into UL_{X}. This implies (26.5.6), because L_{X} is generated as a Lie algebra over k by the image of the natural mapping from X into L_{X}.

Thus Φ is an algebra isomorphism from UL_{X} onto Assoc_{X}, and Ψ is its inverse. This is the first part of Theorem 4.2 on p20 of [25], which corresponds to Exercise 4 on p95 of [14].

Note that

(26.5.8) ϕ maps L_{X} onto the Lie subalgebra of Assoc_{X} generated by the image of the natural mapping from X into Assoc_{X},

because ϕ is a Lie algebra homomorphism from L_{X} into Assoc_{X}, and L_{X} is generated as a Lie algebra over k by the image of the natural mapping from X into L_{X}. This is the easy part of the second part of Theorem 4.2 on p20 of [25].

26.6 Injectivity and free modules

Let us continue with the same notation and hypotheses as in the previous section. Under these conditions, we have that

(26.6.1) ϕ is injective as a mapping from L_{X} into Assoc_{X}.

This means that ϕ is a Lie algebra isomorphism from L_{X} onto the Lie subalgebra of Assoc_{X} generated by the image of the natural mapping from X into Assoc_{X}, because of (26.5.8). This is the second part of Theorem 4.2 on p20 of [25]. The Lie subalgebra of Assoc_{X} generated by the image of the natural mapping from X into Assoc_{X} is used to construct free Lie algebras on p95 of [14].

Part of the third part of Theorem 4.2 on p20 of [25] states that

(26.6.2) L_{X} is free as a module over k.

Of course, this holds automatically when k is a field. Note that i_{UL_{X}} is injective as a mapping from L_{X} into UL_{X} when (26.6.2) holds, by the Poincaré–Birkhoff– Witt theorem. In this case, (26.6.1) follows from (26.5.3) and the fact that Φ is an isomorphism from UL_{X} onto Assoc_{X}, as mentioned on p21 of [25]. Exercise 6 on p95 of [14] asks how the Poincaré–Birkhoff–Witt theorem is used there, perhaps more precisely in the verification of the requirements of free Lie algebras.

Remember that L_{X} is graded as an algebra over k, as in Section 26.3. If n ∈ Z_{+}, then let L_{X}^{n} = L_{X}^{n}(k) be the submodule of L_{X}, as a module over k, consisting of elements of L_{X} that are homogeneous of degree n. The other part of the third part of Theorem 4.2 on p20 of [25] states that

(26.6.3) L_{X}^{n} is free as a module over k for each n ≥ 1.

This implies (26.6.2), because L_{X} corresponds to the direct sum of L_{X}^{n}, n ≥ 1, as a module over k. As before, (26.6.3) holds automatically when k is a field.

If X has only finitely many elements, then L_{X}^{n} is finitely generated as a module over k for each n ≥ 1. More precisely, this follows from the analogous statement for the free algebra A_{X} on X over k. The fourth part of Theorem 4.2 concerns the rank of L_{X}^{n}, as a free module over k, in this case.

Of course, Assoc_{X} = T E is graded as an associative algebra over k, where T^{n}E corresponds to the submodule of T E of elements that are homogeneous of degree n ≥ 0. Let Assoc_{X}^{n} = Assoc_{X}^{n}(k) be the submodule of Assoc_{X} = T E, as a module over k, that corresponds to T^{n}E for each nonnegative integer n. Similarly, Assoc_{X} is graded as a Lie algebra over k in this way, with respect to the commutator bracket corresponding to multiplication. Let us check that

ϕ(L_{X}^{n}) ⊆ Assoc_{X}^{n} (26.6.4)

for every n ≥ 1.

Let f be the natural mapping from X into Assoc_{X}, which corresponds to the natural inclusion mapping of X into E = T^{1}E. Also let A_{X} be the free algebra on X over k, and let A_{X}^{n} be the submodule of A_{X} consisting of elements that are homogeneous of degree n ∈ Z_{+}, as in Section 26.2. There is a unique extension of f to an algebra homomorphism F from A_{X} into Assoc_{X}, considered as a Lie algebra over k with respect to the commutator bracket corresponding to multiplication, as before. It is easy to see that

F (A_{X}^{n}) ⊆ Assoc_{X}^{n} (26.6.5)

for every n ≥ 1. Remember that L_{X} is a quotient of A_{X}, and that L_{X}^{n} is the image of A_{X}^{n} under the quotient mapping. By construction, F is the same as the quotient mapping composed with ϕ. Thus (26.6.4) follows from (26.6.5).

Suppose that k is a field, and that X has only finitely many elements, as in the first step on p21 of [25]. Let {γ_{j}}_{j∈I} be a homogeneous basis for L_{X}, as a graded vector space over k. This means that for each j ∈ I, γ_{j} ∈ L_{X}^{n(j)} for some n(j) ∈ Z_{+}, and for each n ∈ Z_{+}, the collection of γ_{j} with n(j) = n is a basis for L_{X}^{n}, as a vector space over k. Also let ⪯ be a linear ordering on I.

The Poincaré–Birkhoff–Witt theorem implies that the family of products of the form i_{UL_{X}} (γ_{j_{1}}) ⋯ i_{UL_{X}} (γ_{j_{m}}), (26.6.6)

where j_{1}, …, j_{m} ∈ I satisfy j_{1} ⪯ ⋯ ⪯ j_{m} and m is a nonnegative integer, form a basis for UL_{X}. This implies that the corresponding family of products

ϕ(γ_{j_{1}}) ⋯ ϕ(γ_{j_{m}}) (26.6.7)

is a basis for Assoc_{X} as a vector space over k. More precisely, this uses (26.5.3) and the fact that Φ is an algebra isomorphism from UL_{X} onto Assoc_{X}. We also have that ϕ(γ_{j}) ∈ Assoc_{X}^{n(j)} for every j ∈ I, by (26.6.4). This means that (26.6.7) is homogeneous of degree

n(j_{1}) + ⋯ + n(j_{m}) (26.6.8)

in Assoc_{X}, which is interpreted as being equal to 0 when m = 0.

This can be used to express the dimension of Assoc_{X}^{n} in terms of the dimensions of L_{X}^{r} with r ≤ n, as in [25]. This means that the dimension of L_{X}^{n} can be expressed in terms of the dimension of Assoc_{X}^{n} and the dimensions of L_{X}^{r} with r < n, as in the remark on p20 of [25]. In particular, the dimension of L_{X}^{n} does not depend on k.

26.7 Finite sets X, k = Z

Let k be a commutative ring with a multiplicative identity element, and let X be a nonempty set. Also let M_{X} be the free magma on X, and let A_{X}(k) be the free algebra on X over k, as in Sections 26.1 and 26.2. Remember that I_{X} = I_{X}(k) is the two sided ideal in A_{X}(k) generated by elements of the form a a and J(a, b, c), where a, b, c ∈ A_{X}(k) and J(a, b, c) is as in (26.3.2). Equivalently, I_{X}(k) is generated as a two-sided ideal in A_{X}(k) by elements of the form

m m, (26.7.1)

with m ∈ M_{X},

m_{1} m_{2} + m_{2} m_{1} = (m_{1} + m_{2}) (m_{1} + m_{2}) − m_{1} m_{1} − m_{2} m_{2}, (26.7.2)

with m_{1}, m_{2} ∈ M_{X}, and J(m_{1}, m_{2}, m_{3}), (26.7.3)

with m_{1}, m_{2}, m_{3} ∈ M_{X}. This can be seen by expressing a, b, c ∈ A_{X}(k) as linear combinations of elements of M_{X} with coefficients in k.

26.8. ARBITRARY SETS X, K = Z 569

Let us now consider the cases where k is Z or Z/p Z for some prime number p. There is a natural mapping from A_{X}(Z) onto A_{X}(Z/p Z), defined by applying the quotient mapping from Z onto Z/p Z to the coefficients of elements of A_{X}(Z). One can check that this mapping also sends I_{X}(Z) onto I_{X}(Z/p Z). This leads to a mapping from L_{X}(Z) onto L_{X}(Z/p Z).

More precisely, the mapping from A_{X}(Z) onto A_{X}(Z/p Z) mentioned in the preceding paragraph sends A_{X}^{n}(Z) onto A_{X}^{n}(Z/p Z) for every n ∈ Z_{+}, where A_{X}^{n}(k) consists of the elements of A_{X}(k) that are homogeneous of degree n, as in Section 26.2. Similarly, the induced mapping from L_{X}(Z) onto L_{X}(Z/p Z) sends L_{X}^{n}(Z) onto L_{X}^{n}(Z/p Z) for every n ∈ Z_{+}, where L_{X}^{n}(k) consists of elements of L_{X}(k) that are homogeneous of degree n, as in Section 26.3.

Of course, p times any element of L_{X}(Z) is mapped to 0 in L_{X}(Z/p Z). Conversely, one can check that any element of L_{X}(Z) that is mapped to 0 in L_{X}(Z/p Z) can be expressed as p times an element of L_{X}(Z).

Suppose now that X has only finitely many elements. Remember that L_{X}^{n}(Z) is finitely generated as a module over Z for every n ≥ 1 in this case, as in the previous section, which means that L_{X}^{n}(Z) is finitely generated as a commutative group. It follows that L_{X}^{n}(Z) is isomorphic to a direct sum of finitely many cyclic groups for each n ≥ 1.

The dimension of L_{X}^{n}(Z/pZ), as a vector space over Z/p Z, can be determined as in the previous section, and does not depend on p. Using this, one can check that L_{X}^{n}(Z) is isomorphic to the direct sum of finitely many infinite cyclic groups for each n ≥ 1. More precisely, the number of factors is the same as the common value of the dimension of L_{X}^{n}(Z/p Z) as a vector space over Z/p Z.

In particular, L_{X}(Z) is free as a module over Z for every n ≥ 1, which implies that L_{X}(Z) is free as a module over Z too. It follows that the mapping ϕ from L_{X}(Z) into Assoc_{X}(Z) defined in Section 26.5 is injective, as in the previous section. This corresponds to the second step on p22 of [25].

26.8 Arbitrary sets X, k = Z

Let k be a commutative ring with a multiplicative identity element, and let X be a nonempty set. Also let L_{X} be the free Lie algebra on X over k, let E_{X} be the free module on X over k, and let Assoc_{X} = T E_{X} be the free associative algebra on X over k, as in Sections 26.3 and 26.5. Remember that the natural inclusion of X in E_{X} = T^{1}E_{X} leads to a unique Lie algebra homomorphism ϕ = ϕ_{X} from L_{X} into Assoc_{X}, as before.

Let Y be a nonempty subset of X. The free Lie algebra L_{Y} on Y over k, the free module E_{Y} on Y over k, and the free associative algebra Assoc_{Y} = T E_{Y} on Y over k can be defined in the same way as before. Of course, there is a natural inclusion of E_{Y} into E_{X}, which leads to a natural inclusion of Assoc_{Y} into Assoc_{X}. There is a natural Lie algebra homomorphism from L_{Y} into L_{X} as well, as in Section 26.3.

Let ϕ_{Y} be the unique Lie algebra homomorphism from L_{Y} into Assoc_{Y} obtained from the natural inclusion of Y in E_{Y}, as in Section 26.5. One can check that the composition of the natural Lie algebra homomorphism from L_{Y} into L_{X} with ϕ_{X} is the same as the composition of ϕ_{Y} with the natural inclusion from Assoc_{Y} into Assoc_{X}.

Suppose now that k = Z. If Y has only finitely many elements, then ϕ_{Y} is injective, as in the previous section. This implies that ϕ_{X} is injective on the image of L_{Y} in L_{X} under the natural Lie algebra homomorphism.

It is easy to see that every element of L_{X} is in the image of L_{Y} for some nonempty finite subset Y of X. It follows that ϕ_{X} is injective as a mapping from L_{X} into Assoc_{X}.

Note that Assoc_{X} is free as a module over Z, which is to say as a commutative group. It is well known that subgroups of free abelian groups are free abelian groups too.

Using the injectivity of ϕ_{X}, we get that L_{X}^{n} is a free abelian group for every n ≥ 1, and that L_{X} is a free abelian group. Equivalently, these are free as modules over Z. This corresponds to the third step on p22 of [25].

26.9 Arbitrary X, k

Let k be a commutative ring with a multiplicative identity element, and let X be a nonempty set. Also let M_{X} be the free magma on X, let A_{X}(k) be the free algebra on X over k, and let L_{X}(k) be the free Lie algebra on X over k, as in Sections 26.1, 26.2, and 26.3. Remember that L_{X}(k) is the quotient of A_{X}(k) by the two-sided ideal I_{X}(k) discussed in Sections 26.3 and 26.7. Of course, A_{X}(Z), L_{X}(Z), and I_{X}(Z) can be defined in the same way. If n ∈ Z_{+}, then let L_{X}^{n}(k), L_{X}^{n}(Z) be the subsets of L_{X}(k), L_{X}(Z) consisting of elements that are homogeneous of degree n, as before.

One can identify L_{X}(k) with the tensor product of L_{X}(Z) with k over Z. This is stated more broadly on p19 of [25]. Of course, A_{X}(k) corresponds to the tensor product of A_{X}(Z) with k over Z, because A_{X}(Z) is a free module over Z with basis M_{X}. We also have that I_{X}(k) is the image of the tensor product of I_{X}(Z) with k over Z. Similarly, L_{X}^{n}(k) can be identified with the tensor product of L_{X}^{n}(Z) with k over Z for each n ≥ 1.

Remember that L_{X}(Z) is a free module over Z for every n ≥ 1, and that L_{X}(Z) is a free module over Z, as in the previous section. It follows that L^{n}(k) is a free module over k for every n ≥ 1, and that L_{X}(k) is a free module over k. This implies that the Lie algebra homomorphism ϕ from L_{X}(k) into Assoc_{X}(k) defined in Section 26.5 is injective, as in Section 26.6. If X has only finitely many elements, then the rank of L_{X}^{n}(k) as a free module over k is the same as the rank of L_{X}^{n}(Z) as a free module over Z, which is the same as the dimension in the case of fields. This corresponds to the fourth step on p22 of [25].

26.10 Bases from P. Hall families

Let X be a nonempty set, and let M_{X} be the free magma on X, as in Section 26.1. A P. Hall family in M_{X} is a subset H of M_{X} equipped with a linear ordering ⪯ that satisfies the following three properties. First,

X ⊆ H. (26.10.1)

Second, if u, v ∈ H, then

(26.10.2) u ⪯ v when the length of u is strictly less than the length of v.

Third, suppose that u ∈ M_{X} \ X, so that u can be expressed in a unique way as

u = v w, (26.10.3)

with v, w ∈ M_{X}. In this case, u ∈ H if and only if v, w satisfy the following two conditions. The first condition is that

v, w ∈ H, v ⪯ w, v ≠ w. (26.10.4)

The second condition is that either w ∈ X, or

w = w′ w′′, with w′, w′′ ∈ H, w′ ⪯ v. (26.10.5)

This is Definition 5.1 on p22f of [25]. Of course, in (26.10.5), w should satisfy the requirements of the third property of a P. Hall family as well.

Lemma 5.2 on p23 of [25] states that there is always a P. Hall family in M_{X}. Remember that for each positive integer n, X_{n} is the set of elements of M_{X} of length n, as in Section 26.1. We would like to define H^{n} ⊆ X_{n} recursively, which will be the set of elements in the Hall family that are in X_{n}. More precisely, we should also choose a linear ordering on H^{n} for each n, which leads to a linear ordering on their union that satisfies (26.10.2). Put H^{1} = X = X_{1}, which we can take to be equipped with any linear ordering.

Suppose that H^{1}, …, H^{n−1} have been chosen for some n ≥ 2, as well as linear orderings on them. This leads to a linear ordering on their union that satisfies (26.10.2), as before. Suppose also that their union satisfies the three properties of a P. Hall family for elements of length less than or equal to n − 1. Using the third property of a P. Hall family, H^{n} is uniquely determined as a subset of X_{n}. In order to continue the process, we can choose any linear ordering on H^{n}. Thus we can repeat the process, to choose H^{n} and a linear ordering on it for every n ≥ 1. It is easy to see that H = ⋃_{n=1}^{∞} H^{n} is a P. Hall family in M_{X}, as desired.

Let k be a commutative ring with a multiplicative identity element, so that the free algebra A_{X}(k) on X over k can be defined as in Section 26.2. Let L_{X}(k) be the corresponding free Lie algebra on X over k, as in Section 26.3. If H is a P. Hall family in M_{X}, then the elements of H may be considered as elements of A_{X}(k), which are mapped into L_{X}(k) by the quotient mapping from A_{X}(k) onto L_{X}(k). Theorem 5.3 on p23 of [25] states that the images of the elements of H in L_{X}(k) form a basis for L_{X}(k), as a module over k.

26.11 More on L_{X}^{n}

Let k be a commutative ring with a multiplicative identity element, and let X be a nonempty set. Thus the free algebra A_{X} on X over k can be defined as in Section 26.2. Remember that A_{X} is graded as an algebra over k in the strict sense, and let A_{X}^{n} be the submodule of A_{X}, as a module over k, of elements that are homogeneous of degree n ∈ Z_{+}, as before. If n_{1}, n_{2} ∈ Z_{+}, then

A_{X}^{n_{1}} · A_{X}^{n_{2}} ⊆ A_{X}^{n_{1}+n_{2}}, (26.11.1)

where the left side is defined as in Section 9.2. If n ≥ 2, then it is easy to see that every element of A_{X}^{n} can be expressed as a finite sum of elements of A_{X}^{n_{1}} · A_{X}^{n_{2}}, where n_{1} + n_{2} = n.

Similarly, the free Lie algebra L_{X} on X over k is a graded algebra over k, as in Section 26.3. Let L_{X}^{n} be the submodule of L_{X}, as a module over k, consisting of elements that are homogeneous of degree n ∈ Z_{+}, as before. If n_{1}, n_{2} ∈ Z_{+}, then [L_{X}^{n_{1}}, L_{X}^{n_{2}}] ⊆ L_{X}^{n_{1}+n_{2}}, (26.11.2)

where the left side is as defined in Section 9.2. If n ≥ 2, then every element of L_{X}^{n} can be expressed as a finite sum of elements of [L_{X}^{n_{1}}, L_{X}^{n_{2}}], where n_{1}+n_{2} = n. This follows from the analogous statement for A_{X}^{n}, in the preceding paragraph.

If n ≥ 2, then [L_{X}^{1}, L_{X}^{n−1}] = L_{X}^{n}, (26.11.3)

as in Exercise 2 on p29 of [25]. To see this, let n_{1}, n_{2} be positive integers with n_{1}+n_{2} = n, and let u_{1} ∈ L_{X}^{n_{1}} and u_{2} ∈ L_{X}^{n_{2}} be given. We would like to show that [u_{1}, u_{2}] ∈ [L_{X}^{1}, L_{X}^{n−1}], which is obvious when n_{1} or n_{2} is equal to 1. Otherwise, we may as well suppose that n_{2} ≥ n_{1} > 1. Thus u_{1} can be expressed as a finite sum of terms of the form [u_{1,1}, u_{1,2}], where u_{1,1} ∈ L_{X}^{n_{1,1}} and u_{1,2} ∈ L_{X}^{n_{1,2}} for some positive integers n_{1,1} and n_{1,2} with n_{1,1} + n_{1,2} = n_{1}. This means that [u_{1}, u_{2}] can be expressed as a finite sum of terms of the form

[[u_{1,1}, u_{1,2}], u_{2}] = [[u_{1,1}, u_{2}], u_{1,2}] + [u_{1,1}, [u_{1,2}, u_{2}]]. (26.11.4)

The terms on the right side are elements of [L_{X}^{n_{1,1}+n_{2}}, L_{X}^{n_{1,2}}] and [L_{X}^{n_{1,1}}, L_{X}^{n_{1,2}+n_{2}}], respectively. Of course, n_{1,1}, n_{1,2} < n_{1}, and so one can repeat the process as needed to get that [u_{1}, u_{2}] ∈ [L_{X}^{1}, L_{X}^{n−1}], as desired. Equivalently, if n ≥ 2, then L_{X}^{n} is generated, as a module over k, by elements of the form

[x_{1}, [x_{2}, … [x_{n−1}, x_{n}] ⋯]], (26.11.5)

where x_{1}, …, x_{n} ∈ X.

Let L be any Lie algebra over k, and suppose that L is generated, as a Lie algebra over k, by a nonempty subset X. Under these conditions, L is generated as a module over k by the elements of X and elements of the form (26.11.5), where x_{1}, …, x_{n} ∈ X for some n ≥ 2. This could be obtained from the previous remarks for free Lie algebras, or verified directly using the same type of arguments.

Some more Lie algebras

27.1 Generators and relations

Let k be a commutative ring with a mutliplicative identity element, and let X be a nonempty set. Thus the free Lie algebra L_{X} = L_{X}(k) on X over k may be defined as in Section 26.3. If E ⊆ L_{X}, then let R(E) be the ideal in L_{X} generated by E. Equivalently, R(E) consists of finite linear combinations of the elements of E and other elements of L_{X} obtained by taking brackets of elements of E with elements of L_{X} any number of times. The quotient L_{X}/R(E) may be described as the Lie algebra over k generated by the images of the elements of X in the quotient, and with relations given by condition that the images of the elements of E in the quotient be equal to 0, as on p95 of [14].

If B is any Lie algebra over k, then any mapping from X into B leads to a unique Lie algebra homomorphism from L_{X} into B, as before. If this Lie algebra homomorphism maps the elements of E to 0, then it maps the elements of R(E) to 0 as well. This leads to an induced Lie algebra homomorphism from L_{X}/R(E) into B. If B is generated as a Lie algebra over k by the images of the elements of X, then the Lie algebra homomorphism from L_{X} into B just mentioned is surjective. Of course, this implies that the corresponding Lie algebra homomorphism from L_{X}/R(E) into B is surjective as well.

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a reduced root system in V. If α ∈ Φ, then let σ_{α} be the symmetry on V with vector α that maps Φ onto itself, as usual. Remember that σ_{α} can be expressed as σ_{α}(v) = v − λ_{α}(v) α, where λ_{α} is a linear functional on V such that λ_{α}(α) = 2. Let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ. If α ∈ Φ, then σ_{α} is the reflection on V with respect to (·, ·) associated to α, so that λ_{α}(v) = 2 (v, α) (α, α)^{−1} for every v ∈ V. Put

n(α, β) = λ_{β}(α) = 2 (α, β) (β, β)^{−1} (27.1.1)

for every α, β ∈ Φ, as before. Remember that this is an integer, by the definition of a root system. Let Δ be a base for Φ, so that the restriction of (27.1.1) to α, β ∈ Δ is the Cartan matrix of Φ with respect to Δ. If α, β ∈ Δ and α ≠ β, then −3 ≤ n(α, β) ≤ 0, (27.1.2)

as in Section 20.2. We have also seen that the Cartan matrix is invertible, as in Section 21.15.

Let k be a field of characteristic 0, and let X_{Δ}, Y_{Δ}, and H_{Δ} be disjoint sets, with distinct elements x̂_{α}, ŷ_{α}, and ĥ_{α} indexed by α ∈ Δ, respectively. Also let L̂ = L̂(k) be the free Lie algebra on

X_{Δ} ∪ Y_{Δ} ∪ H_{Δ} (27.1.3)

over k. If α, β ∈ Δ, then put δ_{α,β} = 1 when α = β, and equal to 0 when α ≠ β. Let Ê_{0} be the subset of L̂ consisting of the following elements, for α, β ∈ Δ:

[ĥ_{α}, ĥ_{β}], (27.1.4)

[x̂_{α}, ŷ_{β}] − δ_{α,β} ĥ_{α}, (27.1.5)

[ĥ_{α}, x̂_{β}] − n(β, α) x̂_{β}, (27.1.6)

[ĥ_{α}, ŷ_{β}] + n(β, α) ŷ_{β}. (27.1.7)

This leads to an ideal R(Ê_{0}) in L̂, as before.

Let L̃ = L̃(k) = L̂/R(Ê_{0}) (27.1.8)

be the corresponding quotient Lie algebra, and let x̃_{α}, ỹ_{α}, and h̃_{α} be the images of x̂_{α}, ŷ_{α}, and ĥ_{α}, respectively, under the quotient mapping for each α ∈ Δ. Note that L̃ is generated by the x̃_{α}’s, ỹ_{α}’s, and h̃_{α}’s, α ∈ Δ, as a Lie algebra over k. If α, β ∈ Δ, then we have that

(27.1.9)  [h̃_{α}, h̃_{β}] = 0,
(27.1.10)  [x̃_{α}, ỹ_{β}] = δ_{α,β} h̃_{α},
(27.1.11)  [h̃_{α}, x̃_{β}] = n(β, α) x̃_{β},
(27.1.12)  [h̃_{α}, ỹ_{β}] = −n(β, α) ỹ_{β}

in L̃, by construction. This Lie algebra has been analyzed by Chevalley, Harish-Chandra, and Jacobson. Here we are largely following the discussion that begins on p96 of [14].

27.2 A module over L̂

27.2. A MODULE OVER L̂ 575

Let us continue with the same notation and hypotheses as in the previous section, and let Z_{Δ} be a vector space over k with a basis given by distinct elements z_{α}, α ∈ Δ. If m is a positive integer, then the mth tensor power T^{m}Z_{Δ} is a vector space over k, and we can get a basis for T^{m}Z_{Δ} using elements of the form z_{α_{1}} ⊗⋯⊗z_{α_{m}}, where α_{1}, …, α_{m} ∈ Δ. Remember that the tensor algebra T Z_{Δ} is obtained by taking the direct sum of T^{m}Z_{Δ} over all nonnegative integers m, with T^{0}Z_{Δ} = k, as in Section 25.3. We would like to make T Z_{Δ} into a module over L̂ in a certain way, as on p97 of [14]. To do this, we need to define the action of the generators of L̂ on T Z_{Δ}.

Of course, we can get a basis for T Z_{Δ}, as a vector space over k, using the bases for T^{m}Z_{Δ} mentioned in the preceding paragraph for each m ≥ 1, and 1 ∈ k. In order to define the action of the generators of L̂ on T Z_{Δ}, it suffices to define the action on the basis elements of T Z_{Δ}. If α ∈ Δ, then we put

ĥ_{α} · 1 = 0, (27.2.1)

and

ĥ_{α} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}}) (27.2.2)

= −(n(γ_{1}, α) + ⋯ + n(γ_{m}, α)) (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}})

for every m ≥ 1 and γ_{1}, …, γ_{m} ∈ Δ. Similarly, we put

ŷ_{α} · 1 = z_{α}, (27.2.3)

and ŷ_{α} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}}) = z_{α} ⊗ z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}} (27.2.4)

for every m ≥ 1 and γ_{1}, …, γ_{m} ∈ Δ. We also put

x̂_{α} · 1 = 0, (27.2.5)

and x̂_{α} · z_{γ} = 0 (27.2.6)

for every γ ∈ Δ. If m ≥ 2 and γ_{1}, …, γ_{m} ∈ Δ, then we put

x̂_{α} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}}) (27.2.7)

= z_{γ_{1}} ⊗ (x̂_{α} · (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}))
−δ_{γ_{1},α} (n(γ_{2}, α) + ⋯ + n(γ_{m}, α)) (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}).

More precisely, this can be used to define the left side recursively.

Let gl(T Z_{Δ}) be the space of linear mappings from T Z_{Δ} into itself, as a Lie algebra over k with respect to the corresponding commutator bracket, as usual. If α ∈ Δ, then we get elements of gl(T Z_{Δ}) associated to ĥ_{α}, ŷ_{α}, and x̂_{α}, as in the previous paragraph. This leads to a Lie algebra homomorphism ϕ̂ from L̂ into gl(T Z_{Δ}), so that T Z_{Δ} becomes a module over L̂, as a Lie algebra over k. We would like to show that

R(Ê_{0}) ⊆ ker ϕ̂, (27.2.8)

where R(Ê_{0}) is as in the previous section. This corresponds to the proposition on p97 of [14].

576 CHAPTER 27. SOME MORE LIE ALGEBRAS

Of course, it suffices to show that Ê_{0} ⊆ ker ϕ̂, (27.2.9) where Ê_{0} is as in the previous section, because R(Ê_{0}) is the ideal in L̂ generated by Ê_{0}. If α, β ∈ Δ, then it is easy to see that ϕ̂([ĥ_{α}, ĥ_{β}]) = [ϕ̂(ĥ_{α}), ϕ̂(ĥ_{β})] = 0, (27.2.10) which is to say that the actions of ĥ_{α} and ĥ_{β} on T Z_{Δ} commute. Note that the action of ŷ_{α} on T Z_{Δ} is the same as multiplication by z_{α} on the left, with respect to multiplication on the tensor algebra T Z_{Δ}.

Let α, β ∈ Δ be given, and let us check that ϕ̂([x̂_{α}, ŷ_{β}]) = [ϕ̂(x̂_{α}), ϕ̂(ŷ_{β})] = δ_{α,β} ϕ̂(ĥ_{α}). (27.2.11) Observe that x̂_{α} · (ŷ_{β} · 1) = x̂_{α} · z_{β} = 0, so that x̂_{α} · (ŷ_{β} · 1) − ŷ_{β} · (x̂_{α} · 1) = 0, (27.2.12) which is the same as δ_{α,β} ĥ_{α} · 1. If m ≥ 2 and γ_{2}, …, γ_{m} ∈ Δ, then

x̂_{α} · (ŷ_{β} · (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}})) − ŷ_{β} · (x̂_{α} · (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}))

(27.2.13) = x̂_{α} · (z_{β} ⊗ z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}) − z_{β} ⊗ (x̂_{α} · (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}))
= −δ_{β,α} (n(γ_{2}, α) + ⋯ + n(γ_{m}, α)) (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}})
= δ_{β,α} ĥ_{α} · (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}).

More precisely, this uses (27.2.7) in the second step. This shows that (27.2.11) holds.

Similarly, let us verify that ϕ̂([ĥ_{α}, ŷ_{β}]) = [ϕ̂(ĥ_{α}), ϕ̂(ŷ_{β})] = −n(β, α) ϕ̂(ŷ_{β}). (27.2.14) Observe that

ĥ_{α} · (ŷ_{β} · 1) − ŷ_{β} · (ĥ_{α} · 1) = ĥ_{α} · z_{β}
(27.2.15)  = −n(β, α) z_{β} = −n(β, α) ŷ_{β} · 1.

If γ_{1}, …, γ_{m} ∈ Δ for some m ≥ 1, then

ĥ_{α} · (ŷ_{β} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}})) − ŷ_{β} · (ĥ_{α} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}}))

(27.2.16) = ĥ_{α} · (z_{β} ⊗ ·z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}}) − z_{β} ⊗ (ĥ_{α} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}}))
= −n(β, α) (z_{β} ⊗ z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}})
= −n(β, α) ŷ_{β} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}}).

This implies (27.2.14).

Note that ĥ_{α} · (x̂_{β} · 1) = n(β, α) x̂_{β} · 1 = 0. (27.2.17)

We would like to check that

ĥ_{α} · (x̂_{β} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}})) (27.2.18)

= −(n(γ_{1}, α) + ⋯ + n(γ_{m}, α) − n(β, α)) x̂_{β} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}})

for every m ≥ 1 and γ_{1}, …, γ_{m} ∈ Δ. If m = 1, then both sides of the equation are equal to 0. Suppose that m ≥ 2, and that the analogous statement holds when m − 1, by induction, so that

ĥ_{α} · (x̂_{β} · (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}})) (27.2.19)

= −(n(γ_{2}, α) + ⋯ n(γ_{m}, α) − n(β, α)) x̂_{β} · (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}).

Using this, one can verify that

ĥ_{α} · (z_{γ_{1}} ⊗ (x̂_{β} · (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}))) (27.2.20)

= −(n(γ_{1}, α) + ⋯ + n(γ_{m}, α) − n(β, α)) (z_{γ_{1}} ⊗ (x̂_{β} · (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}))).

One can obtain (27.2.18) from this and (27.2.7). More precisely, this also uses the fact that

ĥ_{α} · (δ_{γ_{1},β} (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}})) (27.2.21)

= −δ_{γ_{1},β} (n(γ_{2}, α) + ⋯ + n(γ_{m}, α)) (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}})
= −δ_{γ_{1},β} (n(γ_{1}, α) + n(γ_{2}, α) + ⋯ + n(γ_{m}, α) − n(β, α)) (z_{γ_{2}} ⊗ ⋯ ⊗ z_{γ_{m}}).

Let us check that

ϕ̂([ĥ_{α}, x̂_{β}]) = [ϕ̂(ĥ_{α}), ϕ̂(x̂_{β})] = n(β, α) ϕ̂(x̂_{β}). (27.2.22)

Observe that ĥ_{α} · (x̂_{β} · 1) − x̂_{β} · (ĥ_{α} · 1) = 0, (27.2.23)

which is the same as n(β, α) x̂_{β} ·1. Let γ_{1}, …, γ_{m} ∈ Δ be given for some m ≥ 1, and note that

x̂_{β} · (ĥ_{α} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}})) (27.2.24)

= −(n(γ_{1}, α) + ⋯ + n(γ_{m}, α)) x̂_{β} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}}).

Combining this with (27.2.18), we obtain that

ĥ_{α} · (x̂_{β} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}})) − x̂_{β} · (ĥ_{α} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}})) (27.2.25)

= (−(n(γ_{1}, α) + ⋯ + n(γ_{m}, α) − n(β, α)) + (n(γ_{1}, α) + ⋯ + n(γ_{m}, α)))
x̂_{β} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}})
= n(β, α) x̂_{β} · (z_{γ_{1}} ⊗ ⋯ ⊗ z_{γ_{m}}).

This implies (27.2.22), using (27.2.23) as well.

It follows that (27.2.9) holds, because of (27.2.10), (27.2.11), (27.2.14), and (27.2.22). This implies (27.2.8), as before.

27.3 Some properties of L̃

We continue with the same notation and hypotheses as in the previous two sections. In particular, remember that L̃ = L̂/R(Ê_{0}), and that x̃_{α}, ỹ_{α}, h̃_{α} are the images of x̂_{α}, ŷ_{α}, ĥ_{α} ∈ L̂ in L̃, respectively, under the natural quotient mapping for each α ∈ Δ. In the previous section, a Lie algebra homomorphism ϕ̂ from L̂ into gl(T Z_{Δ}) was defined, whose kernel contains R(Ê_{0}). This leads to a Lie algebra homomorphism ϕ̃ from L̃ into gl(T Z_{Δ}), whose composition with the natural quotient mapping from L̂ onto L̃ is equal to ϕ̂, as in the proposition on p97 of [14]. Thus T Z_{Δ} may be considered as a module over L̃, as a Lie algebra over k.

Of course, ϕ̃(x̃_{α}) = ϕ̂(x̂_{α}), ϕ̃(ỹ_{α}) = ϕ̂(ŷ_{α}), and ϕ̃(h̃_{α}) = ϕ̂(ĥ_{α}) for every α ∈ Δ, by construction. This means that the actions of x̃_{α}, ỹ_{α}, and h̃_{α} on T Z_{Δ} are the same as for x̂_{α}, ŷ_{α}, and ĥ_{α}, respectively, for each α ∈ Δ.

Let H̃ be the linear span of the h̃_{α}’s, α ∈ Δ, in L̃. This is a Lie subalgebra of L̃, that is commutative as a Lie algebra, because the h̃_{α}’s commute with each other, as in (27.1.9). Let X̃ be the Lie subalgebra of L̃ generated by the x̃_{α}’s, α ∈ Δ, and let Ỹ be the Lie subalgebra of L̃ generated by the ỹ_{α}’s, α ∈ Δ. The theorem on p97f of [14] states that

(27.3.1) the h̃_{α}’s, α ∈ Δ, form a basis for H̃ as a vector space over k,

and that

(27.3.2)  L̃ corresponds to the direct sum of X̃, H̃, and Ỹ,

as a vector space over k.

Step (1) on p98 of [14] states that the intersection of the linear span of the ĥ_{α}’s, α ∈ Δ, in L̂ with the kernel of ϕ̂ is trivial. To see this, let a_{α} ∈ k be given for each α ∈ Δ, and put

ĥ = ∑_{α∈Δ} a_{α} ĥ_{α}. (27.3.3)

Thus ĥ ∈ L̂, and

ĥ · z_{γ} = − ∑_{α∈Δ} a_{α} n(γ, α) z_{γ} (27.3.4)

for every γ ∈ Δ, by (27.2.2). If ϕ̂(ĥ) = 0, then (27.3.4) is equal to 0 for each γ ∈ Δ, so that

∑_{α∈Δ} a_{α} n(γ, α) = 0 (27.3.5)

for every γ ∈ Δ. We would like to use this to get that a_{α} = 0 for every α ∈ Δ.

Remember that the Cartan matrix is invertible, as a matrix of real numbers, as in Section 21.15. The entries of the Cartan matrix are integers, so that its determinant is a nonzero integer. If we consider the Cartan matrix as having entries in k, then its determinant is still nonzero, because k has characteristic 0, 27.3. SOME PROPERTIES OF L̃ 579

by hypothesis. This means that the Cartan matrix is also invertible as a matrix with entries in k. Thus (27.3.5) implies that a_{α} = 0 for every α ∈ Δ, as desired.

Of course,

h̃ = ∑_{α∈Δ} a_{α} h̃_{α} (27.3.6)

is the image of (27.3.3) under the natural quotient mapping from L̂ onto L̃. If h̃ = 0, then ϕ̂(ĥ) = ϕ̃(h̃) = 0, so that a_{α} = 0 for every α ∈ Δ, as before. It follows that the restriction of the natural quotient mapping from L̂ onto L̃ to the linear span of the ĥ_{α}’s, α ∈ Δ, in L̂ is injective, as in Step (2) on p98 of [14]. This implies that the dimension of H̃, as a vector space over k, is the same as the number of elements in Δ, as in Step (4) on p98 of [14]. Equivalently, this means that (27.3.1) holds.

In particular,

(27.3.7)  h̃_{α} ≠ 0 for each α ∈ Δ.

This implies that

(27.3.8)  x̃_{α}, ỹ_{α} ≠ 0 for each α ∈ Δ,

because [x̃_{α}, ỹ_{α}] = h̃_{α}, as in (27.1.10).

If β ∈ Δ, then put f_{β}(α) = n(β, α) (27.3.9)

for every α ∈ Δ, which may be considered as a k-valued function on Δ. In fact, the f_{β}’s form a basis for the space of k-valued functions on Δ, as a vector space over k. This follows from the invertibility of the Cartan matrix, as a matrix with entries in k, as before. In particular, for each β ∈ Δ, f_{β} ≢ 0 on Δ. Similarly, if β, β′ ∈ Δ and β ≠ β′, then f_{β} ≠ f_{β′}, as functions on Δ. We also have that f_{β} ≠ −f_{β′}, even when β = β′. This uses the hypothesis that k has characteristic 0, so that 1 ≠ −1 in k.

One can use this to check that

(27.3.10) the set of x̃_{β}’s, ỹ_{β}’s, and h̃_{β}’s, β ∈ Δ,

is linearly independent in L̃,

as in Step (3) on p98 of [14]. This also uses the fact that the x̃_{β}’, ỹ_{β}’s, and h̃_{β}’s are simultaneous eigenvectors for ad_{h_{α}}, α ∈ Δ, as in (27.1.9), (27.1.11), and (27.1.12), with the corresponding eigenvalues given by f_{β}(α), −f_{β}(α), and 0, respectively. Note that the ad_{h_{α}}’s, α ∈ Δ, commute with each other, as linear

mappings on L̃, because the h̃_{α}’s commute in L̃, as in Section 2.4.

More precisely, suppose that we have a linear combination of the x̃_{β}’s, ỹ_{β}’s, and h̃_{β}’s, β ∈ Δ, in L̃ that is equal to 0. One can show that the coefficients of the x̃_{β}’s and ỹ_{β}’s are all equal to 0, and that the corresponding linear combination of the h̃_{β}’s is equal to 0, using the earlier remarks about the f_{β}’s. It follows that the coefficients of the h̃_{β}’s have to be equal to 0 too, by (27.3.1).

580 CHAPTER 27. SOME MORE LIE ALGEBRAS

Equivalently, consider the linear subspace of L̂ spanned by the x̂_{β}’s, ŷ_{β}’s, and ĥ_{β}’s, β ∈ Δ. The restriction of the natural quotient mapping from L̂ onto L̃ to this linear subspace is injective, as in Step (3) on p98 of [14].

27.4 Some brackets in L̃

Let us continue with the discussion in the previous section. In particular, we would like to prove (27.3.2).

If ũ_{1}, …, ũ_{r} ∈ L̃ for some r ≥ 2, then put [ũ_{1}, …, ũ_{r}] = [ũ_{1}, [ũ_{2}, … [ũ_{r−1}, ũ_{r}] ⋯]]. (27.4.1) If α, β_{1}, …, β_{r} ∈ Δ, then [h̃_{α}, [x̃_{β_{1}}, …, x̃_{β_{r}}]] = (n(β_{1}, α) + ⋯ + n(β_{r}, α)) [x̃_{β_{1}}, …, x̃_{β_{r}}], (27.4.2) as in Step (5) on p98 of [14]. This follows from (27.1.11) and the Jacobi identity when r = 2, and one can use induction to get the analogous conclusion for every r ≥ 2. Similarly, [h̃_{α}, [ỹ_{β_{1}}, …, ỹ_{β_{r}}]] = −(n(β_{1}, α) + ⋯ + n(β_{r}, α)) [ỹ_{β_{1}}, …, ỹ_{β_{r}}], (27.4.3) by (27.1.12).

If α, β_{1}, …, β_{r} ∈ Δ for some r ≥ 2 again, then [ỹ_{α}, [x̃_{β_{1}}, …, x̃_{β_{r}}]] (27.4.4) can be expressed as a linear combination of x̃_{β_{1}} and x̃_{β_{2}} when r = 2, and as a linear combination of terms of the form [x̃_{γ_{1}}, …, x̃_{γ_{r−1}}] (27.4.5) when r ≥ 3, where γ_{1}, …, γ_{r−1} ∈ {β_{1}, …, β_{r}}. In particular, this means that (27.4.4) is an element of X̃, as in Step 6 on p98 of [14]. If r = 2, then the previous statement can be obtained from (27.1.10), (27.1.11), and the Jacobi identity. If r ≥ 3, then one can use induction and (27.4.2). Similarly, [x̃_{α}, [ỹ_{β_{1}}, …, ỹ_{β_{r}}]] (27.4.6) can be expressed as a linear combination of ỹ_{β_{1}} and ỹ_{β_{2}} when r = 2, and as a linear combination of terms of the form [ỹ_{γ_{1}}, …, ỹ_{γ_{r−1}}] (27.4.7) when r ≥ 3, where γ_{1}, …, γ_{r−1} ∈ {β_{1}, …, β_{r}}, as before.

Remember that X̃ and Ỹ are the Lie subalgebras of L̃ generated by the x̃_{α}’s and ỹ_{α}’s, respectively, with α ∈ Δ. Note that X̃ is spanned by the x̃_{α}’s, α ∈ Δ, and elements of the form [x̃_{α_{1}}, …, x̃_{α_{r}}], where α_{1}, …, α_{r} ∈ Δ for some r ≥ 2, 27.5. EIGENVECTORS IN L̃ 581

as in Section 26.11. Similarly, Ỹ is spanned by the ỹ_{α}’s, α ∈ Δ, and elements of the form [ỹ_{α_{1}}, …, ỹ_{α_{r}}], with α_{1}, …, α_{r} ∈ Δ for some r ≥ 2.

One can check that the linear subspace X̃ + H̃ + Ỹ of L̃ spanned by elements of X̃, H̃, and Ỹ is a Lie subalgebra of L̃, as in Step (7) on p98 of [14]. More precisely, this uses the fact that H̃ is a Lie subalgebra of L̃, (27.4.2), and (27.4.3), as well as (27.1.10) and the earlier remarks about (27.4.4) and (27.4.6).

It follows that X̃ + H̃ + Ỹ is equal to L̃, because L̃ is generated as a Lie algebra by the x̃_{α}’s, ỹ_{α}’s, and h̃_{α}’s, α ∈ Δ.

In order to verify (27.3.2), we need to show that if a sum of elements of X̃, H̃, and Ỹ is equal to 0, then each term is 0. Note that X̃ is spanned by simultaneous eigenvectors for ad_{h_{α}}, α ∈ Δ, by (27.4.2). The corresponding eigenvalues for the simultaneous eigenvectors are given by nontrivial sums of f_{β}’s, β ∈ Δ. Similarly, Ỹ is spanned by simultaneous eigenvectors for ad_{h_{α}}, α ∈ Δ, by (27.4.3), with the corresponding eigenvalues given by nontrivial sums of −f_{β}’s, β ∈ Δ. Of course, H̃ is contained in the kernel of ad_{h_{α}} for each α ∈ Δ, by (27.1.9). Because k has characteristic 0, nontrivial sums of f_{β}’s or of −f_{β}’s cannot be equal to 0 on all of Δ, or equal to each other on all of Δ. This can be used to get that if a sum of elements of X̃, H̃, and Ỹ is equal to 0, then each term is 0, as in Step (8) on p98 of [14].

27.5 Eigenvectors in L̃

Let us look a bit more at the simultaneous eigenvectors for ad_{h_{α}}, α ∈ Δ, in L̃,

as in the previous two sections. Remember that H̃ is the linear span of the h̃_{α}’s, α ∈ Δ. Let H̃′ be the dual of H̃, as a vector space over k, as usual. If μ̃ ∈ H̃′, then put

(27.5.1)  L̃_{μ} = {ũ ∈ L : [h̃, ũ] = μ̃(h̃) ũ for every h̃ ∈ H̃},

which is a linear subspace of L̃. Of course, the ad_{h}’s, h̃ ∈ H̃, commute as linear mappings on L̃, because of the same property of the ad_{h_{α}}’s, α ∈ Δ.

Suppose for the moment that μ̃_{1}, …, μ̃_{n} are distinct elements of H̃′, and that

ũ_{j} ∈ L̃_{μ_{j}} for j = 1, …, n. If ∑_{j=1}^{n} ũ_{j} = 0, then ũ_{j} = 0 for every j = 1, …, n,

by standard arguments.

Remember that the h̃_{α}’s, α ∈ Δ, form a basis for H̃. If β ∈ Δ, then there is a unique ν̃_{β} ∈ H̃′ such that

ν̃_{β}(h̃_{α}) = n(β, α) (27.5.2)

for every α ∈ Δ. The ν̃_{β}’s, β ∈ Δ, form a basis for H̃′, as a vector space over k. This is essentially the same as the fact that the f_{β}’s in (27.3.9) form a basis for the space of k-valued functions on Δ, as before.

Every element of X̃ can be expressed as the sum of elements of finitely many of the subspaces L̃_{μ}, where μ̃ ∈ H̃′ can be expressed as a nontrivial sum of ν̃_{β}’s, as in the previous section. Similarly, every element of Ỹ can be expressed as the sum of finitely many elements of L̃_{μ}’s, where μ̃ ∈ H̃′ can be expressed as a

nontrivial sum of −ν̃_{β}’s. It follows that X̃ and Ỹ correspond to direct sums of their intersections with the appropriate subspaces L̃_{μ}, as vector spaces over k.

Clearly H̃ ⊆ L̃_{0}, by (27.1.9). Remember that L̃ corresponds to the direct sum of X̃, H̃, and Ỹ, as a vector space over k, as in (27.3.2). Using this, one can check that H̃ = L̃_{0}. (27.5.3)

This also uses the hypothesis that k have characteristic 0, so that nontrivial sums of ν̃_{β}’s cannot be equal to 0.

Similarly, L̃_{μ} ⊆ X̃ when μ̃ ∈ H̃ is a nontrivial sum of ν̃_{β}’s, and L̃_{μ} ⊆ Ỹ when μ̃ is a nontrivial sum of −ν̃_{β}’s. This uses the hypothesis that k have characteristic 0, to get that nontrivial sums of ν̃_{β}’s cannot be equal to nontrivial sums of −ν̃_{β}’s. This implies that X̃ corresponds to the direct sum of the subspaces L̃_{μ},

where μ̃ ∈ H̃′ can be expressed as a nontrivial sum of ν̃_{β}’s, and that Ỹ corresponds to the direct sum of the suspaces L̃_{μ}, where μ̃ ∈ H̃′ can be expressed as a nontrivial sum of −ν̃_{β}’s, as vector spaces over k.

Let μ̃ ∈ H̃′ be given, and suppose that L̃_{μ} ≠ {0}. Under these conditions, one can verify that either μ̃ = 0, or μ̃ is a nontrivial sum of ν̃_{β}’s, or μ̃ is a nontrivial sum of −ν̃_{β}’s.

If μ̃ ∈ H̃′, then L̃_{μ} is finite-dimensional, as a vector space over k. This follows from (27.5.3) when μ̃ = 0, and otherwise we may suppose that μ̃ is a nontrivial sum of ν_{β}’s, or that μ̃ is a nontrivial sum of −ν_{β}’s, as in the preceding paragraph. In both cases, the elements of L̃_{μ} considered in the previous section

are contained in finite-dimensional subspaces of L̃. One can check that all of the elements of L̃_{μ} can be obtained as linear combinations of those considered

before, because L̃ is spanned by elements of this type and the h̃_{α}’s, as in the previous section.

This corresponds to some remarks on p98 of [14], after the proof of the theorem, and on p53 of [24]. In particular, the elements of L̃_{μ} are said to have weight μ̃.

27.6 Some more properties of L̃

Let us continue with the same notation and hypotheses as in the previous five sections. Thus the tensor algebra T Z_{Δ} from Section 27.2 is an associative algebra over k, which may be considered as a Lie algebra over k with respect to the corresponding commutator bracket. Let L_{Y_{Δ}} be the free Lie algebra on Y_{Δ} = {ŷ_{α} : α ∈ Δ} over k, and let ψ_{1} be the Lie algebra homomorphism from L_{Y_{Δ}} into T Z_{Δ} that sends ŷ_{α} to z_{α} ∈ Z_{Δ} for every α ∈ Δ. Note that

(27.6.1) ψ_{1} is injective as a mapping from L_{Y_{Δ}} into T Z_{Δ}, 27.6. SOME MORE PROPERTIES OF L̃ 583

as in Section 26.6. More precisely, this corresponds to identifying T Z_{Δ} with the free associative algebra Assoc_{Y_{Δ}} on Y_{Δ} over k, by identifying ŷ_{α} with z_{α} for each α ∈ Δ.

Of course, each element of T Z_{Δ} determines a linear mapping from T Z_{Δ} into itself, defined by multiplication on the left by the given element of T Z_{Δ}. In particular, this defines a Lie algebra homomorphism from T Z_{Δ} into gl(T Z_{Δ}). Note that this mapping is injective, because T Z_{Δ} has a multiplicative identity element. Let ψ′_{1} be the mapping from L_{Y_{Δ}} into gl(T Z_{Δ}) which is the composition of ψ_{1} with the Lie algebra homomorphism from T Z_{Δ} into gl(T Z_{Δ}) just mentioned. Thus ψ′_{1} is an injective Lie algebra homomorphism from L_{Y_{Δ}} into gl(T Z_{Δ}).

Let ϕ̂ be the Lie algebra homomorphism from L̂ into gl(T Z_{Δ}) defined in Section 27.2. Remember that ϕ̂(ŷ_{α}) is the linear mapping from T Z_{Δ} to itself corresponding to left multiplication by z_{α} for each α ∈ Δ. There is a natural Lie algebra homomorphism from L_{Y_{Δ}} into L̂, which sends ŷ_{α} to itself as an element of L̂ for every α ∈ Δ. Let ϕ̂_{1} be the composition of this homomorphism with ϕ̂, so that ϕ̂_{1} is a Lie algebra homomorphism from L_{Y_{Δ}} into gl(T Z_{Δ}). Observe that ϕ̂_{1} = ψ′_{1} (27.6.2)

on L_{Y_{Δ}}, because both mappings send ŷ_{α} to the linear mapping from T Z_{Δ} into itself that corresponds to left multiplication by z_{α} for each α ∈ Δ.

Remember from Section 27.3 that ϕ̂ is the same as the composition of the natural quotient mapping from L̂ onto L̃ with the Lie algebra homomorphism ϕ̃ from L̃ into gl(T Z_{Δ}). Let ρ_{1} be the composition of the natural mapping from L_{Y_{Δ}} into L̂ with the quotient mapping from L̂ onto L̃. Thus ρ_{1} is a Lie algebra homomorphism from L_{Y_{Δ}} into L̃, and

ϕ̂_{1} = ϕ̃ ∘ ρ_{1} (27.6.3)

on L_{Y_{Δ}}, by construction. It follows that

(27.6.4) ρ_{1} is injective as a mapping from L_{Y_{Δ}} into L̃,

because of (27.6.2) and the fact that ψ′_{1} is injective on L_{Y_{Δ}}, as before.

Note that ρ_{1}(ŷ_{α}) = ỹ_{α} for each α ∈ Δ, by construction. Let Ỹ be the Lie subalgebra of L̃ generated by the ỹ_{α}’s, α ∈ Δ, as in Section 27.3. Thus

ρ_{1}(L_{Y_{Δ}}) = Ỹ, (27.6.5)

because L_{Y_{Δ}} is generated by Y_{Δ} as a Lie algebra over k. This shows that Ỹ is isomorphic to L_{Y_{Δ}} as a Lie algebra over k, as in Exercise 1 on p101 of [14].

Let σ̂ be the Lie algebra homomorphism from L̂ into itself such that

σ̂(x̂_{α}) = −ŷ_{α}, σ̂(ŷ_{α}) = −x̂_{α}, and σ̂(ĥ_{α}) = −ĥ_{α} (27.6.6)

for every α ∈ Δ. The composition of σ̂ with itself sends x̂_{α}, ŷ_{α}, and ĥ_{α} to themselves, respectively, for every α ∈ Δ. Thus σ̂ ∘ σ̂ is the identity mapping on L̂, and in particular σ̂ is a Lie algebra automorphism of L̂. It is easy to see that σ̂(R(Ê_{0})) = R(Ê_{0}), (27.6.7)

where Ê_{0} ⊆ L̂ is as in Section 27.1, and R(Ê_{0}) is the ideal in L̂ generated by Ê_{0}. This implies that there is a Lie algebra automorphism σ̃ of L̃, whose composition with the quotient mapping from L̂ onto L̃ is the same as the composition of the quotient mapping with σ̂.

Of course,

σ̃(x̃_{α}) = −ỹ_{α}, σ̃(ỹ_{α}) = −x̃_{α}, and σ̃(h̃_{α}) = −h̃_{α} (27.6.8)

for every α ∈ Δ. This implies that

σ̃(Ỹ) = X̃, (27.6.9)

which is the Lie subalgebra of L̃ generated by the x̃_{α}’s, α ∈ Δ, as before. In particular, X̃ and Ỹ are isomorphic as Lie algebras over k, as in Exercise 1 on p101 of [14].

27.7 Some more relations in L̃

Let us continue with the same notation and hypotheses as in the previous sections. Remember that n(α, β) ∈ Z for every α, β ∈ Δ, with −3 ≤ n(α, β) ≤ 0 when α ≠ β. Put

x̃_{α,β} = (ad_{x_{α}})^{−n(β,α)+1}(x̃_{β}), ỹ_{α,β} = (ad_{y_{α}})^{−n(β,α)+1}(ỹ_{β}) (27.7.1)

for every α, β ∈ Δ with α ≠ β, which are elements of L̃. We would like to show that

ad_{x_{γ}} (ỹ_{α,β}) = 0 (27.7.2)

for every α, β, γ ∈ Δ with α ≠ β, as in the lemma on p99 of [14]. We also have that

ad_{y_{γ}} (x̃_{α,β}) = 0 (27.7.3)

when α ≠ β, using a similar argument, or the automorphism σ̃ on L̃ discussed in the previous section.

Suppose first that α ≠ γ, so that [x̃_{γ}, ỹ_{α}] = 0, as in (27.1.10). This implies that ad_{x_{γ}} and ad_{y_{α}} commute as linear mappings on L̃, as in Section 2.4. It follows that

ad_{x_{γ}} (ỹ_{α,β}) = (ad_{y_{α}})^{−n(β,α)+1}(ad_{x_{γ}} (ỹ_{β})) (27.7.4)

in this case. If β ≠ γ, then ad_{x_{γ}} (ỹ_{β}) = 0, by (27.1.10) again, so that (27.7.2) holds. If β = γ, then (27.7.4) reduces to

ad_{x_{β}}(ỹ_{α,β}) = (ad_{y_{α}})^{−n(β,α)+1}(h̃_{β}), (27.7.5)

by (27.1.10).

Note that

ad_{y_{α}}(h̃_{β}) = n(α, β) ỹ_{α}, (27.7.6)

by (27.1.12). If n(α, β) = 0, then it follows that (27.7.2) holds, by (27.7.5). Otherwise, if n(α, β) ≠ 0, then n(β, α) ≠ 0, by (27.1.1). More precisely, n(β, α) < 0, because α ≠ β, so that −n(β, α) + 1 ≥ 2. (27.7.7)

This means that (27.7.2) follows from (27.7.5), because ad_{y_{α}}(ỹ_{α}) = 0. Suppose now that α = γ, so that we would like to show that

ad_{x_{α}}(ỹ_{α,β}) = 0 (27.7.8)

when α ≠ β. Remember that x̃_{α}, ỹ_{α}, h̃_{α} ≠ 0, as in (27.3.7) and (27.3.8). The linear span of x̃_{α}, ỹ_{α}, and h̃_{α} in L̃ is a Lie subalgebra of L̃, because of (27.1.10), (27.1.11), and (27.1.12). This subalgebra is isomorphic to sl(2, k) as a Lie algebra over k, because n(α, α) = 2. Thus we may consider L̃ as a module over sl(2, k), as a Lie algebra over k, using ad_{w} when w is in the linear span of x̃_{α}, ỹ_{α}, and h̃_{α}.

Note that ad_{x_{α}}(ỹ_{β}) = 0, as in (27.1.10), because α ≠ β. We also have that ad_{h_{α}}(ỹ_{β}) = −n(β, α) ỹ_{β}, as in (27.1.12). This means that ỹ_{β} is a maximal or

primitive vector of weight λ = −n(β, α) in L̃ as a module over the linear span of x̃_{α}, ỹ_{α}, and h̃_{α}, as in Section 15.2. If j is a positive integer, then

ad_{x_{α}}((ad_{y_{α}})^{j}(ỹ_{β})) = j (λ − j + 1) (ad_{y_{α}})^{j−1}(ỹ_{β}), (27.7.9)

as in (15.3.4). The right side is equal to 0 when j = λ+1 = −n(β, α)+1, which implies (27.7.8), as desired.

27.8 Serre’s theorem

Let us continue with the same notation and hypotheses as in the previous sections, Section 27.1 in particular. If α, β ∈ Δ and α ≠ β, then put

x̂_{α,β} = (ad_{x_{α}})^{−n(β,α)+1}(x̂_{β}), ŷ_{α,β} = (ad_{y_{α}})^{−n(β,α)+1}(ŷ_{β}), (27.8.1)

which are elements of L̂. Of course, ad_{x_{α}} and ad_{y_{α}} refer to the adjoint rep-

resentation on L̂ here. Remember that n(β, α) ≤ 0 when α ≠ β, so that −n(β, α) + 1 ∈ Z_{+}. Clearly the images of x̂_{α,β} and ŷ_{α,β} in L̃ under the natural quotient mapping are the same as x̃_{α,β} and ỹ_{α,β} in (27.7.1), respectively.

Let Ê_{1} be the subset of L̂ consisting of x̂_{α,β} and ŷ_{α,β} for α, β ∈ Δ with α ≠ β. Remember that Ê_{0} ⊆ L̂ was defined in Section 27.1, and let R(Ê_{0} ∪ Ê_{1}) be the ideal in L̂ generated by Ê_{0} ∪ Ê_{1}. Thus the quotient

L̂/R(Ê_{0} ∪ Ê_{1}) (27.8.2) is a Lie algebra over k. A famous theorem of Serre states that this is a finitedimensional semisimple Lie algebra, as on p52 of [24], and p99 of [14]. More precisely, the linear span of the images of the ĥ_{α}’s, α ∈ Δ, in (27.8.2) is a Cartan subalgebra of this Lie algebra, which corresponds to the given root system Φ.

Let Ẽ_{1} be the subset of L̃ consisting of x̃_{α,β} and ỹ_{α,β} for α, β ∈ Δ with α ≠ β, which is the same as the image of Ê_{1} under the natural quotient mapping from L̂ onto L̃. Also let R(Ẽ_{1}) = R_{L}(Ẽ_{1}) be the ideal in L̃ generated by Ẽ_{1}. Consider the quotient L = L(k) = L̃/R(Ẽ_{1}), (27.8.3)

which is a Lie algebra over k. There is a natural Lie algebra homomorphism from L̂ onto L, which is the composition of the quotient mappings from L̂ onto L̃ and from L̃ onto L. The kernel of this homomorphism is R(Ê_{0} ∪ Ê_{1}), which leads to a natural Lie algebra isomorphism from (27.8.2) onto L.

Let x_{α}, y_{α}, and h_{α} be the images of x̃_{α}, ỹ_{α}, and h̃_{α}, respectively, under the natural quotient mapping from L̃ onto L for each α ∈ Δ. Of course, these correspond to the images of x̂_{α}, ŷ_{α}, and ĥ_{α}, respectively, in (27.8.2) for every α ∈ Δ. If α, β ∈ Δ, then

(27.8.4)  [h_{α}, h_{β}] = 0,
(27.8.5)  [x_{α}, y_{β}] = δ_{α,β} h_{α},
(27.8.6)  [h_{α}, x_{β}] = n(β, α) x_{β},
(27.8.7)  [h_{α}, y_{β}] = −n(β, α) y_{β}

in L, because of the analogous properties of x̃_{α}, ỹ_{α}, and h̃_{α} in L̃, as in Section 27.1. If α ≠ β, then we also have that

(ad_{x_{α}})^{−n(β,α)+1}(x_{β}) = (ad_{y_{α}})^{−n(β,α)+1}(y_{β}) = 0 (27.8.8)

in L, by definition of Ẽ_{1}.

The proof of Serre’s theorem will be discussed in the next sections, and continuing in the next chapter.

27.9 Ideals I, J

We continue with the discussion of Serre’s theorem, from the previous section.

Remember that X̃, Ỹ are the Lie subalgebras of L̃ generated by the x̃_{α}’s, ỹ_{α}’s, respectively, with α ∈ Δ, as in Section 27.3. Let Ẽ_{1,1} be the subset of L̃ consisting of x̃_{α,β} for α, β ∈ Δ with α ≠ β, and let Ẽ_{1,2} be the set of ỹ_{α,β} for α, β ∈ Δ with α ≠ β. Thus

Ẽ_{1} = Ẽ_{1,1} ∪ Ẽ_{1,2}, Ẽ_{1,1} ⊆ X̃, and Ẽ_{1,2} ⊆ Ỹ. (27.9.1)

Let I be the ideal in X̃ generated by Ẽ_{1,1}, and let J be the ideal in Ỹ generated by Ẽ_{1,2}. Note that I, J ⊆ R(Ẽ_{1}). (27.9.2)

27.10. SUBALGEBRAS X, H, AND Y OF L 587

We would like to show that

(27.9.3) I, J are ideals in L̃,

as in Step (1) on p99 of [14], and (a) on p53 of [24]. We shall do this for J, since the argument for I is analogous, as in [14]. Observe that

[h̃_{γ}, ỹ_{α,β}] = (−n(β, γ) + (n(β, α) − 1) n(α, γ)) ỹ_{α,β} (27.9.4)

for every α, β, γ ∈ Δ with α ≠ β, by (27.4.3) and the definition (27.7.1) of ỹ_{α,β}. It is easy to see that

ad_{h_{γ}} (Ỹ) ⊆ Ỹ (27.9.5)

for every γ ∈ Δ, using (27.1.12) and the Jacobi identity; a more precise version of this is given by (27.4.3). It follows from (27.9.4) and (27.9.5) that

ad_{h_{γ}} (J) ⊆ J (27.9.6)

for every γ ∈ Δ, using the Jacobi identity again.

Remember that H̃ is the linear span of the h̃_{α}’s, α ∈ Δ, in L̃, as in Section 27.3. Observe that

ad_{x_{γ}} (Ỹ) ⊆ Ỹ + H̃ (27.9.7)

for every γ ∈ Δ, by (27.1.10) and the earlier remarks about (27.4.6). One can check that

ad_{x_{γ}} (J) ⊆ J (27.9.8)

for every γ ∈ Δ, using (27.7.2), (27.9.6), (27.9.7), and the Jacobi identity. This implies that J is an ideal in L̃, because L̃ is generated as a Lie algebra over k by the x̃_{γ}’s and ỹ_{γ}’s, γ ∈ Δ. More precisely, this is the same as saying that the normalizer of J in L̃ is L̃, and the normalizer of J in L̃ is automatically a Lie subalgebra of L̃, by the Jacobi identity.

Step (2) on p99 of [14] states that

I + J = R(Ẽ_{1}), (27.9.9)

which corresponds to a remark just after (a) on p53 of [24]. Of course,

I + J ⊆ R(Ẽ_{1}), (27.9.10)

by (27.9.2). We also have that Ẽ_{1} ⊆ I + J, by construction. Note that I + J is an ideal in L̃, by (27.9.3). This implies that R(Ẽ_{1}) ⊆ I + J, as desired.

27.10 Subalgebras X, H, and Y of L

Let us continue with the same notation and hypotheses as in the previous sections. Let X, H, and Y be the images of X̃, H̃, and Ỹ, respectively, under the natural quotient mapping from L̃ onto L. Thus X, H, and Y are Lie subalgebras of L, and H is commutative as a Lie algebra over k, because of the analogous properties of X̃, H̃, and Ỹ in L̃. More precisely, X is the Lie subalgebra of L generated by the x_{α}’s, α ∈ Δ, and Y is the Lie subalgebra of L generated by the y_{α}’s, α ∈ Δ. Similarly, H is the same as the linear span of the h_{α}’s, α ∈ Δ, in L.

Remember that L̃ corresponds to the direct sum of X̃, H̃, and Ỹ as a vector space over k, as in (27.3.2). Similarly,

(27.10.1) L corresponds to the direct sum of X , H, and Y, as a vector space over k.

This follows from (27.9.9), and the analogous statement for L̃. This is part of Step (3) on p100 of [14], and of (b) on p53 of [24].

The kernels of the restrictions of the natural quotient mapping from L̃ onto L to X̃ and Ỹ are I and J, respectively, because of (27.9.9). Thus we can identify X, Y with the quotients

X̃/I, Ỹ/J , (27.10.2)

respectively. Similarly, the restriction of the natural quotient mapping from L̃ onto L to H̃ is injective, by (27.9.9). This permits us to identify H̃ with H. This is also part of Step (3) on p100 of [14], and (b) on p53 of [24].

Remember that the h̃_{α}’s, α ∈ Δ, form a basis for H̃ as a vector space over k, as in (27.3.1). This means that the h_{α}’s, α ∈ Δ, form a basis for H as a vector space over k. In particular, h̃_{α} ≠ 0 for each α ∈ Δ, as in (27.3.7), so that h_{α} ≠ 0 for every α ∈ Δ too. This implies that x_{α}, y_{α} ≠ 0 for every α ∈ Δ, because [x_{α}, y_{α}] = h_{α}, as in (27.8.5).

Using this, one can verify that

(27.10.3) the set of x_{β}’s, y_{β}’s, and h_{β}’s, β ∈ Δ, is linearly independent in L,

as in (27.3.10). More precisely, the x_{β}’s, y_{β}’s, and h_{β}’s are simultaneous eigenvectors for ad_{h_{α}} on L, α ∈ Δ, by (27.8.4), (27.8.6), and (27.8.7). To get linear independence, one also uses the fact that the eigenvalues for the x_{β}’s and y_{β}’s correspond to distinct functions of α on Δ that are not identically 0, as before. This permits one to reduce to the linear independence of the h_{β}’s, β ∈ Δ. Equivalently, the restriction of the natural quotient mapping from L̃ onto L to the linear subspace of L̃ spanned by the x̃_{β}’s, ỹ_{β}’s, and h̃_{β}’s, β ∈ Δ, is injective, as in Step (4) on p100 of [14].

27.11 Eigenvectors in L

We continue with the same notation and hypotheses as in the previous sections.

27.11. EIGENVECTORS IN L 589

Let H′ be the dual of H, as a vector space over k, as usual, and put

(27.11.1) L_{μ} = {u ∈ L : [h, u] = μ(h) u for every h ∈ H}

for each μ ∈ H′. The elements of L_{μ} are said to have weight μ. If μ ∈ H′, then let μ̃ be the corresponding linear functional on H̃, which is the composition of μ with the restriction to H̃ of the natural quotient mapping from L̃ onto L. Observe that

(27.11.2) the natural quotient mapping from L̃ onto L maps L̃_{μ} into L_{μ},

where L̃_{μ} is as in (27.5.1). Of course, H ⊆ L_{0}, by (27.8.4).

If β ∈ Δ, then there is a unique ν_{β} ∈ H′ such that

ν_{β}(h_{α}) = n(β, α) (27.11.3)

for every α ∈ Δ, because the h_{α}’s form a basis for H. The composition of ν_{β} with the restriction of the natural quotient mapping from L̃ onto L to H̃ is the same as ν̃_{β} ∈ H̃′, defined in (27.5.2). It is easy to see that the ν_{β}’s, β ∈ Δ, form a basis for H′, as a vector space over k, because of the analogous statement for the ν̃_{β}’s in Section 27.5.

Every element of X can be expressed as the sum of finitely elements of the subspaces L_{μ}, where μ ∈ H′ can be expressed as a nontrivial sum of ν_{β}’s. This follows from (27.11.2) and the analogous statement for X̃ in Section 27.5. We also have that every element of Y can be expressed as the sum of finitely many elements of L_{μ}’s, where μ ∈ H′ can be expressed as a nontrivial sum of −ν_{β}’s. This implies that X and Y correspond to the direct sums of their intersections with the appropriate subspaces L_{μ}, as vector spaces over k, as in Section 27.5.

As before, one can use this to check that

H = L_{0}. (27.11.4)

Similarly, X corresponds to the direct sum of the subspaces L_{μ}, where μ ∈ H′ can be expressed as a nontrivial sum of ν_{β}’s, and Y corresponds to the direct sum of the subspaces L_{μ}, where μ ∈ H′ can be expressed as a nontrivial sum of −ν_{β}’s, as vector spaces over k. This uses the hypothesis that k have characteristic 0, so that nontrivial sums of ν_{β}’s or of −ν_{β}’s cannot be equal to 0, or to each other. If μ ∈ H′ and L_{μ} ≠ {0}, then one can verify that μ = 0, or μ is a nontrivial sum of ν_{β}’s, or μ is a nontrivial sum of −ν_{β}’s.

More precisely, every element of X can be expressed as the sum of the images under the natural quotient mapping from L̃ onto L of finitely many elements of L̃_{μ}, where μ̃ is a nontrivial sum of ν̃_{β}’s, and every element of Y can be expressed

as the sum of the images of finitely many elements of L̃_{μ}, where μ̃ is a nontrivial sum of −ν̃_{β}’s. Let μ ∈ H′ be given, and let μ̃ be the corresponding linear functional on H̃′ again, as in (27.11.2). One can check that

(27.11.5) the natural quotient mapping from L̃ onto L maps L̃_{μ} onto L_{μ}.

This follows from (27.11.4) when μ = 0. Otherwise, one can use the previous remarks about X, Y.

Using (27.11.5), we get that L_{μ} is finite-dimensional as a vector space over k for every μ ∈ H′, because of the analogous statement for L̃_{μ} in Section 27.5. This corresponds to Step (5) on p100 of [14], and some remarks on p53 of [24].

27.12 Locally nilpotent linear mappings

Let k be a field, and let V be a vector space over k. A linear mapping T from V into itself is said to be locally nilpotent if for each v ∈ V there is a positive integer l such that T^{l}(v) = 0. If W is a finite-dimensional linear subspace of V, then it follows that there is a positive integer r such that T^{r}(W) = {0}. In this case, the linear subspace W_{1} of V spanned by W, T (W), …, T^{r}(W) is finite-dimensional and satisfies

T (W_{1}) ⊆ W_{1}. (27.12.1)

If T_{1}, T_{2} are commuting locally nilpotent linear mappings from V into itself, then it is easy to see that T_{1} + T_{2} is locally nilpotent on V as well.

Suppose now that k has characteristic 0, and let T be a locally nilpotent linear mapping from V into itself. If v ∈ V, then put

(exp T)(v) = ∑_{j=0}^{∞}(1/j!) T^{j}(v), (27.12.2)

where the sum on the right reduces to a finite sum in V. It is easy to see that this defines a linear mapping from V into itself.

Let T_{1}, T_{2} be commuting locally nilpotent linear mappings from V into itself again. If v ∈ V, then T_{1}^{j}(T_{2}^{l}(v)) = T_{2}^{l}(T_{1}^{j}(v)) = 0 (27.12.3)

when j or l is sufficiently large. One can check that

(27.12.4) (exp(T_{1} + T_{2}))(v) = (exp T_{1})((exp T_{2})(v)),

using the binomial theorem in the usual way. If T is a locally nilpotent linear mapping from V into itself, then one can apply the previous statement to T, −T to get that exp T is invertible on V, with inverse equal to exp(−T). This is related to some remarks just before the theorem on p99 of [14].

Let A be an algebra over k in the strict sense, and let δ be a derivation on A. If δ is locally nilpotent on A, then exp δ is an algebra automorphism of A, as in Section 14.11.

Let us now return to the discussion of Serre’s theorem, as in the previous sections. We would like to show that for each α ∈ Δ, ad_{x_{α}} and ad_{y_{α}} are locally nilpotent as linear mappings from L into itself, as in Step (6) on p100 of [14], and (c) on p53 of [24]. Put

(27.12.5) V_{α} = {u ∈ L : ad_{x_{α}}^{l}(u) = 0 for some l ∈ Z_{+}}, 27.13. SOME PROPERTIES OF θ_{α} 591

which is a linear subspace of L. One can check that V_{α} is a Lie subalgebra of L, using the fact that ad_{x_{α}} is a derivation on L. Observe that x_{β} ∈ V_{α} for every β ∈ Δ, by (27.8.8). One can verify that y_{β} ∈ V_{α} for every β ∈ Δ, using (27.8.5) and (27.8.6). It follows that L ⊆ V_{α}, because L is generated as a Lie algebra over k by the x_{β}’s and y_{β}’s, β ∈ Δ. This means that ad_{x_{α}} is locally nilpotent on L, and the argument for ad_{y_{α}} is analogous.

Let α ∈ Δ be given, and note that − ad_{y_{α}} is locally nilpotent on L. Thus exp ad_{x_{α}} and exp(− ad_{y_{α}}) are Lie algebra automorphisms of L, because ad_{x_{α}}, − ad_{y_{α}} are derivations on L, as before. It follows that

(27.12.6) θ_{α} = (exp ad_{x_{α}}) ∘ (exp −ad_{y_{α}}) ∘ (exp ad_{x_{α}})

is a Lie algebra automorphism of L too. This corresponds to Step (7) on p100 of [14], and to part of (d) on p54 of [24].

If u ∈ L satisfies [x_{α}, u] = [y_{α}, u] = 0, then

(27.12.7) (exp ad_{x_{α}})(u) = (exp −ad_{y_{α}})(u) = u,

and hence θ_{α}(u) = u. (27.12.8)

Remember that n(α, α) = 2, as in Section 27.1. Using the same type of argument as in Section 23.7, we get that

θ_{α}(x_{α}) = −y_{α}, θ_{α}(y_{α}) = −x_{α}, θ_{α}(h_{α}) = −h_{α}. (27.12.9)

27.13 Some properties of θ_{α}

Let us continue with the same notation and hypotheses as in the previous sections. In particular, let α ∈ Δ be given again, and let θ_{α} be the Lie algebra automorphism of L defined in (27.12.6). If h ∈ H, then h can be expressed in a unique way as

h = ∑_{β∈Δ} c_{β} h_{β}, (27.13.1)

with c_{β} ∈ k for each β ∈ Δ. In this case,

(27.13.2)  [h, x_{α}] = ∑_{β∈Δ} c_{β} [h_{β}, x_{α}] = ∑_{β∈Δ} c_{β} n(α, β) x_{α},
(27.13.3)  [h, y_{α}] = ∑_{β∈Δ} c_{β} [h_{β}, y_{α}] = − ∑_{β∈Δ} c_{β} n(α, β) y_{α},

by (27.8.6) and (27.8.7). If

∑_{β∈Δ} c_{β} n(α, β) = 0, (27.13.4)

592 CHAPTER 27. SOME MORE LIE ALGEBRAS then it follows that [h, x_{α}] = [h, y_{α}] = 0, so that θ_{α}(h) = h, (27.13.5) as in (27.12.8).

If h = h_{α}, then (27.13.4) does not hold, because k has characteristic 0, so that n(α, α) = 2 is not zero in k. It follows that (27.13.6) H is spanned by h_{α} and the h ∈ H that satisfy (27.13.4), as a vector space over k. This implies that θ_{α}(H) = H, (27.13.7) by (27.12.9) and (27.13.5). Note that θ_{α}(θ_{α}(h)) = h (27.13.8) for every h ∈ H.

If h ∈ H is as in (27.13.1), then put

ν_{α}(h) = ∑_{β∈Δ} c_{β} n(α, β). (27.13.9)

This defines a linear functional on H, which is the same as the one in (27.11.3), with α and β interchanged. Observe that ν_{α}(h_{α}) = n(α, α) = 2, and that θ_{α}(h) = h − ν_{α}(h) h_{α} (27.13.10) for every h ∈ H, because of (27.12.9) and (27.13.5).

Let μ ∈ H′ be given, and put ζ_{μ,α}(h) = μ(θ_{α}(h)) (27.13.11) for every h ∈ H, which defines a linear functional on H too. Clearly h ∈ H and u ∈ L satisfy [h, u] = ζ_{μ,α}(h) u (27.13.12) if and only if [θ_{α}(h), θ_{α}(u)] = μ(θ_{α}(h)) θ_{α}(u), (27.13.13) because θ_{α} is a Lie algebra automorphism of L. This implies that θ_{α}(L_{ζ_{μ,α}}) = L_{μ}, (27.13.14) where L_{μ}, L_{ζ_{μ,α}} are as in (27.11.1). This corresponds to part of Step (8) on p100 of [14], and of (d) on p54 of [24].

Equivalently, ζ_{μ,α}(h) = μ(h) − ν_{α}(h) μ(h_{α}) (27.13.15)

27.14. ROOTS AND H′ 593

for every h ∈ H, by (27.13.10). This means that

ζ_{μ,α} = μ − μ(h_{α}) ν_{α}, (27.13.16)

as elements of H′. Let γ ∈ Δ be given, and let ν_{γ} ∈ H′ be as in (27.11.3) and (27.13.9), so that ν_{γ}(h_{β}) = n(γ, β) (27.13.17)

for every β ∈ Δ. If we take μ = ν_{γ} in (27.13.16), then we get that

ζ_{ν_{γ},α} = ν_{γ} − ν_{γ}(h_{α}) ν_{α} = ν_{γ} − n(γ, α) ν_{α}. (27.13.18)

This means that ζ_{ν_{γ},α} = ν_{γ} − λ_{α}(γ) ν_{α}, (27.13.19)

where λ_{α}(γ) ∈ Z is as in Section 27.1.

27.14 Roots and H′

We continue with the discussion of Serre’s theorem, as in the previous sections. Remember from Section 27.1 that V is a vector space over the real numbers of positive finite dimension, Φ is a reduced root system in V, and that Δ is a base for Φ. Let Θ_{Φ} be the subgroup of V, as a commutative group with respect to addition, generated by Φ. This is the same as the subgroup of V generated by Δ, which consists of linear combinations of elements of Δ with coefficients in Z.

If α ∈ Φ, then σ_{α} is the symmetry on V with vector α that maps Φ onto itself, as before. This can be expressed as σ_{α}(v) = v − λ_{α}(v) α, where λ_{α} is a linear functional on V such that λ_{α}(α) = 2 and λ_{α}(β) ∈ Z for every β ∈ Φ. Observe that σ_{α} maps Θ_{Φ} onto itself, and that λ_{α} maps Θ_{Φ} into Z. If σ is in the Weyl group W of Φ, then it follows that

σ(Θ_{Φ}) = Θ_{Φ}. (27.14.1)

If γ ∈ Θ_{Φ}, then let ν_{γ} be the linear functional on H such that

ν_{γ}(h_{β}) = λ_{β}(γ) (27.14.2)

for every β ∈ Δ. This is equivalent to (27.13.17) when γ ∈ Δ, by (27.1.1). Note that γ ↦ ν_{γ} (27.14.3)

is a homomorphism from Θ_{Φ} into H′, as commutative groups with respect to addition. Remember that the ν_{γ}’s, γ ∈ Δ, form a basis for H′ as a vector space over k, as in Section 27.11. This implies that (27.14.3) is injective on Θ_{Φ}, because k has characteristic 0.

Let α ∈ Δ be given, and let θ_{α} be the Lie algebra automorphism of L defined in Section 27.12. If γ ∈ Θ_{Φ}, then ζ_{ν_{γ},α} ∈ H′ can be defined as in (27.13.11). Using (27.13.16), we get that

ζ_{ν_{γ},α} = ν_{γ} − ν_{γ}(h_{α}) ν_{α} = ν_{γ} − λ_{α}(γ) ν_{α}. (27.14.4)

Equivalently, this means that

ζ_{ν_{γ},α} = ν_{σ_{α}(γ)}. (27.14.5)

It follows that θ_{α}(L_{ν_{σα(γ)}}) = L_{ν_{γ}} , (27.14.6)

by taking μ = ν_{γ} in (27.13.14).

Remember that L_{μ} is finite-dimensional as a vector space over k for every μ ∈ H′, as in Section 27.11. Using (27.14.6), we get that

dim L_{ν_{γ}} = dim L_{ν_{σα(γ)}} (27.14.7)

for every α ∈ Δ and γ ∈ Θ_{Φ}. This implies that

dim L_{ν_{γ}} = dim L_{ν_{σ(γ)}} (27.14.8)

for every σ ∈ W and γ ∈ Θ_{Φ}. More precisely, this uses the fact that W is generated by σ_{α} with α ∈ Δ, as in Section 19.14. This corresponds to Step (8) on p100 of [14], and (d) on p54 of [24].

27.15 Some dimensions of eigenspaces

We continue with the same notation and hypotheses as in the previous sections. Let β ∈ Δ be given, and let ν̃_{β} be the linear functional on H̃ such that ν̃_{β}(h̃_{α}) = n(β, α) for every α ∈ Δ, as in (27.5.2). Remember that L̃_{ν_{β}} consists of the

ũ ∈ L̃ such that [h̃, ũ] = ν̃_{β}(h̃) ũ for every h̃ ∈ H̃, as in (27.5.1). Equivalently, this means that [h̃_{α}, ũ] = n(β, α) ũ for every α ∈ Δ. In particular, x̃_{β} ∈ L̃_{ν_{β}}, by (27.1.11).

In fact, L̃_{ν_{β}} is spanned by x̃_{β}. Remember that L̃ is spanned by X̃, H̃, and

Ỹ, and more precisely by elements of L̃_{μ} for suitable μ̃ ∈ H̃′. Only multiples of

x̃_{β} by elements of k were needed when μ̃ = ν̃_{β}. This implies that L̃_{ν_{β}} consists of only multiples of x̃_{β}.

Similarly, if r ∈ Z and r ≠ 0, ±1, then L̃_{r _{ν_{β}}} = {0}. Indeed, If |r| ≥ 2, then

the only elements of L̃_{r _{ν_{β}}} that would have been needed before were multiples of [x̃_{β_{1}}, …, x̃_{β_{|r|}}] or [ỹ_{β_{1}}, …, ỹ_{β_{|r|}}] with β_{l} = β for l = 1, …, |r|, as in Section 27.4. Of course, [x̃_{β_{1}}, …, x̃_{β_{|r|}}] and [ỹ_{β_{1}}, …, ỹ_{β_{|r|}}] are equal to 0 in this case. It follows that 0 is the only element of L̃_{r _{ν_{β}}} when |r| ≥ 2.

Let ν_{β} be the linear functional on H such that ν_{β}(h_{α}) = n(β, α) for every α ∈ Δ, as in (27.11.3). Note that ν̃_{β} is the same as the composition of ν_{β} with the restriction to H̃ of the natural quotient mapping from L̃ onto L. If r ∈ Z, then the natural quotient mapping from L̃ onto L maps L̃_{r _{ν_{β}}} onto L_{r ν_{β}}, as in (27.11.5). This implies that

L_{r ν_{β}} = {0} when |r| ≥ 2, (27.15.1) by the remarks in the preceding paragraph.

If we take r = 1, then we get that L_{ν_{β}} is spanned by x_{β}, because of the analogous statement for L̃_{ν_{β}}. Remember that x_{β} ≠ 0, as in Section 27.10. Thus

dim L_{ν_{β}} = 1. (27.15.2)

This corresponds to Step (9) on p100 of [14], and (e) on p54 of [24].

Let Θ_{Φ} be as in the previous section, and if γ ∈ Θ_{Φ}, then let ν_{γ} ∈ H′ be as in (27.14.2). Remember that this is equivalent to the previous definition when γ ∈ Δ. If γ ∈ Φ, then there is an element σ of the Weyl group of Φ such that σ(γ) ∈ Δ, as in Section 19.14. It follows that

dim L_{ν_{γ}} = 1, (27.15.3)

by (27.14.8) and (27.15.2).

If r ∈ Z, then r γ ∈ Θ_{Φ}, and r ν_{γ} = ν_{r γ}. Similarly, ν_{σ(r γ)} = ν_{r σ(γ)} = r ν_{σ(γ)}. If |r| ≥ 2, then we get that dim L_{r ν_{γ}} = 0, (27.15.4)

using (27.14.8) and (27.15.1). This corresponds to Step (10) on p100 of [14], and (f) on p54 of [24].

Some more Lie algebras, 2

28.1 Linear combinations of roots

Let us continue with the discussion of Serre’s theorem, as in the previous chapter. We first need an auxiliary result about root systems. Remember that (·, ·) is an inner product on V that is invariant under the Weyl group of Φ, as in Section 27.1. If w ∈ V, then let V′_{w} be the linear subspace of the dual V′ of V consisting of linear functionals on V that are equal to 0 at w, which is a hyperplane in V′ when w ≠ 0. Of course, one can use (·, ·) to identify V′ with V, and V′_{w} with the set of vectors in V that are orthogonal to w.

Suppose that

(28.1.1) γ ∈ V is not a multiple of any element of Φ by a real number.

This implies that γ ≠ 0, and that V′_{γ} ≠ V′_{α} for each α ∈ Φ. More precisely, V′_{γ} ⊈ V′_{α} for each α ∈ Φ, so that V′_{γ} ∩ V′_{α} is a hyperplane in V′_{γ} for every α ∈ Φ. It follows that the union of V′_{γ} ∩ V′_{α} over α ∈ Φ is a proper subset of V′_{γ}, as in Section 18.12. Equivalently, this means that V′_{γ} is not contained in the union of V′_{α}, α ∈ Φ. Let τ be an element of V′_{γ} that is not in V′_{α} for any α ∈ Φ. Thus τ ∈ V′ satisfies (28.1.2) τ(γ) = 0 and τ(α) ≠ 0 for every α ∈ Φ.

As in Section 19.14, there is an element σ of the Weyl group of Φ such that τ(σ(β)) ≥ 0 for every β ∈ Δ. This means that

(28.1.3) τ(σ(β)) > 0 for every β ∈ Δ,

because σ(β) ∈ Φ, so that τ(σ(β)) ≠ 0, by (28.1.2). Note that σ(Δ) is a basis for V, because Δ is a basis for V. Thus we can express γ as

γ = ∑_{β∈Δ} c_{β} σ(β) (28.1.4)

28.2. THE DIMENSION OF L 597

for some real numbers c_{β}, β ∈ Δ. It follows that

0 = τ(γ) = ∑_{β∈Δ} c_{β} τ(σ(β)). (28.1.5)

Of course, c_{β} ≠ 0 for some β ∈ Δ, because γ ≠ 0. In fact, (28.1.5) implies that

(28.1.6) c_{β} > 0 for some β ∈ Δ, and c_{β} < 0 for some β ∈ Δ,

because of (28.1.3). This corresponds to Exercise 10 on p54 of [14], and (g) on p54 of [24].

Suppose now that γ is also an element of the subgroup Θ_{Φ} of V generated by Φ, as a commutative group with respect to addition. This implies that

σ^{−1}(γ) = ∑_{β∈Δ} c_{β} β (28.1.7)

is an element of Θ_{Φ} too, as in (27.14.1), so that c_{β} ∈ Z for every β ∈ Δ. Let ν_{γ} ∈ H′ be as in (27.14.2), and similarly for ν_{σ^{−1}(γ)}. Remember that L_{μ} ⊆ L is defined for μ ∈ H′ as in (27.11.1). It follows that

dim L_{ν_{γ}} = dim L_{ν_{σ^{−1}(γ)}}, (28.1.8)

as in (27.14.8).

As in Section 27.11, L_{μ} = {0} unless μ = 0, or μ can be expressed as a sum of ν_{β}’s, β ∈ Δ, or μ can be expressed as a sum of −ν_{β}’s, β ∈ Δ. This implies that L_{ν_{σ^{−1}(γ)}} = {0} (28.1.9)

under the conditions described in the previous paragraphs, because of (28.1.6). This means that L_{ν_{γ}} = {0}, (28.1.10)

by (28.1.8). More precisely, this holds when γ ∈ Θ_{Φ} satisfies (28.1.1). This corresponds to part of Step (11) on p100 of [14], and of (h) on p54 of [24].

28.2 The dimension of L

Let us continue with the same notation and hypotheses as in the previous sections. If α ∈ Φ, γ ∈ Θ_{Φ}, and γ = t α (28.2.1)

for some t ∈ R, then t ∈ Z. (28.2.2)

This is easy to see when α is an element of a base for Φ, because γ can be expressed in a unique way as a linear combination of elements of the base, with coefficients in Z. Remember that Φ is supposed to be reduced as a root system in V, as in Section 27.1. This implies that every element of Φ is an element of a base for Φ, as in Section 19.14.

If (28.2.1) and (28.2.2) hold, then (28.1.10) holds when

|t| ≥ 2, (28.2.3)

as in (27.15.4). Combining this with the remarks in the previous section, we get that (28.1.10) holds for every γ ∈ Θ_{Φ} such that γ ≠ 0 and γ ∉ Φ.

Remember that L_{μ} = {0} unless μ ∈ H′ satisfies μ = 0, or μ is a sum of ν_{β}’s, β ∈ Δ, or μ is a sum of −ν_{β}’s, β ∈ Δ. In particular, this means that L_{μ} = {0} unless μ = ν_{γ} for some γ ∈ Θ_{Φ}. It follows from this and the remarks in the preceding paragraph that L_{μ} = {0} unless μ = 0, or μ = ν_{α} for some α ∈ Φ.

Remember that X corresponds to the direct sum of the subspaces L_{μ}, where μ ∈ H′ can be expressed as a nontrivial sum of ν_{β}’s, β ∈ Δ, as a vector space over k, as in Section 27.11. Similarly, Y corresponds to the direct sum of the subspaces L_{μ}, where μ can be expressed as a nontrivial sum of ν_{β}’s, β ∈ Δ, as a vector space over k. Let Φ^{+} = Φ_{Δ}^{+} be the set of α ∈ Φ that can be expressed as a linear combination of β ∈ Δ with nonnegative coefficients, so that Φ = Φ^{+} ∪ (−Φ^{+}). Note that ν_{−α} = −ν_{α} for every α ∈ Φ, by the definition (27.14.2) of ν_{α}. Thus X corresponds to the direct sum of the subspaces L_{ν_{α}}, α ∈ Φ^{+}, as a vector space over k, and Y corresponds to the direct sum of the subspaces L_{−ν_{α}}, α ∈ Φ^{+}, as a vector space over k. These subspaces have dimension equal to 1, as in (27.15.3). This means that X and Y are finitedimensional as vector spaces over k, with dimension equal to the number of elements of Φ^{+}.

Remember that L corresponds to the direct sum of X, H, and Y as a vector space over k, as in (27.10.1). We also have that the h_{α}’s, α ∈ Δ, form a basis for H as a vector space over k, as in Section 27.10. Thus the dimension of H as a vector space over k is equal to the number of elements of Δ, which is the same as the dimension of V as a vector space over R. It follows that the dimension of L, as a vector space over k, is equal to the sum of the number of elements of Φ and the number of elements of Δ. This corresponds to Step (12) on p100 of [14], and to (i) on p54 of [24].

28.3 Automorphisms and the Weyl group

We continue with the same notation and hypotheses as in the previous sections. If α ∈ Φ, then σ_{α} is the symmetry on V with vector α that maps Φ onto itself, as usual. Suppose that α ∈ Δ, and let θ_{α} be the automorphism of L associated to α as in Section 27.12. Remember that θ_{α} maps H onto itself, as in (27.13.7). If μ ∈ H′, then let ζ_{μ,α} ∈ H′ be as in (27.13.11), so that ζ_{μ,α}(h) = μ(θ_{α}(h)) for every h ∈ H.

Suppose that γ is an element of the subgroup Θ_{Φ} of V, as a commutative group with respect to addition, generated by Φ. Thus ν_{γ} ∈ H′ can be defined as in (27.14.2), so that ζ_{ν_{γ},α} can be defined as before. Remember that ζ_{ν_{γ},α} = ν_{σ_{α}(γ)}, as in (27.14.5).

Let α_{1}, …, α_{r} be a finite sequence of elements of Δ, possibly with repetitions. Thus σ = σ_{α_{1}} ∘ ⋯ ∘ σ_{α_{r}} (28.3.1)

is an element of the Weyl group of Φ, and

θ = θ_{α_{r}} ∘ ⋯ ∘ θ_{α_{1}} (28.3.2)

is a Lie algebra automorphism of L. If μ ∈ H′, then put

ζ_{μ,θ}(h) = μ(θ(h)) (28.3.3)

for every h ∈ H, which defines a linear functional on H′. Observe that

ζ_{ν_{γ},θ} = ν_{σ(γ)}, (28.3.4)

because of the analogous statement for σ_{α} mentioned in the preceding paragraph.

If μ ∈ H′, then let L_{μ} be as in (27.11.1). Observe that h ∈ H and u ∈ L satisfy [h, u] = ζ_{μ,θ}(h) u (28.3.5)

if and only if [θ(h), θ(u)] = μ(θ(h)) θ(u). (28.3.6)

This means that θ(L_{ζ_{μ,θ}}) = L_{μ}, (28.3.7)

which could also be obtained from (27.13.14). It follows that

θ(L_{ν_{σ(γ)}}) = L_{ν_{γ}} , (28.3.8)

by (28.3.4).

If β ∈ Δ, then L_{ν_{β}} is spanned by x_{β}, as in Section 27.15. Similarly, one can check that L_{−ν_{β}} is spanned by y_{β}. This implies that [L_{ν_{β}}, L_{−ν_{β}}] is spanned by h_{β}, because of (27.8.5). Remember that x_{β}, y_{β}, h_{β} ≠ 0, as in Section 27.10. The linear subspace of L spanned by L_{ν_{β}}, L_{−ν_{β}}, and [L_{ν_{β}}, L_{−ν_{β}}] is a Lie subalgebra of L isomorphic to sl_{2}(k), because of (27.8.5), (27.8.6), (27.8.7) and the fact that n(β, β) = 2, as in Section 27.1.

If γ ∈ Φ, then there is an element σ of the Weyl group of Φ such that σ(γ) is contained in Δ, as in Section 19.14. Remember that σ can be expressed as in (28.3.1), with α_{1}, …, α_{r} ∈ Δ, as in Section 19.14, so that we get a Lie algebra automorphism θ of L as in (28.3.2). This means that (28.3.8) holds, and similarly θ(L_{−ν_{σ(γ)}}) = L_{−ν_{γ}} . (28.3.9)

It follows that

θ([L_{ν_{σ(γ)}}, L_{−ν_{σ(γ)}}]) = [θ(L_{ν_{σ(γ)}}), θ(L_{−ν_{σ(γ)}})] = [L_{ν_{γ}} , L_{−ν_{γ}}]. (28.3.10)

In particular, dim[L_{ν_{γ}} , L_{−ν_{γ}}] = 1, (28.3.11)

because [L_{ν_{σ(γ)}}, L_{−ν_{σ(γ)}}] is one-dimensional as a linear subspace of L, as in the preceding paragraph.

We also get that the linear subspace of L spanned by L_{ν_{γ}}, L_{−ν_{γ}}, and [L_{ν_{γ}} , L_{−ν_{γ}}] is a Lie subalgebra of L isomorphic to sl_{2}(k). More precisely, this subspace is the image under θ of the linear subspace of L spanned by L_{ν_{σ(γ)}}, L_{−ν_{σ(γ)}}, and [L_{ν_{σ(γ)}}, L_{−ν_{σ(γ)}}]. The latter subspace of L is a Lie subalgebra isomorphic to sl_{2}(k), because σ(γ) ∈ Δ, as before. This corresponds to (j) on p55 of [24].

28.4 Semisimplicity of L

Let us continue with the same notation and hypotheses as in the previous sections. We would like to show that L is semisimple as a Lie algebra over k. It suffices to show that if C is an ideal in L that is commutative as a Lie algebra, then C = {0}, as in Section 9.4.

Remember that L corresponds to the direct sum of the linear subspaces L_{μ}, μ ∈ H′, defined in (27.11.1), as a vector space over k. More precisely, L corresponds to the direct sum of H = L_{0} and the linear subspaces L_{ν_{α}}, α ∈ Φ, as in Section 28.2. If h ∈ H, then ad_{h} maps C into itself, because C is an ideal in L. This implies that C corresponds to the direct sum of C ∩ H and the subspaces C ∩ L_{ν_{α}}, α ∈ Φ, as a vector space over k.

If α ∈ Φ, then the linear subspace of L spanned by

L_{ν_{α}}, L_{−ν_{α}}, and [L_{ν_{α}}, L_{−ν_{α}}] (28.4.1)

is a Lie subalgebra of L isomorphic to sl_{2}(k) as a Lie algebra over k, as in the previous section. The intersection of C with this Lie subalgebra is an ideal in this Lie subalgebra, and hence the intersection is trivial, because sl_{2}(k) is simple. In particular, C ∩ L_{ν_{α}} = {0} for every α ∈ Φ. It follows that C ⊆ H.

If β ∈ Δ, then x_{β} ∈ L_{ν_{β}}, as in Section 27.15. Equivalently, [h, x_{β}] = ν_{β}(h) x_{β} for every h ∈ H. If h ∈ C, then [h, x_{β}] ∈ C, because C is an ideal in L. This implies that [h, x_{β}] = 0 for every h ∈ C, by (27.10.3). Thus ν_{β}(h) = 0 for every h ∈ C. Remember that the ν_{β}’s, β ∈ Δ, form a basis for the dual of H, as a vector space over k, as in Section 27.11. This implies that C = {0}, as desired. This corresponds to Step (13) on p100 of [14], and (k) on p55 of [24].

Of course, H is commutative as a Lie subalgebra of L, and in particular H is nilpotent as a Lie algebra. Let us check that H is self-normalizing in L, which means that H is a Cartan subalgebra of L. Suppose that u ∈ L is in the normalizer N_{L}(H) of H in L, so that ad_{u} maps H into itself. As before, u can be expressed as the sum of an element of H and elements of L_{ν_{α}}, α ∈ Φ. If the component of u in L_{ν_{α}} is not equal to 0 for some α ∈ Φ, then the component of [h_{α}, u] in L_{ν_{α}} is nonzero, and [h_{α}, u] is not contained in H. Thus the component of u in L_{ν_{α}} is equal to 0 for every α ∈ Φ when u ∈ N_{L}(H). This means that N_{L}(H) is contained in H, as desired.

Remember that V is the vector space over the real numbers of positive finite dimension in which Φ is a root system. Consider the linear subspace of V, as a vector space over Q, generated by Φ. It is easy to see that the mapping γ ↦ ν_{γ} defined in Section 27.14 extends to an injective linear mapping from the subspace of V just mentioned into H′, as a vector space over Q.

Remember that L_{μ} ≠ {0} when μ ∈ H′ and μ ≠ {0} exactly when μ = ν_{α} for some α ∈ Φ, as in Section 28.2. Thus we get the usual relationship between Φ and the set of μ ∈ H′ such that μ ≠ 0 and L_{μ} ≠ {0}. This corresponds to Step (14) on p100 of [14], and (l) on p55 of [24].

28.5 An isomorphism theorem

Let V_{1}, V_{2} be vector spaces over the real numbers of equal positive finite dimension, and let Φ_{1}, Φ_{2} be reduced root systems in V_{1}, V_{2}, respectively. If i = 1, 2 and α, β ∈ Φ_{i}, then let n_{i}(α, β) ∈ Z (28.5.1)

be as in (27.1.1), as usual. Let Δ_{i} be a base for Φ_{i} for i = 1, 2, and remember that −3 ≤ n_{i}(α, β) ≤ 0 (28.5.2)

for every α, β ∈ Δ_{i} with α ≠ β, as in (27.1.2).

Let k be a field of characteristice 0, and suppose that (A_{i}, [·, ·]_{A_{i}}) is a finitedimensional Lie algebra over k for i = 1, 2 with the following properties. If i = 1, 2 and α ∈ Δ_{i}, then x_{i,α}, y_{i,α}, and h_{i,α} are elements of A_{i}. If α, β ∈ Δ_{i}, then

(28.5.3)  [h_{i,α}, h_{i,β}]_{A_{i}} = 0,
(28.5.4)  [x_{i,α}, y_{i,β}]_{A_{i}} = δ_{α,β} h_{i,α},
(28.5.5)  [h_{i,α}, x_{i,β}]_{A_{i}} = n_{i}(β, α) x_{i,β},
(28.5.6)  [h_{i,α}, y_{i,β}]_{A_{i}} = −n_{i}(β, α) y_{i,β},

where δ_{α,β} ∈ k is equal to 1 when α = β, and is equal to 0 when α ≠ β, as usual. If α ≠ β, then

(ad_{A_{i},x_{i,α}})^{−n_{i}(β,α)+1}(x_{i,β}) = (ad_{A_{i},y_{i,α}})^{−n_{i}(β,α)+1}(y_{i,β}) = 0. (28.5.7)

We also ask that A_{i} be generated as a Lie algebra over k by

x_{i,α}, y_{i,α}, and h_{i,α}, α ∈ Δ_{i}, (28.5.8)

and that the dimension of A_{i} as a vector space over k be equal to the sum of the number of elements of Φ_{i} and the number of elements of Δ_{i}.

Of course, the number of elements of Δ_{i} is the same as the dimension of V_{i}. Suppose that ϕ is a one-to-one mapping from Δ_{1} onto Δ_{2} such that

n_{2}(ϕ(α), ϕ(β)) = n_{1}(α, β) (28.5.9) for every α, β ∈ Δ_{1}. Thus ϕ has a unique extension to a one-to-one linear mapping from V_{1} onto V_{2} that maps Φ_{1} onto Φ_{2}, as in Section 20.2. Conversely, if a one-to-one linear mapping from V_{1} onto V_{2} maps Φ_{1} onto Φ_{2} and Δ_{1} onto Δ_{2}, then its restriction to Δ_{1} satisfies (28.5.9), as before.

Under these conditions, there is a unique Lie algebra isomorphism from A_{1} onto A_{2} with

x_{1,α} ↦ x_{2,ϕ(α)}, y_{1,α} ↦ y_{2,ϕ(α)}, h_{1,α} ↦ h_{2,ϕ(α)}, (28.5.10)

for every α ∈ Δ_{1}. This corresponds to Theorem 8’ on p50 of [24], and to part (b) of the theorem on p101 of [14]. This is also related to Theorem 7 on p49 of [24], whose proof is discussed on p55 of [24]. Another approach to isomorphisms like these is discussed in Section 14.2 of [14].

Note that Serre’s theorem shows that there are Lie algebras with these properties, as in Sections 27.8 and 28.2. This permits us to reduce to the case where A_{1} is as in Serre’s theorem. It follows from the proof of Serre’s theorem that there is a unique Lie algebra homomorphism from A_{1} into A_{2} that satisfies (28.5.10). This homomorphism is surjective, because A_{2} is generated by (28.5.8) with i = 2, as a Lie algebra over k. This homomorphism is injective as well, because A_{1} and A_{2} have the same dimension, as vector spaces over k.

28.6 Some automorphisms

Let V_{1} be a vector space over the real numbers of positive finite dimension, and let Φ_{1} be a reduced root system in V_{1}. Also let n_{1}(α, β) be as before for α, β ∈ Φ_{1}, and let Δ_{1} be a base for Φ_{1}. Suppose that (A_{1}, [·, ·]_{A_{1}}) is a finitedimensional Lie algebra over a field k of characteristic 0 with the properties mentioned in the previous section. In particular, A_{1} should be generated as a Lie algebra over k by (28.5.8), with i = 1.

In this section, we take V_{2} = V_{1}, Φ_{2} = Φ_{1}, and A_{2} = A_{1}. Thus n_{2}(·, ·) is the same as n_{1}(·, ·). Let Δ_{2} be another base for Φ_{1}, and suppose that we have generators for A_{1} = A_{2} as in (28.5.8), with i = 2, and which satisfy the same conditions as before.

Let ϕ be a one-to-one mapping from Δ_{1} onto Δ_{2} such that

n_{1}(ϕ(α), ϕ(β)) = n_{1}(α, β) (28.6.1)

for every α, β ∈ Δ_{1}, which is the same as (28.5.9) in this case. As before, ϕ has a unique extension to an automorphism of Φ_{1}. Conversely, if T is any automorphism of Φ_{1}, then Δ_{2} = T (Δ_{1}) is a base for Φ_{1}, and the restriction of T to Δ_{1} satisfies (28.6.1).

The isomorphism theorem described in the previous section implies that there is a unique Lie algebra automorphism of A_{1} that satisfies (28.5.10). This is related to the remarks at the beginning of Section 14.3 on p76 of [14].

Remember that the mapping T from V_{1} onto itself defined by

T (v_{1}) = −v_{1} (28.6.2) defines an automorphism of Φ_{1}. In particular,

n_{1}(−α, −β) = n_{1}(α, β) (28.6.3)

for every α, β ∈ Φ_{1}. If α ∈ Δ_{1}, then −α ∈ T (Δ_{1}), and we put

x_{2,−α} = −y_{1,α}, y_{2,−α} = −x_{1,α}, h_{2,−α} = −h_{1,α}. (28.6.4)

One can check that x_{2,−α}, y_{2,−α}, and h_{2,−α}, α ∈ Δ_{1}, satisfy the requirements in the previous section for A_{1} = A_{2}, using (28.6.3) and the analogous properties of x_{1,α}, y_{1,α}, and h_{1,α}.

It follows that there is a unique Lie algebra automorphism σ of A_{1} with

x_{1,α} ↦ −y_{1,α}, y_{1,α} ↦ −x_{1,α}, h_{1,α} ↦ −h_{1,α} (28.6.5)

for every α ∈ Δ_{1}. Observe that σ ∘ σ is the identity mapping on A_{1}, because it sends the generators x_{1,α}, y_{1,α}, and h_{1,α} to themselves for every α ∈ Δ_{1}. This corresponds to the corollary to Theorem 7 on p49 of [24], and to the remark after Theorem 8’ on p50 of [24]. This also corresponds to the proposition on p77 of [14].

Of course, the identity mapping on V_{1} is an automorphism of Φ_{1}. Let t_{α} be a nonzero element of k for every α ∈ Δ_{1}, and put Δ_{2} = Δ_{1} and

x_{2,α} = t_{α} x_{1,α}, y_{2,α} = (1/t_{α}) y_{1,α}, h_{2,α} = h_{1,α} (28.6.6)

for every α ∈ Δ_{1}. It is easy to see that x_{2,α}, y_{2,α}, and h_{2,α}, α ∈ Δ_{1}, satisfy the requirements in the previous section for A_{1} = A_{2}, using the analogous properties of x_{1,α}, y_{1,α}, and h_{1,α}. This implies that there is a unique Lie algebra automorphism of A_{1} with

x_{1,α} ↦ t_{α} x_{1,α}, y_{1,α} ↦ (1/t_{α}) y_{1,α}, h_{1,α} ↦ h_{1,α} (28.6.7)

for every α ∈ Δ_{1}. This corresponds to the diagonal automorphisms mentioned on p87 of [14].

Let T be an automorphism of Φ_{1} such that

T (Δ_{1}) = Δ_{1}. (28.6.8)

Put Δ_{2} = Δ_{1}, and let ϕ be the restriction of T to Δ_{1}. Equivalently, one can take ϕ to be any one-to-one mapping from Δ_{1} onto itself that satisfies (28.6.1). Of course, x_{2,α} = x_{1,α}, y_{2,α} = y_{1,α}, h_{2,α} = h_{1,α} (28.6.9)

automatically satisfy the requirements in the previous section for A_{1} = A_{2}, by hypothesis. Thus there is a unique Lie algebra automorphism of A_{1} with

x_{1,α} ↦ x_{1,ϕ(α)}, y_{1,α} ↦ y_{1,ϕ(α)}, h_{1,α} ↦ h_{1,ϕ(α)} (28.6.10)

for every α ∈ Δ_{1}. More precisely, the collection of automorphisms of Φ_{1} that satisfy (28.6.8) is a group with respect to composition of mappings, and we get an injective homomorphism from this group into the automorphism group of A_{1}. This corresponds to Exercise 6 on p77 of [14].

28.7 Roots and diagonalizability

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a reduced root system in V. As usual, if α ∈ Φ, then we let σ_{α} be the symmetry on V with vector α that maps Φ onto itself. Thus σ_{α} can be expressed as σ_{α}(v) = v − λ_{α}(v) α, where λ_{α} is a linear functional on V with λ_{α}(α) = 2. Remember that λ_{α}(β) ∈ Z for every β ∈ Φ, by the definition of a root system.

Let Θ be the subgroup of V, as a commutative group with respect to addition, generated by Φ. We would like to consider a Lie algebra (A, [·, ·]_{A}) over a field k with properties like those mentioned in Sections 22.11, 23.3, 23.4, and 23.5. In particular, these conditions will hold in the situation described in Section 22.1, as before.

If α ∈ Φ ∪ {0}, then we ask that A_{α} be a linear subspace of A, and that A correspond to the direct sum of A_{α}, α ∈ Φ ∪ {0}, as a vector space over k. More precisely, if α ∈ Φ, then we ask that A_{α} have dimension one. As before, it is convenient to put A_{α} = {0} when α ∈ Θ \ (Φ ∪ {0}), and we ask that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ Θ.

Let A′_{0} be the dual space of linear functionals on A_{0}, as a vector space over k, as usual. Suppose that ϕ is a group homomorphism from Θ into A′_{0}, as a commutative group with respect to addition, such that ad_{w}(x) = ϕ_{α}(w) x for every w ∈ A_{0} and x ∈ A_{α}, α ∈ Φ ∪ {0}, where ϕ_{α} = ϕ(α). In particular, this implies that A_{0} is a Lie subalgebra of A that is commutative as a Lie algebra.

Let Δ be a base for Φ, and let Φ^{+} = Φ^{Δ,+} be the corresponding set of positive roots in Φ. If α ∈ Δ, β ∈ Φ^{+}, and α + β ∈ Φ, then we ask that

[A_{α}, A_{β}] = A_{α+β}, [A_{−α}, A_{−β}] = A_{−α−β}. (28.7.1)

Suppose that x_{A,α}, y_{A,α} are nonzero elements of A_{α}, A_{−α}, respectively, for each α ∈ Δ, and put h_{A,α} = [x_{A,α}, y_{A,α}]_{A}, (28.7.2) which is an element of A_{0}. We ask that ϕ_{β}(h_{A,α}) = λ_{α}(β) · 1 in k for every α ∈ Δ and β ∈ Φ, and that the h_{A,α}’s, α ∈ Δ, form a basis for A_{0}.

Under these conditions, A is generated by x_{A,α}, y_{A,α}, h_{A,α}, α ∈ Δ, as a Lie algebra over k, as in Section 23.4. These generators satisfy the same relations as in Section 28.5, as discussed in Section 23.5. We also have that the dimension of A as a vector space over k is equal to the sum of the numbers of elements of Φ and Δ.

If k has characteristic 0, then all of the conditions mentioned in Section 28.5 are satisfied. It follows that there is a unique Lie algebra isomorphism from the Lie algebra L obtained from Serre’s theorem in Section 27.8 onto A, with

x_{α} ↦ x_{A,α}, y_{α} ↦ y_{A,α}, h_{α} ↦ h_{A,α} (28.7.3)

for every α ∈ Δ. This corresponds to Theorem 7 on p49 of [24]. This is also related to Theorems 8 and 8’ on p50 of [24], and part (b) of the theorem on p101 of [14]. Note that the remarks in the previous section can be used as well in this case.

28.8. QUOTIENTS OF L̃ 605

28.8 Quotients of L̃

Let V be a vector space over the real numbers of positive finite dimension again, and let Φ be a reduced root system in V. Also let σ_{α}, λ_{α} be as before for every α ∈ Φ, and put n(α, β) = λ_{β}(α) for every α, β ∈ Φ, as in (27.1.1). Let Δ be a base for Φ as well, and let k be a field of characteristic 0. Under these conditions, the Lie algebra L̃ = L̃(k) can be defined as in Section 27.1.

Let (B, [·, ·]_{B}) be a Lie algebra over k with the following properties. If α ∈ Δ, then there are elements x_{B,α}, y_{B,α}, and h_{B,α} that satisfy the Weyl relations. This means that for every α, β ∈ Δ, we have that

(28.8.1)  [h_{B,α}, b_{B,β}]_{B} = 0,
(28.8.2)  [x_{B,α}, y_{B,β}]_{B} = δ_{α,β} h_{B,α},
(28.8.3)  [h_{B,α}, x_{B,β}]_{B} = n(β, α) x_{B,β},
(28.8.4)  [h_{B,α}, y_{B,β}]_{B} = −n(β, α) y_{B,β}.

Let us also ask that B be generated by x_{B,α}, y_{B,α}, h_{B,α}, α ∈ Δ, as a Lie algebra over k. Of course, this can be arranged by replacing B with the subalgebra generated by these elements.

Remember that L̃ is generated by elements x̃_{α}, ỹ_{α}, and h̃_{α}, α ∈ Δ, that satisfy the Weyl relations. More precisely, L̃ was defined to be the quotient of a suitable free Lie algebra by the ideal corresponding to the Weyl relations. It follows that there is a unique Lie algebra homomorphism from L̃ onto B with

x̃_{α} ↦ x_{B,α}, ỹ_{α} ↦ y_{B,α}, h̃_{α} ↦ h_{B,α} (28.8.5)

for every α ∈ Δ.

Let α, β ∈ Δ and a nonnegative integer l be given. Observe that

ad_{_{L},_{h_{α}}}((ad_{_{L},_{x_{α}}})^{l}(x̃_{β})) = (l · n(α, α) + n(β, α)) (ad_{_{L},_{x_{α}}})^{l}(x̃_{β}) (28.8.6)

and

ad_{_{L},_{h_{α}}}((ad_{_{L},_{y_{α}}})^{l}(ỹ_{β})) = −(l · n(α, α) + n(β, α)) (ad_{_{L},_{y_{α}}})^{l}(ỹ_{β}), (28.8.7)

by (27.4.2) and (27.4.3), respectively. This implies that

ad_{B,h_{B,α}}((ad_{B,x_{B,α}})^{l}(x_{B,β})) (28.8.8)

= (l · n(α, α) + n(β, α)) (ad_{B,x_{B,α}})^{l}(x_{B,β})

and

ad_{B,h_{B,α}}((ad_{B,y_{B,α}})^{l})(y_{B,β}) (28.8.9)

= −(l · n(α, α) + n(β, α)) (ad_{B,y_{B,α}})^{l}(y_{B,β}).

Of course, these identities could also be obtained more directly, using the same type of argument.

606 CHAPTER 28. SOME MORE LIE ALGEBRAS, 2

Remember that n(α, α) = λ_{α}(α) = 2. If, for each l ≥ 0, (ad_{B,x_{B,α}})^{l}(x_{B,β}) ≠ 0, (28.8.10) then it follows that B has infinite dimension as a vector space over k, because these elements of B are eigenvectors of ad_{B,h_{B,α}} with distinct eigenvalues. Equivalently, if B has finite dimension as a vector space over k, then (ad_{B,x_{B,α}})^{l}(x_{B,β}) = 0 (28.8.11) for some l ≥ 0. Similarly, (ad_{B,y_{B,α}})^{l}(y_{B,β}) = 0 (28.8.12) for some l ≥ 0 in this case. This corresponds to the fact that (ii) implies (iii) in the exercise on p55 of [24], which is related to Execise 3 on p101 of [14].

28.9 Additional identities in B

Let us continue with the same notation and hypotheses as in the previous section. Let α, β ∈ Δ be given again, and suppose that α ≠ β. Remember that

ad_{_{L},_{x_{α}}}((ad_{_{L},_{y_{α}}})^{j}(ỹ_{β})) = j (−n(β, α) − j + 1) (ad_{_{L},_{y_{α}}})^{j−1}(ỹ_{β}) (28.9.1)

for every positive integer j, as in (27.7.9). Similarly,

ad_{_{L},_{y_{α}}}((ad_{_{L},_{x_{α}}})^{j}(x̃_{β})) = j (−n(β, α) − j + 1) (ad_{_{L},_{x_{α}}})^{j−1}(x̃_{β}) (28.9.2)

for every positive integer j. This can be obtained from (28.9.1) using the automorphism σ̃ of L̃ discussed in Section 27.6, which sends x̃_{γ}, ỹ_{γ} to −ỹ_{γ}, −x̃_{γ}, respectively, for each γ ∈ Δ.

It follows that ad_{B,x_{B,α}}((ad_{B,y_{B,α}})^{j}(y_{B,β})) (28.9.3)

= j (−n(β, α) − j + 1) (ad_{B,y_{B,α}})^{j−1}(y_{B,β})

for every positive integer j, using the Lie algebra homomorphism from L̃ onto B mentioned in the previous section. Similarly, ad_{B,y_{B,α}}((ad_{B,x_{B,α}})^{j}(x_{B,β})) (28.9.4)

= j (−n(β, α) − j + 1) (ad_{B,x_{B,α}})^{j−1}(x_{B,β})

for every positive integer j.

Remember that −3 ≤ n(β, α) ≤ 0, as in (27.1.2). If (28.8.11) holds for some l ≥ 0, then (ad_{B,x_{B,α}})^{−n(β,α)+1}(x_{B,β}) = 0. (28.9.5)

28.10. EIGENSPACES IN L 607

More precisely, this is trivial when

l ≤ −n(β, α) + 1, (28.9.6)

and otherwise one can use (28.9.4). Similarly, if (28.8.12) holds for some l ≥ 0, then (ad_{B,y_{B,α}})^{−n(β,α)+1}(y_{B,β}) = 0, (28.9.7)

because of (28.9.3). This corresponds to the fact that (iii) implies (i) in the exercise on p55 of [24]. In particular, (28.9.5) and (28.9.7) hold when B has finite dimension as a vector space over k, by the remarks in the previous section. This corresponds to Exercise 3 on p101 of [14].

Let L = L(k) be the Lie algebra defined as in Serre’s theorem, in Section 27.8. Remember that L is a quotient of L̃, and that the natural quotient mapping sends x̃_{γ}, ỹ_{γ}, and h̃_{γ} to x_{γ}, y_{γ}, and h_{γ}, respectively, for each γ ∈ Δ. If (28.9.5) and (28.9.7) hold for every α, β ∈ Δ with α ≠ β, then there is a unique Lie algebra homomorphism from L onto B with

x_{γ} ↦ x_{B,γ}, y_{γ} ↦ y_{B,γ}, h_{γ} ↦ h_{B,γ} (28.9.8)

for every γ ∈ Δ. More precisely, the Lie algebra homomorphism from L̃ onto B mentioned in the previous section is the same as the composition of the natural quotient mapping from L̃ onto L with this homomorphism from L onto B. This shows that any finite-dimensional quotient of L̃ is in fact a quotient of L.

28.10 Eigenspaces in L

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a reduced root system in V again. If α, β ∈ Φ, then we let n(α, β) ∈ Z be as in (27.1.1), and we let Δ be a base for Φ. Let k be a field of characteristic 0, and let L = L(k) be the Lie algebra over k defined as in Serre’s theorem, in Section 27.8. Remember that L is generated by the elements x_{α}, y_{α}, and h_{α} with α ∈ Δ.

Let H be the linear subspace of L spanned by the h_{α}’s, α ∈ Δ, as in Section 27.10. Also let H′ be the dual of H, as a vector space over k. If μ ∈ H′, then let L_{μ} be the set of u ∈ L such that

[h, u] = μ(h) u (28.10.1)

for every h ∈ H, as in (27.11.1). If μ_{1}, μ_{2} ∈ H′, then it is easy to see that

[L_{μ_{1}}, L_{μ_{2}}] ⊆ L_{μ_{1}+μ_{2}}, (28.10.2)

using the Jacobi identity. Remember that H = L_{0}, as in (27.11.4).

Let Θ_{Φ} be the subgroup of V, as a commutative group with respect to addition, generated by Φ, which is the same as the subgroup generated by Δ.

If γ ∈ Θ_{Φ}, then let ν_{γ} be the linear functional defined on H as in (27.14.2). If γ ∈ Φ, then ν_{γ}(h_{β}) = n(γ, β) (28.10.3)

for every β ∈ Δ, because of (27.1.1). Remember that γ ↦ ν_{γ} is a homomorphism from Θ_{Φ} into H′, as commutative groups with respect to addition.

If γ ∈ Φ, then L_{ν_{γ}} has dimension one as a vector space over k, as in Section 27.15. Let γ ∈ Φ be given, and let σ be an element of the Weyl group of Φ such that σ(γ) ∈ Δ, as in Section 19.14. This leads to an automorphism θ of L, as in Section 28.3. By construction, θ maps L_{ν_{σ(γ)}}, L_{−ν_{σ(γ)}} onto L_{ν_{γ}}, L_{−ν_{γ}}, respectively, and thus [L_{ν_{σ(γ)}}, L_{−ν_{σ(γ)}}] onto [L_{ν_{γ}} , L_{−ν_{γ}}], as before.

Remember that [L_{ν_{σ(γ)}}, L_{−ν_{σ(γ)}}] is spanned by h_{σ(γ)}, because σ(γ) ∈ Δ. Put

h_{γ} = θ(h_{σ(γ)}), (28.10.4)

which is an element of [L_{ν_{γ}} , L_{−ν_{γ}}]. In fact, [L_{ν_{γ}} , L_{−ν_{γ}}] is spanned by h_{γ}, because [L_{ν_{σ(γ)}}, L_{−ν_{σ(γ)}}] is spanned by h_{σ(γ)}.

If α ∈ Θ_{Φ}, then

ν_{α}(h_{γ}) = ν_{α}(θ(h_{σ(γ)})) = ν_{σ(α)}(h_{σ(γ)}), (28.10.5)

using (28.3.4) in the second step. If α ∈ Φ, then σ(α) ∈ Φ, then we get that

ν_{α}(h_{γ}) = n(σ(α), σ(γ)), (28.10.6)

by (28.10.3), and because σ(γ) ∈ Δ. This implies that

ν_{α}(h_{γ}) = n(α, γ), (28.10.7)

because σ is an automorphism of Φ, as in Section 20.2.

In particular, ν_{γ}(h_{γ}) = 2, (28.10.8)

by the definition of n(γ, γ), as in (27.1.1). It follows that h_{γ} does not depend on the choice of σ. More precisely, this also uses the fact that [L_{ν_{γ}} , L_{−ν_{γ}}] is spanned by h_{γ}, as before.

If γ_{1}, γ_{2} ∈ Φ and γ_{1} + γ_{2} ∈ Φ, then

[L_{ν_{γ_{1}}} , L_{ν_{γ_{2}}} ] = L_{ν_{γ_{1}+γ_{2}}} . (28.10.9)

Remember that L is semisimple, as in Section 28.4. This permits us to use many of the remarks in Chapter 17, although a number of these properties have already been established. In particular, (28.10.9) was discussed in Section 17.9. Remember that if γ ∈ Φ, then the linear subspace of L spanned by L_{ν_{γ}}, L_{−ν_{γ}}, and [L_{ν_{γ}} , L_{−ν_{γ}}] is a Lie subalgebra of L that is isomorphic to sl_{2}(k), as in Section 28.3.

Let α ∈ Δ be given, and remember that x_{α} ∈ L_{ν_{α}}, y_{α} ∈ L_{−ν_{α}}, by construction. If γ ∈ Φ and α + γ ∈ Φ, then the restriction of ad_{x_{α}} to L_{ν_{γ}} is a one-to-one mapping onto L_{ν_{α+γ}}. Similarly, the restriction of ad_{y_{α}} to L_{−ν_{γ}} is a one-to-one mapping onto L_{−ν_{α+γ}}. This is the same as (28.10.9) with γ_{1} = α, γ_{2} = γ and γ_{1} = −α, γ_{2} = −γ, respectively, as in Section 17.9. Note that the fact that the linear subspace of L spanned by x_{α}, y_{α}, and h_{α} is a Lie subalgebra of L isomorphic to sl_{2}(k) can be verified more directly, as in Section 28.3.

If u_{1}, …, u_{r} ∈ L for some r ≥ 2, then put

[u_{1}, …, u_{r}] = [u_{1}, [u_{2}, … [u_{r−1}, u_{r}] …]], (28.10.10)

as before. Let α_{1}, …, α_{r} be a finite sequence of elements of Δ, possible with repetitions, such that

∑_{j=1}^{l} α_{j} ∈ Φ (28.10.11)

for every l = 1, …, r. If r ≥ 2, then we get that

[x_{α_{1}}, …, x_{α_{r}}] ≠ 0 (28.10.12)

and [y_{α_{1}}, …, y_{α_{r}}] ≠ 0, (28.10.13)

as in the preceding paragraph.

28.11 Subsets of Δ

Let us continue with the same notation and hypotheses as in the previous section. Let Δ_{0} be a nonempty subset of Δ, and let V_{0} be the linear subspace of V spanned by Δ_{0}. Remember that

Φ_{0} = Φ ∩ V_{0} (28.11.1)

is a root system in V_{0}, as in Section 20.6. If α, β ∈ Φ_{0}, then the analogue n_{0}(α, β) of (27.1.1) for Φ_{0} is the same as n(α, β), as in Section 20.15. Note that Δ_{0} is a base for Φ_{0}, as before.

Let L^{0} = L^{0}(k) be the Lie algebra over k defined as in Serre’s theorem in Section 27.8, using Φ_{0} and Δ_{0}. More precisely, if α ∈ Δ_{0}, then let x_{0,α}, y_{0,α}, and h_{0,α} be the corresponding generators in L^{0}, as before. These generators satisfy the same relations in L^{0} as their analogues in L, because n_{0}(α, β) = n(α, β) for every α, β ∈ Δ_{0}.

The construction used in Serre’s theorem implies that there is a unique Lie algebra homomorphism from L^{0} into L with

x_{0,α} ↦ x_{α}, y_{0,α} ↦ y_{α}, h_{0,α} ↦ h_{α}, (28.11.2)

for every α ∈ Δ_{0}. This homomorphism maps L^{0} onto the Lie subalgebra of L generated by x_{α}, y_{α}, h_{α}, with α ∈ Δ_{0}. We would like to check that this homomorphism is injective. This corresponds to Exercise 8 on p101 of [14].

It suffices to show that the dimension of the Lie subalgebra of L generated by x_{α}, y_{α}, h_{α}, α ∈ Δ_{0}, is at least the dimension of L^{0}. To do this, it is enough to verify that if γ ∈ Φ_{0}, then L_{ν_{γ}} is contained in this subalgebra of L. This is clear when γ ∈ Δ_{0} or −γ ∈ Δ_{0}.

Let γ ∈ Φ_{0} be a positive root with respect to Δ_{0}, so that γ can be expressed as a linear combination of elements of Δ_{0} with nonnegative coefficients. Remember that γ can be expressed as ∑_{j=1}^{r} α_{j}, where α_{j} ∈ Δ_{0} for every j = 1, …, r, and

∑_{j=1}^{l} α_{j} ∈ Φ_{0} (28.11.3)

for every l = 1, …, r, as in Section 19.12. Suppose that γ ∉ Δ_{0}, so that r ≥ 2.

Observe that [x_{α_{1}}, …, x_{α_{r}}] (28.11.4)

and [y_{α_{1}}, …, y_{α_{r}}] (28.11.5)

are elements of this subalgebra of L. These are also elements of L_{ν_{γ}} and L_{−ν_{γ}}, respectively.

Remember that (28.11.4) and (28.11.5) are nonzero under these conditions, as in the previous section. It follows that L_{ν_{γ}} and L_{−ν_{γ}} are spanned by (28.11.4) and (28.11.5), respectively, because L_{ν_{γ}} and L_{−ν_{γ}} are one-dimensional linear subspaces of L, as in Section 27.15. This implies that L_{ν_{γ}} and L_{−ν_{γ}} are contained in this subalgebra of L, as desired.

Chevalley’s normalization

29.1 Linearly independent roots

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a reduced root system in V. As usual, if α ∈ Φ, then we let σ_{α} be the symmetry on V with vector α that maps Φ onto itself. Let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ, so that σ_{α} is the reflection on V with respect to (·, ·) associated to α for every α ∈ Φ. If α, β ∈ Φ, then n(β, α) = 2 (β, α) (α, α)^{−1} ∈ Z, (29.1.1)

as before.

Suppose that α, β ∈ Φ are linearly independent in V, which is to say that they are not proportional in V. Let r, q be the largest nonnegative integers such that β − r α, β + q α ∈ Φ, (29.1.2)

as in Section 20.5. Remember that

2 (β, α) (α, α)^{−1} = r − q, (29.1.3)

as in (20.5.10). This corresponds to part (a) of the proposition on p145 of [14], and was also mentioned on p45 of [14].

Let V_{1} be the linear subspace of V spanned by α and β, and let Φ_{1} be the set of elements of Φ that can be expressed as a linear combination of α and β with integer coefficients. Note that if the sum of two elements of Φ_{1} is an element of Φ, then the sum is also an element of Φ_{1}. It follows that Φ_{1} is a root system in V_{1}, as in Section 20.6. More precisely, Φ_{1} is reduced, because Φ is reduced, by hypothesis. This corresponds to Exercise 7 on p46 of [14], as before.

Under these conditions, there are at most two possible values for the norms of elements of Φ_{1} with respect to (·, ·), as in part (b) of the proposition on p145 of [14], and its proof. More precisely, this follows from the remarks in Section 20.7 when Φ_{1} is irreducible in V_{1}. Otherwise, if Φ_{1} is reducible, then it consists of only ±α, ±β, because Φ_{1} is reduced. In this case, it is clear that there are at most two possible values for the norms of elements of Φ_{1}.

If α + β ∈ Φ, then part (c) of the proposition on p145 of [14] states that

r + 1 = q (α + β, α + β) (β, β)^{−1}. (29.1.4)

To see this, we first use (29.1.3) to get that

r + 1 − q (α + β, α + β) (β, β)^{−1}
(29.1.5) = q + 2 (β, α) (α, α)^{−1} + 1 − q (α + β, α + β) (β, β)^{−1}
= 2 (β, α) (α, α)^{−1} + 1 − q (α, α) (β, β)^{−1} − 2 q (α, β) (β, β)^{−1}.

If we put

(29.1.6)  A = 2 (β, α) (α, α)^{−1} + 1,
(29.1.7)  B = 1 − q (α, α) (β, β)^{−1},

then the right side of (29.1.5) is the same as A B. Thus it suffices to show that A = 0 or B = 0.

Remember that n(β, α) n(α, β) = 0, 1, 2, or 3, (29.1.8)

because α and β are not proportional in V, as in Section 19.9. Suppose for the moment that (α, α) ≥ (β, β), (29.1.9)

so that |n(β, α)| ≤ |n(α, β)|. (29.1.10)

This implies that |n(β, α)| = 0 or 1, (29.1.11)

by (29.1.8). If n(β, α) = −1, then A = 0, as desired.

Otherwise, if n(β, α) = 0 or 1, then (β, α) ≥ 0. This implies that

(β + α, β + α) > (α, α), (β, β). (29.1.12)

Note that α + β ∈ Φ_{1}, because α + β ∈ Φ, by hypothesis. It follows that

(α, α) = (β, β) (29.1.13)

in this case, because there are at most two possible values for the norms of elements of Φ_{1}, as before.

Similarly, (β + 2 α, β + 2 α) > (β + α, β + α), (29.1.14)

because (β, α) ≥ 0. This implies that β + 2 α ∉ Φ_{1}, because there are at most two possible values of the norms of elements of Φ_{1}. This means that β +2 α ∉ Φ. Remember that β + j α ∈ Φ when j ∈ Z and −r < j < q, as in Section 20.5. It follows that q ≤ 1, by the definition q. Thus q = 1, because α + β ∈ Φ. This implies that B = 0, because of (29.1.13).

Suppose now that (α, α) < (β, β). (29.1.15)

Observe that (α + β, α + β) = (α, α) or (β, β), (29.1.16)

because α + β ∈ Φ_{1}, and there are at most two possible values for the norms of elements of Φ_{1}. This implies that (α, β) < 0, so that

n(α, β) < 0. (29.1.17)

We also get that (β − α, β − α) > (β, β). (29.1.18)

It follows that β − α ∉ Φ, because otherwise β − α would be an element of Φ_{1}, and have the same norm as α or β.

This shows that r = 0, because β + j α ∈ Φ when j ∈ Z and −r < j < q, as before. Using (29.1.15) and the fact that (α, β) ≠ 0, we obtain that

|n(α, β)| < |n(β, α)|. (29.1.19)

Combining this with (29.1.8), we get that n(α, β) = −1, 0, or 1. This means that n(α, β) = −1, (29.1.20)

because of (29.1.17).

Because r = 0, (29.1.3) reduces to saying that q = −n(β, α). Equivalently,

q = \frac{n(β, α)}{n(α, β)}, (29.1.21)

by (29.1.20). This implies that

q = \frac{(β, β)}{(α, α),} (29.1.22)

by the definition of n(·, ·). It follows that B = 0 in this case as well.

29.2 An identity for root strings

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a root system in V. Also let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ, and suppose that α, β ∈ Φ are not proportional in V. As before, we take r, q to be the largest nonnegative integers such that β − r α, β + q α ∈ Φ. Similarly, let r′, q′ be the largest nonnegative integers such that α − r′ β, α + q′ β ∈ Φ. (29.2.1)

Thus 2 (α, β) (β, β)^{−1} = r′ − q′, (29.2.2) as in (29.1.3) and Section 20.5. We would like to check that

q (r + 1) (β, β)^{−1} = q′ (r′ + 1) (α, α)^{−1}, (29.2.3)

as in Exercise 10 on p47 of [14]. This is mentioned in Exercise 3 on p150 of [14] as well.

Remember that β + j α ∈ Φ for every integer j with −r ≤ j ≤ q, as in Section 20.5. Similarly, if l ∈ Z and −r′ ≤ l ≤ q′, then α + l β ∈ Φ.

It follows that q = 0 if and only if α + β ∉ Φ. Of course, q′ = 0 if and only if α + β ∉ Φ too, for the same reason. Thus q = 0 if and only if q′ = 0, in which case (29.2.3) holds trivially.

Similarly, r = 0 if and only if β − α ∉ Φ. The same argument shows that r′ = 0 if and only if α − β ∉ Φ. Of course, α − β ∈ Φ if and only if β − α ∈ Φ. This means that r = 0 if and only if r′ = 0. In this case, (29.2.3) follows from (29.1.3) and (29.2.2).

Thus we may suppose from now on that q, q′, r, r′ ≥ 1. Remember that

q + r, q′ + r′ ≤ 3, (29.2.4)

as in Section 20.5.

Observe that q = r if and only if (α, β) = 0, by (29.1.3). Similarly, q′ = r′ if and only if (α, β) = 0. Under these conditions, we get that q = r = q′ = r′ = 1, because of (29.2.4). Thus it suffices to verify that (α, α) = (β, β), to get (29.2.3).

Remember that α + β ∈ Φ, because q, q′ ≥ 1. We also have that

(α, α + β) = (α, α), (β, α + β) = (β, β) (29.2.5)

and (α + β, α + β) = (α, α) + (β, β), (29.2.6) because (α, β) = 0. In particular,

max((α, α), (β, β)) < (α + β, α + β) ≤ 2 max((α, α), (β, β)). (29.2.7)

Note that α + β is not proportional to either α or β in V.

Using the remarks in Section 19.9, we obtain that (α + β, α + β) is either 1, 2, or 3 times (α, α), and similarly for (β, β). In fact, (α + β, α + β) has to be 2 times (α, α) or (β, β), because of (29.2.7). In either case, we get that (α, α) = (β, β), as desired.

Suppose now that (α, β) ≠ 0, so that q ≠ r and q′ ≠ r′. It follows that

|q − r| = |q′ − r′| = 1, (29.2.8)

because of (29.2.4). This implies that (α, α) = (β, β), because of (29.1.3) and (29.2.2). Using this and (29.1.3), (29.2.2) again, we get that

r − q = r′ − q′. (29.2.9)

One can use this and (29.2.4) to verify that q = q′ and r = r′. More precisely, either q = q′ is equal to 1 and r = r′ is equal to 2, or the other way around. This implies (29.2.3), because (α, α) = (β, β).

29.3. RELATED PROPERTIES OF λ_{α}’S 615

29.3 Related properties of λ_{α}’s

Let V be a vector space over the real numbers of positive finite dimension again, and let Φ be a root system in V. If α ∈ Φ, then the symmetry σ_{α} on V with vector α that maps Φ onto itself can be expressed as σ_{α}(v) = v −λ_{α}(v) α, where λ_{α} is a linear functional on V such that λ_{α}(α) = 2, as usual. Let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ, so that σ_{α} is the reflection on V with respect to (·, ·) associated to α for every α ∈ Φ. This means that λ_{α}(v) = 2 (v, α) (α, α)^{−1} for every α ∈ Φ and v ∈ V, as before.

Suppose that α, β ∈ Φ are not proportional in V, and let r, q, r′, q′ ≥ 0 be as in the previous section. If we multiply both sides of (29.2.3) by 2 (α, β), then we get that q (r + 1) λ_{β}(α) = q′ (r′ + 1) λ_{α}(β). (29.3.1)

If (α, β) = 0, then λ_{α}(β) = λ_{β}(α) = 0, q = q′, and r = r′, as in the previous section.

Suppose now that Φ is a reduced root system, and that α + β ∈ Φ too. If v ∈ V, then

λ_{α+β}(v) = 2 (α + β, α + β)^{−1} (v, α + β)
(29.3.2)  = (α + β, α + β)^{−1} (α, α) λ_{α}(v) + (α + β, α + β)^{−1} (β, β) λ_{β}(v).

It follows that

λ_{α+β}(v) = q′ (r′ + 1)^{−1} λ_{α}(v) + q (r + 1)^{−1} λ_{β}(v) (29.3.3)

for every v ∈ V, using (29.1.4) and its analogue for q′, r′. Note that λ_{α} and λ_{β} are not proportional as linear functionals on V, because α and β are not proportional in V. Thus the coefficients of λ_{α} and λ_{β} on the right side of (29.3.3) are uniquely determined by this condition.

29.4 Associated Lie algebras

Let V be a vector space over the real numbers of positive finite dimension again, and let Φ be a reduced root system in V. If α ∈ Φ, then the symmetry σ_{α} on V with vector α that maps Φ onto itself can be expressed as σ_{α}(v) = v − λ_{α}(v) α, where λ_{α} is a linear functional on V with λ_{α}(α) = 2. If α, β ∈ Φ, then

n(β, α) = λ_{α}(β) ∈ Z, (29.4.1)

by the definition of a root system. Of course, this is the same as (29.1.1), where (·, ·) is an inner product on V that is invariant under the Weyl group. Let Θ be the subgroup of V, as a commutative group with respect to addition, that is generated by Φ.

Let k be a field of characteristic 0. We would like to consider a Lie algebra (A, [·, ·]_{A}) over k as in Serre’s theorem in Section 27.8, or equivalently as in Section 28.7. Thus, if α ∈ Φ ∪ {0}, then we ask that A_{α} be a linear subspace of A, and that A correspond to the direct sum of A_{α}, α ∈ Φ ∪ {0}, as a vector space over k. Put A_{α} = {0} when α ∈ Θ \ (Φ ∪ {0}), so that we may ask that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ Θ. We also ask that A_{α} have dimension one as a vector space over k when α ∈ Φ.

As usual, A′_{0} denotes the dual space of linear functionals on A_{0}, as a vector space over k. We ask that there be a group homomorphism ϕ from Θ into A′_{0}, as a commutative group with respect to addition, such that ad_{w}(x) = ϕ_{α}(w) x for every w ∈ A_{0}, x ∈ A_{α}, and α ∈ Φ ∪ {0}, where ϕ_{α} = ϕ(α). Of course, this implies that A_{0} is a Lie subalgebra of A that is commutative as a Lie algebra.

If α, β ∈ Φ and α + β ∈ Φ, then we ask that

[A_{α}, A_{β}] = A_{α+β}. (29.4.2)

We also ask that for each α ∈ Φ there be an element h_{α} of [A_{α}, A_{−α}] ⊆ A_{0} such that ϕ_{β}(h_{α}) = λ_{α}(β) (29.4.3)

for every β ∈ Φ. In particular, this means that ϕ_{α}(h_{α}) = 2 · 1 = 1 + 1 in k, which is not zero, because k has characteristic zero. Of course, [A_{α}, A_{−α}] has dimension at most one as a vector space over k, because A_{α}, A_{−α} are onedimensional. It follows that [A_{α}, A_{−α}] is the one-dimensional linear subspace of A spanned by h_{α}, and that h_{α} is uniquely determined by (29.4.3), with β = α.

Let Δ be a base for Φ. We ask that h_{α}, α ∈ Δ, be a basis for A_{0}, as a vector space over k. If A is as in Section 22.1, then conditions like these were discussed in Section 22.11. We also have that A is isomorphic to the Lie algebra obtained as in Serre’s theorem in Section 27.8, as discussed in Section 28.7. If A is obtained as in Serre’s theorem, then conditions like these follow from properties discussed in Sections 27.11, 27.15, 28.2, and 28.10.

If β ∈ Φ, then (29.4.3) defines a k-valued function of α ∈ Δ, which is the same as (27.3.9) when β ∈ Δ. The collection of these functions with β ∈ Δ is a basis for the space of k-valued functions on Δ, as a vector space over k, as in Section 27.3. This means that ϕ_{β}, β ∈ Δ, is a basis for A′_{0} as a vector space over k. If A is as in Section 22.1, then this can be obtained from the facts that A′_{0} is spanned by ϕ_{β}, β ∈ Δ, as a vector space over k, and the dimension of A′_{0} is the same as the number of elements of Δ, which is the dimension of V. If h ∈ A_{0}, then it follows that h is uniquely determined by ϕ_{β}(h), β ∈ Δ.

Remember that Θ is the same as the subgroup of V, as a commutative group with respect to addition, generated by Δ. The remark in the preceding paragraph implies that ϕ is injective as a mapping from Θ into A′_{0}. This also uses the hypothesis that k have characteristic 0. If α ∈ Φ ∪ {0}, then one can use the previous statement to get that A_{α} consists of all x ∈ A such that ad_{w}(x) = ϕ_{α}(w) x for every w ∈ A_{0}.

If α ∈ Φ, then λ_{α} is an element of the dual V′ of V, as a vector space over R. The set Φ′ of λ_{α}, α ∈ Φ, is a root system in V′, as in Section 19.8. The set Δ′ of λ_{α}, α ∈ Δ, is a base for Φ′, as in Section 19.13, and because Φ is reduced, by hypothesis. Thus, if γ ∈ Φ, then λ_{γ} can be expressed as a linear combination of λ_{α}, α ∈ Δ, with coefficients in Z.

If γ ∈ Φ, then it follows that h_{γ} can be expressed as a linear combination of h_{α}, α ∈ Δ, with the same coefficients in Z as in the preceding paragraph. This uses (29.4.3), and the fact that h_{γ} is uniquely determined by ϕ_{β}(h_{γ}), β ∈ Δ, as before.

29.5 A lemma about brackets

Let us continue with the same notation and hypotheses as in the previous section. Also let α, β ∈ Φ be given, and suppose that α, β are not proportional in V. Let r, q be the largest nonnegative integers such that β − r α and β + q α are elements of Φ, as before. It follows that

β + j α ∈ Φ (29.5.1)

for every j ∈ Z with −r ≤ j ≤ q, as in Section 20.5.

Suppose that x_{α} ∈ A_{α} and x_{−α} ∈ A_{−α} satisfy

[x_{α}, x_{−α}]_{A} = h_{α}. (29.5.2)

Of course, this implies that x_{α}, x_{−α} ≠ 0, and (29.5.2) can always be arranged by multiplying x_{α} or x_{−α} by a suitable element of k. If x_{β} ∈ A_{β}, then

[x_{−α}, [x_{α}, x_{β}]_{A}]_{A} = q (r + 1) x_{β}. (29.5.3)

This corresponds to the lemma on p146 of [14].

Note that [x_{α}, x_{β}]_{A} ∈ A_{α+β}, so that the left side of (29.5.3) is an element of A_{β}. If α + β ∉ Φ, then A_{α+β} = {0}, which means that [x_{α}, x_{β}]_{A} = 0. We also have that q = 0 in this case, so that (29.5.3) holds.

Let E be the linear subspace of A spanned by A_{β+j α}, j ∈ Z. More precisely, we may as well restrict our attention to −r ≤ j ≤ q, because otherwise β + j α is not in Φ, and A_{β+j α} = {0}. We also have that E corresponds to the direct sum of A_{β+j α}, −r ≤ j ≤ q, because A corresponds to the direct sum of A_{γ}, γ ∈ Φ ∪ {0}, as a vector space over k. It follows that

dim E = q + r + 1 (29.5.4)

as a vector space over k, because A_{γ} is one-dimensional for every γ ∈ Φ.

The linear span of x_{α}, x_{−α}, and h_{α} in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k). If we consider A as a module over this subalgebra, then E is a submodule of A. This is basically the same as in Section 17.9, with y_{α} = x_{−α}.

The weights of h_{α} on E are given by

ϕ_{β+j α}(h_{α}) = ϕ_{β}(h_{α}) + j ϕ_{α}(h_{α}) = ϕ_{β}(h_{α}) + 2 j, (29.5.5)

with −r ≤ j ≤ q. One can verify that E is irreducible as a module over sl_{2}(k), as in Section 17.9.

Under these conditions, E is as in Section 15.3 as a module over sl_{2}(k), with m = q + r, as in Section 17.9. To get (29.5.3), one can consider v_{q} in Section 15.3, as on p146 of [14].

618 CHAPTER 29. CHEVALLEY’S NORMALIZATION

29.6 Some remarks about automorphisms

Let us continue with the same notation and hypotheses as in Section 29.4. Suppose that σ is a Lie algebra automorphism of A such that σ(w) = −w (29.6.1) for every w ∈ A_{0}. Let w ∈ A_{0} and x ∈ A be given, and observe that [w, σ(x)]_{A} = −[σ(w), σ(x)]_{A} = −σ([w, x]_{A}). (29.6.2) Also let α ∈ Φ be given, and note that ϕ_{−α}(w) = −ϕ_{α}(w). It follows that [w, x]_{A} = ϕ_{α}(w) x if and only if [w, σ(x)]_{A} = ϕ_{−α}(w) σ(x). (29.6.3) This implies that σ(A_{α}) = A_{−α}, (29.6.4) because A_{α} consists of all x ∈ A such that ad_{w}(x) = ϕ_{α}(w) x for every w ∈ A_{0}, as in Section 29.4, and similarly for −α. Of course, (29.6.4) holds for α = 0 too, by hypothesis.

Let x_{α} ∈ A_{α} and y_{α} ∈ A_{−α} be given, with x_{α}, y_{α} ≠ 0. Observe that there are ξ_{α}, η_{α} ∈ k such that σ(x_{α}) = ξ_{α} y_{α}, σ(y_{α}) = η_{α} x_{α}, (29.6.5) because of (29.6.4) and the analogous statement for −α. This also uses the fact that A_{α}, A_{−α} are one-dimensional linear subspaces of A, and we have that ξ_{α}, η_{α} ≠ 0, because σ is injective on A. It follows that σ([x_{α}, y_{α}]_{A}) = [σ(x_{α}), σ(y_{α})]_{A} = −ξ_{α} η_{α} [x_{α}, y_{α}]_{A}. (29.6.6) Remember that [x_{α}, y_{α}]_{A} ∈ A_{0}, so that σ([x_{α}, y_{α}]_{A}) = −[x_{α}, y_{α}]_{A}, (29.6.7) by (29.6.1).

Note that [x_{α}, y_{α}]_{A} ≠ 0, because [A_{α}, A_{−α}]_{A} is one-dimensional as a linear subspace of A. Thus we get that ξ_{α} η_{α} = 1 (29.6.8) in k. This implies that σ(σ(x_{α})) = x_{α}, σ(σ(y_{α})) = y_{α}. (29.6.9) It follows that σ ∘ σ is the identity mapping on A, because α ∈ Φ is arbitrary, and using (29.6.1) on A_{0}. Remember that an automorphism on A with these properties was discussed in Section 28.6.

Let α ∈ Φ be given again, and let x_{α} be a nonzero element of A_{α}. Also let ζ_{α} be a nonzero element of k, and put

z_{α} = ζ_{α} x_{α}, z_{−α} = −σ(z_{α}) = −ζ_{α} σ(x_{α}). (29.6.10)

Thus [z_{α}, z_{−α}]_{A} = −ζ_{α}^{2} [x_{α}, σ(x_{α})]_{A}, (29.6.11)

and [x_{α}, σ(x_{α})]_{A} ≠ 0, as before. If k is algebraically closed, then we can choose ζ_{α} so that [z_{α}, z_{−α}]_{A} = h_{α}. (29.6.12)

Note that −ζ_{α} would also work, and that only these two elements of k would have this property.

29.7 Diagonal automorphisms

We continue with the same notation and hypotheses as in Section 29.4. Suppose now that τ is a Lie algebra automorphism of A such that

τ(w) = w (29.7.1)

for every w ∈ A_{0}. If w ∈ A_{0} and x ∈ A, then

[w, τ(x)]_{A} = [τ(w), τ(x)]_{A} = τ([w, x]_{A}). (29.7.2)

If α ∈ Φ, then it follows that [w, x]_{A} = ϕ_{α}(w) x if and only if

[w, τ(x)]_{A} = ϕ_{α}(w) τ(x). (29.7.3)

This means that τ(A_{α}) = A_{α} (29.7.4)

for every α ∈ Φ, as before.

In this case, τ is said to be a diagonal automorphism of A, as on p87 of [14]. If α ∈ Φ, then there are nonzero elements τ_{α}, τ_{−α} such that τ corresponds to multiplication by τ_{α}, τ_{−α} on A_{α}, A_{−α}, respectively, because A_{α}, A_{−α} are one-dimensional linear subspaces of A. It is easy to see that

τ_{α} τ_{−α} = 1, (29.7.5)

because [A_{α}, A_{−α}] is a nontrivial linear subspace of A_{0}.

Of course, any Lie algebra automorphism of A is uniquely determined by its values on a set of generators of A as a Lie algebra over k. If A is as in Serre’s theorem in Section 27.8, then the diagonal automorphisms of A can be obtained as in Section 28.6.

Let σ be a Lie algebra automorphism of A that satisfies (29.6.1), and thus (29.6.4) for every α ∈ Φ. If τ is a diagonal automorphism of A, then one can check that σ ∘ τ = τ^{−1} ∘ σ. (29.7.6)

Observe that σ ∘ τ is a Lie algebra automorphism of A that sends every w ∈ A_{0} to −w. If σ̃ is any Lie algebra automorphism of A that sends every w ∈ A_{0} to −w, then σ^{−1} ∘ σ̃ (29.7.7)

is a diagonal automorphism on A.

Let τ be a diagonal automorphism on A again, and for each α ∈ Φ, let τ_{α} be the nonzero element of k such that τ corresponds to multiplication by τ_{α} on A_{α}, as before. Similarly, we may as well put τ_{0} = 1. If α, β ∈ Φ and α + β ∈ Φ, then it is easy to see that τ_{α+β} = τ_{α} τ_{β}, (29.7.8)

because [A_{α}, A_{β}] = A_{α+β} ≠ {0}.

Let Δ be a base for Φ, and let β ∈ Φ be given. We would like to check that τ_{β} is uniquely determined by the values of τ_{α} with α ∈ Δ. Suppose first that β is a positive root with respect to Δ, so that β can be expressed as a linear combination of elements of Δ with nonnegative coefficients. Remember that β can be expressed ∑_{j=1}^{r} α_{j}, where α_{j} ∈ Δ for every j = 1, …, r, and ∑_{j=1}^{l} α_{j} ∈ Φ for every l = 1, …, r, as in Section 19.12. This implies that

τ_{β} = τ_{α_{1}} ⋯ τ_{α_{r}}, (29.7.9)

by (29.7.8). Otherwise, if −β is a positive root, then one can reduce to the previous case, using (29.7.5).

Let Θ be the subgroup of V, as a commutative group with respect to addition, generated by Φ, which is the same as the subgroup of V generated by Δ. Every mapping from Δ into k \ {0} has a unique extension to a group homomorphism from Θ into the multiplicative group of nonzero elements of k. In particular, α ↦ τ_{α}, α ∈ Δ, has a unique extension to a group homomorphism from Θ into k \ {0}. If β ∈ Φ, then τ_{β} is the same as the value of this extension at β, by the remarks in the preceding paragraph. Conversely, every group homomorphism from Θ into k \ {0} corresponds to a diagonal automorphism on A in this way.

29.8 Basis vectors in A

Let us continue with the same notation and hypotheses as in Section 29.4. As before, we let Δ be a base for Φ, and ask that h_{α}, α ∈ Δ, be a basis for A_{0}, as a vector space over k. Let z_{α} be a nonzero element of A_{α} for each α ∈ Φ. Note that the collection of z_{α}’s, α ∈ Φ, together with h_{α}, α ∈ Δ, is a basis for A as a vector space over k.

If α, β ∈ Φ and α + β ∈ Φ, then there is a unique element c_{α,β} of k such that

[z_{α}, z_{β}]_{A} = c_{α,β} z_{α+β}, (29.8.1)

because [z_{α}, z_{β}]_{A} is an element of A_{α+β}, which is spanned by z_{α+β}. Of course, if α, β ∈ Φ and α+β ∉ Φ∪{0}, then [z_{α}, z_{β}]_{A} = 0, because [z_{α}, z_{β}] ∈ A_{α+β} = {0}.

29.8. BASIS VECTORS IN A 621

Observe that [h_{α}, z_{β}]_{A} = ϕ_{β}(h_{α}) z_{β} = λ_{α}(β) z_{β} (29.8.2)

for every α, β ∈ Φ, as in Section 29.4. If γ ∈ Φ, then h_{γ} can be expressed as a linear combination of h_{α}, α ∈ Δ, with coefficients in Z obtained from the dual root system Φ′, as in Section 29.4.

As in Section 28.6, there is a Lie algebra automorphism σ on A such that σ(w) = −w for every w ∈ A_{0}, and σ ∘ σ is the identity mapping on A. It follows that σ(A_{α}) = A_{−α} for every α ∈ Φ, as in Section 29.6. Let Φ^{+} be the set of positive roots with respect to Δ, which is to say the set of α ∈ Φ that can be expressed as linear combinations of elements of Δ with nonnegative coefficients.

Suppose for the moment that k is algebraically closed. If α ∈ Φ^{+}, then there is a nonzero element z_{α} of A_{α} such that

z_{−α} = −σ(z_{α}) (29.8.3)

satisfies [z_{α}, z_{−α}]_{A} = h_{α}, (29.8.4)

as in Section 29.6. Note that z_{−α} is a nonzero element of A_{−α}. By doing this for every α ∈ Φ^{+}, we get a nonzero element z_{α} of A_{α} for every α ∈ Φ. We also have that the analogues of (29.8.3) and (29.8.4) hold with α replaced with −α, because σ ∘ σ is the identity mapping on A, and h_{−α} = −h_{α}.

Let z_{α} be a nonzero element of A_{α} for each α ∈ Φ again, and suppose that (29.8.4) holds for every α ∈ Φ. Also let c_{α,β} ∈ k be defined for α, β ∈ Φ with α + β ∈ Φ as in (29.8.1), as before. Note that [·, ·]_{A} is uniquely determined on A by the c_{α,β}’s under these conditions.

Suppose that (29.8.3) holds for every α ∈ Φ. If α, β ∈ Φ and α + β ∈ Φ, then

[z_{−α}, z_{−β}]_{A} = [−σ(z_{α}), −σ(z_{β})]_{A} = σ([z_{α}, z_{β}]_{A})
(29.8.5)  = c_{α,β} σ(z_{α+β}) = −c_{α,β} z_{−α−β}.

Of course, −α, −β, −α − β ∈ Φ in this case, so that

[z_{−α}, z_{−β}]_{A} = c_{−α,−β} z_{−α−β}, (29.8.6)

as in (29.8.1). It follows that

c_{−α,−β} = −c_{α,β} (29.8.7)

for every α, β ∈ Φ with α + β ∈ Φ. This corresponds to Theorem 10 on p51 of [24], and to parts (a) and (b) of the proposition on p146 of [14].

Let z_{α} be a nonzero element of A_{α} for each α ∈ Φ again, and suppose that (29.8.4) holds for every α ∈ Φ. If α, β ∈ Φ and α + β ∈ Φ, then let c_{α,β} ∈ k be as in (29.8.1), and suppose that (29.8.7) holds. Under these conditions, the the collection of z_{α}’s, α ∈ Φ, together with the h_{α}’s, α ∈ Δ, is said to be a Chevalley basis for A, as on p147 of [14].

29.9 Some properties of the c_{α,β}’s

We continue with the same notation and hypotheses as in Section 29.4. Let z_{α} be a nonzero element of A_{α} for every α ∈ Φ again, and suppose that (29.8.4) holds for every α ∈ Φ. If α, β ∈ Φ and α + β ∈ Φ, then let c_{α,β} ∈ k be as in (29.8.1).

Let α, β ∈ Φ with α + β ∈ Φ be given, so that −α, −β, −α − β ∈ Φ too. Observe that

(29.9.1) [[z_{α}, z_{β}]_{A}, [z_{−α}, z_{−β}]_{A}]_{A} = [c_{α,β} z_{α+β}, c_{−α,−β} z_{−α−β}]_{A}
= c_{α,β} c_{−α,−β} h_{α+β}.

We can also use the Jacobi identity to get that

[[z_{α}, z_{β}]_{A}, [z_{−α}, z_{−β}]_{A}]_{A} (29.9.2)

= [z_{α}, [z_{β}, [z_{−α}, z_{−β}]_{A}]_{A}]_{A} − [z_{β}, [z_{α}, [z_{−α}, z_{−β}]_{A}]_{A}]_{A}
= −[z_{α}, [z_{β}, [z_{−β}, z_{−α}]_{A}]_{A}]_{A} − [z_{β}, [z_{α}, [z_{−α}, z_{−β}]_{A}]_{A}]_{A}.

Note that α and β are not proportional in V, because α + β ∈ Φ, and Φ is reduced. Let r, q be the largest nonnegative integers such that β − r α and β + q α are elements of Φ again. Similarly, let r′, q′ be the largest nonnegative integers such that α − r′ β and α + q′ β are elements of Φ. Equivalently, r, q are the largest nonnegative integers such that

(−β) − r (−α), (−β) + q (−α) ∈ Φ, (29.9.3)

and r′, q′ are the largest nonnegative integers such that

(−α) − r′ (−β), (−α) + q′ (−β) ∈ Φ. (29.9.4)

Of course, q, q′ ≥ 1, (29.9.5)

because α + β ∈ Φ.

Note that [z_{α}, [z_{−α}, z_{−β}]_{A}]_{A} = q (r + 1) z_{−β}, (29.9.6)

as in (29.5.3). Similarly,

[z_{β}, [z_{−β}, z_{−α}]_{A}]_{A} = q′ (r′ + 1) z_{−α}. (29.9.7)

Combining these identites with (29.9.2), we get that

[[z_{α}, z_{β}]_{A}, [z_{−α}, z_{−β}]_{A}]_{A} = −q′ (r′ + 1) [z_{α}, z_{−α}]_{A} − q (r + 1) [z_{β}, z_{−β}]_{A}
(29.9.8)  = −q′ (r′ + 1) h_{α} − q (r + 1) h_{β}.

It follows that

c_{α,β} c_{−α,−β} h_{α+β} = −q′ (r′ + 1) h_{α} − q (r + 1) h_{β}, (29.9.9) by (29.9.1).

If γ ∈ Φ, then we can evaluate ϕ_{γ} at both sides of (29.9.9), using (29.4.3), to get that

c_{α,β} c_{−α,−β} λ_{α+β}(γ) = −q′ (r′ + 1) λ_{α}(γ) − q (r + 1) λ_{β}(γ). (29.9.10)

In particular, we can take γ = α + β, to get that 2 c_{α,β} c_{−α,−β} corresponds to an integer under the natural embedding of Q into k. This means that

c_{α,β} c_{−α,−β} λ_{α+β} = −q′ (r′ + 1) λ_{α} − q (r + 1) λ_{β} (29.9.11)

as linear functionals on V, because V is spanned by Φ. Comparing this with (29.3.3), we obtain that

c_{α,β} c_{−α,−β} = −(r + 1)^{2} = −(r′ + 1)^{2}. (29.9.12)

This also uses (29.9.5), and corresponds to a remark on p148 of [14]. If (29.8.7) holds, then we get that

c_{α,β}^{2} = (r + 1)^{2} = (r′ + 1)^{2}. (29.9.13)

This means that c_{α,β} = ±(r + 1) = ±(r′ + 1), (29.9.14)

which is part of a famous theorem of Chevalley, as in part (d) of the theorem on p147 of [14], and Theorem 11 on p51 of [24]. More precisely, some of the previous arguments correspond to the proof of part (c) of the proposition on p146f of [14].

Remember Δ is a base for Φ, and that the z_{α}’s, α ∈ Φ, together with the h_{α}’s, α ∈ Δ, form a basis for A as a vector space over k, which is a Chevalley basis, as in the previous section. Observe that the brackets of any two elements of this basis can be expressed as a linear combination of elements of the basis with integer coefficients, because of (29.9.14). This corresponds to the theorem on p147 of [14].

Let A(Z) be the collection of elements of A that can be expressed as linear combinations of the z_{α}’s, α ∈ Φ, and the h_{α}’s, α ∈ Δ, with integer coefficients. This is a subgroup of A, as a commutative group with respect to addition, and in fact a Lie subalgebra of A, as a Lie algebra over Z. This is part of Remark (1) on p51 of [24], and is also mentioned on p149 of [14].

Note that h_{γ} ∈ A(Z) for every γ ∈ Φ, as in Section 29.4. This implies that A(Z) does not depend on the base Δ for Φ, as mentioned on p149 of [14].

29.10 Uniqueness of Chevalley bases

Let us continue with the same notation and hypotheses as in Section 29.4. Let z_{α}, z̃_{α} be nonzero elements of A_{α} for every α ∈ Φ. Thus, for each α ∈ Φ, there is a nonzero element η_{α} of k such that

z̃_{α} = η_{α} z_{α}. (29.10.1)

624 CHAPTER 29. CHEVALLEY’S NORMALIZATION

Suppose that [z_{α}, z_{−α}]_{A} = h_{α} for every α ∈ Φ, as before. Observe that [z̃_{α}, z̃_{−α}]_{A} = h_{α} (29.10.2) for every α ∈ Φ if and only if η_{α} η_{−α} = 1 (29.10.3) for every α ∈ Φ, as on p148 of [14].

If α, β ∈ Φ and α + β ∈ Φ, then there is a unique c_{α,β} ∈ k such that [z_{α}, z_{β}]_{A} = c_{α,β} z_{α+β}, as before. Similarly, there is a unique c̃_{α,β} ∈ k such that [z̃_{α}, z̃_{β}]_{A} = c̃_{α,β} z̃_{α+β}. (29.10.4) More precisely, c_{α,β}, c̃_{α,β} ≠ 0, because of (29.4.2). Using (29.10.1), we get that c̃_{α,β} η_{α+β} = η_{α} η_{β} c_{α,β}. (29.10.5)

Suppose that c_{−α,−β} = −c_{α,β} for every α, β ∈ Φ with α + β ∈ Φ, as before. Suppose also that (29.10.3) holds for every α ∈ Φ. Under these conditions, one can check that c̃_{−α,−β} = −c̃_{α,β} (29.10.6) for every α, β ∈ Φ with α + β ∈ Φ if and only if η_{α+β} = ±η_{α} η_{β} (29.10.7) for every α, β ∈ Φ with α + β ∈ Φ, as on p148 of [14].

Let Δ be a base for Φ, and remember that Θ is the subgroup of V, as a commutative group with respect to addition, generated by Φ. This is the same as the subgroup of V generated by Δ, and every mapping from Δ into k \ {0} has a unique extension to a group homomorphism from Θ into the multiplicative group of nonzero elements of k.

Suppose now that η_{α} is a nonzero element of k for each α ∈ Φ that satisfies (29.10.3) for every α ∈ Φ, and (29.10.7) for every α, β ∈ Φ with α + β ∈ Φ. Let α ↦ τ_{α} be the unique group homomorphism from Θ into the multiplicative group of nonzero elements of k such that (29.10.8) τ_{α} = η_{α} for every α ∈ Δ. If α ∈ Φ, then let ρ_{α} be the nonzero element of k determined by η_{α} = ρ_{α} τ_{α}. (29.10.9) Observe that ρ_{α} ρ_{−α} = 1 (29.10.10) for every α ∈ Φ, and that ρ_{α+β} = ±ρ_{α} ρ_{β} (29.10.11) for every α, β ∈ Φ with α + β ∈ Φ, because of the analogous properties of the η_{α}’s and τ_{α}’s. We would like to verify that

ρ_{β} = ±1 (29.10.12)

for every β ∈ Φ.

Of course, ρ_{α} = 1 for every α ∈ Δ, by construction. Suppose that β ∈ Φ is a positive root with respect to Δ, so that β can be expressed as a linear combination of elements of Δ with nonnegative coefficients. In this case, β can be expressed as ∑_{j=1}^{r} α_{j}, where α_{j} ∈ Δ for every j = 1, …, r, and ∑_{j=1}^{l} α_{j} ∈ Φ for every l = 1, …, r, as in Section 19.12. This permits one to obtain (29.10.12) using (29.10.11). If −β is a positive root, then (29.10.12) can be obtained from (29.10.10) and the previous case.

Let the z_{α}’s and z̃_{α}’s be as before, so that the z_{α}’s, α ∈ Φ, together with the h_{α}’s, α ∈ Δ, is a Chevalley basis for A, and similarly for the z̃_{α}’s. The corresponding η_{α}’s can be defined for α ∈ Φ as in (29.10.1), and satisfy (29.10.3) and (29.10.7). Thus we can define τ_{α} and ρ_{α} as in (29.10.8) and (29.10.9), respectively. Using the τ_{α}’s, we get a diagonal automorphism τ of A, as in Section 29.7. Note that τ(h) = h for every h ∈ A_{0}, and τ(z_{α}) = τ_{α} z_{α} for every α ∈ Φ, as before.

If α ∈ Φ, then we get that

z̃_{α} = ρ_{α} τ(z_{α}). (29.10.13)

Let A(Z) be the collection of elements of A that can be expressed as linear combinations of the z_{α}’s, α ∈ Φ, and h_{α}’s, α ∈ Δ, with coefficients in Z, as in the previous section. Similarly, let Ã(Z) be the collection of elements of A that can be expressed as linear combinations of the z̃_{α}’s, α ∈ Φ, and h_{α}’s, α ∈ Δ, with coefficients in Z. It is easy to see that

Ã(Z) = τ(A(Z)), (29.10.14)

using (29.10.12) and (29.10.13). In particular, the restriction of τ to A(Z) defines an isomorphism from A(Z) onto Ã(Z) as Lie algebras over Z, as in Exercise 5 on p150 of [14].

29.11 The complex case

Let us continue with the same notation and hypotheses as in Section 29.4, with k = C. As before, we let Δ be a base for Φ, so that h_{α}, α ∈ Δ, forms a basis for A_{0}, as a vector space over C. If α ∈ Φ, then we let z_{α} be a nonzero element of A_{α}, and ask that z_{α}, α ∈ Φ, together with h_{α}, α ∈ Δ, be a Chevalley basis for A. Let Φ^{+} = Φ^{Δ,+} be the set of positive roots with respect to Δ, which is to say the set of elements of Φ that can be expressed as linear combinations of elements of Δ with nonnegative coefficients.

If α ∈ Φ^{+}, then put

u_{α} = z_{α} − z_{−α}, v_{α} = i (z_{α} + z_{−α}). (29.11.1)

This corresponds to Remark (3) on p52 of [24], and Exercise 7 on p151 of [14]. It is easy to see that

{u_{α}, v_{α} : α ∈ Φ^{+}} ∪ {i h_{α} : α ∈ Δ} (29.11.2)

is a basis for A, as a vector space over C. In particular, the elements of (29.11.2) are linearly independent in A, as a vector space over R. Let B be the linear span of (29.11.2) in A, as a vector space over R.

Let i B be the set of elements of A that can be expressed as i times an element of B, as usual. Observe that

B + (i B) = A, (29.11.3)

because (29.11.2) is a basis for A as a vector space over C. Similarly,

B ∩ (i B) = {0}. (29.11.4)

This means that A may be considered as the complexification of B, as a vector space over R.

Let α ∈ Δ and β ∈ Φ^{+} be given, and observe that

(29.11.5) [i h_{α}, u_{β}]_{A} = i [h_{α}, z_{β}]_{A} − i [h_{α}, z_{−β}]_{A}
= i λ_{α}(β) z_{β} − i λ_{α}(−β) z_{−β} = λ_{α}(β) v_{β}.

Similarly,

(29.11.6) [i h_{α}, v_{β}]_{A} = −[h_{α}, z_{β}]_{A} − [h_{α}, z_{−β}]_{A}
= −λ_{α}(β) z_{β} − λ_{α}(−β) z_{−β} = −λ_{α}(β) u_{β}.

Note that the right sides of (29.11.5) and (29.11.6) are elements of B. If α ∈ Φ^{+}, then

[u_{α}, v_{α}]_{A} = 2 i [z_{α}, z_{−α}]_{A} = 2 i h_{α}. (29.11.7)

This is an element of B, because h_{α} can be expressed as a linear combination of h_{β}, β ∈ Δ, with integer coefficients, as in Section 29.4.

If α, β ∈ Φ and α + β ∈ Φ, then there is a unique c_{α,β} ∈ C such that [z_{α}, z_{β}]_{A} = c_{α,β} z_{α+β}, as before. More precisely, c_{α,β} ∈ R, by (29.9.14).

Let α, β ∈ Φ^{+} be given, with α ≠ β. Note that α + β ≠ 0 in this case. If α + β ∉ Φ, then −α − β ∉ Φ, and

[z_{α}, z_{β}]_{A} = [z_{−α}, z_{−β}]_{A} = 0. (29.11.8)

Similarly, if α − β ∉ Φ, then β − α ∉ Φ, and

[z_{−α}, z_{β}]_{A} = [z_{α}, z_{−β}]_{A} = 0. (29.11.9)

If α + β ∈ Φ, then

[z_{α}, z_{β}]_{A} + [z_{−α}, z_{−β}]_{A} = c_{α,β} z_{α+β} + c_{−α,−β} z_{−α−β}
(29.11.10)  = c_{α,β}z_{α+β} − c_{α,β} z_{−α−β} = c_{α,β} u_{α+β},

using (29.8.7) in the second step. Similarly, if α − β ∈ Φ, then

(29.11.11) [z_{−α}, z_{β}]_{A} + [z_{α}, z_{−β}]_{A} = c_{−α,β} z_{−α+β} + c_{α,−β} z_{α−β}
= c_{α,−β} (z_{α−β} − z_{β−α}).

This is the same as c_{α,−β} u_{α−β} when α − β ∈ Φ^{+}, and −c_{α,−β} u_{β−α} when β − α ∈ Φ^{+}.

If α + β ∈ Φ, then we also have that

i [z_{α}, z_{β}]_{A} − i [z_{−α}, z_{−β}]_{A} = i c_{α,β} z_{α+β} − i c_{−α,−β} z_{−α−β}
(29.11.12)  = i c_{α,β} z_{α+β} + i c_{α,β} z_{−α−β} = c_{α,β} v_{α+β}.

If α − β ∈ Φ, then

(29.11.13) i [z_{−α}, z_{β}]_{A} − i [z_{α}, z_{−β}]_{A} = i c_{−α,β} z_{−α+β}
−i c_{α,−β} z_{α−β} = −i c_{α,−β} (z_{α−β} + z_{β−α}).

This is the same as −c_{α,−β} v_{α−β} when α − β ∈ Φ^{+}, and −c_{α,−β} v_{β−α} when β − α ∈ Φ^{+}.

Using these remarks, one can check that

[u_{α}, u_{β}]_{A}, [u_{α}, v_{β}]_{A}, [v_{α}, v_{β}]_{A} ∈ A (29.11.14)

for every α, β ∈ Φ^{+}. It follows that B is a Lie subalgebra of A, as a Lie algebra over R, as in Remark (3) on p52 of [24], and Exercise 7 on p151 if [14]. Thus A may be considered as the complexification of B as a Lie algebra over R.

29.12 The corresponding Killing forms

Let us continue with the same notation and hypotheses as in the previous section. If x ∈ A, then ad_{A,x} defines a linear mapping from A into itself, as a vector space over C, as in Section 2.4. Similarly, if x ∈ B, then ad_{B,x} defines a linear mapping from B into itself, as a vector space over R. Of course, ad_{B,x} is the same as the restriction of ad_{A,x} to B in this case.

Remember that the Killing form on A is defined by

b_{A}(x, y) = tr_{A}(ad_{A,x} ∘ ad_{A,y}) (29.12.1)

for every x, y ∈ A, where the right side is the trace of ad_{A,x} ∘ ad_{A,y} as a linear mapping from A into itself, as a complex vector space. Similarly, the Killing form on B is defined by

b_{B}(x, y) = tr_{B}(ad_{B,x} ∘ ad_{B,y}) (29.12.2)

for every x, y ∈ B, where the right side is the trace of ad_{B,x} ∘ ad_{B,y} as a linear mapping from B into itself, as a real vector space. If x, y ∈ B, then

b_{A}(x, y) = b_{B}(x, y), (29.12.3)

628 CHAPTER 29. CHEVALLEY’S NORMALIZATION

as in Exercise 7 on p151 of [14]. This uses the fact that a basis for B as a real vector space is also a basis for A as a complex vector space, as in the previous section.

If α, β ∈ Φ and α + β ≠ 0, then b_{A}(z_{α}, z_{β}) = 0, (29.12.4) as in Section 17.3. Note that the Lie subalgebra of A that is denoted B in Section 17.3 may be taken to be A_{0} here. Similarly, if α ∈ Φ and w ∈ A_{0}, then b_{A}(z_{α}, w) = 0, (29.12.5) as before.

If α, β ∈ Φ^{+} and α ≠ β, then it is easy to see that b_{A}(u_{α}, u_{β}) = b_{A}(u_{α}, v_{β}) = b_{A}(v_{α}, v_{β}) = 0, (29.12.6) using (29.12.4). This implies that b_{B}(u_{α}, u_{β}) = b_{B}(u_{α}, v_{β}) = b_{B}(v_{α}, v_{β}) = 0, (29.12.7) by (29.12.3). Similarly, if α ∈ Φ^{+} and β ∈ Δ, then b_{A}(u_{α}, i h_{β}) = b_{A}(v_{α}, i h_{β}) = 0, (29.12.8) by (29.12.5). This implies that b_{B}(u_{α}, i h_{β}) = b_{B}(v_{α}, i h_{β}) = 0, (29.12.9) by (29.12.3).

If α ∈ Φ^{+}, then b_{A}(u_{α}, v_{α}) = i b_{A}(z_{α} − z_{−α}, z_{α} + z_{−α}), (29.12.10) by the definition (29.11.1) of u_{α}, v_{α}. Of course, b_{A}(z_{α}, z_{α}) = b_{A}(z_{−α}, z_{−α}) = 0, (29.12.11) by (29.12.4). This implies that b_{A}(u_{α}, v_{α}) = 0, (29.12.12) because b_{A}(·, ·) is symmetric on A. It follows that b_{B}(u_{α}, v_{α}) = 0, (29.12.13) by (29.12.3).

Remember that for each α ∈ Φ there is a linear functional ϕ_{α} on A_{0} such that ad_{A,w}(z_{α}) = ϕ_{α}(w) for every w ∈ A_{0}. We also have that ϕ_{α}(h_{β}) = λ_{β}(α) ∈ R for every β ∈ Φ, as in (29.4.3). If w ∈ A_{0} ∩ B, then w can be expressed as a linear combination of i h_{β}, β ∈ Δ, with coefficients in R, by construction. This implies that ϕ_{α}(w) ∈ i R (29.12.14)

for every α ∈ Φ and w ∈ A_{0} ∩ B.

If w, w′ ∈ A_{0}, then

b_{A}(w, w′) = ∑_{α∈Φ} ϕ_{α}(w) ϕ_{α}(w′). (29.12.15)

This follows from the fact that the z_{α}’s, α ∈ Φ, together with the h_{β}’s, β ∈ Δ, form a basis for A as a vector space over C. If w, w′ ∈ A_{0} ∩ B, then

b_{B}(w, w′) = ∑_{α∈Φ} ϕ_{α}(w) ϕ_{α}(w′), (29.12.16)

by (29.12.3). In particular,

b_{B}(w, w) = ∑_{α∈Φ} ϕ_{α}(w)^{2} (29.12.17)

for every w ∈ A_{0} ∩ B.

If w ∈ A_{0} ∩ B, then every term in the sum on the right side of (29.12.17) is less than or equal to 0, by (29.12.14). If each of these terms is equal to 0, then w = 0, as in Section 29.4. This implies that b_{B}(·, ·) is negative-definite on A_{0} ∩ B.

Let α ∈ Φ^{+} be given, and observe that

b_{B}([u_{α}, v_{α}]_{A}, i h_{α}) = 2 b_{B}(i h_{α}, i h_{α}), (29.12.18)

by (29.11.7). Remember that h_{α} ≠ 0, as in Section 29.4, and that i h_{α} ∈ B, as in the previous section. Thus

b_{B}([u_{α}, v_{α}]_{A}, i h_{α}) < 0, (29.12.19)

as in the preceding paragraph.

Of course, b_{B}([u_{α}, v_{α}]_{A}, i h_{α}) = −b_{B}(v_{α}, [u_{α}, i h_{α}]_{A}), (29.12.20)

as in Section 7.9. We also have that

b_{B}(v_{α}, [u_{α}, i h_{α}]_{A}) = −λ_{α}(α) b_{B}(v_{α}, v_{α}) = −2 b_{B}(v_{α}, v_{α}), (29.12.21)

by (29.11.5). This implies that

b_{B}(v_{α}, v_{α}) < 0, (29.12.22)

by (29.12.19).

Similarly,

b_{B}([u_{α}, v_{α}]_{A}, i h_{α}) = −b_{B}([v_{α}, u_{α}]_{A}, i h_{α}) = b_{B}(u_{α}, [v_{α}, i h_{α}]_{A}), (29.12.23) as in Section 7.9. Note that

b_{B}(u_{α}, [v_{α}, i h_{α}]_{A}) = λ_{α}(α) b_{B}(u_{α}, u_{α}) = 2 b_{B}(u_{α}, u_{α}), (29.12.24)

by (29.11.6). This means that

b_{B}(u_{α}, u_{α}) < 0, (29.12.25)

because of (29.12.19). One can check more directly that b_{B}(u_{α}, u_{α}) = b_{B}(v_{α}, v_{α}) too.

It follows that b_{B}(·, ·) is negative-definite on B, as in Remark (3) on p52 of [24], and Exercise 7 on p151 of [14].

29.13 Using other commutative rings

Let us return to the same notation and hypotheses as in Section 29.4. Thus Δ is a base for Φ, and we let z_{α} be a nonzero element of A_{α} for every α ∈ Φ. Suppose that z_{α}, α ∈ Φ, together with h_{α}, α ∈ Δ, is a Chevalley basis for A, as in Section 29.8. Let A(Z) be the collection of elements of A that can be expressed as a linear combination of the z_{α}’s, α ∈ Φ, and h_{α}’s, α ∈ Δ, with integer coefficients. This is a Lie subalgebra of A, as a Lie algebra over Z, as in Section 29.9.

Let k_{1} be a commutative ring with a multiplicative identity element. Let us define A(k_{1}) initially as a free module over k_{1} with basis consisting of z_{α}, α ∈ Φ, and h_{α}, α ∈ Δ. These basis elements may also be denoted z_{α,k_{1}}, α ∈ Φ, and h_{α,k_{1}}, α ∈ Δ, to be more precise. One may consider A(k_{1}) to be the same as the tensor product of A(Z) and k_{1}, as modules over Z. One can use multiplication on k_{1} to define scalar multiplication by elements of k_{1} on this tensor product over Z, to get a module over k_{1}.

If n is a positive integer, then we can consider the sum of n 1’s in k_{1}, where 1 is the multiplicative identity element in k_{1}. This leads to a natural ring homomorphism from Z into k_{1}, as usual. Using this, we get a mapping from A(Z) into A(k_{1}) that is linear over Z.

It is easy to define the Lie bracket [·, ·]_{A(k_{1})} initially as a mapping from A(k_{1}) × A(k_{1}) into A(k_{1}) that is bilinear over k_{1}. The brackets of basis elements can be defined using the corresponding brackets in A(Z), and the natural mapping from A(Z) into A(k_{1}) described in the preceding paragraph. This can be extended in a unique way to a mapping from A(k_{1}) × A(k_{1}) into A(k_{1}) that is bilinear over k.

Clearly [·, ·]_{A(k_{1})} is antisymmetric on A(k_{1}), because [·, ·]_{A} is antisymmetric on A. One can check that [a, a]_{A(k_{1})} = 0 for every a ∈ A(k_{1}), using the antisymmetry of [·, ·]_{A(k_{1})} on A(k_{1}), and the fact that this property holds when a is a basis element, because of the corresponding property of [·, ·]_{A} on A. Similarly, [·, ·]_{A(k_{1})} satisfies the Jacobi identity on A(k_{1}), because [·, ·]_{A} satisfies the Jacobi identity on A. Thus A(k_{1}) is a Lie algebra over k_{1} with respect to [·, ·]_{A(k_{1})}. This may be called a Chevalley algebra, as on p149 of [14]. This is also related to Remark (1) on p51 of [24].

29.14. SOME MAPPINGS ON A(Z) 631

Of course, A(k_{1}) may be considered as a Lie algebra over Z as well. The natural mapping from A(Z) into A(k_{1}) mentioned earlier is a homomorphism as a mapping between Lie algebras over Z.

Remember that A(Z) does not depend on the base Δ for Φ, as in Section 29.9. This used the fact that for each γ ∈ Φ, h_{γ} can be expressed as a linear combination of h_{α}, α ∈ Δ, with integer coefficients, as in Section 29.4. Similarly, if α ∈ Δ, then h_{α} can be expressed as a linear combination of h_{β}, with β in any other base for Φ, and with integer coefficients. One can use this to check that A(k_{1}) does not depend on Δ, up to isomorphism, as a Lie algebra over k_{1}. Another base for Φ essentially only leads to another basis for A(k_{1}) as a free module over k_{1}. If A(k_{1}) is considered as the tensor product of A(Z) and k_{1} over Z, then this clearly does not depend on Δ, because A(Z) does not depend on Δ.

Let z̃_{α} be another nonzero element of A_{α} for each α ∈ Φ, and suppose that z̃_{α}, α ∈ Φ, together with h_{α}, α ∈ Δ, is another Chevalley basis for A. Under these conditions, there is a diagonal automorphism τ of A such that for every α ∈ Φ, z̃_{α} = ρ_{α} τ(z_{α}) with ρ_{α} = ±1, as in Section 29.10. If Ã(Z) is the collection of elements of A that can be expressed as linear combinations of the z̃_{α}’s, α ∈ Φ, and h_{α}’s, α ∈ Δ, then Ã(Z) = τ(A(Z)), as before. Let Ã(k_{1}) be the Lie algebra over k_{1} defined using the z̃_{α}’s in the same way as before. One can check that τ leads to an isomorphism from A(k_{1}) onto Ã(k_{1}), as Lie algebras over k_{1}. If A(k_{1}), Ã(k_{1}) are considered as tensor products of A(Z), Ã(Z), respectively, and k_{1} over Z, then this follows from the fact that the restriction of τ to A(Z) defines an isomorphism from A(Z) onto Ã(Z), as Lie algebras over Z. This corresponds to a remark on p149 of [14].

29.14 Some mappings on A(Z)

Let us continue with the same notation and hypotheses as in Section 29.4 again. Remember that Δ is a base for Φ, and let z_{α} be a nonzero element of A_{α} for every α ∈ Φ. Suppose that z_{α}, α ∈ Φ, together with h_{α}, α ∈ Δ, is a Chevalley basis for A, as in Section 29.8, and let A(Z) be the collection of elements of A that can be expressed as a linear combination of the z_{α}’s, α ∈ Φ, and h_{α}’s, α ∈ Δ, with integer coefficients. This is a Lie subalgebra of A, as a Lie algebra over Z, as in Section 29.9.

Let α ∈ Φ and a positive integer m be given. We would like to check that (1/m!) (ad_{z_{α}})^{m} maps A(Z) into itself, as in the proposition on p149 of [14]. It suffices to show that (1/m!) (ad_{z_{α}})^{m}(h_{β}) ∈ A(Z) (29.14.1)

for every β ∈ Δ, and that

(1/m!) (ad_{z_{α}})^{m}(z_{β}) ∈ A(Z) (29.14.2)

for every β ∈ Φ.

If β ∈ Δ, then

ad_{z_{α}}(h_{β}) = [z_{α}, h_{β}]_{A} = −λ_{β}(α) z_{α}, (29.14.3)

which is an element of A(Z). It follows that (ad_{z_{α}})^{m}(h_{β}) = 0 when m ≥ 2, so that (29.14.1) holds for every m ≥ 1.

Note that (ad_{z_{α}})^{m}(z_{α}) = 0 for every m ≥ 1, so that (29.14.2) holds trivially when α = β. Remember that ad_{z_{α}}(z_{−α}) = [z_{α}, z_{−α}]_{A} = h_{α}, as in Section 29.8. This implies that (29.14.2) holds when m = 1 and β = −α, because h_{α} ∈ A(Z), as in Section 29.9. It follows that

(1/2!) (ad_{z_{α}})^{2}(z_{−α}) = (1/2) [z_{α}, h_{α}]_{A} = −(λ_{α}(α)/2) z_{α} = −z_{α}, (29.14.4)

so that (29.14.2) holds when m = 2 and β = −α. If m ≥ 3, then we get that (ad_{z_{α}})^{m}(z_{−α}) = 0, so that (29.14.2) holds when β = −α.

Suppose now that β ∈ Φ, β ≠ ±α. Thus α, β are linearly independent in V, because Φ is reduced, by hypothesis. If γ ∈ Φ is not proportional to α, then let r(γ), q(γ) be the largest nonnegative integers such that

γ − r(γ) α, γ + q(γ) α ∈ Φ, (29.14.5)

as in Section 20.5. Remember that γ +j α ∈ Φ when j is an integer that satisfies −r(γ) ≤ j ≤ q(γ), as before. If α + γ ∈ Φ, then

[z_{α}, z_{γ}]_{A} = c_{α,γ} z_{α+γ} = ±(r(γ) + 1) z_{α+γ}, (29.14.6)

as in Sections 29.8 and 29.9.

Let j be a nonnegative integer less than or equal to q(β), and observe that

r(β + j α) = r(β) + j. (29.14.7)

If 0 ≤ j < q(β), then β + j α and β + (j + 1) α are elements of Φ, and

(29.14.8)  [z_{α}, z_{β+j α}]_{A} = ±(r(β + j α) + 1) z_{β+(j+1) α}
= ±(r(β) + j + 1) z_{β+(j+1) α}.

If m is a positive integer less than or equal to q(β), then we get that

(ad_{z_{α}})^{m}(z_{β}) = ±( ∏_{j=0}^{m−1}(r(β) + j + 1)) z_{β+m α}. (29.14.9)

This means that

(1/m!) (ad_{z_{α}})^{m}(z_{β}) = ±\binom{r(β) + m}{m} z_{β+m α}, (29.14.10)

where \binom{r(β)+m}{m} is the usual binomial coefficient. It follows that (29.14.2) holds in this case, because the binomial coefficient is an integer.

29.15. SOME RELATED MAPPINGS ON A(K_{1}) 633

Note that (ad_{z_{α}})^{q(β)+1}(z_{β}) = 0, because β + (q(β) + 1) α ∉ Φ. This implies that (ad_{z_{α}})^{m}(z_{β}) = 0 when m ≥ q(β) + 1, so that (29.14.2) holds for every m ≥ 1, as desired.

If c ∈ Z, then

exp(c ad_{z_{α}}) = ∑_{m=0}^{∞}(c^{m}/m!) (ad_{z_{α}})^{m} (29.14.11)

defines a Lie algebra automorphism of A, as in Section 14.11, because ad_{z_{α}} is nilpotent on A. We also have that

(exp(c ad_{z_{α}}))(A(Z)) ⊆ A(Z), (29.14.12)

because (1/m!) (ad_{z_{α}})^{m} maps A(Z) into itself for every m, as before. This implies that (exp(c ad_{z_{α}}))(A(Z)) = A(Z), (29.14.13)

by applying the previous statement to −c. This corresponds to some remarks on p150 of [14], and is related to Remark (1) on p51 of [24].

29.15 Some related mappings on A(k_{1})

We continue with the same notation and hypotheses as in the previous section. Let α ∈ Φ be given again, and for each positive integer m, put

Z_{α,m} = (1/m!) (ad_{z_{α}})^{m}. (29.15.1)

This defines a linear mapping from A into itself, as a vector space over k, which may be interpreted as the identity mapping on A when m = 0. Remember that

Z_{α,m}(A(Z)) ⊆ A(Z) (29.15.2)

for every m ≥ 0, as before.

If l, m are nonnegative integers, then

\binom{l + m}{l} Z_{α,l+m} = Z_{α,l} ∘ Z_{α,m}, (29.15.3)

by construction. If n is a nonnegative integer and a, b ∈ A, then

(ad_{z_{α}})^{n}([a, b]_{A}) = ∑_{j=0}^{n} \binom{n}{j} [(ad_{z_{α}})^{j}(a), (ad_{z_{α}})^{n−j}(b)]_{A}, (29.15.4)

because ad_{z_{α}} is a derivation on A, as in Section 14.10. Equivalently, this means that

Z_{α,n}([a, b]_{A}) = ∑_{j=0}^{n}[Z_{α,j}(a), Z_{α,n−j}(b)]_{A}. (29.15.5)

Let k_{1} be a commutative ring with a multiplicative identity element, and let A(k_{1}) be as in Section 29.13. Note that every mapping from A(Z) into itself that is linear over Z leads to a linear mapping from A(k_{1}) into itself that is linear over k_{1}. This can be seen in terms of the bases for A(Z), A(k_{1}) as free modules over Z, k_{1}, respectively, described earlier, or by considering A(k_{1}) as the tensor product of A(Z) and k_{1}, as modules over Z. This correspondence between mappings on A(Z) and on A(k_{1}) is linear over Z, and preserves compositions.

If m is a nonnegative integer, then let Z_{α,m,k_{1}} be the mapping on A(k_{1}) that corresponds to the restriction of Z_{α,m} to A(Z) in this way. If l is another nonnegative integer, then

\binom{l + m}{l} Z_{α,l+m,k_{1}} = Z_{α,l,k_{1}} ∘ Z_{α,m,k_{1}}, (29.15.6)

by (29.15.3). Similarly, if n is a nonnegative integer and a, b ∈ A(k_{1}), then

Z_{α,n,k_{1}}([a, b]_{A(k_{1})}) = ∑_{j=0}^{n}[Z_{α,j,k_{1}}(a), Z_{α,n−j,k_{1}}(b)]_{A(k_{1})}, (29.15.7)

by (29.15.5). More precisely, it suffices to verify this when a, b are basis elements of A(k_{1}), which can be obtained from the previous case.

If t ∈ k_{1}, then put

E_{α,k_{1}}(t) = ∑_{l=0}^{∞} t^{l} Z_{α,l,k_{1}}. (29.15.8)

Of course, Z_{α,l} = 0 for all but finitely many l ≥ 0, because z_{α} is ad-nilpotent in A. This implies that Z_{α,l,k_{1}} = 0 for the same l ≥ 0, so that the right side of (29.15.8) reduces to a finite sum. Thus (29.15.8) defines E_{α,k_{1}}(t) as a linear mapping from A(k_{1}) into itself.

If r ∈ k_{1} too, then

E_{α,k_{1}}(r + t) = ∑_{n=0}^{∞}(r + t)^{n} Z_{α,n,k_{1}} = ∑_{n=0}^{∞} ∑_{l=0}^{n} \binom{n}{l}r^{l} t^{n−l} Z_{α,n,k_{1}}, (29.15.9)

by the binomial theorem. This implies that

E_{α,k_{1}}(r + t) = ∑_{n=0}^{∞} ∑_{l=0}^{n} r^{l} t^{n−l} Z_{α,l,k_{1}} ∘ Z_{α,n−l,k_{1}}, (29.15.10)

by (29.15.6). It follows that

E_{α,k_{1}}(r + t) = E_{α,k_{1}}(r) ∘ E_{α,k_{1}}(t), (29.15.11)

because the right side of (29.15.10) corresponds to the Cauchy product of the series that defined E_{α,k_{1}}(r) and E_{α,k_{1}}(t). In particular, E_{α,k_{1}}(t) is invertible on A(k_{1}), with E_{α,k_{1}}(t)^{−1} = E_{α,k_{1}}(−t), (29.15.12)

because E_{α,k_{1}}(0) is the identity mapping on A(k_{1}).

If a, b ∈ A(k_{1}), then

(E_{α,k_{1}}(t))([a, b]_{A(k_{1})}) = ∑_{n=0}^{∞} t^{n} Z_{α,n,k_{1}}([a, b]_{A(k_{1})})

(29.15.13)  = ∑_{n=0}^{∞} t^{n} ∑_{l=0}^{n}[Z_{α,l,k_{1}}(a), Z_{α,n−l,k_{1}}]_{A(k_{1})},

using (29.15.7) in the second step. This implies that

(E_{α,k_{1}}(t))([a, b]_{A(k_{1})}) = ∑_{n=0}^{∞} ∑_{l=0}^{n}[t^{l} Z_{α,l,k_{1}}(a), t^{n−l} Z_{α,n−l,k_{1}}(b)]_{A(k_{1})}
(29.15.14)  = [(E_{α,k_{1}}(t))(a), (E_{α,k_{1}}(t))(b)]_{A(k_{1})},

because the right side is equal to the Cauchy product with respect to [·, ·]_{A(k_{1})} of the series that define (E_{α,k_{1}}(t))(a) and (E_{α,k_{1}}(t))(b). Thus E_{α,k_{1}}(t) is an automorphism of A(k_{1}), as a Lie algebra over k_{1}.

This corresponds to some remarks on p150 of [14], and to Exercise 11 on p151 of [14]. This is also related to Remark (1) on p51 of [24].

Roots and abstract weights

30.1 Abstract weights

Let V be a vector space over the real numbers of positive finite dimension, and let Φ be a root system in V. If α ∈ Φ, then let σ_{α} be the unique symmetry on V with vector α that maps Φ onto itself, as usual. Thus σ_{α} can be expressed as σ_{α}(v) = v − λ_{α}(v) α, where λ_{α} is a linear functional on V with λ_{α}(α) = 2.

As before, we let Θ = Θ_{Φ} be the subgroup of V, as a commutative group with respect to addition, generated by Φ. Equivalently, this is the subgroup of V generated by any base for Φ. This may be called the root lattice in V associated to Φ, as on p67 of [14].

Put (30.1.1) Υ = Υ_{Φ} = {v ∈ V : λ_{α}(v) ∈ Z for every α ∈ Φ}.

The elements of Υ may be called (abstract) weights with respect to Φ, as on p67 of [14]. This corresponds to P on p63 of [24], where the root system is associated to a finite-dimensional semisimple Lie algebra. Note that Υ is a subgroup of V, as a commutative group with respect to addition. We also have that

Θ ⊆ Υ, (30.1.2)

by the definition of a root system.

Let V′ be the dual space of linear functionals on V, and remember that

Φ′ = {λ_{α} : α ∈ Φ} (30.1.3)

is a root system in V′, as in Section 19.8. If α ∈ Φ and 2 α ∈ Φ, then σ_{2 α} = σ_{α}, and λ_{2 α} = λ_{α}/2, as in Section 19.13. Let Δ be a base for Φ, and put

Δ_{1} = {β ∈ Δ : 2 β ∉ Φ}, Δ_{2} = {β ∈ Δ : 2 β ∈ Φ}. (30.1.4)

Note that Δ_{2} = ∅ when Φ is reduced, and put

Δ′_{1} = {λ_{β} : β ∈ Δ_{1}}, Δ_{2}^{∗} = {λ_{β}/2 : β ∈ Δ_{2}}. (30.1.5)

Remember that Δ′_{1} ∪ Δ_{2}^{∗} is a base for Φ′, as in Section 19.13.

It follows that

(30.1.6)  Υ = {v ∈ V : λ_{β}(v) ∈ Z for every β ∈ Δ_{1},
and λ_{β}(v)/2 ∈ Z for every β ∈ Δ_{2}}.

Equivalently,

(30.1.7) Υ = {v ∈ V : λ(v) ∈ Z for every λ ∈ Δ′_{1} ∪ Δ_{2}^{∗}}.

In particular, Δ′_{1} ∪ Δ_{2}^{∗} is a basis for V′, as a vector space over R. If v ∈ V, then λ ↦ λ(v) (30.1.8)

defines a real-valued function on Δ′_{1} ∪ Δ_{2}^{∗}. The mapping from v ∈ V to (30.1.8) defines a one-to-one linear mapping from V onto the space of real-valued functions on Δ′_{1} ∪ Δ_{2}^{∗}. Thus, for each μ ∈ Δ′_{1} ∪ Δ_{2}^{∗}, there is a unique v_{μ} ∈ V such that

(30.1.9)  λ(v_{μ}) = 1 when λ = μ
= 0 for every λ ∈ Δ′_{1} ∪ Δ_{2}^{∗} with λ ≠ μ.

The v_{μ}’s, μ ∈ Δ′_{1} ∪ Δ_{2}^{∗}, form a basis for V, which is dual to Δ′_{1} ∪ Δ_{2}^{∗}.

Using (30.1.7), we get that v_{λ} ∈ Υ (30.1.10)

for every λ ∈ Δ′_{1} ∪ Δ′_{2}. The v_{λ}’s, λ ∈ Δ′_{1} ∪ Δ_{2}^{∗}, are called the fundamental (dominant) weights with respect to Δ, as on p67 of [14], and Remark (2) on p62 of [24]. Note that Υ consists exactly of linear combinations of the v_{λ}’s, λ ∈ Δ′_{1} ∪ Δ_{2}^{∗}, with integer coefficients.

30.2 Some remarks about lattices

Let V be a vector space over the real numbers of positive finite dimension n, and let u_{1}, …, u_{n} and w_{1}, …, w_{n} be bases for V. Also let A and B be the subgroups of V, as a commutative group with respect to addition, generated by the u_{j}’s and w_{j}’s, respectively. Equivalently, A and B consist of the elements of V that can be expressed as linear combinations of the u_{j}’s and w_{j}’s, respectively, with integer coefficients.

Suppose that B ⊆ A. (30.2.1)

This means that for each l = 1, …, n, w_{l} can be expressed as a linear combination of the u_{j}’s with integer coefficients. Let T be the linear mapping from V onto itself with T (u_{l}) = w_{l} (30.2.2) for every l = 1, …, n. Note that T is invertible, so that det T ≠ 0. We also have that T (A) = B, (30.2.3)

by construction.

Of course, T corresponds to an n × n matrix, with respect to the basis u_{1}, …, u_{n} of V. The entries of this matrix are obtained by expressing w_{l} as a linear combination of the u_{j}’s for each l = 1, …, n. In particular, the entries of the matrix associated to T are integers in this case. This implies that

det T ∈ Z. (30.2.4)

Using Cramer’s rule, we get that (det T) T^{−1} corresponds to a matrix with integer entries as well. This means that

(det T) T^{−1}(A) ⊆ A. (30.2.5)

It follows that (det T) A ⊆ T (A) = B. (30.2.6)

This implies that the quotient group A/B has only finitely many elements.

Because V is isomorphic to R^{n} as a vector space over R, one can define n-dimensional volumes on V as on R^{n}. These volumes are determined up to multiplication by a positive real number, depending on the isomorphism with R^{n}. These volumes are invariant under translations on V, and are changed by T by a factor of | det T |, because of the analogous properties on R^{n}.

It is well known that #(A/B) = | det T |, (30.2.7)

where the left side is the number of elements of A/B. To see this, one can consider the quotients V/A and V/B of V by A and B, respectively. There is a natural quotient mapping from V/B onto V/A, because of (30.2.1). The natural quotient mapping from V onto V/A is the same as the composition of the natural quotient mapping from V onto V/B with the quotient mapping from V/B onto V/A. One can define n-dimensional volumes on V/A and V/B that correspond locally to volumes on V under the quotient mappings. In particular, volumes on V/A correspond locally to volumes on V/B under the natural quotient mapping. This implies that the ratio of the volumes of V/B and V/A is equal to #(A/B).

There is also a natural one-to-one mapping from V/A onto V/B induced by T. Because volumes on V are changed by T by a factor of | det T |, volumes on the quotients are changed in the same way by the mapping induced by T, where the volumes on the quotients correspond locally to volumes on V as before. This means that the ratio of the volumes of V/B and V/A is equal to | det T |, which implies (30.2.7).

Let us return now to the situation considered in the previous section. Note that Θ is generated, as a subgroup of V, by Δ, which is a basis for V. We also have that Υ is generated, as a subgroup of V, by the v_{λ}’s, λ ∈ Δ′_{1} ∪ Δ_{2}^{∗}, which form a basis for V too. Remember that Θ ⊆ Υ, so that the quotient Υ/Θ is defined as a commutative group, and has only finitely many elements, as before. This is known as the fundamental group of Φ, as on p68 of [14].

Suppose that Φ is reduced, as a root system in V. This means that Δ_{1} = Δ and Δ_{2} = ∅, and we put Δ′ = {λ_{β} : β ∈ Δ}, (30.2.8)

which is the same as Δ′_{1} in this case. If β ∈ Δ, then there is a unique v_{β} = v_{λ_{β}} ∈ V such that

(30.2.9)  λ_{α}(v_{β}) = 1 when α = β
= 0 for every α ∈ Δ with α ≠ β,

as in (30.1.9). The v_{β}’s, β ∈ Δ, form a basis for V, which is dual to Δ′, as a basis for V′, as before. If v ∈ V, then

v = ∑_{β∈Δ} λ_{β}(v) v_{β}. (30.2.10)

In particular, if α ∈ Δ, then

α = ∑_{β∈Δ} λ_{β}(α) v_{β}. (30.2.11)

Remember that n(α, β) = λ_{β}(α), α, β ∈ Δ, is the Cartan matrix of Φ with respect to Δ, as in Section 20.2. The determinant of the Cartan matrix is a positive integer, as in Section 21.15. In fact,

#(Υ/Θ) = det(n(α, β))_{α,β∈Δ}, (30.2.12)

as in (30.2.7). This corresponds to some remarks on p68 of [14].

30.3 Related positivity conditions

Let us go back to the notation and hypotheses in Section 30.1. In particular, Δ is a base for Φ, and we put

(30.3.1) Υ^{+} = Υ_{Φ,Δ}^{+} = {v ∈ Υ : λ_{α}(v) ≥ 0 for every α ∈ Δ}.

The elements of Υ^{+} are said to be dominant weights with respect to Δ, as on p67 of [14]. If v ∈ Υ satisfies

(30.3.2) λ_{α}(v) > 0 for every α ∈ Δ,

then v is said to be strongly dominant.

Equivalently, it is easy to see that

(30.3.3) Υ^{+} = {v ∈ Υ : λ(v) ≥ 0 for every λ ∈ Δ′_{1} ∪ Δ_{2}^{∗}}.

Similarly, (30.3.2) holds if and only if

(30.3.4) λ(v) > 0 for every λ ∈ Δ′_{1} ∪ Δ_{2}^{∗}.

If μ ∈ Δ′_{1} ∪ Δ_{2}^{∗} and v_{μ} ∈ Υ is as in (30.1.9), then we get that

v_{μ} ∈ Υ^{+}, (30.3.5)

by (30.3.3).

Let Φ^{+} = Φ^{Δ,+} be the set of α ∈ Φ that can be expressed as linear combinations of elements of Δ with nonnegative coefficients, as usual. Put

(Φ′)^{+} = {λ_{α} : α ∈ Φ^{+}}. (30.3.6)

Remember that Δ′_{1} ∪ Δ_{2}^{∗} is a base for Φ′, as a root system in V′. One can check directly that Δ′_{1} ∪ Δ_{2}^{∗} ⊆ (Φ′)^{+}, (30.3.7)

because Δ ⊆ Φ^{+}.

In fact, (Φ′)^{+} consists exactly of the elements of Φ′ that can be expressed as linear combinations of elements of Δ′_{1} ∪ Δ_{2}^{∗} with nonnegative coefficients. This is basically implicit in the proof of the fact that Δ′_{1} ∪ Δ_{2}^{∗} is a base for Φ′, as in Section 19.13. More precisely, in the proof an inner product on V that is invariant under the Weyl group of Φ was used to identify V and V′. Using this identification, Δ′_{1} ∪ Δ_{2}^{∗} was shown to correspond to the construction of a base for a root system in Section 19.11. The set of positive roots with respect to this base is also determined in this construction, as before. In this case, the set of positive roots corresponds exactly to (Φ′)^{+}. This is because of the way that the linear functional τ is defined in the proof in Section 19.13.

If v ∈ Υ^{+}, then it follows that

(30.3.8) λ_{α}(v) ≥ 0 for every α ∈ Φ^{+}.

This corresponds to the statement following Theorem 3 on p60 of [24]. Similarly, if v ∈ Υ is strongly dominant, then

(30.3.9) λ_{α}(v) > 0 for every α ∈ Φ^{+}.

Let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ. If α ∈ Φ, then σ_{α} is the reflection on V associated to α with respect to (·, ·), so that λ_{α}(v) = 2 (v, α) (α, α)^{−1}, as before. Under these conditions,

(30.3.10) Υ^{+} = {v ∈ Υ : (v, α) ≥ 0 for every α ∈ Δ}.

Similarly, v ∈ Υ is strongly dominant if and only if

(30.3.11) (v, α) > 0 for every α ∈ Δ.

This corresponds to some remarks about Weyl chambers on p67 of [14].

If v ∈ V, then (30.3.8) holds if and only if

(30.3.12) (v, α) ≥ 0 for every α ∈ Φ^{+}.

In particular, this holds when v ∈ Υ^{+}, as before. Alternatively, (30.3.12) holds when v ∈ V satisfies (v, α) ≥ 0 for every α ∈ Δ, by definition of Φ^{+}.

Similarly, v ∈ V satisfies (30.3.9) if and only if

(30.3.13) (v, α) > 0 for every α ∈ Φ^{+}.

This holds when v ∈ Υ is strongly dominant, as before. Alternatively, (30.3.11) implies (30.3.13), by definition of Φ^{+}.

30.4 Automorphisms and Υ

Let us continue with the notation and hypotheses in Section 30.1 again. Remember that a one-to-one linear mapping T from V onto itself is an automorphism of Φ when T (Φ) = Φ, and that Aut(Φ) denotes the group of these automorphisms. If T ∈ Aut(Φ), then T (Θ) = Θ. (30.4.1)

In this case, λ_{T (α)}(T (v)) = λ_{α}(v) (30.4.2)

for every α ∈ Φ and v ∈ V, as in Section 19.5. One can use this to check that

T (Υ) = Υ. (30.4.3)

Remember that the quotient group Υ/Θ has only finitely many elements, as in Section 30.2. If T ∈ Aut(Φ), then T induces an automorphism T̂ of Υ/Θ, by (30.4.1) and (30.4.3).

If α ∈ Φ and v ∈ Υ, then

σ_{α}(v) − v = −λ_{α}(v) α ∈ Θ. (30.4.4)

Of course, σ_{α} ∈ Aut(Φ), so that σ_{α} induces an automorphism σ̂_{α} on Υ/Θ, as in the preceding paragraph. It follows from (30.4.4) that σ̂_{α} is the identity mapping on Υ/Θ. If σ is any element of the Weyl group of Φ, then σ ∈ Aut(Φ), and we get that σ̂ is the identity mapping on Υ/Θ. Equivalently, this means that σ(v) − v ∈ Θ (30.4.5)

for every v ∈ Υ.

Let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ. Remember that Δ is a base for Φ, and suppose for the rest of the section that Φ is reduced, as a root system in V. If v ∈ V, then there is an element σ of the Weyl group of Φ such that

(30.4.6) (σ(v), α) ≥ 0 for every α ∈ Δ, as in Section 20.9. If v ∈ Υ, then σ(v) ∈ Υ, and it follows that

σ(v) ∈ Υ^{+}, (30.4.7)

as in(30.3.10). This corresponds to part of Lemma A on p68 of [14].

As in Section 21.12, a linear functional μ on V is said to be nonnegative with respect to Δ if (30.4.8) μ(α) ≥ 0 for every α ∈ Δ.

If v ∈ V, then put μ_{v}(w) = (v, w) (30.4.9)

for every w ∈ V, which defines a linear functional on V. If σ is in the Weyl group of Φ, then μ_{σ(v)}(w) = (σ(v), w) = (v, σ^{−1}(w)) (30.4.10)

for every w ∈ V, which is to say that

μ_{σ(v)} = μ ∘ σ^{−1}. (30.4.11)

Suppose that μ_{v} is nonnegative with respect to Δ, and that μ_{σ(v)} is nonnegative with respect to Δ for some σ in the Weyl group of Φ. Under these conditions, μ_{σ(v)} = μ_{v}, (30.4.12)

as in Section 21.12. Equivalently, this means that

σ(v) = v. (30.4.13)

In particular, this holds when v ∈ Υ^{+} and σ(v) ∈ Υ^{+} for some σ in the Weyl group of Φ. This is another part of Lemma A on p68 of [14].

Suppose again that μ_{v} is nonnegative with respect to Δ for some v ∈ V, and that μ_{σ(v)} is nonnegative with respect to Δ for some σ in the Weyl group of Φ. As in Section 21.12, this implies that σ can be expressed as the composition of finitely many reflections σ_{α} with α ∈ Δ and

μ_{v} = μ_{v} ∘ σ_{α}. (30.4.14)

Observe that (30.4.14) implies that

μ_{v}(α) = −μ_{v}(α), (30.4.15)

so that μ_{v}(α) = 0. This is not possible when μ_{v} is strictly positive with respect to Δ, in the sense that

(30.4.16) μ_{v}(α) = (v, α) > 0 for every α ∈ Δ.

In this case, we get that σ is the identity mapping on V.

If v ∈ Υ is strongly dominant and σ(v) ∈ Υ^{+} for some σ in the Weyl group of Φ, then it follows that σ is the identity mapping on V. This is another part of Lemma A on p68 of [14].

30.5. SOME PARTIAL ORDERINGS ON V 643

30.5 Some partial orderings on V

Let us continue with the usual notation and hypotheses in Section 30.1. Thus Δ is a base for Φ, and a basis for V in particular. Consider the binary relation ⪯=⪯_{Δ} defined on V by putting v ⪯ w when v, w ∈ V and either v = w, or w − v can be expressed as a finite sum of elements of Δ. This defines a partial ordering on V, as on p47 of [14].

Similarly, consider the binary relation ⪯^{R}=⪯_{Δ}^{R} defined on V by putting v ⪯^{R} w when v, w ∈ V and w − v can be expressed as a linear combination of elements of Δ with nonnegative coefficients. This defines a partial ordering on V too, as mentioned in Exercise 14 on p55 of [14]. Of course,

(30.5.1) v ⪯ w implies v ⪯^{R} w

for every v, w ∈ V. Conversely, if v, w ∈ V, v ⪯^{R} w, and w−v ∈ Θ, then v ⪯ w. Let v ∈ V and α ∈ Δ be given, and note that

v − σ_{α}(v) = λ_{α}(v) α. (30.5.2)

In this case, σ_{α}(v) ⪯^{R} v when λ_{α}(v) ≥ 0, and v ⪯^{R} σ_{α}(v) when λ_{α}(v) ≤ 0. If v ∈ Υ, then we get that σ_{α}(v) ⪯ v when λ_{α}(v) ≥ 0, and v ⪯ σ_{α}(v) when λ_{α}(v) ≤ 0.

Let v ∈ V be given again, and suppose that σ is an element of the Weyl group of Φ such that σ(v) is maximal with respect to ⪯^{R} among the images of v under elements of the Weyl group. If α ∈ Δ and σ(v) ⪯^{R} σ_{α}(σ(v)), then it follows that σ_{α}(σ(v)) = σ(v). This means that

(30.5.3) λ_{α}(σ(v)) ≥ 0 for every α ∈ Δ,

as in the preceding paragraph.

Let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ. Observe that (30.5.3) is the same as saying that (30.4.6) holds.

Of course, there are only finitely many images of v in V under elements of the Weyl group of Φ, because the Weyl group has only finitely many elements. This implies that there is an element σ of the Weyl group such that σ(v) is maximal with respect to ⪯^{R}, because every nonempty finite partially-ordered set has a maximal element. This gives another way to get the existence of an element σ of the Weyl group such that (30.4.6) holds, as in Exercise 14 on p55 of [14].

Let σ^{0} be any element of the Weyl group of Φ. Consider the set of elements σ^{1} of the Weyl group such that

σ^{0}(v) ⪯^{R} σ^{1}(v). (30.5.4)

Let σ be an element of this set such that σ(v) is maximal with respect to ⪯^{R} among the images of v under the elements of this set, which exists as before. Thus σ^{0}(v) ⪯^{R} σ(v), (30.5.5) and it is easy to see that σ(v) is maximal among all images of v under elements of the Weyl group with respect to ⪯^{R}. This implies that (30.4.6) holds, as before.

Suppose from now on in this section that Φ is reduced, as a root system in V. Suppose also that v ∈ V satisfies

(30.5.6) (v, α) ≥ 0 for every α ∈ Δ.

This is the same as saying that the linear functional μ_{v} on V associated to v as in (30.4.9) is nonnegative with respect to Δ. If σ is an element of the Weyl group that satisfies (30.4.6), then μ_{σ(v)} is nonnegative with respect to Δ too. This implies that σ(v) = v, as before.

If σ^{0} is any element of the Weyl group, then there is an element σ of the Weyl group that satisfies (30.4.6) and (30.5.5). Under these conditions, we get that σ^{0}(v) ⪯^{R} v, (30.5.7)

because σ(v) = v, as in the preceding paragraph.

Suppose that v is a dominant weight, so that v ∈ Υ and (30.5.6) holds, as in (30.3.10). Note that v − σ^{0}(v) ∈ Θ, as in (30.4.5). In this case, (30.5.7) implies that σ^{0}(v) ⪯ v. (30.5.8)

This corresponds to another part of Lemma A on p68 of [14].

30.6 Some more properties of Υ

Let us continue with the usual notation and hypotheses in Section 30.1 again. In particular, Δ is a base for Φ, so that Υ^{+} can be defined as in Section 30.3. Let ⪯ be defined on V as in the previous section, and let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ.

Let v ∈ Υ^{+} be given, and suppose that

(30.6.1) w ∈ Υ^{+} satisfies w ⪯ v.

This implies that v + w ∈ Υ^{+}, and that v − w can be expressed as a sum of elements of Δ. It follows that

(v + w, v − w) ≥ 0, (30.6.2)

because (v + w, α) ≥ 0 for every α ∈ Δ. This means that

(v, v) − (w, w) ≥ 0, (30.6.3)

which is to say that (w, w) ≤ (v, v). Using this, one can check that there are only finitely many w as in (30.6.1), which is Lemma B at the top of p70 of [14].

Let Φ^{+} be the set of elements of Φ that can be expressed as linear combinations of elements of Δ with nonnegative coefficients, as before. Suppose from now on in this section that Φ is reduced as a root system in V, and put

ρ = \frac{1}{2} ∑_{γ∈Φ^{+}} γ, (30.6.4)

as in Section 19.12. If α ∈ Δ, then

σ_{α}(ρ) = ρ − α, (30.6.5)

as before. Equivalently, this means that

λ_{α}(ρ) = 1 (30.6.6)

for every α ∈ Δ. This shows that

ρ ∈ Υ^{+}, (30.6.7)

and more precisely that ρ is strongly dominant with respect to Δ, as in Lemma A around the middle of p70 of [14].

Because Φ is reduced, we have that Δ_{1} = Δ and Δ_{2} = ∅, in the notation of Section 30.1. If β ∈ Δ, then let v_{β} = v_{λ_{β}} ∈ Υ be as in Sections 30.1 and 30.2. Using (30.2.10) and (30.6.6), we get that

ρ = ∑_{β∈Δ} v_{β}. (30.6.8)

This is another part of Lemma A around the middle of p70 of [14].

Let u ∈ Υ^{+} be given, and suppose that w = σ^{−1}(u) for some σ in the Weyl group of Φ. Under these conditions, Lemma B around the middle of p70 of [14] states that (w + ρ, w + ρ) ≤ (u + ρ, u + ρ), (30.6.9)

with equality only when u = w. To see this, observe first that

(w + ρ, w + ρ) = (σ(w + ρ), σ(w + ρ)) = (u + σ(ρ), u + σ(ρ)). (30.6.10)

This implies that

(w + ρ, w + ρ) = (u + ρ, u + ρ) − 2 (u, ρ − σ(ρ)), (30.6.11)

because (σ(ρ), σ(ρ)) = (ρ, ρ).

Note that σ(ρ) ⪯ ρ, (30.6.12)

by (30.5.8) and (30.6.7). Thus ρ−σ(ρ) is either equal to 0, or a sum of elements of Δ. It follows that (u, ρ − σ(ρ)) ≥ 0, (30.6.13) because u ∈ Υ^{+}. This implies (30.6.9), using (30.6.11). We also get that equality holds in (30.6.9) if and only if

(u, ρ − σ(ρ)) = 0. (30.6.14)

Clearly

(30.6.15)  (u, ρ − σ(ρ)) = (u, ρ) − (u, σ(ρ))
= (u, ρ) − (σ^{−1}(u), ρ) = (u − w, ρ).

We also have that w ⪯ u, (30.6.16)

as in (30.5.8), because u ∈ Υ^{+}. This means that either u = w, or u − w is a sum of elements of Δ. In the second case, we would get that

(u, ρ − σ(ρ)) = (u − w, ρ) > 0, (30.6.17)

because ρ is strongly dominant with respect to Δ, as before. Thus (30.6.14) holds only when u = w, as desired.

30.7 Saturated subsets of Υ

Let us continue with the same notation and hypotheses as in Sections 30.1 and 30.3, and let ⪯ be defined on V as in Section 30.5. A subset Π of Υ is said to be saturated if for every v ∈ Π and α ∈ Φ we have that

v − j α ∈ Π (30.7.1)

for every integer j between 0 and λ_{α}(v), as on p70 of [14]. Of course, this holds trivially when j = 0, and it is also supposed to hold when j = λ_{α}(v).

If α ∈ Φ, then it follows that σ_{α}(Π) ⊆ Π. This implies that σ_{α}(Π) = Π, because σ_{α} is its own inverse on V. Thus

σ(Π) = Π (30.7.2)

for every σ in the Weyl group of Φ when Π is saturated, as on p70 of [14]. A saturated set Π ⊆ Υ is said to have highest weight v ∈ Υ^{+} if v ∈ Π and

(30.7.3) w ⪯ v for every w ∈ Π,

as on p70 of [14]. It is easy to see that Π = {0} is saturated, with highest weight 0, as in Example (1) on p70 of [14].

Let us check that Π = Φ∪{0} is saturated, as in Example (2) on p70 of [14]. If v = 0, then λ_{α}(v) = 0 for every α ∈ Φ, and (30.7.1) holds trivially. Otherwise, let v, α ∈ Φ be given, and note that (30.7.1) holds when j = λ_{α}(v), because σ_{α}(v) ∈ Φ, by the definition of a root system. If v and α are not proportional in V, then it follows that (30.7.1) holds when j is between 0 and λ_{α}(v), as in Section 20.5. If v and α are proportional in V, then this can be verified directly.

Let us say that an element of Φ is maximal with respect to Δ if it is maximal with respect to ⪯. This is equivalent to the definition of a maximal root in Section 20.8. If α ∈ Φ, then the set of β ∈ Φ such that α ⪯ β is a nonempty finite set. It follows that this set has a maximal element β_{0} with respect to ⪯, and it is easy to see that β_{0} is also maximal in Φ.

Suppose for the moment that Φ is irreducible as a root system in V. This implies that Φ has a unique maximal element α_{0}, as in Section 20.8. If α ∈ Φ, then α ⪯ α_{0}, (30.7.4)

by the remarks in the preceding paragraph, and which was also mentioned in Section 20.8. It is easy to see that (30.7.4) holds with α = 0 too.

Let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ. If γ ∈ Δ, then (α_{0}, γ) ≥ 0, (30.7.5)

as in Section 20.8. This means that

α_{0} ∈ Υ^{+}, (30.7.6)

as in Section 30.3. It follows that Π = Φ ∪ {0} has highest weight α_{0} under these conditions, as on p70 of [14].

Suppose that Π ⊆ Υ is saturated, and has highest weight v ∈ Υ^{+}. Using the remarks near the beginning of the previous section, we get that Π ∩ Υ^{+} has only finitely many elements. Suppose also that Φ is reduced, as a root system in V. If w ∈ Π, then there is an element σ in the Weyl group of Φ such that σ(w) ∈ Υ^{+}, as in Section 30.4. We also have that σ(w) ∈ Π, as in (30.7.2). It follows that Π has only finitely many elements under these conditions, because Π ∩ Υ^{+} and the Weyl group of Φ have only finitely many elements. This is Lemma A near the bottom of p70 of [14].

30.8 More on highest weights

We continue with the same notation and hypotheses as in Sections 30.1 and 30.3, and to let ⪯ be defined on V as in Section 30.5. We also ask that Φ be reduced as a root system in V in this section. Let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ.

Suppose that Π ⊆ Υ is saturated, with highest weight v ∈ Υ^{+}. If u ∈ Υ^{+} and u ⪯ v, then Lemma B at the bottom of p70 of [14] states that

u ∈ Π. (30.8.1)

To see this, we shall consider elements w of Π of the form

w = u + ∑_{α∈Δ} c_{α} α, (30.8.2)

where c_{α} is a nonnegative integer for each α ∈ Δ. Of course, if this happens with c_{α} = 0 for each α ∈ Δ, then (30.8.1) holds, as desired. Note that w = v can be expressed as in (30.8.2), by hypothesis.

Suppose that w ∈ Π is of the form (30.8.2), and that c_{α} > 0 for some α ∈ Δ. Thus ∑_{α∈Δ} c_{α} α ≠ 0, which implies that

( ∑_{α∈Δ} c_{α} v_{α}, ∑_{β∈Δ} c_{β} β) > 0. (30.8.3)

It follows that there is a β_{0} ∈ Δ such that c_{β_{0}} > 0 and

( ∑_{α∈Δ} c_{α} α, β_{0}) > 0. (30.8.4)

Equivalently,

λ_{β_{0}}( ∑_{α∈Δ} c_{α} α) > 0. (30.8.5)

We also have that λ_{β_{0}}(u) ≥ 0, (30.8.6)

because u ∈ Υ^{+}, by hypothesis. Combining this with (30.8.5), we obtain that

λ_{β_{0}}(w) > 0. (30.8.7)

This means that λ_{β_{0}}(w) ≥ 1, because w ∈ Π ⊆ Υ, so that λ_{β_{0}}(w) ∈ Z. It follows that w − β ∈ Π, (30.8.8)

because w ∈ Π and Π is saturated.

Note that c_{β_{0}} ≥ 1, because c_{β_{0}} > 0 and c_{β_{0}} ∈ Z. This means that w − β can be expressed as in (30.8.2) as well. Thus we can start with w = v and repeat the process, as needed, to get that (30.8.1) holds.

If u ∈ Υ^{+} and u ⪯ v, then σ(u) ∈ Π (30.8.9)

for every element σ of the Weyl group of Φ, because of (30.8.1) and (30.7.2). Conversely, if u_{1} ∈ Π, then there is an element σ_{1} of the Weyl group of Φ such that σ_{1}^{−1}(u_{1}) ∈ Υ^{+}, (30.8.10)

as in Section 30.4. We also have that σ_{1}^{−1}(u_{1}) ∈ Π, by (30.7.2), and that σ_{1}^{−1}(u_{1}) ⪯ v, because Π has highest weight v, by hypothesis.

It follows that

(30.8.11)  Π = {σ(u) : u ∈ Υ^{+}, u ⪯ v, and σ is an element

of the Weyl group of Φ},

as on p71 of [14]. In particular, this means that Π is uniquely determined by v.

30.9 More on saturated sets

We continue with the same notations and hypotheses as in Sections 30.1 and 30.3 again, and to let ⪯ be defined on V as in Section 30.5. We ask that Φ be reduced as a root system in V in this section too.

Let Π be a subset of Υ that is invariant under the Weyl group of Φ, so that σ(Π) = Π for every element σ of the Weyl group. Suppose that for every w ∈ Π ∩ Υ^{+} and α ∈ Φ we have that

w − j α ∈ Π (30.9.1)

for every integer j between 0 and λ_{α}(w). We would like to check that Π is saturated as a subset of Υ under these conditions. This is related to a remark on p71 of [14].

To do this, let v ∈ Π be given. Remember that there is an element σ of the Weyl group such that σ(v) ∈ Υ^{+}, as in Section 30.4. Note that σ(v) ∈ Π, by hypothesis. Let α ∈ Φ be given, and remember that

λ_{σ(α)}(σ(v)) = λ_{α}(v), (30.9.2)

as in (30.4.2).

If j is an integer between 0 and λ_{σ(α)}(σ(v)), then

σ(v) − j σ(α) ∈ Π, (30.9.3)

by hypothesis, and because σ(α) ∈ Φ. This implies that v − j α ∈ Π, because σ(Π) = Π. Equivalently, this holds for every integer j between 0 and λ_{α}(v), by (30.9.2), as desired.

Let v ∈ Υ^{+} be given, and let Π = Π_{v} be as in (30.8.11). We would like to verify that Π is saturated as a subset of Υ, as in Exercise 10 on p72 of [14]. Of course, Π is invariant under the Weyl group of Φ, by construction. Let w ∈ Π∩Υ^{+} and α ∈ Φ be given. It suffices to show that (30.9.1) holds for every integer j between 0 and λ_{α}(w), as before.

By definition of Π, there is a u ∈ Υ^{+} and an element σ of the Weyl group such that w = σ(u) and u ⪯ v. Under these conditions, we have that u = w, as in Section 30.4. Thus w ⪯ v.

If j ∈ Z, then w − j α ∈ Υ, and so there is an element σ_{j} of the Weyl group such that σ_{j}(w − j α) ∈ Υ^{+}, (30.9.4)

as in Section 30.4. We would like to check that

σ_{j}(w − j α) ⪯ v (30.9.5)

when j is between 0 and λ_{α}(w).

Remember that σ_{j}(w) ⪯ w, because w ∈ Υ^{+}, as in Section 30.5. If

j σ_{j}(α) ⪰ 0, (30.9.6) then we get that

σ_{j}(w − j α) = σ_{j}(w) − j σ_{j}(α) ⪯ σ_{j}(w) ⪯ w ⪯ v. (30.9.7)

Observe that w − j α = σ_{α}(w) − (j − λ_{α}(w)) α, (30.9.8)

so that σ_{j}(w − j α) = σ_{j}(σ_{α}(w)) − (j − λ_{α}(w)) σ_{j}(α). (30.9.9)

As before, σ_{j}(σ_{α}(w)) ⪯ w, because w ∈ Υ^{+}, as in Section 30.5. If

(j − λ_{α}(w)) σ_{j}(α) ⪰ 0, (30.9.10)

then it follows that

σ_{j}(w − j α) ⪯ σ_{j}(σ_{α}(w)) ⪯ w ⪯ v. (30.9.11)

Because σ_{j}(α) ∈ Φ, we automatically have that σ_{j}(α) ⪰ 0 or σ_{j}(α) ⪯ 0. If j is between 0 and λ_{α}(w), then it is easy to see that one of j and j − λ_{α}(w) is greater than or equal to 0, and the other is less than or equal to 0. This implies that (30.9.6) or (30.9.10) holds. It follows that (30.9.5) holds in either case, by (30.9.7) and (30.9.11). This shows that Π is saturated in Υ, as desired.

It is easy to see that Π has highest weight v, as on p71 of [14]. More precisely, v ∈ Π by construction, because v ∈ Υ^{+}, by hypothesis. Let u ∈ Υ^{+} with u ⪯ v be given, and let σ be an element of the Weyl group of Φ. Note that σ(u) ⪯ u, as in Section 30.5. This implies that σ(u) ⪯ v, as desired.

30.10 Another look at saturated sets

Let us continue with the same notation and hypotheses as in the previous section. Also let v ∈ Υ^{+} be given, and put

(30.10.1)  Π^{v} = {u ∈ Υ : σ(u) ⪯ v for every element σ

of the Weyl group of Φ}.

Observe that v ∈ Π^{v}, (30.10.2)

because v ∈ Υ^{+}, as in Section 30.5. Of course, Π^{v} is automatically invariant under every element of the Weyl group of Φ.

Let Π_{v} = Π be the subset of Υ defined as in (30.8.11) again, and let us check that Π^{v} = Π_{v}. (30.10.3)

By construction, every element of Π_{v} is of the form τ(u), where u ∈ Υ^{+}, u ⪯ v, and τ is an element of the Weyl group of Φ. Let σ be any element of the Weyl group of Φ, so that σ ∘ τ is an element of the Weyl group too. This implies that

σ(τ(u)) ⪯ u, (30.10.4) because u ∈ Υ^{+}, as in Section 30.5. It follows that

σ(τ(u)) ⪯ v, (30.10.5)

so that τ(u) ∈ Π^{v}.

Now let u be any element of Π^{v}, and let σ be an element of the Weyl group of Φ such that σ(u) ∈ Υ^{+}, as in Section 30.4. We also have that σ(u) ⪯ v, by definition of Π^{v}. This implies that u ∈ Π_{v}, because the inverse of σ is in the Weyl group of Φ.

Let us verify directly that Π^{v} is saturated as a subset of Υ, as a variant of the argument in the previous section. Let u ∈ Π^{v} and α ∈ Φ be given, and let us check that u − j α ∈ Π^{v} (30.10.6)

for every integer j between 0 and λ_{α}(u). To do this, let an element σ of the Weyl group of Φ be given, and let us show that

σ(u − j α) ⪯ v. (30.10.7)

Of course, σ(u) ⪯ v, because u ∈ Π^{v}. If

j σ(α) ⪰ 0, (30.10.8)

then σ(u − j α) = σ(u) − j σ(α) ⪯ σ(u) ⪯ v. (30.10.9)

We also have that

u − j α = σ_{α}(u) − (j − λ_{α}(u)) α, (30.10.10)

so that σ(u − j α) = σ(σ_{α}(u)) − (j − λ_{α}(u)) σ(α). (30.10.11)

Note that σ(σ_{α}(u)) ⪯ v, because u ∈ Π^{v} and σ ∘ σ_{α} is in the Weyl group of Φ. If (j − λ_{α}(u)) σ(α) ⪰ 0, (30.10.12)

then we get that σ(u − j α) ⪯ σ(σ_{α}(u)) ⪯ v. (30.10.13)

As before, we automatically have that σ(α) ⪰ 0 or σ(α) ⪯ 0, because σ(α) is in Φ. We also have that one of j and j − λ_{α}(u) is greater than or equal to 0, and the other is less than or equal to 0, because j is between 0 and λ_{α}(u), by hypothesis. Using this, it is easy to see that (30.10.8) or (30.10.12) holds. In either case, we get that (30.10.7) holds, as desired. Note that Π^{v} has highest weight v, by construction.

30.11 A property of saturated sets

Let us continue with the same notations and hypotheses as in Section 30.1 and 30.3, and let ⪯ be defined on V as in Section 30.5. In this section, we continue to suppose that Φ is reduced as a root system in V. Let (·, ·) be an inner product on V that is invariant under the Weyl group of Φ.

Let Π be a saturated subset of Υ, with highest weight v ∈ Υ^{+}. If w ∈ Π and ρ is as in Section 30.6, then Lemma C on p71 of [14] states that

(w + ρ, w + ρ) ≤ (v + ρ, v + ρ), (30.11.1)

with equality only when w = v. To see this, let σ be an element of the Weyl group of Φ such that u = σ(w) ∈ Υ^{+}, as in Section 30.4. Remember that

(w + ρ, w + ρ) ≤ (u + ρ, u + ρ), (30.11.2)

with equality only when u = w, as in Section 30.6. Thus it suffices to show that

(u + ρ, u + ρ) ≤ (v + ρ, v + ρ), (30.11.3)

with equality only when u = v.

Note that u ∈ Π, because σ(Π) = Π, as in Section 30.7. This means that u ⪯ v, because Π has highest weight v. Observe that

(v + ρ, v + ρ) − (u + ρ, u + ρ) (30.11.4)

= ((v + ρ, v + ρ) − (v + ρ, u + ρ)) + ((v + ρ, u + ρ) − (u + ρ, u + ρ))
= (v + ρ, v − u) + (v − u, u + ρ).

Remember that ρ ∈ Υ^{+}, and in fact ρ is strongly dominant, as in Section 30.6. This implies that u + ρ and v + ρ are strongly dominant too, because u, v ∈ Υ^{+}.

It follows that each of the terms on the right side of (30.11.4) is greater than or equal to 0, with equality only when u = v, because u ⪯ v. Combining this with (30.11.4), we get that (30.11.3) holds, with equality only when u = v, as desired.

Regular elements and polynomials

31.1 Integral domains and polynomials

As usual, a commutative ring A with a nonzero multiplicative identity element is said to be an integral domain if it has no nontrivial zero-divisors. This means that if a, b ∈ A and a, b ≠ 0, then a b ≠ 0.

Let A be an integral domain, and let T be an indeterminate. It is well known and easy to see that the ring A[T] of formal polynomials in T with coefficients in A is an integral domain too.

Similarly, let T_{1}, …, T_{n} be n commuting indeterminates, for some positive integer n. It is well known that the ring A[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in A is an integral domain. To see this, remember that A[T_{1}, …, T_{n}] is isomorphic to the ring

(A[T_{1}, …, T_{n−1}])[T_{n}] (31.1.1)

of formal polynomials in T_{n} with coefficients in A[T_{1}, …, T_{n−1}], as in Section 5.8. This permits one to reduce to the n = 1 case, using induction.

Let T be an indeterminate again, and let f(T) be a nonzero element of A[T]. This leads to a polynomial function f(a) on A with values in A, as in Section 5.7. It is well known and not difficult to show that f(a) = 0 for only finitely many a ∈ A. More precisely, the number of such elements a of A is less than or equal to the degree of f(T).

Let T_{1}, …, T_{n} be n commuting indeterminates again, and let f(T_{1}, …, T_{n}) be a nonzero element of A[T_{1}, …, T_{n}]. This leads to a polynomial function f(a_{1}, …, a_{n}) on the space A^{n} of n-tuples of elements of A with values in A, as in Section 5.9.

Suppose that E_{1}, …, E_{n} are infinite subsets of A, and put

E = ∏_{j=1}^{n} E_{j}. (31.1.2)

Under these conditions, there is an (a_{1}, …, a_{n}) ∈ E such that f(a_{1}, …, a_{n}) ≠ 0. In particular, if A has infinitely many elements, then f(a_{1}, …, a_{n}) ≠ 0 for some (a_{1}, …, a_{n}) ∈ A^{n}.

To see this, it is helpful to consider f(T_{1}, …, T_{n}) as an element of (31.1.1), as before. More precisely, we can express f(T_{1}, …, T_{n}) as

f(T_{1}, …, T_{n}) = ∑_{j=1}^{l} f_{j}(T_{1}, …, T_{n−1}) T_{n}^{j}, (31.1.3)

where l is a nonnegative integer and f_{j}(T_{1}, …, T_{n−1}) is an element of

A[T_{1}, …, T_{n−1}] (31.1.4)

for each j = 0, 1, …, l. The hypothesis that f(T_{1}, …, T_{n}) ≠ 0 means that f_{j}(T_{1}, …, T_{n−1}) ≠ 0 for some j, and one may as well suppose that this holds with j = l.

If a_{n} ∈ A, then

f(T_{1}, …, T_{n−1}, a_{n}) = ∑_{j=0}^{l} f_{j}(T_{1}, …, T_{n−1}) a_{n}^{j} (31.1.5)

defines an element of (31.1.4). Equivalently, if we consider (31.1.3) as a formal polynomial in T_{n} with coefficients in (31.1.4), then we can evaluate the corresponding polynomial function at a_{n} to get an element of (31.1.4).

Of course, (31.1.3) is nonzero as an element of (31.1.1), by hypothesis. This implies that (31.1.5) can be equal to 0 as an element of (31.1.4) for only finitely many a_{n} ∈ A, as before. In particular, there is an a_{n} ∈ E_{n} such that (31.1.5) is nonzero as an element of (31.1.4), because E_{n} has infinitely many elements, by hypothesis.

If n ≥ 2, then we may as well suppose that the polynomial function on A^{n−1} associated to (31.1.5) is nonzero at some element (a_{1}, …, a_{n−1}) of ∏_{j=1}^{n−1} E_{j}, by induction. The value of this function at (a_{1}, …, a_{n−1}) is the same as the value of the polynomial function on A^{n} associated to f(T_{1}, …, T_{n−1}, T_{n}) at (a_{1}, …, a_{n−1}, a_{n}). Thus f(a_{1}, …, a_{n}) ≠ 0, as desired.

31.2 The Zariski topology

Let k be an integral domain, let n be a positive integer, and let T_{1}, …, T_{n} be n commuting indeterminates. Also let k^{n} be the set of n-tuples of elements of k, as usual, so that every element of k[T_{1}, …, T_{n}] determines a polynomial function on k^{n}, as in Section 5.9. If C is a nonempty subset of k[T_{1}, …, T_{n}], then let

V(C) = V_{k^{n}}(C) = {(x_{1}, …, x_{n}) ∈ k^{n} : f(x_{1}, …, x_{n}) = 0

(31.2.1) for every f(T_{1}, …, T_{n}) ∈ C}

be the set of points in k^{n} at which all of the polynomial functions associated to elements of C vanish.

It is easy to see that (31.2.1) is the same as for the ideal in k[T_{1}, …, T_{n}] generated by C. If k is Noetherian, then it is well known that k[T_{1}, …, T_{n}] is Noetherian, so that every ideal in k[T_{1}, …, T_{n}] is finitely generated.

If C_{1}, C_{2} are nonempty subsets of k[T_{1}, …, T_{n}], then the collection C_{1} · C_{2} of all products of elements of C_{1} and C_{2} is a nonempty subset of k[T_{1}, …, T_{n}] as well. One can check that

V(C_{1} · C_{2}) = V(C_{1}) ∪ V(C_{2}), (31.2.2)

because k is an integral domain.

Similarly, let I be a nonempty set, and let C_{j} be a nonempty subset of k[T_{1}, …, T_{n}] for each j ∈ I. Observe that

V( ⋃_{j∈I} C_{j}) = ⋂_{j∈I} V(C_{j}). (31.2.3)

A subset of k^{n} of the form (31.2.1) is said to be Zariski closed. Note that the empty set and k^{n} are Zariski-closed sets. Thus the collection of Zariski-closed subsets of k^{n} is the collection of closed sets with respect to a topology on k^{n}, which is the Zariski topology.

It is easy to see that a subset of k^{n} with only one element is Zariski closed, so that k^{n} satisfies the first separation condition with respect to the Zariski topology. If k has only finitely many elements, then the Zariski topology on k^{n} is the same as the discrete topology.

Of course, the Zariski topology can be defined on k, by taking n = 1 in the previous definition. The Zariski-closed subsets of k are the finite subsets of k, as well as k itself. Note that polynomial functions on k^{n} are continuous with respect to the Zariski topologies on k and k^{n}.

A subset of k^{n} is said to be irreducible if it is not contained in the union of two Zariski-closed sets, neither of which contains the given set. If k has infinitely many elements, then it is well known that

(31.2.4) k^{n} is irreducible,

as in Lemma A on p133 of [14].

Indeed, suppose that C_{1}, C_{2} are nonempty subsets of k[T_{1}, …, T_{n}] such that

k^{n} ⊆ V(C_{1}) ∪ V(C_{2}), (31.2.5)

and that k^{n} is not contained in either V(C_{1}) or V(C_{2}). This implies that there are elements f(T_{1}, …, T_{n}), g(T_{1}, …, T_{n}) of C_{1}, C_{2}, respectively, such that the corresponding polynomial functions are nonzero at some elements of k^{n}. Of course, this means that the corresponding formal polynomials are nonzero, so that their product f(T_{1}, …, T_{n}) g(T_{1}, …, T_{n}) (31.2.6)

is nonzero as an element of k[T_{1}, …, T_{n}] too, as in the previous section. It follows that the polynomial function on k^{n} corresponding to (31.2.6) is nonzero at an element of k^{n}, because k has infinitely many elements, as in the previous section. This contradicts (31.2.5), which implies that the product of the polynomial functions corresponding to f(T_{1}, …, T_{n}) and g(T_{1}, …, T_{n}) vanishes everywhere on k^{n}.

If U, V are two nonempty Zariski-open subsets of k^{n}, then (31.2.4) implies that U ∩ V ≠ ∅ (31.2.7)

when k has infinitely many elements. This means that nonempty Zariski-open subsets of k^{n} are dense in k^{n} with respect to the Zariski topology when k has infinitely many elements, as in the corollary on p133 of [14].

31.3 Polynomial mappings

Let us continue with the same notation and hypotheses as in the previous section, and let m be another positive integer. Also let Z_{1}, …, Z_{m} be m commuting indeterminates, so that elements of k[Z_{1}, …, Z_{m}] determine polynomial functions on k^{m}, as before.

Suppose that ϕ_{1}(Z_{1}, …, Z_{m}), …, ϕ_{n}(Z_{1}, …, Z_{m}) (31.3.1)

are n elements of k[Z_{1}, …, Z_{m}]. These determine n polynomial functions on k^{m}, which can be used to define a mapping ϕ from k^{m} into k^{n}. More precisely, if x ∈ k^{m}, then ϕ(x) is the element of k^{n} whose jth coordinate is ϕ_{j}(x_{1}, …, x_{m}) for every j = 1, …, n.

It is easy to see that the composition of a polynomial function on k^{n} with ϕ is a polynomial function on k^{m}. Indeed, if f(T_{1}, …, T_{n}) ∈ k[T_{1}, …, T_{n}], then one can get an element of k[Z_{1}, …, Z_{m}] by replacing T_{j} with ϕ_{j}(Z_{1}, …, Z_{m}) for every j = 1, …, n. This defines a homomorphism from k[T_{1}, …, T_{n}] into k[Z_{1}, …, Z_{m}], as algebras over k, with

T_{j} ↦ ϕ_{j}(Z_{1}, …, Z_{m}) (31.3.2)

for each j = 1, …, n. Of course, this homomorphism takes elements of k to themselves, as formal polynomials in T_{1}, …, T_{n} or Z_{1}, …, Z_{m}. The composition of the polynomial function on k^{n} associated to f(T_{1}, …, T_{n}) with ϕ is the same as the polynomial function on k^{m} associated to the corresponding element of k[Z_{1}, …, Z_{m}].

Let C be a nonempty subset of k[T_{1}, …, T_{n}], and let C_{0} be the corresponding collection of elements of k[Z_{1}, …, Z_{m}], as in the preceding paragraph. Observe that ϕ^{−1}(V_{k^{n}}(C)) = V_{k^{m}}(C_{0}), (31.3.3)

where V_{k^{n}}(C), V_{k^{m}}(C_{0}) are as in (31.2.1). This implies that ϕ is continuous with respect to the Zariski topologies on k^{m}, k^{n}.

Remember that k^{m}, k^{n} may be considered as modules over k, with respect to coordinatewise addition and scalar multiplication. Let a = (a_{j,l}) be an n×m matrix with entries in k, and consider

ϕ_{j}(Z_{1}, …, Z_{m}) = ∑_{l=1}^{m} a_{j,l} Z_{l} (31.3.4)

for each j = 1, …, n. The corresponding mapping ϕ from k^{m} into k^{n} is a module homomorphism in this case.

Suppose that m = n, and that a is invertible as an n × n matrix with entries in k. This implies that ϕ is a one-to-one mapping from k^{n} onto itself, whose inverse corresponds to the inverse of a. It follows that ϕ is a homeomorphism from k^{n} onto itself, with respect to the Zariski topology.

If V is any module over k that is free of rank n, then V is isomorphic to k^{n}, as a module over k. In particular, we can use this to define the Zariski topology on V. Using the remarks in the preceding paragraph, we get that this topology does not depend on the particular isomorphism between V and k^{n}.

Suppose now that m = 1, so that we can use a single indeterminate Z. If a_{1}, …, a_{n} and b_{1}, …, b_{n} are elements of k, then we can take

ϕ_{j}(Z) = a_{j} Z + b_{j} (31.3.5)

for j = 1, …, n. The corresponding mapping ϕ from k into k^{n} maps k onto an affine line in k^{n}, at least if a_{j} ≠ 0 for some j.

31.4 More on the Zariski topology

Let X be a topological space, and let E be a subset of X. As in Section 31.2, we may say that E is irreducible if E is not contained in the union of two closed subsets of X, neither of which contains E. Let us check that this implies that E is connected, as mentioned on p133 of [14].

If E is not connected, then E can be expressed as the union of two nonempty separated subsets E_{1}, E_{2} of X. This means that

\overline{E_{1}} ∩ E_{2} = E_{1} ∩ \overline{E_{2}} = ∅, (31.4.1)

where \overline{E_{1}}, \overline{E_{2}} are the closures of E_{1}, E_{2} in X, respectively. It follows that

E ⊆ \overline{E_{1}} ∪ \overline{E_{2}}, (31.4.2)

and that E is not contained in either \overline{E_{1}} or \overline{E_{2}}. This contradicts irreducibility, as desired.

Note that X is irreducible as a subset of itself if and only if X cannot be expressed as the union of two proper closed subsets of itself. This is the same as saying that the intersection of any two nonempty open subsets of X is nonempty. Equivalently, this means that any nonempty open subset of X is dense in X.

Suppose that X is irreducible, as a subset of itself. If U is any open subset of X, then one can verify that U is irreducible in X.

Let us continue now with the same notation and hypotheses as in Section 31.2. Suppose for the moment that k has infinitely many elements, so that k^{n} is irreducible, as before. If U is a Zariski-open subset of k^{n}, then U is irreducible, as in the preceding paragraph. This implies that U is connected with respect to the Zariski topology.

Let k be a field equipped with an absolue value function |·|. Note that polynomial functions on k^{n} are continuous, with respect to the topology determined on k by the metric associated to |·|, and the corresponding product topology on k^{n}. This implies that Zariski-closed subsets of k^{n} are closed sets with respect to this product topology.

Suppose that | · | is not the trivial absolute value function on k. This means that open balls in k with respect to the metric associated to | · | have infinitely many elements. It follows that any nonempty open subset of k^{n} with respect to the corresponding product topology contains the Cartesian product of n infinite subsets of k. If a polynomial on k^{n} vanishes on a nonempty open subset W of k^{n} with respect to this product topology, then the polynomial vanishes on all of k^{n}, as in Section 31.1. Thus W is dense in k^{n} with respect to the Zariski topology under these conditions.

Equivalently, a proper Zarski-closed subset of k^{n} has empty interior with respect to this product topology. This implies that a nonempty Zariski-open subset of k^{n} is dense in k^{n} with respect to this product topology in this case.

Of course, any two distinct elements of k^{n} are contained in a unique affine line in k^{n}. Let U be a Zariski-open subset of k^{n}. The intersection of U with any affine line in k^{n} is either empty, or contains all but finitely many elements of the line. One can use this to check that any Zariski-open subset of C^{n} is path connected, with respect to the product topology on C^{n} corresponding to the standard topology on C.

31.5 Regular elements of Lie algebras

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k of positive finite dimension n, as a vector space over k. Also let T be an indeterminate, and let I = I_{A} be the identity mapping on A. If x ∈ A, then let

Q_{x}(T) = det(ad_{x} − T I) (31.5.1)

be the characteristic polynomial of ad_{x}, as a linear mapping from A into itself, as in Section 18.3.

Of course, we can use a basis for A to identify A with k^{n}, as a vector space over k. This permits us to express (31.5.1) as

Q_{x}(T) = ∑_{j=0}^{n} q_{j}(x) T^{j}, (31.5.2)

where q_{j}(x) corresponds to a homogeneous polynomial of degree n − j in the coordinates of x for every j = 0, 1, …, n, as in Section 18.3 again.

Let us suppose from now on in this section that k has infinitely many elements. This implies that formal polynomials in n commuting indeterminates with coefficients in k are uniquely determined by the corresponding polynomial functions on k^{n}, as in Section 31.1.

Remember that q_{n}(x) corresponds to the constant polynomial (−1)^{n}, by construction. Let n_{0} be the smallest integer j = 0, 1, …, n such that q_{j}(x) is not identically zero on A. This may be called the rank of A, as in Definition 2 on p11 of [24], or the ρ-rank of A, as on p133 of [14].

Of course, ad_{x}(x) = 0 for every x ∈ A, which implies that q_{0}(x) = 0, because n ≥ 1, by hypothesis. This means that

n_{0} ≥ 1, (31.5.3)

as on p11 of [24].

If x ∈ A, then let n_{0}(x) be the smallest integer j = 0, 1, …, n such that q_{j}(x) ≠ 0. This was initially defined another way in Section 18.3, and the equivalence with this formulation was mentioned just afterwards. Note that

n_{0}(x) ≤ n, (31.5.4)

because q_{n}(x) = (−1)^{n}, and that

n_{0} = min{n_{0}(x) : x ∈ A}, (31.5.5)

by construction.

Thus n_{0} = n if and only if n_{0}(x) = n for every x ∈ A. This happens exactly when A is nilpotent as a Lie algebra, as on p11 of [24], and mentioned in Section 18.3.

Observe that x ∈ A satisfies

n_{0}(x) = n_{0} (31.5.6)

if and only if q_{n_{0}}(x) ≠ 0, (31.5.7)

because q_{j}(x) = 0 when j < n_{0}, by definition of n_{0}. In this case, x is said to be regular, as in Definition 2 on p11 of [24], or ρ-regular, as on p133 of [14]. The set R(A) of regular elements of A is nonempty, by (31.5.5).

31.6 The set of regular elements

Let us continue with the same notation and hypotheses as in the previous section. As before, we can use a basis for A to identify A with k^{n}, as a vector space over k. Using this, we can define the Zariski topology on A as in Section 31.2. The Zariski topology on A does not depend on the choice of basis, as in Section 31.3.

The set R(A) of regular elements of A is an open set with respect to the Zariski topology on A, because of the characterization (31.5.7) of its elements. We also have that R(A) ≠ ∅, by definition of n_{0}, as in the previous section. It follows that R(A) is dense in A with respect to the Zariski topology, as in Section 31.2, because k has infinitely many elements. This corresponds to some remarks on p133 of [14].

Let | · | be an absolute value function on k. As usual, the metric on k associated to |·| determines a topology on k, which leads to the corresponding product topology on k^{n}. This defines a topology on A, that we shall call the topology associated to | · |, because A is identified with k^{n}. It is easy to see that this topology does not depend on the choice of basis for A, because linear mappings from k^{n} into itself are continuous with respect to this product topology.

The topology on A associated to | · | is at least as strong as the Zariski topology, as in Section 31.4. This means that R(A) is an open set in A with respect to the topology associated to | · |. If | · | is not the trivial absolute value function on k, then R(A) is dense in A with respect to the topology associated to | · |, as in Section 31.4 again.

Note that R(A) is connected with respect to the Zariski topology on A, because k has infinitely many elements, as in Section 31.4. If k = C with the standard absolute value function, then R(A) is path connected with respect to the topology on A associated to | · |, as before. These properties of R(A) correspond to Proposition 1 on p11 of [24].

If x ∈ A, then let A_{x,0} be the set of y ∈ A such that (ad_{x})^{l}(y) = 0 for some positive integer l, as in Section 18.1. Remember that this is a Lie subalgebra of A, which is called an Engel subalgebra, as in Section 18.4. We also have that

n_{0}(x) = dim A_{x,0}, (31.6.1)

as in Section 18.3.

Thus n_{0} = min{dim A_{x,0} : x ∈ A}, (31.6.2)

by (31.5.5). Similarly,

R(A) = {x ∈ A : dim A_{x,0} = n_{0}}. (31.6.3)

If x ∈ R(A), then A_{x,0} is minimal with respect to inclusion among Engel subalgebras of A, because the dimension of A_{x,0} is minimal among the dimensions of Engel subalgebras of A. This implies that

(31.6.4) A_{x,0} is a Cartan subalgebra of A, as in the theorem on p80 of [14], and as mentioned in Section 18.8. This corresponds to Theorem 1 on p12 of [24].

31.7 Regularity and automorphisms

Let us continue with the same notation and hypotheses as in the previous two sections. Suppose for the moment that ϕ is an automorphism of A, as a Lie algebra over k. If x, y ∈ A, then

ϕ(ad_{x}(y)) = ϕ([x, y]_{A}) = [ϕ(x), ϕ(y)]_{A} = ad_{ϕ(x)}(ϕ(y)). (31.7.1)

This means that ϕ ∘ ad_{x} = ad_{ϕ(x)} ∘ϕ, so that

ad_{ϕ(x)} = ϕ ∘ ad_{x} ∘ ϕ^{−1}. (31.7.2)

It follows that ad_{x} and ad_{ϕ(x)} have the same characteristic polynomials, which is to say that Q_{ϕ(x)}(T) = Q_{x}(T). (31.7.3)

This implies that q_{j}(ϕ(x)) = q_{j}(x) (31.7.4)

for every j = 0, 1, …, n. One can also verify that

ϕ(A_{x,0}) = A_{ϕ(x),0}. (31.7.5)

Note that n_{0}(ϕ(x)) = n_{0}(x), (31.7.6)

using either the characterization of n_{0}(x) in Section 31.5, or (31.6.1). Similarly,

ϕ(R(A)) = R(A). (31.7.7)

Let x be an element of R(A), so that A_{x,0} is a Cartan subalgebra of A. Note that x ∈ A_{x,0}, so that every element of R(A) is contained in a Cartan subalgebra of A.

Suppose for the moment that k is an algebraically closed field of characteristic 0. If C is any Cartan subalgebra of A, then there is a Lie algebra automorphism ϕ of A such that

C = ϕ(A_{x,0}), (31.7.8)

as in Section 24.9. Of course, this means that

C = A_{ϕ(x),0}, (31.7.9)

by (31.7.5). We also have that ϕ(x) ∈ R(A), by (31.7.7). This corresponds to Corollary 2 on p13 of [24].

Observe that dim C = dim A_{x,0} = n_{0} (31.7.10)

under these conditions. This corresponds to Corollary 1 on p13 of [24]. It follows that the definition of the rank of A in Section 24.13 is equivalent to the one in Section 31.5 when k is an algebraically closed field of characteristic 0.

31.8 Regular semisimple elements

Let us continue with the same notation and hypotheses as in the previous three sections. If x ∈ A is ad-diagonalizable, then A_{x,0} is the same as the kernel of ad_{x} on A. This is the same as the centralizer C_{A}(x) = C_{A}({x}) of x in A, so that A_{x,0} = C_{A}(x), (31.8.1)

as in Section 18.9.

Of course, if x is any element of A, then C_{A}(x) contains any commutative Lie subalgebra of A that contains x. If C_{A}(x) is commutative as a Lie subalgebra of A, then C_{A}(x) is maximal among commutative Lie subalgebras of A.

Suppose that C_{A}(x) is a toral subalgebra of A, as in Section 17.1. Remember that toral subalgebras of A are commutative as Lie subalgebras of A. This implies that C_{A}(x) is maximal among toral subalgebras of A, as in the preceding paragraph.

If C_{A}(x) is a toral subalgebra of A, then x is ad-diagonalizable in A, because x ∈ C_{A}(x). This means that (31.8.1) holds, so that

n_{0}(x) = dim C_{A}(x), (31.8.2)

by (31.6.1).

Suppose from now on in this section that k is an algebraically closed field of characteristic 0, and that A is semisimple as a Lie algebra over k. If x ∈ A has the property that C_{A}(x) is a toral subalgebra of A, and thus a maximal toral subalgebra of A, then x may be called regular semisimple in A, as on p80 of [14].

If B is any maximal toral subalgebra of A, then

B = C_{A}(x) (31.8.3)

for some x ∈ A, which is therefore regular semisimple in A, as on p80 of [14]. Indeed, B is a Cartan subalgebra of A, as in Section 18.10. This implies that

B = A_{x,0} (31.8.4)

for some x ∈ A, as in Section 18.8. Of course, x ∈ A_{x,0}, so that x ∈ B, and thus x is ad-diagonalizable in A. This means that (31.8.3) follows from (31.8.1) and (31.8.4), as desired.

Alternatively, one can get x ∈ A such that (31.8.3) holds as in Section 18.13, as mentioned on p80 of [14]. This uses the fact that B is equal to its centralizer C_{A}(B) in A, as in Section 17.4.

Suppose that x ∈ A and that A_{x,0} is a Cartan subalgebra of A. Every Cartan subalgebra of A is of this form, as in Section 18.8 again. Under these conditions, A_{x,0} is a maximal toral subalgebra of A, as in Section 18.10. This implies that x is ad-diagonalizable in A, because x ∈ A_{x,0}, so that (31.8.1) holds. It follows that x is regular semisimple in A.

If x ∈ R(A), then A_{x,0} is a Cartan subalgebra of A, as before. In particular, this implies that x is ad-diagonalizable in A, as in Corollary 2 on p15 of [24]. This is also mentioned on p134 of [14], using another argument. More precisely, x is regular semisimple in A, as in the preceding paragraph.

31.9 Regularity and semisimplicity

We continue with the same notation and hypotheses as in the previous four sections. Let us also continue to ask that k be an algebraically closed field of characteristic 0, and that A be semisimple as a Lie algebra over k.

Let x ∈ A be given, and let x = x_{1}+x_{2} be the abstract Jordan decomposition of x in A, as in Section 14.3. Thus x_{1}, x_{2} ∈ A, x_{1} is ad-diagonalizable in A, x_{2} is ad-nilpotent in A, and [x_{1}, x_{2}]_{A} = 0. This means that ad_{x} = ad_{x_{1}} + ad_{x_{2}} is the usual Jordan decomposition of ad_{x}, as a linear mapping from A into itself.

It follows that ad_{x} and ad_{x_{1}} have the same characteristic polynomials, as linear mappings from A into itself. This means that

Q_{x}(T) = Q_{x_{1}}(T), (31.9.1)

in the notation of Section 31.5. It follows that

q_{j}(x) = q_{j}(x_{1}) (31.9.2)

for every j = 0, 1, …, n. This is related to some remarks on p133 of [14]. Similarly, n_{0}(x) = n_{0}(x_{1}), (31.9.3)

using the characterization of n_{0}(·) in Section 31.5.

Alternatively, one can check directly that

A_{x,0} = A_{x_{1},0}, (31.9.4)

as in Section 18.9. This implies (31.9.3), because of (31.6.1).

This gives another way to see that there are elements of R(A) that are addiagonalizable in A, as on p133 of [14]. More precisely, x ∈ R(A) if and only if x_{1} ∈ R(A), by (31.9.3).

If y ∈ A is ad-diagonalizable, then y is contained in a maximal toral subalgebra B of A. If z is any element of B, then z is ad-diagonalizable in A, so that A_{z,0} = C_{A}(z), as in (31.8.1). We also have that B ⊆ C_{A}(z), because B is commutative as a Lie subalgebra of A, as in Section 17.1. This implies that

(31.9.5) n_{0}(z) = dim A_{z,0} = dim C_{A}(z) ≥ dim B.

Note that there are z ∈ B such that B = C_{A}(z), so that equality holds in (31.9.5), as in the previous section.

Of course, we can take z = y in (31.9.5), to get that

n_{0}(y) = dim A_{y,0} ≥ dim B. (31.9.6)

If y ∈ R(A) too, then we get that

n_{0} = n_{0}(y) = dim B, (31.9.7)

because there are z ∈ B for which equality holds in (31.9.5). This implies that A_{y,0} = C_{A}(y) = B, so that y is regular semisimple in A, in the sense described in the previous section. If z ∈ B and C_{A}(z) = B, so that equality holds in (31.9.5), then we obtain that n_{0}(z) = n_{0}, which means that z ∈ R(A). This corresponds to some remarks on p133 of [14].

Suppose that x ∈ R(A), so that the ad-diagonalizable part x_{1} of x is in R(A) too, as before. This implies that x_{1} is regular semisimple in A, as in the preceding paragraph. Thus C_{A}(x_{1}) is a toral subalgebra of A that contains the ad-nilpotent part x_{2} of x. This means that x_{2} is also ad-diagonalizable, so that ad_{x_{2}} = 0. It follows that x_{2} = 0, because A is semisimple, by hypothesis. Equivalently, x = x_{1}, so that x is ad-diagonalizable. This gives another way to see that every element of R(A) is ad-diagonalizable in A, as on p133f of [14]. More precisely, this gives another way to see that every element of R(A) is regular semisimple in A.

If w is any regular semisimple element of A, then n_{0}(w) = dim C_{A}(w), as in (31.8.2). In this case, C_{A}(w) is a maximal toral subalgebra of A, and thus a Cartan subalgebra of A, as in Section 18.10. This implies that dim C_{A}(w) = n_{0}, by (31.7.10). It follows that n_{0}(w) = n_{0}, (31.9.8)

so that w ∈ R(A). This shows that R(A) is the same as the set of all regular semisimple elements of A, as on p134 of [14].

31.10 Symmetric algebras and homomorphisms

Let k be a commutative ring with a multiplicative identity element, and let V be a module over k. Also let SV be a symmetric algebra of V, as in Section 25.8. Thus SV is a commutative associative algebra over k with a multiplicative identity element e_{SV}, and SV comes with a homomorphism i_{SV} from V into SV, as modules over k.

Let B be a commutative associative algebra over k with a multiplicative identity element e_{B}. If ϕ is a homomorphism from V into B, as modules over k, then there is a unique algebra homomorphism ψ from SV into B such that ψ(e_{SV}) = e_{B} and ψ ∘ i_{SV} = ϕ, (31.10.1)

as in Section 25.8. Note that SV is uniquely determined, up to isomorphic equivalence, by this property.

Let W be another module over k, and let SW be a symmetric algebra of W, with multiplicative identity element e_{SW}, and module homomorphism i_{SW} from W into SW. If ξ_{1} is a homomorphism from V into W, as modules over k, then i_{SW} ∘ ξ_{1} (31.10.2)

31.11. SOME POLYNOMIAL FUNCTIONS ON V′ 665

is a homomorphism from V into SW, as modules over k. This leads to a unique algebra homomorphism η_{1} from SV into SW such that η_{1}(e_{SV}) = e_{SW} and

η_{1} ∘ i_{SV} = i_{SW} ∘ ξ_{1}, (31.10.3)

as before.

Similarly, let Z be a third module over k, and let SZ be a symmetric algebra of Z, with multiplicative identity element e_{SZ}, and module homomorphism i_{SZ} from Z into SZ. Also let ξ_{2} be a module homomorphism from W into Z, so that i_{SZ} ∘ ξ_{2} (31.10.4)

is a module homomorphism from W into SZ. Using this, we get a unique algebra homomorphism η_{2} from SW into SZ such that η_{2}(e_{SW}) = e_{SZ} and

η_{2} ∘ i_{SW} = i_{SZ} ∘ ξ_{2}, (31.10.5)

as usual.

Under these conditions, ξ_{2}∘ξ_{1} is a homomorphism from V into Z, as modules over k, and i_{SZ} ∘ ξ_{2} ∘ ξ_{1} (31.10.6)

is a module homomorphism from V into SZ. Of course, η_{2} ∘ η_{1} is an algebra homomorphism from SV into SZ that maps e_{SV} to e_{SZ}, and

η_{2} ∘ η_{1} ∘ i_{SV} = η_{2} ∘ i_{SW} ∘ ξ_{1} = i_{SZ} ∘ ξ_{2} ∘ ξ_{1}, (31.10.7)

by (31.10.3) and (31.10.5). In fact, η_{2} ∘ η_{1} is uniquely determined by these properties, as before.

In particular, we can take V = Z, SV = SZ, and i_{SV} = i_{SZ}. If ξ_{2} ∘ ξ_{1} is the identity mapping on V, then we get that η_{2} ∘ η_{1} is the identity mapping on SV.

Similarly, if ξ_{1} ∘ ξ_{2} is the identity mapping on W, then η_{1} ∘ η_{2} is the identity mapping on SW. If ξ_{1} is a module isomorphism from V onto W, then it follows that η_{1} is an algebra isomorphism from SV onto SW.

31.11 Some polynomial functions on V′

Let k be a commutative ring with a multiplicative identity element again, and let V be a module over k. Put

V′ = Hom_{k}(V, k), (31.11.1)

which is the space of module homomorphisms from V into k, considered as a module over itself. Remember that V′ is a module over k too, with respect to pointwise addition and scalar multiplication of functions on V.

Let SV be a symmetric algebra of V, with multiplicative identity element e_{SV}, and module homomorphism i_{SV} from V into SV. If ϕ_{0} ∈ V′, then there is a unique algebra homomorphism ψ_{0} from SV into k such that ψ_{0}(e_{SV}) = 1 and ψ_{0} ∘ i_{SV} = ϕ_{0}. (31.11.2)

This corresponds to taking B = k in the previous section.

Let c(V′, k) be the set of all functions on V′ with values in k. This is a module over k with respect to pointwise addition and scalar multiplication of functions on V′, and a commutative algebra over k with respect to pointwise multiplication of functions on V′. The multiplicative identity element in c(V′, k) is the constant function on V′ equal to 1 ∈ k.

Note that V′′ = Hom_{k}(V′, k) may be considered as a submodule of c(V′, k), as a module over k. If v ∈ V, then let Φ(v) be the k-valued function on V′ defined by Φ(v)(ϕ_{0}) = ϕ_{0}(v) (31.11.3)

for every ϕ_{0} ∈ V′. This defines Φ as a module homomorphism from V into c(V′, k). More precisely, Φ maps V into V′′.

It follows that there is a unique algebra homomorphism Ψ from SV into c(V′, k) such that Ψ(e_{SV}) is the constant function equal to 1 on V′ and

Ψ ∘ i_{SV} = Φ. (31.11.4)

This corresponds to taking B = c(V′, k) in the previous section. Note that Ψ maps SV onto the subalgebra of c(V′, k) generated by Φ(V) and the constant functions.

Let ϕ_{0} ∈ V′ be given, and put

E_{ϕ_{0}}(f) = f(ϕ_{0}) (31.11.5)

for every f ∈ c(V′, k). This defines an algebra homomorphism from c(V′, k) into k. Observe that E_{ϕ_{0}} ∘ Φ = ϕ_{0}, (31.11.6)

as a mapping from V into k, by construction.

Clearly E_{ϕ_{0}} ∘Ψ is an algebra homomorphism from SV into k that sends e_{SV} to 1. We also have that

E_{ϕ_{0}} ∘ Ψ ∘ i_{SV} = E_{ϕ_{0}} ∘ Φ = ϕ_{0}, (31.11.7)

by (31.11.4) and (31.11.6). This implies that

E_{ϕ_{0}} ∘ Ψ = ψ_{0}, (31.11.8)

where ψ_{0} is as in (31.11.2).

Suppose now that V is free as a module over k, with positive finite rank n, so that V can be identified with k^{n}, as a module over k. In this case, every element of V′ can be expressed in a unique way as a linear combination of the coordinates of an element of k^{n} with coefficients in k, so that V′ can be identified with k^{n} too, as a module over k.

Similarly, the elements of V′′ can be expressed in a unique way as a linear combination of the coordinates of an element of V′ with coefficients in k. This means that the mapping Φ defined earlier is an isomorphism from V onto V′′, as modules over k. Thus Ψ maps SV onto the subalgebra of c(V′, k) generated by V′′ and the constant functions.

Under these conditions, SV can be identified with the algebra of formal polynomials in n commuting indeterminates with coefficients in k, as in Section 25.8. With these identifications, Ψ corresponds to the usual mapping from formal polynomials to polynomial functions on k^{n}.

31.12 Polynomials and symmetric algebras

Let k be a commutative ring with a multiplicative identity element, let V be a module over k, and let c(V, k) be the set of all functions on V with values in k. Thus c(V, k) is a module over k with respect to pointwise addition and scalar multiplication, and a commutatitive associative algebra over k with respect to pointwise multiplication of functions. Put V′ = Hom_{k}(V, k) again, which may be considered as a submodule of c(V, k), as a module over k.

Let SV′ be the symmetric algebra associated to V′, as in Section 25.8. This is a commutative associative algebra over k, with a multiplicative identity element e_{SV′}. This also comes with a homomorphism i_{SV′} from V′ into SV′, as modules over k, as before.

Let B be a commutative associative algebra over k, with a multiplicative identity element e_{B}, and let ϕ be a homomorphism from V′ into B, as modules over k. Under these conditions, there is a unique algebra homomorphism ψ from SV′ into B such that ψ(e_{SV′}) = e_{B} and

ψ ∘ i_{SV′} = ϕ, (31.12.1)

as in Section 25.8.

Let us take B = c(V, k), whose multiplicative identity element is the constant function equal to 1 ∈ k on V. The natural inclusion mapping from V′ into c(V, k) leads to an algebra homomorphism from SV′ into c(V, k), as in the preceding pargraph.

One may consider SV′ as the algebra of polynomials associated to V, as on p126 of [14]. The algebra homomorphism from SV′ into c(V, k) mentioned in the previous paragraph maps SV′ onto the subalgebra of c(V, k) generated by V′ and the constant functions.

Suppose for the moment that V is a free module over k of positive finite rank n, so that V can be identified with k^{n}, as a module over k. In this case, V′ can be identified with k^{n} too, as a module over k, as in the previous section. Equivalently, V′ is freely generated by the n coordinate functions on k^{n}, as a module over k.

Using this, one can identify SV′ with the algebra of formal polynomials in n commuting indeterminates with coefficients in k, as in Section 25.8. With these identifications, the algebra homomorphism from SV′ into c(V, k) corresponds to the usual mapping from formal polynomials to polynomial functions on k^{n}.

Let V be any module over k again, let W be another module over k, and put W′ = Hom_{k}(W, k), as before. If ξ is a module homomorphism from W into V, then there is a natural dual homomorphism ξ′ from V′ into W′. This is defined by composing a module homomorphism from V into k with ξ, to get a module homomorphism from W into k.

Let SW′ be the symmetric algebra associated to W′, with multiplicative identity element e_{SW′} and module homomorphism i_{SW′} from W′ into SW′. Note that i_{SW′} ∘ ξ′ (31.12.2)

is a homomorphism from V′ into SW′, as modules over k. This leads to a unique algebra homomorphism η from SV′ into SW′ such that η(e_{SV′}) = e_{SW′} and

η ∘ i_{SV′} = i_{SW′} ∘ ξ′, (31.12.3)

as before.

If ξ is a module isomorphism from W onto V, then ξ′ is a module isomorphism from W′ onto V′. This implies that η is an algebra isomorphism from SV′ onto SW′, as in Section 31.10.

Suppose that V, W are free as modules over k, with positive finite ranks n, m, respectively. Thus V, V′ and W, W′ can be identified with k^{n} and k^{m}, respectively, as modules over k, as before.

In this case, a module homomorphism ξ from W into V corresponds exactly to an n × m matrix with entries in k in the usual way. The corresponding dual homomorphism ξ′ corresponds to the transpose of this matrix in the same way. The algebra homomorphism η from SV′ into SW′ obtained from ξ as in (31.12.3) corresponds to the analogous algebra homomorphism obtained from an n × m matrix with entries in k in Section 31.3.

31.13 Polynomials on Lie algebras

Let k be a field with infinitely many elements, and let (A, [·, ·]_{A}) be a Lie algebra over k of positive finite dimension n. Remember that the set R(A) of regular elements of A is dense in A with respect to the Zariski topology, as in Section 31.6. Of course, this uses the identification of A with k^{n}, as a vector space over k, as before.

Let C(A) be the union of all of the Cartan subalgebras of A. Observe that

R(A) ⊆ C(A), (31.13.1)

because every element of R(A) is contained in a Cartan subalgebra of A, as mentioned in Section 31.7. It follows that C(A) is dense in A with respect to the Zariski topology.

Suppose from now on in this section that k is an algebraically closed field of characteristic 0. Let E(A) be the subgroup of the group of all Lie algebra automorphisms of A generated by exponentials of strongly ad-nilpotent elements of A, as in Section 24.3. Also let f be a polynomial function on A that is invariant under E(A).

If C is a Cartan subalgebra of A and

(31.13.2) f(x) = 0 for every x ∈ C,

then (31.13.3) f(x) = 0 for every x ∈ A.

Indeed, remember that every Cartan subalgebra of A is conjugate to C by an element of E(A), as in Section 24.9. Thus (31.13.2) implies that

(31.13.4) f(x) = 0 for every x ∈ C(A),

because f is invariant under E(A), by hypothesis. This implies (31.13.3), because C(A) is Zariski dense in A, as before. This corresponds to some remarks on p132, 134 of [14].

Some remarks and helpful facts

32.1 Roots and vector spaces

Let k be a field of characteristic 0, and let A_{0} be a vector space over k of positive finite dimension n. Also let A′_{0} be the dual space of linear functionals on A_{0}, as usual, and let Φ be a finite set of nonzero elements of A′_{0} whose linear span is A′_{0}. We would like to consider conditions on A_{0} and Φ like those in Sections 22.11, 28.7, 29.4, and 33.1, without the larger Lie algebra.

Of course, we may consider A_{0} and A′_{0} as vector spaces over Q, using the natural embedding of Q into k. Let E_{Q} be the linear subspace of A′_{0}, as a vector space over Q, spanned by Φ. We can use E_{Q} to get a vector space E_{R} over R, as in Section 17.13. Thus E_{Q} corresponds to a linear subspace of E_{R}, as a vector space over Q, and any basis for E_{Q}, as a vector space over Q, is a basis for E_{R}, as a vector space over R, by construction.

We ask that Φ be a root system in E_{R}. If α ∈ Φ, then let σ_{α} be the symmetry on E_{R} with vector α that maps Φ onto itself, as usual. Note that

σ_{α}(E_{Q}) = E_{Q}. (32.1.1)

Let λ_{α} be the linear functional on E_{R} corresponding to σ_{α}, so that σ_{α} is the identity mapping on E_{R} minus λ_{α} times α. Remember that λ_{α} takes integer values on Φ, by definition of a root system, which means that λ_{α} takes values in Q on E_{Q}.

Let Δ be a base for Φ as a root system in E_{R}. The elements of Δ are automatically linearly independent in E_{Q}, as a vector space over Q, because they are linearly independent in E_{R}, as a vector space over R. Observe that E_{Q} is spanned by Δ, as a vector space over Q, because E_{Q} is spanned by Φ, and every element of Φ can be expressed as a linear combination of elements of Δ with integer coefficients. This means that

(32.1.2) Δ is a basis for E_{Q}, as a vector space over Q.

Similarly, A′_{0} is spanned by Δ, as a vector space over k, because A′_{0} is spanned by Φ, by hypothesis. We ask that

(32.1.3) Δ be a basis for A′_{0},

as a vector space over k. Equivalently, this means that

(32.1.4) dim_{k}A′_{0} = dim_{Q}E_{Q},

where the subscripts indicate the fields over which the dimensions of the corresponding vector spaces are taken. This is the same as saying that

dim_{Q}E_{Q} = n, (32.1.5)

because dim_{k}A′_{0} = dim_{k}A_{0} = n, (32.1.6)

by hypothesis.

Let T be an automorphism of Φ in E_{R}, so that T is a one-to-one linear mapping from E_{R} onto itself such that T (Φ) = Φ. This implies that

T (E_{Q}) = E_{Q}. (32.1.7)

The restriction of T to E_{Q} is a one-to-one linear mapping from E_{Q} onto itself, as a vector space over Q. Every linear mapping from E_{Q} into itself, as a vector space over Q, has a unique extension to a linear mapping from E_{R} into itself, as a vector space over R. A one-to-one linear mapping from E_{Q} onto itself extends to a one-to-one linear mapping from E_{R} onto itself.

Similarly, every linear mapping from E_{Q} into itself, as a vector space over Q, has a unique extension to a linear mapping from A′_{0} into itself, as a vector space over k, because of (32.1.3). The extension of a one-to-one linear mapping from E_{Q} onto itself is a one-to-one linear mapping from A′_{0} onto itself. If R is a one-to-one linear mapping from A′_{0} onto itself, as a vector space over k, and R(Φ) = Φ, then R(E_{Q}) = E_{Q}. (32.1.8)

The restriction of R to E_{Q} is a one-to-one linear mapping from E_{Q} onto itself, as a vector space over Q.

It follows that the group Aut(Φ) of automorphisms of Φ in E_{R} can be identified with the group of one-to-one linear mappings from E_{Q} onto itself, as a vector space over Q, that send Φ onto itself. This can also be identified with the group of one-to-one linear mappings from A′_{0} onto itself, as a vector space over k, that send Φ onto itself. Similarly, the Weyl group of Φ can be identified with a group of one-to-one linear mappings from E_{Q} onto itself, or a group of one-to-one linear mappings from A′_{0} onto itself.

32.2 Roots and dual spaces

Let us continue with the same notation and hypotheses as in the previous section. If α ∈ Φ, then λ_{α} is an element of the dual E′_{R} of E_{R}, as a vector space over R. Put Φ′ = {λ_{α} : α ∈ Φ}, (32.2.1)

which is a root system in E′_{R}, as in Section 19.8.

Similarly, let E′_{Q} be the dual of E_{Q}, as a vector space over Q. Every element of E′_{Q} has a unique extension to an element of E′_{R}, so that E′_{Q} may be identified with a subset of E′_{R}. Equivalently, the elements of E′_{Q} correspond to the elements of E′_{R} that take values in Q on E_{Q}. Thus

(32.2.2) E′_{Q} corresponds to a linear subspace of E′_{R}, as a vector space over Q.

If α ∈ Φ, then λ_{α} takes values in Q on E_{Q}, as in the previous section, so that Φ′ corresponds to a subset of E′_{Q}. Let Δ be a base of Φ, and put

Δ′ = {λ_{β} : β ∈ Δ}. (32.2.3)

If Φ is reduced as a root system in E_{R}, then Δ′ is a base for Φ′, as in Section 19.13. Otherwise, one can get a base for Φ by replacing λ_{β} with λ_{β}/2 when β ∈ Δ and 2 β ∈ Φ, as before. In either case,

(32.2.4) Δ′ is a basis for E′_{R},

as a vector space over R.

In particular, the elements of Δ′ are linearly independent in E′_{R}, as a vector space over R. Of course, this means that the elements of Δ′ are linearly independent in E′_{R} as a vector space over Q. This implies that Δ′ corresponds to a linearly independent set in E′_{Q}, as a vector space over Q. More precisely, this uses the fact that if a linear combination of elements of Δ′ is equal to 0 on E_{Q}, then it is equal to 0 on E_{R} too.

Remember that (32.2.5) dim_{Q}E_{Q} = dim_{R}E_{R},

by construction. This implies that

(32.2.6) dim_{Q}E′_{Q} = dim_{R}E′_{R}.

It follows that (32.2.7) Δ′ corresponds to a basis for E′_{Q},

as a vector space over Q.

Let A′′_{0} be the dual of A′_{0}, as a vector space over k. Every element of E′_{Q} has a unique extension to an element of A′′_{0}, because Δ is a basis for A′_{0}, as a vector space over k. Thus E′_{Q} corresponds to a subset of A′′_{0}, whose elements are the linear functionals on A′_{0} that take values in Q on E_{Q}, with respect to the natural embedding of Q into k. More precisely,

(32.2.8) E′_{Q} corresponds to a linear subspace of A′′_{0}, as a vector space over Q.

Remember that there is a natural isomorphism from A_{0} onto A′′_{0}, as vector spaces over k. This isomorphism sends w ∈ A_{0} to the linear functional on A′_{0} defined by evaluating an element of A′_{0} at w.

If α ∈ Φ, then there is a unique element h_{α} of A_{0} such that

β(h_{α}) = λ_{α}(β) (32.2.9)

for every β ∈ Φ. The right side is an integer, by definition of a root system, which may be considered as an element of k, using the natural embedding of Q into k. The uniqueness of h_{α} follows from the fact that A′_{0} is spanned by Φ, as a vector space over k.

To get the existence of h_{α}, we consider λ_{α} as an element of E′_{Q}, which can be extended to a unique element of A′′_{0}, as before. This element of A′′_{0} can be expressed in terms of evaluation at an element of A_{0}, which is h_{α}. More precisely, (32.2.9) holds for every β ∈ E_{Q}. By construction,

β ↦ β(h_{α}) (32.2.10)

is the linear functional on A′_{0} whose restriction to E_{Q} is the same as λ_{α}.

Remember that σ_{α} sends E_{Q} onto itself, as in the previous section. The restriction of σ_{α} to E_{Q} has a unique extension to a one-to-one linear mapping from A′_{0} onto itself, as a vector space over k, as before. This extension is given by β ↦ β − β(h_{α}) α, (32.2.11)

by the remarks in the previous two paragraphs.

32.3 Dual spaces and abstract weights

Let us continue with the same notation and hypotheses as in the previous two sections. Remember that E_{Q} is the linear span of Φ in A′_{0}, as a vector space over Q. We may also consider E_{Q} to be the linear span of Φ in E_{R}, as a vector space over Q.

Remember that the restrictions of the λ_{α}’s, α ∈ Δ, to E_{Q} form a basis for E′_{Q}, as a vector space over Q. This implies that

(32.3.1) any function of α ∈ Δ with values in Q can be obtained by evaluating λ_{α} at an element of E_{Q}.

If z ∈ E_{R}, then z is uniquely determined by λ_{α}(z), α ∈ Δ, because the λ_{α}’s, α ∈ Δ, form a basis for E′_{R}. If λ_{α}(z) ∈ Q for every α ∈ Δ, then there is an element of E_{Q} for which the values of λ_{α}, α ∈ Δ, are the same, as in the preceding paragraph. This means that z ∈ E_{Q}, because z is determined by λ_{α}(z), α ∈ Δ. Thus

(32.3.2) E_{Q} = {z ∈ E_{R} : λ_{α}(z) ∈ Q for every α ∈ Δ},

because λ_{α} maps E_{Q} into Q for every α ∈ Φ, as before. Of course, it follows that (32.3.3) E_{Q} = {z ∈ E_{R} : λ_{α}(z) ∈ Q for every α ∈ Φ}.

If β ∈ A′_{0}, then put f_{β}(α) = β(h_{α}) (32.3.4)

for every α ∈ Δ, so that f_{β} defines a k-valued function on Δ. Of course, the set of k-valued functions on Δ is a vector space over k, with respect to pointwise addition and scalar multiplication. Clearly

{f_{β} : β ∈ A′_{0}} (32.3.5)

is a linear subspace of the space of all k-valued functions on Δ, because A′_{0} is a vector space over k, and f_{β} is linear in β over k.

If β ∈ E_{Q}, then f_{β}(α) = λ_{α}(β) (32.3.6)

for every α ∈ Δ, as in (32.2.9), where the right side is considered as an element of k, using the natural embedding of Q into k. Every Q-valued function on Δ corresponds to some β ∈ E_{Q} in this way, as in (32.3.1). This implies that every k-valued function on Δ can be expressed as f_{β} for some β ∈ A′_{0}, because (32.3.5) is a linear subspace of the space of all k-valued functions on Δ, as in the preceding paragraph.

It follows that

(32.3.7) the h_{α}’s, α ∈ Δ, are linearly independent in A_{0},

because otherwise the values of the f_{β}’s, β ∈ A′_{0}, on Δ would satisfy a nontrivial linear relation. Note that the dimension of A_{0}, as a vector space over k, is equal to the number of elements of Δ, by the remarks in Section 32.1. This means that (32.3.8) {h_{α} : α ∈ Δ} is a basis for A_{0},

as a vector space over k.

In particular, every element of A′_{0} is uniquely determined by its values on the h_{α}’s, α ∈ Δ. If ν ∈ A′_{0} and ν(h_{α}) corresponds to an element of Q for every α ∈ Δ, under the natural embedding of Q into k, then it follows that ν ∈ E_{Q}, because of (32.3.1). This shows that

(32.3.9) E_{Q} = {ν ∈ A′_{0} : ν(h_{α}) ∈ Q for every α ∈ Δ}.

We also have that

(32.3.10) E_{Q} = {ν ∈ A′_{0} : ν(h_{α}) ∈ Q for every α ∈ Φ}, 32.3. DUAL SPACES AND ABSTRACT WEIGHTS 675 because ν(h_{α}) = λ_{α}(ν) ∈ Q for every α ∈ Φ when ν ∈ E_{Q}, as in (32.2.9).

Put (32.3.11) Υ = Υ_{Φ} = {z ∈ E_{R} : λ_{α}(z) ∈ Z for every α ∈ Φ}, as in Section 30.1. If Φ is reduced as a root system in E_{R}, then (32.3.12) Υ = {z ∈ E_{R} : λ_{α}(z) ∈ Z for every α ∈ Δ}, because the collection Δ′ of λ_{α}, α ∈ Δ, is a base for Φ′, as before. Note that Υ ⊆ E_{Q}, (32.3.13) by (32.3.3). Thus (32.3.14) Υ = {z ∈ E_{Q} : λ_{α}(z) ∈ Z for every α ∈ Φ}, and (32.3.15) Υ = {z ∈ E_{Q} : λ_{α}(z) ∈ Z for every α ∈ Δ} when Φ is reduced.

Let us check that (32.3.16) Υ = {ν ∈ A′_{0} : ν(h_{α}) ∈ Z for every α ∈ Φ}. If ν ∈ A′_{0} satisfies ν(h_{α}) ∈ Z for every α ∈ Φ, then ν ∈ E_{Q}, by (32.3.10). This means that ν(h_{α}) = λ_{α}(ν) for every α ∈ Φ, as in (32.2.9). It follows that ν ∈ Υ, by (32.3.14). It is easy to see that Υ is contained in the right side of (32.3.16), using (32.3.14) and the fact that E_{Q} ⊆ A′_{0}, by construction. Similarly, (32.3.17) Υ = {ν ∈ A′_{0} : ν(h_{α}) ∈ Z for every α ∈ Δ} when Φ is reduced as a root system in E_{R}. This uses (32.3.9) and (32.3.15), to get that the right side is contained in Υ.

As in Section 30.3, we put (32.3.18) Υ^{+} = Υ_{Φ,Δ}^{+} = {z ∈ Υ : λ_{α}(z) ≥ 0 for every α ∈ Δ}. This is the same as (32.3.19) Υ^{+} = {z ∈ Υ : λ_{α}(z) ≥ 0 for every α ∈ Φ^{+}}, where Φ^{+} is the set of positive roots with respect to Δ, as usual. Equivalently,

(32.3.20)  Υ^{+} = {ν ∈ Υ : ν(h_{α}) ≥ 0 for every α ∈ Δ}
= {ν ∈ Υ : ν(h_{α}) ≥ 0 for every α ∈ Φ^{+}},

by (32.2.9) and (32.3.13). An element of A′_{0} may be called integral if it is in Υ, and dominant integral if it is in Υ^{+}, as on p112 of [14].

32.4 Abstract weights and group rings

We continue with the same notation and hypotheses as in the previous three sections. More precisely, in this section it is enough to have Φ as a root system in E_{R}, and to let Υ be as in (32.3.11). However, it will be helpful later on to consider Φ and Υ as being contained in A′_{0}.

Remember that Υ is a commutative group with respect to addition, which may be considered as a subgroup of E_{Q}, E_{R}, or A′_{0}. We would like to consider the group ring Z[Υ] of Υ with coefficients in Z, which may also be considered as the group algebra of Υ over Z. Normally the elements of Z[Υ] might be defined as formal linear combinations of elements of Υ with integer coefficients, where multiplication in Z[Υ] is defined using the group operation on Υ. Because the group operation on Υ is addition, it is better to express the embedding of Υ into Z[Υ] another way.

Thus we take Z[Υ] to be the free module over Z with distinct basis elements denoted e_{α}, α ∈ Υ. If α, β ∈ Υ, then we put

e_{α} e_{β} = e_{α+β}, (32.4.1)

using addition in Υ on the right. This can be used to define multiplication in Z[Υ], which is bilinear over Z. Note that e_{0} is the multiplicative identity element in Z[Υ]. This corresponds to some remarks on p124 of [14], and on p63 of [24].

Alternatively, consider the set c_{00}(Υ, Z) of all Z-valued functions on Υ with finite support, which is to say that they are equal to 0 at all but finitely many elements of Υ. This is a commutative group with respect to pointwise addition of functions. If f, g ∈ c_{00}(Υ, Z), then their convolution product is the Z-valued function on Υ defined by

(f ∗ g)(γ) = ∑_{α+β=γ} f(α) g(β) (32.4.2)

for every γ ∈ Υ. More precisely, the sum on the right is taken over all α, β ∈ Υ with α + β = γ. This reduces to a finite sum when either f or g has finite support in Υ. If f and g both have finite support in Υ, then f ∗ g has finite support in Υ too, and is thus an element of c_{00}(Υ, Z). One can check that c_{00}(Υ, Z) is a commutative ring with respect to convolution.

If α ∈ Υ, then one can take e_{α} to be the k-valued function on Υ equal to 1 at α, and to 0 at all other elements of Υ. This is an element of c_{00}(Υ, Z), and c_{00}(Υ, Z) is a free module over Z with basis e_{α}, α ∈ Υ. It is easy to see that (32.4.1) holds for every α, β ∈ Υ, using the convolution product on the left side. Note that automorphisms of Υ as a commutative group lead to ring automorphisms of Z[Υ].

32.5 Homomorphisms and Lie algebras

Let k be a commutative ring with a multiplicative identity element, and let (A_{1}, [·, ·]_{A_{1}}), (A_{2}, [·, ·]_{A_{2}}) be Lie algebras over k. Also let UA_{1}, UA_{2} be universal enveloping algebras of A_{1}, A_{2}, respectively, as in Section 25.4. Thus UA_{1}, UA_{2} are associative algebras over k with multiplicative identity elements e_{UA_{1}}, e_{UA_{2}}, respectively. We have Lie algebra homomorphisms i_{UA_{1}}, i_{UA_{2}} from A_{1}, A_{2} into UA_{1}, UA_{2}, respectively, as well.

Suppose that ξ_{1} is a Lie algebra homomorphism from A_{1} into A_{2}. Of course, this implies that i_{UA_{2}} ∘ ξ_{1} (32.5.1)

is a Lie algebra homomorphism from A_{1} into UA_{2}. It follows that there is a unique algebra homomorphism η_{1} from UA_{1} into UA_{2} such that η_{1}(e_{UA_{1}}) = e_{UA_{2}} and η_{1} ∘ i_{UA_{1}} = i_{UA_{2}} ∘ ξ_{1}, (32.5.2)

as in Section 25.4.

Let (A_{3}, [·, ·]_{A_{3}}) be another Lie algebra over k, and let UA_{3} be a universal enveloping algebra of A_{3}, with multiplicative identity element e_{UA_{3}} and Lie algebra homomorphism i_{UA_{3}} from A_{3} into UA_{3}. If ξ_{2} is a Lie algebra homomorphism from A_{2} into A_{3}, then

i_{UA_{3}} ∘ ξ_{2} (32.5.3)

is a Lie algebra homomorphism from A_{2} into UA_{3}. This leads to a unique algebra homomorphism η_{2} from UA_{2} into UA_{3} such that η_{2}(e_{UA_{2}}) = e_{UA_{3}} and

η_{2} ∘ i_{UA_{2}} = i_{UA_{3}} ∘ ξ_{2}, (32.5.4)

as before.

It follows that ξ_{2} ∘ ξ_{1} is a Lie algebra homomorphism from A_{1} into A_{3}, so that i_{UA_{3}} ∘ ξ_{2} ∘ ξ_{1} (32.5.5)

is a Lie algebra homomorphism from A_{1} into UA_{3}. Observe that η_{2} ∘ η_{1} is an algebra homomorphism from UA_{1} into UA_{3} with

(η_{2} ∘ η_{1})(e_{UA_{1}}) = η_{2}(η_{1}(e_{UA_{1}})) = η_{2}(e_{UA_{2}}) = e_{UA_{3}} (32.5.6)

and η_{2} ∘ η_{1} ∘ i_{UA_{1}} = η_{2} ∘ i_{UA_{2}} ∘ ξ_{1} = i_{UA_{3}} ∘ ξ_{2} ∘ ξ_{1}. (32.5.7)

More precisely, η_{2} ∘ η_{1} is uniquely determined by these properties, because UA_{1} is a universal enveloping algebra of A_{1}.

Suppose that A_{1} = A_{3}, UA_{1} = UA_{3}, and i_{UA_{1}} = i_{UA_{3}}. If ξ_{2} ∘ ξ_{1} is the identity mapping on A_{1}, then it follows that η_{2} ∘ η_{1} is the identity mapping on UA_{1}, by uniqueness.

If ξ_{1} ∘ ξ_{2} is the identity mapping on A_{2}, then η_{1} ∘ η_{2} is the identity mapping on UA_{2}, for the same reasons. If ξ_{1} is a Lie algebra isomorphism from A_{1} onto A_{2}, then we get that η_{1} is an algebra isomorphism from UA_{1} onto UA_{2}. Of course, these remarks reduce to those in Section 31.10 when the Lie algebras are commutative.

Let us now take A_{1} = A_{2}, UA_{1} = UA_{2}, and i_{UA_{1}} = i_{UA_{2}}. If ξ_{1} is a Lie algebra automorphism of A_{1}, then η_{1} is an algebra automorphism of UA_{1}.

32.6 Submodules of UA

Let k be a commutative ring with a multiplicative identity element again, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let UA be a universal enveloping algebra of A, as in Section 25.4. As usual, UA is an associative algebra over k with a multiplicative identity element e = e_{UA}, and we have a Lie algebra homomorphism i = i_{UA} from A into UA.

If a ∈ A and w ∈ UA, then put

δ_{a}(w) = [i(a), w] = i(a) w − w i(a). (32.6.1)

This defines a derivation on UA, as an associative algebra over k, as in Section 2.5. Of course, δ_{a}(i(x)) = [i(a), i(x)] = i([a, x]_{A}) (32.6.2)

for every a, x ∈ A, because i is a Lie algebra homomorphism from A into UA.

In fact, UA is a module over A, as a Lie algebra over k, with respect to (32.6.1). Remember that UA is a Lie algebra over k, with respect to the commutator bracket defined using multiplication in UA, because UA is an associative algebra over k. Thus UA may be considered as a module over itself, as a Lie algebra over k, with respect to the adjoint representation. Equivalently, the adjoint representation on UA is a Lie algebra homomorphism from UA into the Lie algebra of homomorphisms from UA into itself, as a module over k. By construction, a ↦ δ_{a} is the same as the composition of i with the Lie algebra homomorphism just mentioned, and thus defines a Lie algebra homomorphism from A into the the Lie algebra of module homomorphisms from UA into itself, as a module over k.

If n is a nonnegative integer, then let U_{n}A be the submodule of UA, as a module over k, generated by products of at most n elements of i(A), as in Section 25.9. We interpret e as being a product of 0 elements of i(A), so that e ∈ U_{n}A for every n ≥ 0, as before. If x_{1}, …, x_{m} ∈ A, then

δ_{a}(i(x_{1}) ⋯ i(x_{m})) (32.6.3)

= ∑_{j=1}^{m} i(x_{1}) ⋯ i(x_{j−1}) δ_{a}(i(x_{j})) i(x_{j+1}) ⋯ i(x_{m}),

because δ_{a} is a derivation on UA. This implies that

δ_{a}(i(x_{1}) ⋯ i(x_{m})) (32.6.4)

= ∑_{j=1}^{m} i(x_{1}) ⋯ i(x_{j−1}) i([a, x_{j}]_{A}) i(x_{j+1}) ⋯ i(x_{m}),

by (32.6.2). It follows that

(32.6.5) U_{n}A is a submodule of UA, as a module over A,

for every n ≥ 0.

32.7. SOME AUTOMORPHISMS OF UA 679

Remember that ⋃_{n=0}^{∞} U_{n}A = UA, as in Section 25.9. If A is finitelygenerated as a module over k, then it is easy to see that U_{n}A is finitely-generated as a module over k for every n ≥ 0. In particular, if k is a field, and A has finite dimension as a vector space over k, then U_{n}A has finite dimension as a vector space over k for every n ≥ 0. This corresponds to Exercise 3 on p95 of [14].

32.7 Some automorphisms of UA

Let k be a field of characteristic 0, and let (A, [·, ·]_{A}) be a Lie algebra over k. Suppose that a ∈ A is ad-nilpotent, so that ad_{A,a} is nilpotent as a linear mapping on A. Thus exp ad_{A,a} defines a Lie algebra automorphism of A, as in Section 14.11.

Let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA}, and Lie algebra homomorphism i = i_{UA} from A into UA, as in Section 25.4 again. There is a unique algebra automorphism η of UA such that η(e) = e and η ∘ i = i ∘ (exp ad_{A,a}), (32.7.1)

as in Section 32.5.

Let δ_{a} be defined on UA as in the previous section. If x_{1}, …, x_{m} ∈ A, then it is easy to see that (δ_{a})^{l}(i(x_{1}) ⋯ i(x_{m})) = 0 (32.7.2)

when l is sufficiently large, depending on m. This implies that for each nonnegative integer n, the restriction of δ_{a} to U_{n}A is nilpotent, where U_{n}A is as in the previous section.

In particular, δ_{a} is locally nilpotent on UA, as in Section 27.12. It follows thst exp δ_{a} can be defined as a linear mapping from UA into itself, as before. More precisely, exp δ_{a} is invertible as a linear mapping on UA, with inverse equal to exp(−δ_{a}). In fact, exp δ_{a} is an algebra automorphism of UA, because δ_{a} is a derivation on UA, as in the previous section. Note that exp δ_{a} maps e to itself.

Clearly δ_{a} ∘ i = i ∘ ad_{A,a}, by construction. Using this, one can check that

(32.7.3) (exp δ_{a}) ∘ i = i ∘ (exp ad_{A,a}).

Thus η = exp δ_{a}, by uniqueness.

32.8 Polarization

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. If a ∈ k, then let v(a) be the element of k^{n} defined by

v(a) = (1, a, a^{2}, …, a^{n−1}). (32.8.1)

If a_{1}, …, a_{n} are n elements of k, then v(a_{1}), …, v(a_{n}) make up an n × n matrix with entries in k. The determinant of this matrix is the well-known Vandermonde determinant

∏_{1≤j<l≤n}(a_{l} − a_{j}). (32.8.2)

Suppose now that k is a field, so that k^{n} is a vector space over k, with respect to coordinatewise addition and scalar multiplication. If a_{1}, …, a_{n} are distinct elements of k, then (32.8.2) is not equal to 0. This implies that v(a_{1}), …, v(a_{n}) form a basis for k^{n}, as a vector space over k.

Let X, Y be commuting indeterminates, and remember that

(a X + Y)^{n} = ∑_{j=0}^{n} \binom{n}{j} a^{j} X^{j} Y^{n−j} (32.8.3)

for every a ∈ k, by the binomial theorem. If k has characteristic 0 and a_{1}, …, a_{n}, a_{n+1} are n+1 distinct elements of k, then every homogeneous formal polynomial in X, Y with coefficients in k and of degree n can be expressed as a linear combination of (a_{j} X + Y)^{n}, 1 ≤ j ≤ n + 1. This can be obtained from the remarks in the preceding paragraph, with n replaced by n + 1, and using the fact that \binom{n}{j} is nonzero for every j = 0, …, n.

Let r be a positive integer, and let T_{1}, …, T_{r} be r commuting indeterminates. If k has characteristic 0, then it is well known that every formal polynomial in T_{1}, …, T_{r} with coefficients in k can be expressed as a linear combination of powers of linear polynomials. This is known as polarization, and can be seen using induction on r and the remarks in the previous paragraph, as in Exercise 5 on p134 of [14].

More precisely, if E_{1}, …, E_{r} are infinite subsets of k, then it suffices to use powers of linear polynomials for which the coefficient of T_{j} is an element of E_{j} for each j = 1, …, r. In particular, it is enough to use powers of linear polynomials for which the coefficient of T_{j} corresponds to an integer for each j, under the natural embedding of Q into k.

Suppose that k has characteristic 0, and let V be a vector space over k of dimension r. Thus the dual space V′ of linear functionals on V has dimension r as well, as a vector space over k. Polynomial functions on V can be obtained using a basis for V, and formal polynomials in r commuting indeterminates with coefficients in k. Equivalently, the algebra of polynomial functions on V is generated by V′ and the constant functions on V.

Every polynomial function on V can be expressed as a linear combination of powers of elements of V′. This follows from the previous remarks about formal polynomials.

Let λ_{1}, …, λ_{r} be a basis for V′, as a vector space over k. Of course, polynomial functions on V can be expressed as polynomials in λ_{1}, …, λ_{r}, with coefficients in k.

Let Λ be the subgroup of V′, as a group with respect to addition, generated by λ_{1}, …, λ_{r}. This consists of the linear combinations of λ_{1}, …, λ_{r} whose coef-32.9. THE CENTER OF UA 681

ficients correspond to integers, with respect to the natural embedding of Q into k.

Every polynomial function on V can be expressed as a linear combination of powers of elements of Λ, by the analogous statement for formal polynomials mentioned earlier. This corresponds to a remark on p126 of [14].

32.9 The center of UA

Let k be a field of characteristic 0, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA}, and Lie algebra homomorphism i = i_{UA} from A into UA, as in Section 25.4. Consider the center Z of UA, as an associative algebra over k. Of course, Z is a subalgebra of UA that contains e.

If a ∈ A, then let δ_{a} be as in Section 32.6. Note that

δ_{a}(z) = 0 (32.9.1)

for every a ∈ A and z ∈ Z. More precisely, Z consists exactly of the z ∈ UA such that (32.9.1) holds for every a ∈ A, because UA is generated as an associative algebra over k by e and i(A).

Suppose that a is ad-nilpotent in A, so that δ_{a} is locally nilpotent on UA, as in Section 32.7. If z ∈ Z, then

(exp δ_{a})(z) = z, (32.9.2)

by (32.9.1).

Remember that Int A is the subgroup of the group of Lie algebra automorphisms of A generated by automorphisms of the form exp ad_{A,a}, where a ∈ A is ad-nilpotent, as in Section 14.11. We have also seen that every Lie algebra automorphism of A leads to an algebra automorphism of UA, as in Section 32.5. If a ∈ A is ad-nilpotent, then the algebra automorphism of UA corresponding to exp ad_{A,a} is exp δ_{a}, as in Section 32.7. If z ∈ Z, then z is fixed by these automorphisms on UA, as in (32.9.2). This means that z is fixed by all of the automorphisms of UA corresponding to elements of Int A, as on p128 of [14].

Suppose that a ∈ A is ad-nilpotent again, so that t a is ad-nilpotent in A for every t ∈ k. Similarly, δ_{t a} = t δ_{a} is locally nilpotent on UA for every t ∈ k. If w ∈ UA satisfies (exp t δ_{a})(w) = w (32.9.3)

for every t ∈ k, then we would like to show that

δ_{a}(w) = 0. (32.9.4)

This corresponds to some remarks on p128f of [14].

Let U_{n}A be as in Section 32.6 for each nonnegative integer n, and let n_{0} be a nonnegative integer such that w ∈ U_{n_{0}}A. Remember that the restriction of δ_{a} to U_{n_{0}}A is nilpotent, as in Section 32.7. Thus there is a positive integer l_{0} such that (δ_{a})^{l_{0}} = 0 on U_{n_{0}}A. If t ∈ k, then we get that

exp t δ_{a} = ∑_{j=0}^{l_{0}−1}(t^{j}/j!) (δ_{a})^{j} (32.9.5)

on U_{n_{0}}A.

Let t_{1}, …, t_{l_{0}} be l_{0} distinct elements of k. Under these conditions, there are l_{0} elements c_{1}, …, c_{l_{0}} of k such that

∑_{r=1}^{l_{0}} c_{r} (exp t_{r} δ_{a}) = δ_{a} (32.9.6)

on U_{n_{0}}A. This uses (32.9.5) and the remarks at the beginning of the previous section.

It follows that

δ_{a}(w) = ( ∑_{r=1}^{l_{0}} c_{r}) w, (32.9.7)

by (32.9.3), and because w ∈ U_{n_{0}}A. This implies (32.9.4), because (δ_{a})^{l_{0}}(w) = 0.

Suppose that w ∈ UA is invariant under all of the automorphisms of UA that correspond to elements of Int A, as before. This implies that (32.9.3) holds for every a ∈ A that is ad-nilpotent, and every t ∈ k. This means that (32.9.4) holds for every a ∈ A that is ad-nilpotent.

Remember that UA is a module over A, as a Lie algebra over k, with respect to the action of a ∈ A defined by δ_{a}, as in Section 32.6. If (32.9.4) holds for some collection of a ∈ A, then it holds for every a in the Lie subalgebra of A generated by this collection.

Suppose that A is generated, as a Lie algebra over k, by ad-nilpotent elements. If w ∈ UA is invariant under all of the automorphisms of UA that correspond to elements of Int A, then we get that (32.9.4) holds for every a ∈ A. This means that w ∈ Z, as in the lemma on p128 of [14].

32.10 Constant-coefficient differential operators

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Also let ∂_{1}, …, ∂_{n} be commuting formal symbols, which may be used to represent partial derivatives in some related commuting indeterminates, as in Section 5.11. A formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in k can be expressed as

∑_{|α|≤N} a^{α} ∂^{α}, (32.10.1)

32.10. CONSTANT-COEFFICIENT DIFFERENTIAL OPERATORS 683

where N is a nonnegative integer, the sum is taken over all multi-indices α of length n with |α| ≤ N, and a^{α} ∈ k for each such α. This is basically the same as a formal polynomial in ∂_{1}, …, ∂_{n}, considered as n commuting indeterminates, with coefficients in k. The usual way of multiplying formal differential operators corresponds to multiplication of formal polynomials in ∂_{1}, …, ∂_{n} with coefficients in k in this case.

Let T_{1}, …, T_{n} be n commuting indeterminates, so that formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] may be defined as in Section 5.11. Now ∂_{1}, …, ∂_{n} may be used to represent partial derivatives in T_{1}, …, T_{n}, as before, so that multiplication on k[[T_{1}, …, T_{n}]] may be extended to

(32.10.2) the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]],

as before. Note that

(32.10.3) the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k

corresponds to a commutative subalgebra of (32.10.2), with respect to multiplication of formal differential operators. Multiplication of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k, as formal differential operators with coefficients in k[[T_{1}, …, T_{n}]], corresponds exactly to multiplication of formal polynomials in ∂_{1}, …, ∂_{n} with coefficients in k, as in the preceding paragraph.

If ∂_{1}, …, ∂_{n} are simply considered as n commuting indeterminates, then (32.10.2) may be defined as a module over k as the space

(k[[T_{1}, …, T_{n}]])[∂_{1}, …, ∂_{n}] (32.10.4)

of formal polynomials in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]], as a module over k, as in Section 5.8. This is essentially the same as in Section 5.11, with some additional notation. Using this identification, (32.10.3) corresponds to

k[∂_{1}, …, ∂_{n}], (32.10.5)

as a submodule of (32.10.4), as a module over k. One may consider (32.10.4) as a module over k[[T_{1}, …, T_{n}]] too, as a commutative algebra over k, with respect to multiplication on the left by elements of k[[T_{1}, …, T_{n}]]. This corresponds to multiplication of an element of k[[T_{1}, …, T_{n}]], considered as an element of (32.10.2) of order 0, with another element of (32.10.2), as in Section 5.11.

We can also use multiplication on k[[T_{1}, …, T_{n}]] to define multiplication on (32.10.4) as in Section 5.8, so that ∂_{1}, …, ∂_{n} are not considered as being related to differentiation in T_{1}, …, T_{n}. This defines a commutative algebra over k, which is a commutative algebra over k[[T_{1}, …, T_{n}]] as well. If ∂_{1}, …, ∂_{n} may be used to represent partial derivatives in some related commuting indeterminates, then this would basically correspond to the derivatives of T_{1}, …, T_{n} being equal to 0. Of course, (32.10.5) is a subalgebra of (32.10.4) with respect to this definition of multiplication on (32.10.4). This definition of multiplication on (32.10.5) corresponds to multiplication of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k, as before.

Let L_{1}, L_{2} be formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] of order N_{1}, N_{2} ≥ 0, (32.10.6)

respectively. This means that the coefficients of ∂^{α} in L_{1}, L_{2} are equal to zero when α is a multi-index of length n with |α| strictly larger than N_{1}, N_{2}, respectively. If N_{1} = N_{2} = 0, then L_{1}, L_{2} correspond to elements of k[[T_{1}, …, T_{n}]], and the product of L_{1} and L_{2} as formal differential operators corresponds to their product as formal polynomials in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]]. Otherwise, the product of L_{1} and L_{2} as formal differential operators corresponds to their product as formal polynomials in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]], plus a formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] of order

N_{1} + N_{2} − 1. (32.10.7)

In particular, the commutator of L_{1} and L_{2} with respect to multiplication of formal differential operators is a formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in k[[T_{1}, …, T_{n}]] of order (32.10.7).

Part V

Some representation theory Representations and semisimplicity

33.1 Diagonalizability and roots

Let k be a field of characteristic 0, and let (A, [·, ·]_{A}) be a Lie algebra over k of positive finite dimension. We would like to consider conditions on A like those in Section 29.4. More precisely, we shall start here with conditions related to diagonalizability, and consider additional conditions related to roots. In particular, these conditions hold when k is algebraically closed, and A is semisimple.

Suppose that A_{0} is a Lie subalgebra of A, and that every element of A_{0} is ad-diagonalizable as an element of A. This implies that A_{0} is commutative as a Lie algebra, as in Section 17.1. Thus A_{0} is contained in its centralizer C_{A}(A_{0}) in A, and we ask that C_{A}(A_{0}) = A_{0}. (33.1.1)

Remember that this holds when k is algebraically closed, A is semisimple, and A_{0} is a maximal toral subalgebra of A, as in Section 17.4. Note that (33.1.1) implies that A_{0} ≠ {0}, because A ≠ {0}.

Let A′_{0} be the dual space of linear functionals on A_{0}, as a vector space over k, as usual. If α ∈ A′_{0}, then let A_{α} be the set of x ∈ A such that ad_{w}(x) = α(w) x for every w ∈ A_{0}, which is a linear subspace of A. The condition (33.1.1) says exactly that A_{α} is equal to A_{0} when α = 0.

Let Φ = Φ_{A_{0}} be the set of α ∈ A′_{0} such that α ≠ 0 and A_{α} ≠ {0}. Thus A corresponds to the direct sum of A_{α}, α ∈ Φ∪{0}, as a vector space over k, as in Section 17.2. Remember that [A_{α}, A_{β}] ⊆ A_{α+β} for every α, β ∈ A′_{0}, as before.

Under the conditions mentioned so far, the center Z(A) of A as a Lie algebra is contained in A_{0}, and in fact is the intersection of the kernels of the elements of Φ. We ask that Z(A) = {0}, which means that A′_{0} is spanned by Φ, as a vector space over k. This implies that Φ ≠ ∅, because A_{0} ≠ {0}.

We ask that for each α ∈ Φ, A_{α} have dimension one, as a vector space over k. We also ask that for every α ∈ Φ there be an element h_{α} of [A_{α}, A_{−α}] ⊆ A_{0} such that α(h_{α}) = 2 = 1 + 1 in k. This determines h_{α} uniquely, because k has characteristic 0, and [A_{α}, A_{−α}] has dimension at most one as a vector space over k. Note that A is semisimple under these conditions, as in Section 17.15. If α, β ∈ Φ and α + β ∈ Φ, then we ask that [A_{α}, A_{β}] = A_{α+β}.

Let E_{Q} be the linear subspace of A′_{0}, as a vector space over Q, spanned by Φ. This leads to a vector space E_{R} over R, as in Section 17.13. More precisely, E_{Q} corresponds to a linear subspace of E_{R}, as a vector space over Q, and any basis for E_{Q}, as a vector space over Q, is a basis for E_{R}, as a vector space over R.

We ask that Φ be a reduced root system in E_{R}, and we let Δ be a base for Φ. In particular, this means that Δ is a basis for E_{R}, as a vector space over R. This implies that Δ is a basis for E_{Q}, as a vector space over Q, because E_{Q} is spanned by Δ, as a vector space over Q.

Note that A′_{0} is spanned by Δ, as a vector space over k, because A′_{0} is spanned by Φ, as before. We ask that Δ be a basis for A′_{0}, as a vector space over k, which means that Δ is linearly independent in A′_{0}, as a vector space over k. We also ask that h_{α}, α ∈ Δ, form a basis for A_{0}, as a vector space over k.

Let α ∈ Φ be given, and let σ_{α} be the symmetry on E_{R} with vector α that maps Φ onto itself. Thus there is a linear functional λ_{α} on E_{R} so that σ_{α} minus the identity mapping on E_{R} is equal to −λ_{α} times α. We ask that

β(h_{α}) = λ_{α}(β) (33.1.2)

for every β ∈ Φ, where the right side is an integer by the definition of a root system, which may be considered as an element of k.

If Ψ_{0} ⊆ Φ ∪ {0}, then A(Ψ_{0}) denotes the linear subspace of A spanned by A_{α}, α ∈ Ψ_{0}, as in Section 22.4. Let Φ^{+} = Φ^{Δ,+} be the set of positive roots in Φ with respect to Δ, which is to say the elements of Φ that can be expressed as linear combinations of elements of Δ with nonnegative coefficients. Put

B_{Δ} = A(Φ^{+} ∪ {0}), (33.1.3)

which is the standard Borel subalgebra of A associated to A_{0} and Δ, as in Section 22.12.

In particular, the conditions considered in Section 32.1 hold here. Thus the remarks in Sections 32.2 and 32.3 can be used here too.

33.2 Using a larger field

Let k, k_{1} be fields, with k ⊆ k_{1}. If V is a vector space over k, then we can get a vector space Ṽ over k_{1} from V in a well-known way. Of course, Ṽ may also be considered as a vector space over k, and V may be considered as a linear subspace of Ṽ, as a vector space over k. If {v_{j}}_{j∈I} is a basis for V as a vector space over k, then {v_{j}}_{j∈I} corresponds to a basis for Ṽ as a vector space over k_{1} too. In particular, the dimension of Ṽ, as a vector space over k_{1}, is the same as the dimension of V, as a vector space over k.

One can use a basis for V to obtain Ṽ in this way, and verify that Ṽ does not depend on the choice of basis for V, up to isomorphic equivalence. In particular, every element of Ṽ can be expressed as a finite linear combination of elements of V, with coefficients in k_{1}. Any collection of linearly independent vectors in V, as a vector space over k, is linearly independent in Ṽ as well, as a vector space over k_{1}.

If W_{1} is a vector space over k_{1}, then W_{1} may be considered as a vector space over k. Every linear mapping from V into W_{1} as a vector space over k has a unique extension to a linear mapping from Ṽ into W_{1}, as vector spaces over k_{1}. Of course, the restriction of any linear mapping from Ṽ into W_{1}, as vector spaces over k_{1}, to V is linear as a mapping into W_{1}, as a vector space over k.

If Z is a linear subspace of V, then one can get a vector space Z̃ over k_{1} in the same way. More precisely, Z̃ corresponds to a linear subspace of Ṽ, whose intersection with V is Z. If V is the direct sum of some family of vector spaces over k, then Ṽ is the direct sum of the corresponding family of vector spaces over k_{1}.

Let Z be any vector space over k, and let Z̃ be the corresponding vector space over k_{1}. If W_{1} is a vector space over k_{1} again, then any mapping from V × Z into W_{1} that is bilinear over k has a unique extension to a mapping from Ṽ × Z̃ into W_{1} that is bilinear over k_{1}.

Let (A, [·, ·]_{A}) be a Lie algebra over k, so that A is a vector space over k in particular. Thus we can get a corresponding vector space Ã over k_{1} as before. The Lie bracket [·, ·]_{A} has a unique extension to a mapping [·, ·]_{A} from Ã × Ã into Ã that is bilinear over k_{1}, as in the previous paragraph. One can check that Ã is a Lie algebra over k_{1} with respect to [·, ·]_{A}. Note that Ã may be considered as a Lie algebra over k, and that A is a Lie subalgebra of Ã, as a Lie algebra over k.

If B is a Lie subalgebra of A, then B is a linear subspace of A in particular, and B̃ corresponds to a linear subspace of Ã, as a vector space over k_{1}. It is easy to see that B̃ is a Lie subalgebra of Ã, as a Lie algebra over k_{1}.

Let V be a vector space over k again, and suppose that V is a module over A, as a Lie algebra over k. If Ã, Ṽ are as before, then the action of A on V has a unique extension to a mapping from Ã × Ṽ into Ṽ that is bilinear over k_{1}, as before. One can verify that this makes Ṽ into a module over Ã, as a Lie algebra over k_{1}.

33.3 Related spaces of linear mappings

Let k, k_{1} be fields with k ⊆ k_{1} again, let V be a vector space over k, and let W_{1} be a vector space over k_{1}. Thus W_{1} may also be considered as a vector space over k, and we let L_{k}(V, W_{1}) be the space of linear mappings from V into W_{1}, as a vector space over k. Of course, the space of all mappings from any nonempty set into W_{1} may be considered as a vector space over k_{1}, with respect to pointwise addition and scalar multiplication. It is easy to see that L_{k}(V, W_{1}) is a linear subspace of the space of all mappings from V into W_{1}, as a vector space over k_{1}. This means that L_{k}(V, W_{1}) may be considered as a vector space over k_{1}.

There is a natural mapping from L_{k}(V, W_{1}) into the space L_{k_{1}}(Ṽ, W_{1}) of all linear mappings from Ṽ into W_{1}, as vector spaces over k_{1}. This mapping sends an element of L_{k}(V, W_{1}) to its unique extension to an element of L_{k_{1}}(Ṽ, W_{1}), as before. Similarly, there is a natural mapping from L_{k_{1}}(Ṽ, W_{1}) into L_{k}(V, W_{1}), which sends an element of L_{k_{1}}(Ṽ, W_{1}) to its restriction to V. Of course, these two mappings are inverses of each other. These mappings are also linear over k_{1}, so that we get

(33.3.1) a natural isomorphism from L_{k}(V, W_{1}) onto L_{k_{1}}(Ṽ, W_{1}), as vector spaces over k_{1}.

Let W be another vector space over k, and let W̃ be the corresponding vector space over k_{1}. As before, W̃ may be considered as a vector space over k that contains W as a linear subspace. A linear mapping from V into W, as vector spaces over k, may be considered as a linear mapping from V into W̃, as vector spaces over k. This may be extended to a unique linear mapping from Ṽ into W̃, as vector spaces over k_{1}.

We can take W_{1} = W̃ in (33.3.1), to get

(33.3.2) a natural isomorphism from L_{k}(V, W̃) onto L_{k_{1}}(Ṽ, W̃), as vector spaces over k_{1}.

We may also consider W̃ and L_{k}(V, W̃) as vector spaces over k, and the space L_{k}(V, W) of all linear mappings from V into W, as vector spaces over k, is a linear subspace of L_{k}(V, W̃), as a vector space over k. Thus

(33.3.3) L_{k}(V, W) corresponds to a linear subspace of L_{k_{1}}(Ṽ, W̃), as a vector space over k,

using the natural isomorphism mentioned in (33.3.2).

Suppose that {v_{j}}_{j∈I} is a basis for V as a vector space over k, which may also be considered as a basis for Ṽ as a vector space over k_{1}. We may as well suppose that I ≠ ∅, which is to say that V ≠ {0}. Let c(I, W), c(I, W̃) be the spaces of functions on I with values in W, W̃, respectively. These are vector spaces over k, k_{1}, respectively, with respect to pointwise addition and scalar multiplication.

A linear mapping from V into W is determined by its values on the basis vectors, and any element of c(I, W) determines such a linear mapping in this way. This defines

(33.3.4) an isomorphism from L_{k}(V, W) onto c(I, W), as vector spaces over k.

Similarly, we get

(33.3.5) an isomorphism from L_{k}(V, W̃) onto c(I, W̃), as vector spaces over k_{1}.

In the same way, we obtain

(33.3.6) an isomorphism from L_{k_{1}}(Ṽ, W̃) onto c(I, W̃), as vector spaces over k_{1}.

Of course, (33.3.5) and (33.3.6) correspond to each other as in (33.3.2). Note that c(I, W̃) may be considered as a vector space over k, by considering W̃ as a vector space over k. Clearly

(33.3.7) c(I, W) is a linear subspace of c(I, W̃), as a vector space over k,

because W is a linear subspace of W̃, as a vector space over k. This corresponds to the fact that L_{k}(V, W) may be considered as a linear subspace of L_{k}(V, W̃), as a vector space over k, using (33.3.4) and (33.3.5). Similarly, (33.3.3) corresponds to (33.3.7), using (33.3.4) and (33.3.6).

Let V′, Ṽ′ be the duals of V, Ṽ, as vector spaces over k, k_{1}, respectively. Thus V′ consists of the linear functionals on V, which are linear mappings from V into k, and Ṽ consists of mappings from Ṽ into k_{1} that are linear over k_{1}. Of course, W = k may be considered as a one-dimensional vector space over k, for which we get W̃ = k_{1}, as a one-dimensional vector space over k_{1}.

The space L_{k}(V, k_{1}) of linear mappings from V into k_{1}, as a vector space over k, may be considered as a vector space over k_{1}, as before. There is a natural isomorphism between this space and Ṽ′, as vector spaces over k_{1}, as in (33.3.2). As a vector space over k, L_{k}(V, k_{1}) contains V′ as a linear subspace. This means that

(33.3.8) V′ corresponds to a linear subspace of Ṽ′, as a vector space over k,

in a natural way, as in (33.3.3).

Suppose that V has positive finite dimension n, and let v_{1}, …, v_{n} be a basis for V. This may also be considered as a basis for Ṽ, as a vector space over k_{1}, as before. Using this basis, we can identify V′ with k^{n}, and Ṽ′ with k_{1}^{n}. One can use this to see that the vector space ˜(V′) over k_{1} associated to V′ can be identified with Ṽ′.

33.4 Larger fields and toral subalgebras

Let k, k_{1} be fields with k ⊆ k_{1} again, and let (A, [·, ·]_{A}) be a Lie algebra over k of positive finite dimension. This leads to a vector space Ã over k_{1} of the same dimension, with A a linear subapce of Ã as a vector space over k, as in Section 33.2. As before, [·, ·]_{A} has a unique extension to a mapping [·, ·]_{A} from Ã × Ã into Ã that is bilinear over k_{1}, and which makes (Ã, [·, ·]_{A}) into a Lie algebra over k_{1}.

Suppose that A_{0} is a Lie subalgebra of A such that every element of A_{0} is ad-diagonalizable as an element of A, so that A_{0} is commutative as a Lie algebra, as in Section 17.1. If α is an element of the dual A′_{0} of A_{0}, as a vector space over k, then let A_{α} be the set of x ∈ A such that ad_{A,w}(x) = α(w) x for every w ∈ A_{0}, as before. Suppose that A_{0} is its own centralizer in A, as in (33.1.1), so that A_{0} is the same as A_{α} with α = 0.

Let Φ = Φ_{A_{0}} be the set of α ∈ A′_{0} such that α ≠ 0 and A_{α} ≠ {0}, as before. Remember that A corresponds to the direct sum of the linear subspaces A_{α}, α ∈ Φ ∪ {0}, as a vector space over k, as in Section 17.2. If α ∈ A′_{0}, so that A_{α} is a linear subspace of A, then let ˜(A_{α}) be the linear subspace of Ã as a vector space over k_{1} obtained from A_{α} as before. Observe that Ã corresponds to the direct sum of ˜(A_{α}), α ∈ Φ ∪ {0}, as a vector space over k_{1}, as in Section 33.2.

More precisely, ˜(A_{0}) is a Lie subalgebra of Ã, as a Lie algebra over k_{1}. If α ∈ A′_{0}, then let α̃ be the unique extension of α to a mapping from ˜(A_{0}) into

k_{1} that is linear over k_{1}, so that α̃ is an element of the dual ˜(A_{0})′ of ˜(A_{0}), as a vector space over k_{1}. It is easy to see that

ad_{_{A},w}(x) = [w, x]_{A} = α̃(w) x (33.4.1)

for every w ∈ ˜(A_{0}) and x ∈ ˜(A_{α}), because this holds when w ∈ A_{0} and x ∈ A_{α}, by construction.

In particular, the elements of ˜(A_{0}) are ad-diagonalizable in Ã. If β ∈ ˜(A_{0})′, then we can take Ã_{β} to be the set of x ∈ Ã such that ad_{_{A},w}(x) = β(w) x for

every w ∈ ˜(A_{0}), as usual. If α ∈ A′_{0} and α̃ is the corresponding element of ˜(A_{0})′, then

˜(A_{α}) ⊆ Ã_{α}, (33.4.2)

by (33.4.1). Let Φ̃ = Φ̃_{(A_{0})} be the set of β ∈ ˜(A_{0})′ such that β ≠ 0 and

Ã_{β} ≠ {0}, as usual. Thus {α̃ : α ∈ Φ} ⊆ Φ̃, (33.4.3)

by (33.4.2).

In fact, Φ̃ = {α̃ : α ∈ Φ}, (33.4.4)

and

Ã_{α} = ˜(A_{α}) (33.4.5)

for every α ∈ A′_{0}. This follows from (33.4.1) and the fact that Ã corresponds to the direct sum of ˜(A_{α}), α ∈ Φ ∪ {0}, as a vector space over k_{1}. In particular, if

α = 0, then α̃ = 0 as an element of ˜(A_{0})′, and we get that

Ã_{0} = ˜(A_{0}). (33.4.6)

Note that ˜(A_{0}) is commutative as a Lie algebra over k_{1}, and that it is its own centralizer in Ã, by (33.4.6).

If α, β ∈ A′_{0}, then [A_{α}, A_{β}] ⊆ A_{α+β} in A, as in Section 17.2. This implies that [ ˜(A_{0}), ˜(A_{β})] ⊆ ˜ (A_{α+β}) in Ã. If [A_{α}, A_{β}] = A_{α+β} in A, then we get that

[ ˜(A_{α}), ˜(A_{β})] = ˜ (A_{α+β}) (33.4.7)

in Ã.

Every element of ˜(A_{0})′ can be expressed as a linear combination of elements of the form α̃, α ∈ A′_{0}. Suppose that A′_{0} is spanned by Φ, as a vector space over k. This means that the center Z(A) of A as a Lie algebra over k is trivial, as in Section 33.1. Under these conditions, we get that ˜(A_{0})′ is spanned by Φ̃, as a vector space over k_{1}. This means that Z(Ã) = {0}, as before.

Suppose that for each α ∈ Φ, A_{α} has dimension one, as a vector space over k. This implies that ˜(A_{α}) has dimension one, as a vector space over k_{1}. Suppose also that for every α ∈ Φ there is an element h_{α} of [A_{α}, A_{−α}] ⊆ A_{0} such that α(h_{α}) = 2. Of course, h_{α} may also be considered as an element of [ ˜(A_{α}), ˜ (A_{−α})] in Ã.

Suppose that k has characteristic 0, so that k_{1} has characteristic 0 as well. The linear subspace E_{Q} of A′_{0}, as a vector space over Q, corresponds exactly to the linear subspace of ˜(A_{0})′ spanned by Φ̃, as a vector space over Q, because of (33.4.4). This means that the associated vector space E_{R} over R corresponds exactly to the analogous vector space over R for ˜(A_{0})′.

Suppose that Φ is a reduced root system in E_{R}, so that Φ̃ is a reduced root system in the analogous vector space over R for ˜(A_{0})′. Let Δ be a base for Φ, which means that Δ̃ = {α̃ : α ∈ Δ} is a base for Φ̃. If h_{α}, α ∈ Δ, is a basis for A_{0}, as a vector space over k, then this is also a basis for ˜(A_{0}), as a vector space over k_{1}.

If Z(A) = {0}, then A′_{0} is spanned by Φ as a vector space over k, and ˜(A_{0})′ is spanned by Φ̃ as a vector space over k_{1}, as before. In this case, we get that A′_{0} is spanned by Δ as a vector space over k, and that ˜(A_{0})′ is spanned by Δ̃ as a vector space over k_{1}. If Δ is a basis for A′_{0} as a vector space over k, then Δ̃ is a basis for ˜(A_{0})′, as a vector space over k_{1}.

If α, β ∈ Φ, then one may ask that β(h_{α}) can be expressed in terms of Φ as a root system as in (33.1.2). This implies the analogous condition for Φ̃.

If Ψ_{0} ⊆ Φ ∪ {0}, then A(Ψ_{0}) is the linear subspace of A spanned by A_{α}, α ∈ Ψ_{0}, as in Section 33.1. Similarly, if Ψ_{1} ⊆ Φ̃ ∪ {0}, then let Ã(Ψ_{1}) be the linear subspace of Ã, as a vector space over k_{1}, spanned by Ã_{γ}, γ ∈ Ψ_{1}, as before. If Ψ_{0} ⊆ Φ ∪ {0}, then

Ψ_{1} = {α̃ : α ∈ Ψ_{0}} (33.4.8) is contained in Φ̃ ∪ {0}, and

Ã(Ψ_{1}) = ˜ (A(Ψ_{0})), (33.4.9)

by (33.4.5).

Let Φ^{+} = Φ^{Δ,+} be the set of positive roots in Φ with respect to Δ, and let Φ̃^{+} = Φ̃^{_{Δ},+} be the set of positive roots in Φ̃ with respect to Δ̃. Observe that

Φ̃^{+} = {β̃: β ∈ Φ^{+}}. (33.4.10)

Remember that, under the conditions on A and A_{0} considered in Section 33.1, B_{Δ} = A(Φ^{+} ∪ {0}) is the standard Borel subalgebra of A associated to A_{0} and Δ, as in Section 22.12. In this case, Ã and ˜(A_{0}) satisfy the analogous conditions over k_{1}, and

B̃_{Δ} = Ã(Φ̃^{+} ∪ {0}) (33.4.11)

is the standard Borel subalgebra of Ã, as a Lie algebra over k_{1}, associated to ˜(A_{0}) and Δ̃. We also have that

B̃_{Δ} = ˜(B_{Δ}), (33.4.12)

by (33.4.9) and (33.4.10).

33.5 Weights and modules

Let us return to the same notation and hypotheses as in Section 33.1. Let V be a vector space over k which is a module over A, as a Lie algebra over k. If μ ∈ A′_{0}, then put

(33.5.1) V_{μ} = {v ∈ V : w · v = μ(w) v for every w ∈ A_{0}},

which is a linear subspace of V. An element of V_{μ} is said to have weight μ with respect to A_{0}, as on p56 of [24].

The elements of V_{μ} may also be called eigenvectors of A_{0} in V with weight μ, as on p57 of [25]. More precisely, Chapter VII of [25] deals with representations of sl_{n}(k), n ≥ 2. However, all but one of the results discussed there extend to arbitrary semisimple Lie algebras, as in Remark 1 on p61 of [25].

The dimension of V_{μ} is called the multiplicity of μ in V, as on p56 of [24], and Definition 2.3 on p58 of [25]. In particular, μ is said to be a weight of V with respect to A_{0} when V_{μ} ≠ {0}, as on p107 of [14], p56 of [24], and Definition 2.3 on p58 of [25]. In this case, V_{μ} may be called a weight space of V with respect to μ, as on p107 of [14].

Let α, μ ∈ A′_{0} be given, and suppose that x ∈ A_{α}, v ∈ V_{μ}. If w ∈ A_{0}, then

w · (x · v) = x · (w · v) + ([w, x]_{A}) · v = μ(w) (x · v) + α(w) (x · v). (33.5.2)

This shows that x · v ∈ V_{μ+α}, (33.5.3) as in part (a) of the lemma on p107 of [14], part (a) of Proposition 1 on p56 of [24], and Proposition 2.1 on p57 of [25]. Of course, this can be seen more directly when α = 0, and it is trivial when α ∉ Φ∪{0}, in which case A_{α} = {0}.

Let V̂ be the linear subspace of V spanned by V_{μ}, μ ∈ A′_{0}. In fact,

(33.5.4)  V̂ corresponds to the direct sum of V_{μ}, μ ∈ A′_{0},

as a vector space over k.

This means that if μ_{1}, …, μ_{r} are finitely many distinct elements of A′_{0}, v_{j} ∈ V_{μ_{j}} for each j = 1, …, r, and

∑_{j=1}^{r} v_{j} = 0, (33.5.5)

then v_{j} = 0 for every j = 1, …, r. This can be obtained from the fact that eigenvectors of a linear mapping with distinct eigenvalues are linearly independent. This corresponds to the first part of part (b) of the lemma on p107 of [14], the first part of part (b) of Proposition 1 on p56 of [24], and part of Proposition 2.2 on p57 of [25].

It is easy to see that V̂ is a submodule of V, as a module over A, using (33.5.3). This corresponds to the second part of part (b) of the lemma on p107 of [14], the second part of part (b) of Proposition 1 on p56 of [24], and a remark in the proof of Proposition 2.2 on p57 of [25].

As an example, consider A as a module over itself, with respect to the adjoint representation. In this case, the weights are the roots α ∈ Φ, with the corresponding weight spaces A_{α} of dimension 1, and 0. This is Example (1) on p107 of [14], and Example 2.4 of p58 of [25].

33.6 Primitive or maximal vectors

Let us continue to use the same notation and hypotheses as in Section 33.1. In particular, Δ is a base for Φ as a root system, and Φ^{+} is the set of positive roots in Φ with respect to Δ. Let V be a vector space over k that is a module over A as a Lie algebra over k again.

If v ∈ V and E ⊆ A, then put

E · v = {x · v : x ∈ E}, (33.6.1)

which is a linear subspace of V when E is a linear subspace of A. Suppose that

E_{1} · v = E_{2} · v = {0} (33.6.2)

for some linear subspaces E_{1}, E_{2} of A. If x_{1} ∈ E_{1} and x_{2} ∈ E_{2}, then we get that ([x_{1}, x_{2}]_{A}) · v = x_{1} · (x_{2} · v) − x_{2} · (x_{1} · v) = 0. (33.6.3)

This means that ([E_{1}, E_{2}]) · v = {0} (33.6.4) when (33.6.2) holds. If α, β ∈ Φ satisfy A_{α} · v = A_{β} · v = {0} and α + β ∈ Φ, then it follows that (A_{α+β}) · v = {0}, (33.6.5)

because [A_{α}, A_{β}] = A_{α+β}.

If (33.6.6) A_{α} · v = {0} for every α ∈ Δ,

then (33.6.7) A_{γ} · v = {0} for every γ ∈ Φ^{+}.

This is implicit in a remark on p108 of [14], and on p57 of [24]. To see this, let γ ∈ Φ^{+} be given, and remember that γ can be expressed as ∑_{j=1}^{r} α_{j}, where α_{j} ∈ Δ for each j = 1, …, r, and ∑_{j=1}^{l} α_{j} ∈ Φ for every l = 1, …, r, as in Section 19.12. Under these conditions, one can check that

(A_{( ∑_{j=1}^{l} α_{j})}) · v = {0} (33.6.8)

for every l = 1, …, r, using (33.6.5).

Let μ be a linear functional on A_{0}. We say that v ∈ V_{μ} is a primitive or maximal vector of weight μ in V if v ≠ 0 and (33.6.6) or equivalently (33.6.7) holds, as on p108 of [14], and p57 of [24]. We may simply say that v ∈ V is a primitive or maximal vector in V if v is primitive or maximal of weight μ for some μ ∈ A′_{0}.

Remember that B_{Δ} is the standard Borel subalgebra of A associated to A_{0} and Δ, as in Section 33.1. Remember also that N_{Δ} = A(Φ^{+}) is a Lie subalgebra of A that is nilpotent as a Lie algebra over k and whose elements are ad-nilpotent in A, and that [B_{Δ}, B_{Δ}] = N_{Δ}, as in Sections 22.6 and 24.10. By construction, B_{Δ} corresponds to the direct sum of A_{0} and N_{Δ}, as a vector space over k.

Let us say that v ∈ V is an eigenvector of x ∈ A if x · v is the multiple of v by an element of k, which is the corresponding eigenvalue. Of course, this is the same as saying that v is an eigenvector for the linear mapping corresponding to x on V, as a module over A. If E ⊆ A and v is an eigenvector of every x ∈ E, then v is said to be an eigenvector of E. If v is an eigenvector of x_{1}, x_{2} ∈ A with eigenvalues c_{1}, c_{2} ∈ k, then

([x_{1}, x_{2}]_{A}) · v = x_{1} · (x_{2} · v) − x_{2} · (x_{1} · v)
(33.6.9)  = c_{2} (x_{1} · v) − c_{1} (x_{2} · v) = c_{1} c_{2} v − c_{1} c_{2} v = 0.

This means that v is an eigenvector of [x_{1}, x_{2}]_{A}, with eigenvalue 0.

If v ∈ V is an eigenvector of B_{Δ}, then v is an eigenvector of every element of N_{Δ} with eigenvalue 0, because [B_{Δ}, B_{Δ}] = N_{Δ}. This is the same as saying that (33.6.7) holds. More precisely, v ∈ V is an eigenvector of B_{Δ} if and only if v is an eigenvector of A_{0} and (33.6.7) holds, as in Proposition 2.5 on p58 of [25].

Note that a nonzero v ∈ V is an eigenvector of A_{0} if and only if v ∈ V_{μ} for some μ ∈ A′_{0}, and that μ is unique. Thus a nonzero v ∈ V is primitive or maximal if and only if it is an eigenvector of A_{0} and satisfies (33.6.7), or equivalently v is an eigenvector of B_{Δ}. This corresponds to Definition 2.6 on p58 of [25], and is also mentioned on p57 of [24].

33.7 Representations and UA

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Also let V be a module over k, which is a module over A, as a Lie algebra over k. Suppose that UA is a universal enveloping algebra of A with multiplicative identity element e = e_{UA} and associated mapping i = i_{UA} from A into UA, as in Section 25.4.

Remember that the space Hom_{k}(V, V) of homomorphisms from V into itself, as a module over k, is an associative algebra over k with respect to composition of mappings. In particular, Hom_{k}(V, V) is a Lie algebra over k, with respect to the commutator bracket. To say that V is a module over A as a Lie algebra over k means that we have a Lie algebra homomorphism from A into Hom_{k}(V, V).

Because UA is a universal enveloping algebra of A, there is a corresponding homomorphism from UA into Hom_{k}(V, V), as associative algebras over k, that sends e to the identity mapping on V. This defines a representation of UA, as an associative algebra over k, on V. Equivalently, V may be considered as a (left) module over UA, as an associative algebra over k.

If x ∈ A and v ∈ V, then the action of x on v may be denoted x · v, as usual. Similarly, if u ∈ UA and v ∈ V, then let the action of u on v be denoted u · v. It should be clear in practice which is intended, and the two are compatible in the sense that x · v is the same as i(x) · v for every x ∈ A.

Let v ∈ V be given, and define E · v ⊆ V as in (33.6.1) when E ⊆ A. This is a submodule of V, as a module over k, when E is a submodule of A, as a module over k. If E ⊆ UA, then E · v can be defined as a subset of V analogously, and is a submodule of V as a module over k when E is a submodule of UA as a module over k. Note that v ∈ E · v when e ∈ E ⊆ UA.

Clearly every submodule of V, as a module over UA, is a submodule of V, as a module over A. Conversely, any submodule of V as a module over A is a submodule of V as a module over UA, because UA is generated by e and i(A), as an associative algebra over k.

If v ∈ V, then it is easy to see that (UA)·v is a submodule of V, as a module over UA, that contains v, and in fact it is the smallest such submodule of V. Equivalently, (UA) · v is a submodule of V, as a module over A, that contains v, and is the smallest such submodule.

Remember that i is injective as a mapping from A into UA when A is free as a module over k, as in Section 25.12. In this case, we may identify A with i(A) ⊆ UA. In particular, this condition holds when k is a field.

33.8 Spanning standard cyclic modules

Let us return now to the notation and hypotheses in Section 33.1, so that in particular k is a field of characteristic 0, and (A, [·, ·]_{A}) is a Lie algebra over k of positive finite dimension. Let V be a module over A again, as a Lie algebra over k, and let UA be a universal enveloping algebra of A, so that V may also be considered as a module over UA, as an associative algebra over k. Suppose that v ∈ V is a primitive or maximal vector of weight μ ∈ A′_{0}, as in Section 33.6, and that V = (UA) · v. (33.8.1)

In this case, V is said to be standard cyclic of weight μ, as on p108 of [14]. Of course, if (33.8.1) does not hold, (UA) · v is a submodule of V, as a module over A, which is standard cyclic of weight μ as a module over A.

If α ∈ Φ^{+}, then let x_{α} be a nonzero element of A_{α}, so that A_{α} consists of multiples of x_{α} by elements of k. Also let y_{α} be the unique element of A_{−α} such that [x_{α}, y_{α}]_{A} = h_{α}. Thus y_{α} ≠ 0, because h_{α} ≠ 0, so that A_{−α} consists exactly of multiples of y_{α} by elements of k.

Remember that A(−Φ^{+}) is the linear subspace of A spanned by A_{−α}, with α ∈ Φ^{+}, as in Sections 22.4 and 33.1. More precisely, A(−Φ^{+}) is a Lie subalgebra of A, as in Section 22.6. Because A(−Φ^{+}) and B_{Δ} are Lie subalgebras of A, there are natural injective homomorphisms from universal enveloping algebras UA(−Φ^{+}) and UB_{Δ} of A(−Φ^{+}) and B_{Δ}, respectively, into UA, as associative algebras over k, as in Section 25.12. Thus we may identify UA(−Φ^{+}) and UB_{Δ} with subalgebras of UA in this way.

By construction, A corresponds to the direct sum of A(−Φ^{+}) and B_{Δ}, as a vector space over k. This leads to an isomorphism between UA and (UA(−Φ^{+})) ⨂(UB_{Δ}), as vector spaces over k, as in Section 25.12. In particular,

(33.8.2) every element of UA can be expressed as a finite sum, where each term corresponds to the product of an element of UA(−Φ^{+}) and an element of UB_{Δ}.

Observe that (UB_{Δ}) · v = {t v : t ∈ k}, (33.8.3)

because v is an eigenvector of B_{Δ}, by hypothesis. This also uses the fact that UB_{Δ} is generated as an associative algebra over k by its multiplicative identity element and the image of B_{Δ} in UB_{Δ}. It follows that

V = (UA(−Φ^{+})) · v, (33.8.4)

by (33.8.1) and (33.8.2).

Let β_{1}, …, β_{r} be a list of the elements of Φ^{+}, so that y_{β_{1}}, …, y_{β_{r}} is a basis for A(−Φ^{+}), as a vector space over k. If m_{1}, …, m_{r} are nonnegative integers, then y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}} (33.8.5) corresponds to an element of UA(−Φ^{+}) ⊆ UA. More precisely, y_{β_{j}} is identified with an element of UA(−Φ^{+}) for each j = 1, …, r, so that y_{β_{j}}^{m_{j}} corresponds to an element of UA(−Φ^{+}) as well. Of course, this is interpreted as being the multiplicative identity element when m_{j} = 0. Thus the product of y_{β_{j}}^{m_{j}} over j = 1, …, r corresponds to an element of UA(−Φ^{+}) too.

The elements of UA(−Φ^{+}) of the form (33.8.5), where m_{j} is a nonnegative integer for each j = 1, …, r, form a basis for UA(−Φ^{+}) as a vector space over k, by the Poincaré–Birkhoff–Witt theorem. In particular,

(33.8.6) UA(−Φ^{+}) is spanned by elements of the form (33.8.5), as a vector space over k.

If m_{j} is a nonnegative integer for each j = 1, …, r, then (33.8.5) acts on v to get an element (y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}) · v (33.8.7)

of V. In fact,

(33.8.8) V is spanned by elements of the form (33.8.7), as a vector space over k.

This follows from (33.8.4) and (33.8.6). This corresponds to the first part of part (a) of the theorem on p108 of [14], part (1) of Proposition 2 on p57 of [24], and part of the proof of Theorem 3.1 on p58 of [25].

If m_{j} is a nonnegative integer for each j = 1, …, r, then

(33.8.9) (33.8.7) has weight μ − ∑_{j=1}^{r} m_{j} β_{j}

as an element of V, in the sense of Section 33.5. This follows from (33.5.3), because v ∈ V_{μ}, by hypothesis. This implies that (33.8.7) has weight of the form

μ − ∑_{α∈Δ} c_{α} α, (33.8.10)

where c_{α} is a nonnegative integer for each α ∈ Δ, because β_{j} ∈ Φ^{+} for every j = 1, …, r.

In particular, V is spanned by its weight spaces, by (33.8.8). This means that

(33.8.11) V corresponds to the direct sum of its weight spaces, as a vector space over k,

as in Section 33.5. We also get that

all of the weights of V are of the form mentioned in (33.8.9),

(33.8.12)

and thus (33.8.10).

This corresponds to the second part of part (a) and part (b) of the theorem on p108 of [14], the first part of part (2) of Proposition 2 on p57 of [24], and part (b) of Theorem 3.1 on p58 of [25].

Remember that Δ is supposed to be a basis for A′_{0}, as a vector space over k, as in Section 33.1. In particular, the elements of Δ are linearly independent in A′_{0}, as a vector space over k. If c_{α} is a nonnegative integer for each α ∈ Δ, then there are only finitely many nonnegative integers m_{1}, …, m_{r} such that

∑_{j=1}^{r} m_{j} β_{j} = ∑_{α∈Δ} c_{α} α. (33.8.13)

This uses the facts that β_{j} ∈ Φ^{+} for every j = 1, …, r, and k has characteristic 0, by hypothesis. Similarly, if c_{α} = 0 for every α ∈ Δ, and m_{1}, …, m_{r} are nonnegative integers, then (33.8.13) holds only when m_{j} = 0 for every j = 1, …, r.

It follows that

(33.8.14) the weight spaces in V have finite dimension,

as in the first part of part (c) of the theorem on p108 of [14], and the second part of part (2) of Proposition 2 on p57 of [24]. We also get that

dim V_{μ} = 1, (33.8.15)

as in the second part of part (c) of the theorem on p108 of [14], part (3) of Proposition 2 on p57 of [24], and part (c) of Theorem 3.1 on p58 of [25]. Note that μ may be called the highest weight of V, because all of the weights of V are of the form (33.8.10), with c_{α} ≥ 0 for every α ∈ Δ.

33.9 Indecomposability

Let us continue with the same notation and hypotheses as in the previous section. We would like to show that V is indecomposable as a module over A, which is to say that it cannot be expressed is the direct sum of two nontrivial submodules of V. This corresponds to the first part of part (e) of the theorem on p108 of [14], and to part (4) of Proposition 2 on p57 of [24].

Suppose that V corresponds to the direct sum of two linear subspaces Y and Z, and that Y and Z are submodules of V, as a module over A. Thus we get linear projection mappings from V onto Y and Z whose kernels are Z and Y, respectively, and these projections commute with the action of A on V. If ν ∈ A′_{0}, then one can use this to check that these projection mappings send V_{ν} into Y_{ν} and Z_{ν}, respectively, where V_{ν}, Y_{ν} and Z_{ν} are as in Section 33.5. This means that V_{ν} corresponds to the direct sum of Y_{ν} and Z_{ν}, as a vector space over k.

In particular, if we take ν = μ, then we get that one of Y_{μ} or Z_{μ} has dimension one, by (33.8.15). This implies that v is an element of Y or Z, so that Y or Z is all of V, because V is generated by v as a module over A, as in (33.8.1). Of course, this means that the other of Y or Z is {0}, so that V is indecomposable, as desired. This is the argument near the top of p58 of [24].

If V has finite dimension as a vector space over k, then it follows that V is irreducible as a module over A, by Weyl’s theorem. This uses the hypothesis that k have characteristic 0, as well as the semisimplicity of A, as in Section 33.1. This corresponds to part (d) of Proposition 3 on p60 of [24], and part (a) of Theorem 3.1 on p58 of [25].

Let Z be any submodule of V as a module over A, so that Z is a linear subspace of V, and the action of A on V maps Z into itself. Part (d) of the theorem on p108 of [14] states that

(33.9.1) Z corresponds to the direct sum of its weight spaces, as a vector space over k.

Of course, it suffices to check that Z is spanned by its weight spaces, as in Section 33.5.

Suppose for the sake of a contradiction that there is an element of Z that is not in the linear span of the weight spaces of Z. Because V is spanned by its weight spaces, every element of Z can be expressed as

z = ∑_{j=1}^{n} z_{j}, (33.9.2)

where for each j = 1, …, n, z_{j} ∈ V_{ν_{j}} for some ν_{j} ∈ A′_{0}. Let n be the smallest positive integer for which there is a z ∈ Z that is not in the linear span of the weight spaces of Z, and which can be expressed as in (33.9.2). Of course, this means that n ≥ 2, and that the ν_{j}’s are distinct elements of A′_{0}. We also have that z_{j} ∉ Z for each j = 1, …, n.

Because ν_{1} ≠ ν_{2}, there is a w ∈ A_{0} such that ν_{1}(w) ≠ ν_{2}(w). Observe that

w · z = ∑_{j=1}^{n} w · z_{j} = ∑_{j=1}^{n} ν_{j}(w) z_{j}, (33.9.3)

so that

w · z − ν_{1}(w) z = ∑_{j=2}^{n}(ν_{j}(w) − ν_{1}(w)) z_{j}. (33.9.4)

The left side is an element of Z, and the minimality of n implies that it is in the linear span of the weight spaces of Z. Remember that the linear span of the weight spaces in V corresponds to the direct sum of the weight spaces in V, as a vector space over k, as in Section 33.5. It follows that the terms in the sum on the right side of (33.9.4) are elements of Z. This implies that z_{2} ∈ Z, because ν_{2}(w) ≠ ν_{1}(w), by construction. This is a contradiction, as in the preceding paragraph.

Let Z be a submodule of V as a module over A again, and suppose that Z ≠ V. This implies that v ∉ Z, by (33.8.1), and thus

Z ∩ V_{μ} = {0}, (33.9.5)

by (33.8.15). Let V^{0} be the linear subspace of V spanned by the weight spaces V_{ν} with ν ∈ A′_{0} and ν ≠ μ. This is a proper linear subspace of V, because V corresponds to the direct sum of its weight spaces, and V_{μ} ≠ {0}, by hypothesis. Using (33.9.5), we get that Z ⊆ V^{0}, (33.9.6)

because Z is spanned by its weight spaces, as before.

Let V^{1} be the linear subspace of V spanned by all of the proper submodules of V, as a module over A. Thus

V^{1} ⊆ V^{0}, (33.9.7)

by (33.9.6). In particular, this implies that V is indecomposable as a module over A, as on p109 of [14]. More precisely, V is not spanned by proper submodules, as a module over A.

It is easy to see that V^{1} is a submodule of V, as a module over A. In fact, V^{1} is the unique maximal proper submodule of V, as a module over A, as in part (e) of the theorem on p108 of [14]. The quotient V/V^{1} is a module over A in a natural way, because V^{1} is a submodule of V. Observe that

(33.9.8) V/V^{1} is irreducible, as a module over A,

as in part (e) of the theorem on p108 of [14], because V^{1} is a maximal proper submodule of V. Of course, V/V^{1} ≠ {0}, because V^{1} ≠ V.

If the quotient of V by a proper submodule is irreducible as a module over A, then that submodule of V is maximal. This means that V/V^{1} is the only nontrivial irreducible quotient of V, as a module over A, as in part (e) of the theorem on p108 of [14].

33.10 Irreducible standard cyclic modules

Let us use the same notation and hypotheses as in Section 33.1 again, and let V be a module over A, as a Lie algebra over k. Suppose that v ∈ V is a primitive or maximal vector of weight μ ∈ A′_{0}, and that V is irreducible as a module over A. Let UA be a universal enveloping algebra of A, so that (UA) · v is a submodule of V that contains v. Because V is irreducible, we get that V = (UA) · v, so that V is standard cyclic of weight μ. This corresponds to the first step in the proof of Theorem 1 on p58 of [24].

Thus V satisfies the hypotheses in Section 33.8, and so has the same properties as before. This corresponds to part (b) of Theorem 1 on p58 of [24].

Suppose that v_{1} ∈ V is another primitive or maximal vector, with weight μ_{1} ∈ A′_{0}. It follows that V = (UA) · v_{1}, (33.10.1) because V is irreducible, as before. This means that the analogues of the statements in Section 33.8 for v_{1} and μ_{1} hold as well.

Because μ_{1} is a weight of V, there are nonnegative integers c_{α}, α ∈ Δ, such that

μ_{1} = μ − ∑_{α∈Δ} c_{α} α, (33.10.2)

as in (33.8.12). Similarly, there are nonnegative integers c_{1,α}, α ∈ Δ, such that

μ = μ_{1} − ∑_{α∈Δ} c_{1,α} α, (33.10.3)

by the analogue of (33.8.12) for μ_{1}, and the fact that μ is a weight of V. It follows that c_{1,α} = −c_{α} for every α ∈ Δ, because Δ is a basis for A′_{0}, and k has characteristic 0, as in Section 33.1. This implies that c_{α} = c_{1,α} = 0 for every α ∈ Δ, so that μ_{1} = μ. (33.10.4)

We also get that v_{1} = t_{1} v for some t_{1} ∈ k \ {0}, (33.10.5)

by (33.8.15).

This corresponds to the corollary on p109 of [14], part (a) of Theorem 1 on p58 of [24], and part (1) of Theorem 3.2 on p59 of [25]. Note that μ may be called the highest weight of V, as in [24, 25]. This terminology may also be used for any standard cyclic module of weight μ, as in [14]. Of course, this refers to the expression of other weights as in (33.8.10).

Let Z be another irreducible module over A with a primitive or maximal vector z. If Z is isomorphic to V as a module over A, then Z has highest weight μ too, because of (33.10.4). Conversely, suppose that Z has highest weight μ, and let us show that Z is isomorphic to V, as modules over A. This corresponds to Theorem A on p109 of [14], part (c) of Theorem 1 on p58 of [24], and part (2) of Theorem 3.2 on p59 of [25].

Let Y be the direct sum of V and Z, which is the same as the Cartesian product of V and Z, where addition and scalar multiplication are defined coordinatewise. More precisely, Y is a module over A, where the action of A is defined coordinatewise as well. Observe that

y = (v, z) (33.10.6)

is a primitive or maximal vector of weight μ in Y. Put

E = (UA) · y, (33.10.7)

which is the submodule of Y, as a module over A, generated by y. Thus E is a standard cyclic module of weight μ, by construction.

Let p_{V}, p_{Z} be the obvious projections from Y onto V, Z, respectively, and let ϕ_{V}, ϕ_{Z} be their restrictions to E. Of course, these mappings are module homomorphisms, and ϕ_{V} (y) = v, ϕ_{Z}(y) = z. (33.10.8)

This implies that ϕ_{V} (E) = V, ϕ_{Z}(E) = Z, (33.10.9)

because V, Z are standard cyclic with respect to v, z, respectively, as before. The kernels of ϕ_{V}, ϕ_{Z} are

ker ϕ_{V} = ({0} × Z) ∩ E, ker ϕ_{Z} = (V × {0}) ∩ E. (33.10.10)

Of course, V × {0} and {0} × Z may be identified with V and Z, respectively, as modules over A.

Note that the only elements of E of weight μ are multiples of y by elements of k, as in (33.8.15). Thus there are no nonzero elements of the kernels of ϕ_{V} or ϕ_{Z} of weight μ. It follows that (0, z) ∉ ker ϕ_{V}, (v, 0) ∉ ker ϕ_{Z}, so that

ker ϕ_{V} ≠ {0} × Z, ker ϕ_{Z} ≠ V × {0}. (33.10.11)

We can identify ker ϕ_{V}, ker ϕ_{Z} with submodules of Z, V, respectively, as modules over A. These submodules are proper, by (33.10.11).

This implies that these submodules are trivial, because V and Z are irreducible, by hypothesis. This means that the kernels of ϕ_{V} and ϕ_{Z} are trivial, so that ϕ_{V}, ϕ_{Z} are isomorphisms from E onto V, Z, respectively, as modules over A. Of course, this leads to an isomorphism from V onto Z. This is the argument on p58f of [24].

The argument on p109 of [14] uses the fact that V and Z are isomorphic to irreducible quotients of E, while there is only one irreducible quotient of E, as in Section 33.9.

If V and Z have finite dimension as vector spaces over k, then E has finite dimension too. This implies that E is irreducible as a module over A, because E is indecomposable, as in the previous section. In this case, one can get that the kenrels of ϕ_{V} and ϕ_{Z} are trivial because they are proper submodules of E. This is the argument used on p59 of [25], where the modules are taken to be finite-dimensional.

33.11 Representations and left ideals

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over A. Also let UA be a universal enveloping algebra of A with multiplicative identity element e = e_{UA} and associated mapping i = i_{UA} from A into UA, as in Section 25.4.

Suppose that I is a left ideal in UA, as an associative algebra over k. The quotient (UA)/I can be defined as a module over k, and in fact as a left module over UA. Let q_{I} be the natural quotient mapping from UA onto (UA)/I, so that q_{I}(e) is an element of the quotient. Of course,

(UA) · q_{I}(e) = (UA)/I, (33.11.1)

by construction. Note that (UA)/I may be considered as a module over A too, as a Lie algebra over k.

Suppose that V is a module over k that is a module over A as a Lie algebra over k. Thus V may be considered as a (left) module over UA, as an associative algebra over UA, as before. If v ∈ V, then

I_{v} = {x ∈ UA : x · v = 0} (33.11.2)

is a left ideal in UA. This means that the quotient (UA)/I_{v} is defined as a left module over UA. The mapping

x ↦ x · v (33.11.3)

from UA into V leads to an injective homomorphism from (UA)/I_{v} into V, as left modules over UA. More precisely, this mapping sends (UA)/I_{v} onto (UA) · v, which is a submodule of V, as a module over UA. If V = (UA) · v, then we get an isomorphism from (UA)/I_{v} onto V, as left modules over UA.

Note that quotients of (UA)/I, as a left module over UA, or as a module over A as a Lie algebra over k, correspond to quotients of UA by left ideals in UA that contain I. Thus (UA)/I is nontrivial and irreducible as a left module over UA, or as a module over A as a Lie algebra over k, exactly when I ≠ UA is maximal as a proper left ideal in UA.

33.12 Tensor products over algebras

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k, with a multiplicative identity element e = e_{A}. Also let V, W be modules over k, where V is a right module over A, and W is a left module over A. The tensor product of V and W over A is a module V ⨂_{A} W over k with the following two properties.

First, the tensor product is equipped with a mapping from V × W into V ⨂_{A} W that is bilinear over k. If v ∈ V and w ∈ W, then the image of (v, w) under this mapping is denoted v ⊗w, as usual. This mapping should also satisfy

(v · a) ⊗ w = v ⊗ (a · w) (33.12.1)

for every a ∈ A, v ∈ V, and w ∈ W.

Second, let Z be any module over k, and let b be a mapping from V × W into Z that is bilinear over k. Suppose that

b(v · a, w) = b(v, a · w) (33.12.2)

for every a ∈ A, v ∈ V, and w ∈ W. Under these conditions, there should be a unique homomorphism c from V ⨂_{A} W into Z, as modules over k, such that

b(v, w) = c(v ⊗ w) (33.12.3)

for every v ∈ V and w ∈ W.

The tensor product can be obtained using a standard construction, and it is unique up to a suitable isomorphic equivalence. Note that V ⨂_{A} W is generated as a module over k by elements of the form v ⊗ w, v ∈ V, w ∈ W.

Let I be a nonempty set, and let V_{j} be a right module over A for every j ∈ I. Thus ⨁_{j∈I} V_{j} is a right module over A too, and one can check that

( ⨁_{j∈I} V_{j})⨂_{A}W = ⨁_{j∈I}(V_{j}⨂_{A}W). (33.12.4)

Similarly, if W_{j} is a left module over A for every j ∈ I, then ⨁_{j∈I} W_{j} is a left module over A, and one can verify that

V ⨂_{A}( ⨁_{j∈I} W_{j}) = ⨁_{j∈I}(V ⨂_{A}W_{j}). (33.12.5)

Let V_{1}, V_{2} be right modules over A, and let W_{1}, W_{2} be left modules over A. Suppose that ϕ_{V_{1}} is a homomorphism from V_{1} into V_{2}, as right modules over A, and that ϕ_{W_{1}} is a homomorphism from W_{1} into W_{2}, as left modules over A. Consider the mapping from V_{1} × W_{1} into V_{2} ⨂_{A} W_{2} defined by

(v_{1}, w_{1}) ↦ ϕ_{V_{1}}(v_{1}) ⊗ ϕ_{W_{1}}(w_{1}). (33.12.6)

This mapping is bilinear over k, and satisfies

ϕ_{V_{1}}(v_{1} · a) ⊗ ϕ_{W_{1}}(w_{1}) = (ϕ_{V_{1}}(v_{1}) · a) ⊗ ϕ_{W_{1}}(w_{1})
(33.12.7)  = ϕ_{V_{1}}(v) ⊗ (a · ϕ_{W_{1}}(w_{1})) = ϕ_{V_{1}}(v_{1}) ⊗ ϕ_{W_{1}}(a · w_{1})

for every a ∈ A, v_{1} ∈ V_{1}, and w_{1} ∈ W_{1}. This leads to a unique homomorphism ϕ from V_{1} ⨂_{A} W_{1} into V_{2} ⨂_{A} W_{2}, as modules over k, such that

ϕ(v_{1} ⊗ w_{1}) = ϕ_{V_{1}}(v_{1}) ⊗ ϕ_{W_{1}}(w_{1}) (33.12.8)

for every v_{1} ∈ V_{1} and w_{1} ∈ W_{1}.

Let V_{3} be another right module over A, and let W_{3} be another left module over A. Suppose that ψ_{V_{2}} is a homomorphism from V_{2} into V_{3}, as right modules over A, and that ψ_{W_{2}} is a homomorphism from W_{2} into W_{3}, as left modules over A. This leads to a homomorphism ψ from V_{2} ⨂_{A} W_{2} into V_{3} ⨂_{A} W_{3}, as modules over k, as in the previous paragraph. Of course, ψ_{V_{2}} ∘ ϕ_{V_{1}} is a homomorphism from V_{1} into V_{3}, as right modules over A, and ψ_{W_{2}} ∘ ϕ_{W_{1}} is a homomorphism from W_{1} into W_{3}, as left modules over A. One can verify that ψ ∘ ϕ is the same as the homomorphism from V_{1} ⨂_{A} W_{1} into V_{3} ⨂_{A} W_{3}, as modules over k, obtained from ψ_{V_{2}} ∘ ϕ_{V_{1}}, ψ_{W_{2}} ∘ ϕ_{W_{1}} as before.

33.13 Using modules over other algebras

Let us continue with the same notation and hypotheses as in the previous section. Suppose now that A_{1} is another associative algebra over k with a multiplicative identity element e_{1} = e_{A_{1}}. Let V, W be modules over k, where V is a left module over A_{1}, V is a right module over A, and W is a left module over A. More precisely, we suppose that the left and right actions on V commute with each other, so that

(a_{1} · v) · a = a_{1} · (v · a) (33.13.1)

for every a ∈ A, a_{1} ∈ A_{1}, and v ∈ V. Of course, V ⨂_{A} W can be defined as a module over k, as in the previous section.

If a_{1} ∈ A_{1}, then v ↦ a_{1} · v (33.13.2)

defines a mapping from V into itself that is linear over k. In fact, this is a homomorphism from V into itself, as a right module over A, because of (33.13.1). Using this and the identity mapping on W, we get a homomorphism from V ⨂_{A} W into itself, as a module over k, as in the previous section. This homomorphism defines an action of a_{1} on the left on V ⨂_{A} W, which is characterized by the property that a_{1} · (v ⊗ w) = (a_{1} · v) ⊗ w (33.13.3)

for every a_{1} ∈ A_{1}, v ∈ V, and w ∈ W. One can check that

(33.13.4) V ⨂_{A}W becomes a left module over A_{1}

in this way.

Let Z be a module over k that is a left module over A_{1}, and let b be a mapping from V × W into Z that is bilinear over k and satisfies (33.12.2). Thus there is a unique homomorphism c from V ⨂_{A} W into Z, as modules over k, that satisfies (33.12.3), as in the previous section. Suppose now in addition that

a_{1} · b(v, w) = b(a_{1} · v, w) (33.13.5)

for every a_{1} ∈ A_{1}, v ∈ V, and w ∈ W. This means that

a_{1} · c(v ⊗ w) = c(a_{1} · (v ⊗ w)) (33.13.6)

for every a_{1} ∈ A_{1}, v ∈ V, and w ∈ W. One can use this to check that c is a homomorphism from V ⨂_{A} W into Z, as left modules over A_{1}.

Of course, there are analogous statements when W is also a right module over an associative algebra A_{2} over k, where the left action on W by elements of A commutes with the right action by elements of A_{2}.

Note that A_{1} may be considered as a left and right module over itself. The left and right actions of A_{1} on itself commute, by associativity.

Suppose that A_{1} contains A as a subalgebra, with e_{1} = e. We may consider A_{1} as a left and right module over itself, and over A. Let us consider A_{1} as a left module over itself and as a right module over A, and let W be a left module over A again. Under these conditions,

A_{1}⨂_{A}W (33.13.7) becomes a left module over A_{1}, as before.

In particular, we can take A_{1} = A here, so that A ⨂_{A} W may be considered as a left module over A. Of course, we already have a mapping from A × W into W that is bilinear over k, given by the action of A on W as a left module over A. It is easy to see that W satisfies the requirements of the tensor product A ⨂_{A} W, as a module over k, with respect to this mapping from A × W into W. One can verify that the action of A on A ⨂_{A} W on the left just mentioned corresponds exactly to the given action of A on W on the left.

33.14 Existence of standard cyclic modules

Let us return to the notation and hypotheses in Section 33.1. Let μ ∈ A′_{0} be given. We would like to show that there is a standard cyclic module over A with weight μ.

Remember that Δ is a base for the root system Φ, and that B_{Δ} is the standard Borel subalgebra of A associated to A_{0} and Δ. We also have that B_{Δ} corresponds to the direct sum of A_{0} and N_{Δ} = A(Φ^{+}) as a vector space over k, and that [B_{Δ}, B_{Δ}] = N_{Δ}, as before.

Let L(μ) be a one-dimensional vector space over k. We can define an action of B_{Δ} on L(μ) by putting

(w + x) · u = μ(w) u (33.14.1)

for every w ∈ A_{0}, x ∈ N_{Δ}, and u ∈ L(μ). It is easy to see that this makes L(μ) into a module over B_{Δ}, as a Lie algebra over k.

Let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA} and associated mapping i = i_{UA} from A into UA, as in Section 25.4. We may as well take the universal enveloping algebra UB_{Δ} of B_{Δ} to be the subalgebra of UA generated by e and i(B_{Δ}), as in Section 25.12. We may also consider L(μ) as a left module over UB_{Δ}, as an associative algebra over k. Remember that i is injective, as in Section 25.12, so that we may as well identify A with its image in UA.

Of course, UA may be considered as a right module over itself, and thus a right module over UB_{Δ}. Let us take

Z(μ) = (UA)⨂_{UB_{Δ}}L(μ), (33.14.2)

where the right side is as in the previous two sections. More precisely, this is a module over k, and in fact a left module over UA. This means that Z(μ) may be considered as a module over A, as a Lie algebra over k.

Let u_{0} be a nonzero element of L(μ), and put

v = e ⊗ u_{0} ∈ Z(μ). (33.14.3)

One can check that (UA) · v = Z(μ), (33.14.4) because L(μ) has dimension one, as a vector space over k. If w ∈ A_{0}, then

(33.14.5) w · v = w · (e ⊗ u_{0}) = w ⊗ u_{0} = e ⊗ (w · u_{0})
= μ(w) (e ⊗ u_{0}) = μ(w) v.

Similarly, if x ∈ N_{Δ}, then

x · v = x · (e ⊗ u_{0}) = x ⊗ u_{0} = e ⊗ (x · u_{0}) = 0. (33.14.6)

Remember that A(−Φ^{+}) is a Lie subalgebra of A, and that A corresponds to the direct sum of A(−Φ^{+}) and B_{Δ}, as linear subspaces of A. This leads to an isomorphism between UA and (UA(−Φ^{+})) ⨂(UB_{Δ}), as vector spaces over k, as in Section 25.12. One can use this to get that UA is free as a right module over UB_{Δ}. This corresponds to part of Corollary D on p92 of [14], aside from using right modules here.

More precisely, let x_{α} ∈ A_{α} and y_{α} ∈ A_{−α} be as in Section 33.8 for each α in Φ^{+}, and let β_{1}, …, β_{r} be a list of the elements of Φ^{+} again. Thus y_{β_{1}}, …, y_{β_{r}} form a basis for A(−Φ^{+}), as a vector space over k, and

{y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}} : m_{1}, …, m_{r} ∈ Z_{+} ∪ {0}} (33.14.7)

is a basis for UA(−Φ^{+}), as a vector space over k, by the Poincaré–Birkhoff–Witt theorem, as before. One can use the Poincaré–Birkhoff–Witt theorem again to get that UA is free as a right module over UB_{Δ}, with basis (33.14.7).

Indeed, the Poincaré–Birkhoff–Witt theorem implies that UA corresponds to the direct sum of the linear subspaces

y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}} (UB_{Δ}) (33.14.8)

as a vector space over k, where m_{1}, …, m_{r} are nonnegative integers. The Poincaré–Birkoff–Witt theorem also ensures that

z ↦ y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}} z (33.14.9)

is a one-to-one linear mapping from UB_{Δ} onto (33.14.8) for all m_{1}, …, m_{r} ≥ 0. This can be seen by combining y_{β_{1}}, …, y_{β_{r}} with a basis for B_{Δ}, as a vector space over k, to get a basis for A, as a vector space over k.

Clearly (33.14.8) is a submodule of UA, as a right module over UB_{Δ}, for every m_{1}, …, m_{r} ≥ 0. In fact, (33.14.9) is an isomorphism from UB_{Δ} onto (33.14.8), as right modules over UB_{Δ}, for every m_{1}, …, m_{r} ≥ 0. This means that UA corresponds to the direct sum of the submodules (33.14.8) as a right module over UB_{Δ}, and that UA is free as a right module over UB_{Δ}.

It follows that Z(μ) corresponds to the direct sum of

(y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}(UB_{Δ})) ⨂_{UB_{Δ}} L(μ) (33.14.10)

as a vector space over k, where m_{1}, …, m_{r} are nonnegative integers. Note that (33.14.10) is isomorphic to L(μ) as a vector space over k for every m_{1}, …, m_{r} ≥ 0, as in the previous section. This means that (33.14.10) is a one-dimensional vector space over k spanned by

(y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}) · (e ⊗ u_{0}) = (y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}) · v (33.14.11)

for every m_{1}, …, m_{r} ≥ 0, where v is as in (33.14.3).

In particular, (33.14.11) is nonzero for all nonnegative integers m_{1}, …, m_{r}. It follows that v ≠ 0, by taking m_{1} = ⋯ = m_{r} = 0. This shows that v is a primitive or maximal vector in Z(μ) with weight μ. Thus Z(μ) is standard cyclic of weight μ, as a module over A. We also get that the elements (33.14.11) of Z(μ), with m_{1}, …, m_{r} ≥ 0, form a basis for Z(μ), as a vector space over k. This corresponds to some of the remarks on p110 of [14], and on p59 of [24]. This corresponds to part of Exercise 6 on p62 of [25] as well.

Observe that z ↦ z · (e ⊗ u_{0}) (33.14.12)

is a one-to-one linear mapping from UA(−Φ^{+}) onto Z(μ), because (33.14.7) is a basis for UA(−Φ^{+}), as a vector space over k. Note that Z(μ) may be considered as a left module over UA(−Φ^{+}), where UA(−Φ^{+}) is identitifed with a subalgebra of UA in the usual way. In fact, (33.14.12) is an isomorphism from UA(−Φ^{+}) onto Z(μ), as left modules over UA(−Φ^{+}), as mentioned on p110 of [14].

As in Section 33.9, Z(μ) has a unique nontrivial irreducible quotient, as a module over A. It is easy to see that the quotient is also standard cyclic of weight μ, using the image of v in the quotient. This corresponds to Theorem B on p110 of [14], Theorem 2 on p59 of [24], and part of Exercise 6 on p62 of [25].

More precisely, Z(μ) is universal as a standard cyclic module of weight μ over A, as in the next section. These modules are known as Verma modules, as mentioned on p167 of [14].

33.15 Homomorphisms into other modules

Let us continue with the same notation and hypotheses as in the previous section, including those from Section 33.1. Let V be a vector space over k that is a module over A, as a Lie algebra over k, and suppose that v_{0} ∈ V is a primitive or maximal vector of weight μ. Of course, V may also be considered as a module over B_{Δ}, as a Lie algebra over k. The linear span V_{0} of v_{0} in V is a submodule of V, as a module over B_{Δ}.

There is a unique linear mapping from L(μ) onto V_{0} that sends u_{0} to v_{0}. This mapping is a homomorphism from L(μ) into V_{0}, as modules over B_{Δ}.

We may consider V_{0} as a left module over UB_{Δ}, and V as a left module over UA, as associative algebras over k. If z ∈ UA and t ∈ k, then put

b(z, t u_{0}) = z · (t v_{0}), (33.15.1)

which is an element of V. This defines b as a mapping from (UA) × L(μ) into V that is bilinear over k.

If a ∈ UB_{Δ} and t ∈ k, then a · (t u_{0}) ∈ L(μ), because L(μ) is a left module over UB_{Δ}. If z ∈ UA, then we get that

b(z a, t u_{0}) = (z a) · (t v_{0}) = z · (a · (t v_{0})) = b(z, a · (t u_{0})). (33.15.2)

This implies that there is a unique mapping c from Z(μ) = (UA) ⨂_{UB_{Δ}} L(μ) into V that is linear over k and satisfies

b(z, t u_{0}) = c(z ⊗ (t u_{0})) (33.15.3)

for every t ∈ k and z ∈ UA, as in Section 33.12.

If a, z ∈ UA and t ∈ k, then

a · b(z, t u_{0}) = a · (z · (t v_{0})) = (a z) · (t v_{0}) = b(a z, t u_{0}). (33.15.4)

It follows that c is a homomorphism from Z(μ) into V as left modules over UA, as in Section 33.13. This means that c is a homomorphism from Z(μ) into V as modules over A, as a Lie algebra over k.

Remember that v = e ⊗ u_{0}, as in (33.14.3). Thus

c(v) = b(e, u_{0}) = e · v_{0} = v_{0}. (33.15.5)

This implies that c(Z(μ)) = c((UA) · v) = (UA) · v_{0}, (33.15.6)

using (33.14.4) in the first step. It follows that c maps Z(μ) onto V exactly when (UA) · v_{0} = V, (33.15.7)

which means that V is standard cyclic of weight μ, as in Section 33.8. If m_{1}, …, m_{r} are nonnegative integers, then

c((y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}) · v) = (y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}) · c(v) = (y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}) · v_{0}. (33.15.8)

Note that c is injective if and only if these vectors are linearly independent in V, as a vector space over k, because the vectors (33.14.11) form a basis for Z(μ) as a vector space over k, as in the previous section.

One can also obtain Z(μ) as the quotient of UA by a suitable left ideal, as on p110 of [14]. This gives another way to get the module homomorphism c from Z(μ) into V.

Some related examples and properties

34.1 Some remarks about sl_{2}(k)

Let k be a commutative ring with a multiplicative identity element, and let us take A = sl_{2}(k), as a Lie algebra over k with respect to the usual commutator bracket. As usual, we put x = \binom{0 1}{0 0}, y = \binom{0 0}{1 0}, and h = \binom{1 0}{0 −1}, so that sl_{2}(k) is freely generated as a module over k by x, y, and h. Remember that [x, y] = h, [h, x] = 2 · x, and [h, y] = −2 · y.

Let A_{0} be the subset of sl_{2}(k) consisting of multiples of h by elements of k, which is a commutative Lie subalgebra of sl_{2}(k). Suppose now that k is a field of characteristic 0, and let us consider the conditions discussed in Section 33.1 in this case. Of course, the elements of A_{0} are ad-diagonalizable in A, and it is easy to see that the centralizer of A_{0} in A is equal to A_{0}. Similarly, the center of A is trivial.

Let A′_{0} be the dual space of linear functionals on A_{0}, which correspond to multiplication by an element of k, because A_{0} is one-dimensional as a vector space over k. If α ∈ A′_{0}, then

A_{α} = {z ∈ A : [h, z] = α(h) z} (34.1.1)

here, which is the same as A_{0} when α = 0. If α_{x}, α_{y} ∈ A′_{0} are determined by α_{x}(h) = 2, α_{y}(h) = −2, then A_{α_{x}}, A_{α_{y}} are the linear subspaces of A spanned by x, y, respectively. Note that α_{y} = −α_{x}, and that A corresponds to the direct sum of A_{0}, A_{α_{x}}, and A_{α_{y}}, as a vector space over k. Thus

Φ = Φ_{A_{0}} = {α_{x}, α_{y}} (34.1.2)

is the set of nonzero α ∈ A′_{0} such that A_{α} ≠ {0}.

The linear subspace E_{Q} of A′_{0}, as a vector space over Q, spanned by Φ consists of linear functionals on A_{0} defined by multiplication by an element of Q, with respect to the natural embedding of Q into k. This can be identified with Q, as a vector space over itself, in the obvious way, so that the corresponding vector space E_{R} over R can be identified with R. Of course, Φ is a reduced root system in E_{R}, and Δ = {α_{x}} (34.1.3)

is a base for Φ.

Using this base, we have that Φ^{+} = Φ^{Δ,+} = {α_{x}} is the set of positive roots in Φ. The standard Borel subalgebra B_{Δ} = A(Φ^{+} ∪ {0}) of A associated to A_{0} and Δ is the same as the linear span of x and h in A.

With this notation, we have that h_{α_{x}} = h and h_{α_{y}} = −h. The symmetry σ_{α_{x}} = σ_{α_{y}} on E_{R} corresponds to multiplication by −1. The associated linear functionals λ_{α_{x}}, λ_{α_{y}} on E_{R} are determined by λ_{α_{x}}(α_{x}) = λ_{α_{y}}(α_{y}) = 2.

The other conditions mentioned in Section 33.1 can be verified directly here. If V is a module over A, then the basic notions concerning weights and primitive or maximal vectors in Sections 33.5, 33.6 correspond to those discussed in Sections 15.1, 15.2, respectively.

34.2 Some modules over sl_{2}(k)

Let k be a commutative ring with a multiplicative identity element, and let μ_{0} ∈ k be given. Also let Z_{0}(μ_{0}) be a free module over k with generators v_{0}, v_{1}, v_{2}, v_{3}, …, and put v_{−1} = 0, for convenience. Of course, this does not depend on μ_{0}, as a module over k.

Put

(34.2.1)  H(v_{j}) = (μ_{0} − 2 j) · v_{j},
(34.2.2)  Y (v_{j}) = (j + 1) · v_{j+1},
(34.2.3)  X(v_{j}) = (μ_{0} − j + 1) · v_{j−1}

for each j ≥ 0. More precisely, one can use the natural ring homomorphism from Z into k here, or interpret multiplication of elements of Z_{0}(μ_{0}) by integers in the usual way. Let H, Y, and X be the unique homomorphisms from Z_{0}(μ_{0}) into itself, as a module over k, that satisfy these conditions. We may use H_{μ_{0}}, Y_{μ_{0}}, and X_{μ_{0}} to indicate the dependence on μ_{0}, and that these are considered as mappings on Z_{0}(μ_{0}). Note that (34.2.1) and (34.2.2) hold trivially when j = −1.

The definitions of H, Y, and X are analogous to those in Section 15.4, and we shall say more about that later. As before, we have that

H(X(v_{j})) − X(H(v_{j})) = (μ_{0} − j + 1) · H(v_{j−1}) − (μ_{0} − 2 j) · X(v_{j})
(34.2.4)  = (μ_{0} − j + 1) (μ_{0} − 2 (j − 1)) · v_{j−1}
−(μ_{0} − 2 j)X(v_{j})
= ((μ_{0} − 2 (j − 1)) − (μ_{0} − 2 j)) · X(v_{j}) = 2 · X(v_{j})

34.2. SOME MODULES OVER SL_{2}(K) 713

for every j ≥ 0. In the same way,

H(Y (v_{j})) − Y (H(v_{j})) = (j + 1) · H(v_{j+1}) − (μ_{0} − 2 j) · Y (v_{j})
(34.2.5)  = (j + 1) (μ_{0} − 2 (j + 1)) · v_{j+1} − (μ_{0} − 2 j) · Y (v_{j})
= ((μ_{0} − 2 (j + 1)) − (μ_{0} − 2 j)) · Y (v_{j}) = −2 · Y (v_{j})

for each j ≥ 0. Similarly,

X(Y (v_{j})) − Y (X(v_{j})) = (j + 1) · X(v_{j+1}) − (μ_{0} − j + 1) · Y (v_{j−1})
(34.2.6)  = (j + 1) (μ_{0} − (j + 1) + 1) · v_{j}
−(μ_{0} − j + 1) ((j − 1) + 1) · v_{j}
= ((j + 1) (μ_{0} − j) − (μ_{0} − j + 1) j) · v_{j}
= ((μ_{0} − j) − j) · v_{j} = H(v_{j})

for each j ≥ 0. Thus

[H, X] = 2 · X, [H, Y] = −2 · Y, and [X, Y] = H, (34.2.7)

as mappings from Z_{0}(μ_{0}) into itself.

Using this, we may consider Z_{0}(μ_{0}) as a module over sl_{2}(k), as a Lie algebra over k. More precisely, if x, y, h ∈ sl_{2}(k) are as in the previous section, then their actions on Z_{0}(μ_{0}) are defined by X, Y, and H, respectively. This defines a representation of sl_{2}(k) on Z_{0}(μ_{0}), by (34.2.7). This corresponds to the first part of part (a) of Exercise 7 on p34 of [14].

Let m be a nonnegative integer, and let Z_{0,m}(μ_{0}) be the set of z ∈ Z_{0}(μ_{0}) such that for each j = 0, …, m, the coefficient of v_{j} in z is equal to 0. This is a submodule of Z_{0}(μ_{0}), as a module over k. Thus the quotient Z_{0}(μ_{0})/Z_{0,m}(μ_{0}) is defined as a module over k too. Let q_{m} be the natural quotient mapping from Z_{0}(μ_{0}) onto Z_{0}(μ_{0})/Z_{0,m}(μ_{0}). It is easy to see that Z_{0}(μ_{0})/Z_{0,m}(μ_{0}) is free as a module over k, with generators q_{m}(v_{j}), j = 0, …, m.

Suppose that μ_{0} = m · 1, (34.2.8)

as an element of k. This implies that

x · v_{m+1} = X(v_{m+1}) = 0, (34.2.9)

by (34.2.3). In this case, one can check that Z_{0,m}(μ_{0}) is a submodule of Z_{0}(μ_{0}), as a module over sl_{2}(k).

It follows that Z_{0}(μ_{0})/Z_{0,m}(μ_{0}) is defined as a module over sl_{2}(k) under these conditions. Observe that

y · q_{m}(v_{m}) = q_{m}(y · v_{m}) = q_{m}(Y (v_{m})) = 0 (34.2.10)

in Z_{0}(μ_{0})/Z_{0,m}(μ_{0}), by (34.2.2). One can use this to get that Z_{0}(μ_{0})/Z_{0,m}(μ_{0}) is isomorphic to the module W (m) discussed in Section 15.4, as modules over sl_{2}(k). More precisely, q_{m}(v_{j}), j = 0, …, m, correspond to the generators of W (m), as a module over k, in Section 15.4. This is related to part (b) of Exercise 7 on p34 of [14].

34.3 Some module homomorphisms

Let k be a field of characteristic 0, and let V be a vector space over k that is a module over A = sl_{2}(k), as a Lie algebra over k. Also let A_{0} be the linear span of h in A, as in Section 34.1. If μ_{0} ∈ k, then there is a unique linear functional μ on A_{0} such that μ(h) = μ_{0}, (34.3.1)

and every linear functional on A_{0} is of this form.

Let ρ = ρ^{V} be the given representation of sl_{2}(k) on V. An eigenvector of ρ_{h} in V with eigenvalue μ_{0} ∈ k is the same as an eigenvector of A_{0} in V with weight μ ∈ A′_{0} as in (34.3.1), as defined in Section 33.5. A nonzero eigenvector v of this type is primitive or maximal of weight μ_{0} in the sense of Section 15.2 if x · v = 0, which is the same as saying that v is primitive or maximal of weight μ in the sense of Section 33.6. Of course, this uses Δ as in (34.1.3).

Let μ_{0} ∈ k be given, and let Z_{0}(μ_{0}) be as in the previous section, as a vector space over k, and a module over sl_{2}(k). Note that v_{0} is a primitive or maximal vector of weight μ_{0} in Z_{0}(μ_{0}), as in Section 15.2. It is easy to see that Z_{0}(μ_{0}) is standard cyclic with respect to v_{0}, as in Section 33.8.

Suppose that v ∈ V is primitive or maximal of weight μ_{0} in the sense of Section 15.2. Observe that there is a unique linear mapping ϕ from Z_{0}(μ_{0}) into V such that ϕ(v_{j}) = (1/j!) (ρ_{y})^{j}(v) (34.3.2)

for every nonnegative integer j. In fact, ϕ is a homomorphism from Z_{0}(μ_{0}) into V, as modules over sl_{2}(k). This follows from the remarks in Section 15.3, with λ = μ_{0}.

Of course, ϕ(Z_{0}(μ_{0})) is the same as the linear subspace of V spanned by the vectors (34.3.2). This is the same as the submodule of V, as a module over sl_{2}(k), generated by v. Thus ϕ(Z_{0}(μ_{0})) = V exactly when V is standard cyclic with respect to v, as in Section 33.8.

If ϕ is injective as a mapping from Z_{0}(μ_{0}) into V, then (34.3.2) is nonzero for every j ≥ 0. Conversely, if (34.3.2) is nonzero for every j ≥ 0, then these vectors are linearly independent in V. This is because these vectors are eigenvectors for ρ_{h} with distinct eigenvalues, as in Section 15.3. In this case, ϕ is injective on Z_{0}(μ_{0}).

If j = 0, then (34.3.2) is equal to v, which is nonzero by hypothesis. Suppose now that (34.3.2) is equal to 0 for some j ≥ 1, and thus all larger j. Let m be the largest nonnegative integer such that (34.3.2) is nonzero when j = m, which implies that (34.3.2) is nonzero when j ≤ m, and equal to 0 when j > m. Under these conditions, μ_{0} = m, (34.3.3)

with respect to the natural embedding of Q into k, as in Section 15.3.

The vectors (34.3.2) with j = 0, …, m are linearly independent in V, because they are nonzero eigenvectors of ρ_{h} with distinct eigenvalues, as before. This implies that the kernel of ϕ is equal to the linear subspace Z_{0,m}(μ_{0}) of Z_{0}(μ_{0})

34.4. SOME PROPERTIES OF Z_{0}(μ_{0}) 715

defined in the previous section. Remember that Z_{0,m}(μ_{0}) is a submodule of Z_{0}(μ_{0}), as a module over sl_{2}(k), in this case, which also follows from the fact that ϕ is a module homomorphism.

Using (34.3.3), we get that (34.2.9) holds, as before. This means that v_{m+1} is primitive or maximal of weight

μ_{0} − 2 (m + 1) = −m − 2 (34.3.4)

in Z_{0}(μ_{0}). Note that v_{m+1} ∈ Z_{0,m}(μ_{0}), so that v_{m+1} may be considered as primitive or maximal in Z_{0,m}(μ_{0}), as a module over sl_{2}(k).

If j is a nonnegative integer, then Y^{j}(v_{m+1}) is a nonzero multiple of v_{m+1+j}, by (34.2.2). Thus Z_{0,m}(μ_{0}) is spanned by Y^{j}(v_{m+1}), j ≥ 0, as a vector space over k. This means that Z_{0,m}(μ_{0}) is standard cyclic with respect to v_{m+1}, as a module over sl_{2}(k).

Let Z_{0}(−m − 2) be as in the previous section, using the natural embedding of Q into k to identify −m − 2 with an element of k. The same argument as before leads to an isomorphism from Z_{0}(−m − 2) onto Z_{0,m}(μ_{0}), as modules over sl_{2}(k). This corresponds to part of part (b) of Exercise 7 on p34 of [14].

34.4 Some properties of Z_{0}(μ_{0})

Let k be a field of characteristic 0 again, let μ_{0} ∈ k be given, and let Z_{0}(μ_{0}) be as in Section 34.2. If z is any element of Z_{0}(μ_{0}), then X^{j}(x) = 0 when j is large enough. Suppose that z ≠ 0, and let j_{1} be the largest nonnegative integer such that z_{1} = X^{j_{1}}(z) ≠ 0. (34.4.1)

Thus X(z_{1}) = X^{j_{1}+1}(z) = 0. (34.4.2)

Let Z_{1} be a submodule of Z_{0}(μ_{0}), as a module over sl_{2}(k). If Z_{1} ≠ {0}, then there is a z_{1} ∈ Z_{1} such that z_{1} ≠ 0 and X(z_{1}) = 0, by the remarks in the preceding paragraph.

Suppose for the moment that μ_{0} does not correspond to a nonnegative integer, with respect to the natural embedding of Q into k. If z ∈ Z_{0}(μ_{0}) satisfies

x · z = X(z) = 0, (34.4.3)

then it is easy to see that z is a multiple of v_{0}, using (34.2.3).

If Z_{1} is a nontrivial submodule of Z_{0}(μ_{0}), as a module over sl_{2}(k), then it follows that Z_{1} contains a nonzero multiple of v_{0}. It is easy to see that Z_{1} = Z_{0}(μ_{0}) in this case, using the action of y on Z_{1}. This means that Z_{0}(μ_{0}) is irreducible, as a module over sl_{2}(k). This corresponds to part (c) of Exercise 7 on p34 of [14], and is related to the second part of part (a) of Exercise 7.

Let m be a nonnegative integer, and suppose now that μ_{0} corresponds to m, under the natural embedding of Q into k. If z ∈ Z_{0}(μ_{0}) satisfies (34.4.3), then one can check that z is a linear combination of v_{0} and v_{m+1}, using (34.2.3).

Let Z_{1} be a submodule of Z_{0}(μ_{0}) again, as a module over sl_{2}(k). Suppose that Z_{1} ≠ {0}, and let z_{1} be a nonzero element of Z_{1} such that X(z_{1}) = 0, as before. Thus z_{1} is a linear combination of v_{0} and v_{m+1}, as in the previous paragraph. Remember that v_{0} and v_{m+1} are eigenvectors of H with distinct eigenvalues, by (34.2.1). One can use this to get that both the v_{0} and v_{m+1} components of z_{1} are in Z_{1}, because Z_{1} is a submodule of Z_{0}(μ_{0}).

If Z_{1} contains a nonzero multiple of v_{0}, then Z_{1} = Z_{0}(μ_{0}), as before. If Z_{1} ≠ Z_{0}(μ_{0}), then it follows that Z_{1} contains a nonzero multiple of v_{m+1}. This is related to the second part of part (a) of Exercise 7 on p34 of [14].

Let Z_{0,m}(μ_{0}) be the linear subspace of Z_{0}(μ_{0}) defined in Section 34.2, which is a submodule of Z_{0}(μ_{0}), as a module over sl_{2}(k), in this case. If Z_{1} is a submodule of Z_{0}(μ_{0}), as a module over sl_{2}(k), that contains a nonzero multiple of v_{m+1}, then one can check that

Z_{0,m}(μ_{0}) ⊆ Z_{1}, (34.4.4)

using the action of y on Z_{1}.

In particular, one can use this to get that Z_{0,m}(μ_{0}) is irreducible as a module over sl_{2}(k), which corresponds to part of part (b) of Exercise 7 on p34 of [14]. This could also be obtained from the fact that Z_{0,m}(μ_{0}) is isomorphic to Z_{0}(−m − 2), as modules over sl_{2}(k), as in the previous section.

Let Z_{1} be a nontrivial submodule of Z_{0}(μ_{0}) again, as a module over sl_{2}(k), so that (34.4.4) holds, as before. If Z_{1} ≠ Z_{0,m}(μ_{0}), then Z_{1} = Z_{0}(μ_{0}). Equivalently, this means that Z_{0}(μ_{0})/Z_{0,m}(μ_{0}) is irreducible as a module over sl_{2}(k). As in Section 34.2, the quotient is isomorphic to the module W (m) in Section 15.4, whose irreducibility was discussed in Section 15.5. This corresponds to another part of part (b) of Exercise 7 on p34 of [14].

Alternatively, if Z_{1} ≠ Z_{0,m}(μ_{0}), then Z_{1} contains a nonzero element for which the coefficient of v_{j} is equal to 0 when j ≥ m + 1. One can apply X to this element repeatedly, if necessary, to get that Z_{1} contains a nonzero multiple of v_{0}. This implies that Z_{1} = Z_{0}(μ_{0}), as before. One could also start with any z ∈ Z_{1} \ Z_{0,m}(μ_{0}), and apply X to it repeatedly, as needed, to get z_{1} ∈ Z_{1} \ Z_{0,m}(μ_{0}) such that X(z_{1}) = 0. This implies that the v_{0} component of z_{1} is nonzero, and that it is contained in Z_{1}, as before.

Note that Z_{0}(μ_{0}) does not correspond to the direct sum of Z_{0,m}(μ_{0}) and some other submodule, as a module over sl_{2}(k). This is mentioned in part (b) of Exercise 7 on p34 of [14] as well.

34.5 Comparison with Z(μ)

Let k be a field of characteristic 0, and let us return to the same notation and hypotheses as in Section 34.1. Also let μ_{0} ∈ k be given, and let μ be the linear functional on the linear span A_{0} of h in A = sl_{2}(k) with μ(h) = h_{0}, as before. We would like to review the construction of Z(μ) in Section 33.14 in this case, and to compare it with Z_{0}(μ_{0}) in Section 34.2.

34.5. COMPARISON WITH Z(μ) 717

As in Section 34.1, α_{x} ∈ A′_{0} is defined by α_{x}(h) = 2, Δ = {α_{x}} is a base for Φ, Φ^{+} = Φ^{Δ,+} = {α_{x}}, and the standard Borel subalgebra B_{Δ} = A(Φ^{+} ∪ {0}) of A associated to A_{0} and Δ is the same as the linear span of x and h in A. Observe that N_{Δ} = A(Φ^{+}) is the linear span of x in A, B_{Δ} corresponds to the direct sum of A_{0} and N_{Δ} as a vector space over k, and that [B_{Δ}, B_{Δ}] = N_{Δ}.

As in Section 33.14, we let L(μ) be a one-dimensional vector space over k, and we can define an action of B_{Δ} on L(μ) by putting

(w + t x) · u = μ(w) u (34.5.1)

for every w ∈ A_{0}, t ∈ k, and u ∈ L(μ). Of course, this means that h · u = μ_{0} u and x · u = 0. This makes L(μ) into a module over B_{Δ}, as a Lie algebra over k, as before.

Let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA} and associated mapping i = i_{UA} from A into UA, as before. The subalgebra of UA generated by e and i(B_{Δ}) is a universal enveloping algebra of B_{Δ}, and we denote it UB_{Δ}. Remember that i is injective, and let us identify A with its image i(A) in UA, as usual. Thus UB_{Δ} is the same as the subalgebra of UA generated by e, x, and h in this case. We may consider L(μ) as a left module over UB_{Δ}, as an associative algebra over k, as before.

As in Section 33.14, we take

Z(μ) = (UA)⨂_{UB_{Δ}}L(μ), (34.5.2)

where UA is considered as a right module over UB_{Δ}, and the tensor product is as in Sections 33.12 and 33.13. This is a vector space over k, a left module over UA, and thus a module over A, as a Lie algebra over k.

Let u_{0} be a nonzero element of L(μ), and put v = e ⊗ u_{0} ∈ Z(μ), as before. Remember that Z(μ) is generated by v, as a left module over UA. If w ∈ A_{0}, then w · v = μ(w) v, (34.5.3)

as in Section 33.14. This means that that h · v = μ_{0} v, and we also have that x · v = 0, as before.

Remember from Section 34.1 that α_{y} = −α_{x}, so that −Φ^{+} = {α_{y}}. Thus A(−Φ^{+}) is spanned by y, and is a Lie subalgebra of A. The elements of Z(μ) of the form y^{m} · v, (34.5.4)

where m is a nonnegative integer, form a basis for Z(μ) as a vector space over k, as in Section 33.14.

It follows that v is a primitive or maximal vector in Z(μ) with weight μ, and that Z(μ) is standard cyclic of weight μ, as a module over A, as before. This leads to an isomorphism ϕ from Z_{0}(μ_{0}) onto Z(μ), as modules over A, as in Section 34.3. Alternatively, one can get a module isomorphism from Z(μ) onto Z_{0}(μ_{0}) as in Section 33.15. This corresponds to Exercise 4 on p111 of [14].

34.6 Looking at x in UA

Let k be a commutative ring with a multiplicative identity element and k ≠ {0}, and take A = sl_{2}(k), as in Section 34.1. Also let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA} and associated mapping i = i_{UA}, as in Section 25.4.

Remember that sl_{2}(k) is freely generated, as a module over k, by the usual elements x, y, and h. As in Section 25.12, the Poincaré–Birkhoff–Witt theorem implies that i is an injective mapping from A into UA. Thus we may identify A with i(A), as before.

In particular, we may consider x, y, and h as elements of UA. If w ∈ UA, then we would like to check that

w (e − x) ≠ e, (34.6.1)

so that e − x does not have a left inverse in UA. This corresponds to part of part (c) of Exercise 2 on p111 of [14].

Let us use the ordering y, h, x, for the basis elements of A. The Poincaré– Birkhoff–Witt theorem implies that the collection of products of the form

y^{j_{1}} h^{j_{2}} x^{j_{3}}, (34.6.2)

where j_{1}, j_{2}, j_{3} are nonnegative integers, is a basis for UA as a module over k, as in Section 25.10. If w ∈ UA is expressed as a linear combination of these basis elements, then w x can be expressed as a linear combination of these basis elements in a simple way. One can use this to get (34.6.1), as desired.

Let I be a left ideal in UA, as an associative algebra over k. Thus the quotient (UA)/I can be defined as a module over k, and as a left module over UA. Let q = q_{I} be the natural quotient mapping from UA onto (UA)/I, so that (UA) · q(e) = (UA)/I, (34.6.3)

by construction, as in Section 33.11. Of course, we may also consider (UA)/I as a module over A, as a Lie algebra over k, as before.

Suppose that e − x ∈ I. (34.6.4)

Equivalently, this means that

q(x) = q(e). (34.6.5)

Note that q(x) = q(x e) = x · q(e), (34.6.6)

using the fact that (UA)/I is a left module over UA in the second step. Thus (34.6.5) is the same as saying that

x · q(e) = q(e) (34.6.7)

in (UA)/I.

34.7. SOME RELATED PROPERTIES OF X AND H 719

Observe that q(e) ≠ 0 (34.6.8)

if and only if I ≠ UA, (34.6.9)

because e ∈ I if and only if I = UA. Clearly

I_{0} = {w (e − x) : w ∈ UA} (34.6.10)

is a left ideal in UA, which does not contain e, by (34.6.1).

34.7 Some related properties of x and h

Let us continue with the same notation and hypotheses as in the previous section. Observe that

(34.7.1) (x − e) h = h (x − e) − (h x − x h)
= h (x − e) − 2 · x = h (x − e) − 2 · (x − e) − 2 · e,

using the fact that [h, x] = 2 · x in the second step. If r is a positive integer, then we get that

(x − e)^{r} h = (x − e)^{r−1} h (x − e) − 2 · (x − e)^{r} − 2 · (x − e)^{r−1}. (34.7.2)

It follows that

(x − e)^{r} h = h (x − e)^{r} − (2 r) · (x − e)^{r} − (2 r) · (x − e)^{r−1} (34.7.3)

for every r ≥ 1, by using (34.7.2) repeatedly, or induction.

If r, m are positive integers with r > m, then one can use (34.7.3) to check that (x − e)^{r} h^{m} ∈ I_{0}, (34.7.4)

where I_{0} is as in (34.6.10). Similarly, one can use (34.7.3) to verify that

(x − e)^{r} h^{r} − ((−2)^{r} r!) · e ∈ I_{0} (34.7.5)

for every r ≥ 1. This corresponds to part of part (c) of Exercise 2 on p111 of [14].

Let V be a module over k that is a module over A = sl_{2}(k), as a Lie algebra over k. Thus V may be considered as a left module over UA, as an associative algebra over k. Let v_{0} be an element of V, and suppose that

x · v_{0} = v_{0}. (34.7.6)

Equivalently, this means that

(e − x) · v_{0} = 0. (34.7.7)

It follows that I_{0} · v_{0} = {0}. (34.7.8)

If r and m are positive integers with r > m, then we get that

((x − e)^{r} h^{m}) · v_{0} = 0, (34.7.9)

by (34.7.4). We also have that

((x − e)^{r} h^{r}) · v_{0} = ((−2)^{r} r!) · v_{0} (34.7.10)

for every r ≥ 1, by (34.7.5).

Suppose now that k is a field of characteristic 0, and that v_{0} ≠ 0. Under these conditions, one can check that the elements of V of the form h^{r} · v_{0}, where r is a nonnegative integer, are linearly independent, using (34.7.10). This corresponds to another part of part (c) of Exercise 2 on p111 of [14].

If I is a proper left ideal in UA that satisfies (34.6.4), then we can take V = (UA)/I, as in the previous section, with v_{0} = q(e). If I is also maximal as a proper left ideal in UA, then (UA)/I is irreducible as a left module over UA, or a module over A as a Lie algebra over k, as in Section 33.11.

34.8 Irreducibility and weights

Let us return to the notation and hypotheses in Section 33.1, so that k is a field of characteristic 0, and (A, [·, ·]_{A}) is a Lie algebra over k with the properties mentioned before. Remember that A_{0} is a Lie subalgebra of A that is commutative as a Lie algebra. Let V be a vector space over k that is a module over A, as a Lie algebra over k.

Let V̂ be the linear subspace of V spanned by the weight spaces of V with respect to A_{0}, as in Section 33.5. Thus V̂ is a submodule of V, as a module over A, as before. If V̂ ≠ {0}, and V is irreducible as a module over A, then we get that V̂ = V. (34.8.1)

This corresponds to part (a) of Exercise 2 on p110 of [14].

If v ∈ V is an eigenvector of A_{0} of some weight, then the linear span of v in V is mapped into itself by every element of A_{0}. If v ∈ V̂, then v can be expressed as the sum of finitely many eigenvectors of A_{0}. This implies that

(34.8.2) there is a finite-dimensional linear subspace of V that contains v and is mapped into itself by every element of A_{0}.

If (34.8.1) holds, then it follows that (34.8.2) holds for every v ∈ V. This corresponds to part of part (b) of Exercise 2 on p111 of [14].

Suppose for the moment that A = sl_{2}(k), as in Section 34.1, and that V is as in the previous section. If v_{0} ≠ 0, then we have seen that (34.8.2) does not hold with v = v_{0}. This means that

V̂ = {0} (34.8.3)

34.9. WEIGHT SPACES IN Z(μ) 721

when V is irreducible, as before. This corresponds to part of part (c) of Exercise 2 on p111 of [14].

Remember that the actions of the elements of A_{0} on V commute with each other, because A_{0} is commutative as a Lie algebra, and V is a module over A, as a Lie algebra over k. Suppose that U is a linear subspace of V of positive finite dimension that is mapped into itself by every element of A_{0}. If k is algebraically closed, then the action of any element of A_{0} on U has a nontrival eigenspace. One can use this to get a nontrivial linear subspace of U consisting of eigenvectors for the actions of all of the elements of A_{0}, because the actions of the elements of A_{0} on V commute, as before. This means that

U ∩ V̂ ≠ {0}, (34.8.4)

which corresponds to part of part (b) of Exercise 2 on p111 of [14]. Suppose that v ∈ V has the property that

(34.8.5) for every w ∈ A_{0} there is a finite-dimensional linear subspace of V that contains v and is mapped into itself by the action of w on V.

Under these conditions, one can check that (34.8.2) holds. This uses the fact that A_{0} has finite dimension, as a vector space over k, and that the actions of the elements of A_{0} on V commute. This corresponds to another part of part (b) of Exercise 2 on p111 of [14].

Of course, (34.8.2) implies (34.8.5) automatically.

34.9 Weight spaces in Z(μ)

Let us go back to the notation and hypotheses in Section 33.1 again. More precisely, let μ ∈ A′_{0} be given, and let Z(μ) be the universal standard cyclic module of weight μ over A constructed in Section 33.14.

Remember that v = e ⊗ u_{0} ∈ Z(μ) is a primitive or maximal vector in Z(μ) of weight μ, and that (UA) · v = Z(μ). Let β_{1}, …, β_{r} be a list of the elements of Φ^{+}, so that y_{β_{1}}, …, y_{β_{r}} is a basis for A(−Φ^{+}), as a vector space over k, as in Section 33.14. We have seen that

{(y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}) · v : m_{1}, …, m_{r} ∈ Z_{+} ∪ {0}} (34.9.1)

form a basis for Z(μ), as a vector space over k.

If m_{1}, …, m_{r} are nonnegative integers, then

(34.9.2) (y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}) · v has weight μ − ∑_{j=1}^{r} m_{j} β_{j}

as an element of Z(μ) in the sense of Section 33.5, as in Section 33.8. All of the weights of Z(μ) are of this form, as before.

Remember that Δ is a base for the root system Φ, and that Δ is a basis for A′_{0}, as a vector space over k, as in Section 33.1. If m_{1}, …, m_{r} are nonnegative integers, then there are nonnegative integers c_{α}, α ∈ Δ, such that

∑_{j=1}^{r} m_{j} β_{j} = ∑_{α∈Δ} c_{α} α, (34.9.3)

because β_{j} ∈ Φ^{+} for each j = 1, …, r, by hypothesis.

If c_{α} is a nonnegative integer for each α ∈ Δ, then there are only finitely many families of nonnegative integers m_{1}, …, m_{r} such that (34.9.3) holds, as in Section 33.8. More precisely, this is considered as an equality in A′_{0}, and uses the fact that k is a field of characteristic 0.

If ν ∈ A′_{0}, then let P(ν) be the number of families of nonnegative integers m_{1}, …, m_{r} such that

ν = ∑_{j=1}^{r} m_{j} β_{j}. (34.9.4)

Of course, this number may be 0, but it is finite, as in the preceding paragraph.

Thus P(μ − ν) is the same as the number of families of nonnegative integers m_{1}, …, m_{r} such that

ν = μ − ∑_{j=1}^{r} m_{j} β_{j}. (34.9.5)

Equivalently, this is the number of families of nonnegative integers m_{1}, …, m_{r} such that (34.9.6) (y_{β_{1}}^{m_{1}} ⋯ y_{β_{r}}^{m_{r}}) · v has weight ν,

by (34.9.2).

Let Z(μ)_{ν} be the linear subspace of Z(μ) consisting of vectors of weight ν, as in Section 33.5. Under these conditions, we get that

dim Z(μ)_{ν} = P(μ − ν), (34.9.7)

as a vector space over k. More precisely, Z(μ)_{ν} ≠ {0} exactly when P(μ−ν) > 0, in which case we get a basis for Z(μ)_{ν} using the vectors mentioned in (34.9.6). This corresponds to Exercise 5 on p111 of [14].

Formal Laurent series

35.1 Formal Laurent polynomials and series

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, let n be a positive integer, and let T_{1}, …, T_{n} be n commuting indeterminates. If α = (α_{1}, …, α_{n}) is an n-tuple of integers, then the degree of α is defined by

deg(α) = ∑_{j=1}^{n} α_{j}. (35.1.1)

This is the same as the degree of the corresponding formal Laurent monomial

T^{α} = T_{1}^{α_{1}} ⋯ T_{n}^{α_{n}}. (35.1.2)

A formal Laurent series in T_{1}, …, T_{n} with coefficients in A can be expressed as

f(T) = f(T_{1}, …, T_{n}) = ∑_{α∈Z^{n}} f_{α} T^{α}, (35.1.3)

where f_{α} ∈ A for every α ∈ Z^{n}. A formal power series in T_{1}, …, T_{n} with coefficients in A may be considered as a formal Laurent series as in (35.1.3), with f_{α} ≠ 0 only when α_{j} ≥ 0 for each j = 1, …, n.

The space LS_{A}(T_{1}, …, T_{n}) = LS(A)(T_{1}, …, T_{n}) (35.1.4)

of all formal Laurent series in T_{1}, …, T_{n} with coefficients in A can be defined as the set of all A-valued functions on Z^{n}. This is a module over k with respect to pointwise addition and scalar multiplication of A-valued functions on Z^{n}, which corresponds to termwise addition and scalar multiplication of formal Laurent series as in (35.1.3). Of course, this is the same as the direct product of copies of A indexed by Z^{n}, as a module over k. The space A[[T_{1}, …, T_{n}]] of formal power series in T_{1}, …, T_{n} with coefficients in A corresponds to a submodule of LS_{A}(T_{1}, …, T_{n}), as a module over k.

A formal Laurent polynomial in T_{1}, …, T_{n} with coefficients in A may be defined as a formal Laurent series as in (35.1.3) such that f_{α} = 0 for all but finitely many α ∈ Z^{n}. The space

LP_{A}(T_{1}, …, T_{n}) = LP (A)(T_{1}, …, T_{n}) (35.1.5)

of formal Laurent polynomials in T_{1}, …, T_{n} with coefficients in k may be defined as the space of A-valued functions on Z^{n} with finite support. This is a submodule of LS_{A}(T_{1}, …, T_{n}), as a module over k, which corresponds to the direct sum of copies of A indexed by Z^{n}. A formal polynomial in T_{1}, …, T_{n} with coefficients in A may be considered as a formal Laurent polynomial as in (35.1.3), with f_{α} ≠ 0 only when α_{j} ≥ 0 for each j = 1, …, n, and for only finitely many such α. The space A[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in A corresponds to a submodule of LP_{A}(T_{1}, …, T_{n}), as a module over k.

Let l, m be positive integers, and let X_{1}, …, X_{l}, Y_{1}, …, Y_{m} be commuting indeterminates. Let us identify Z^{l} × Z^{m} with Z^{l+m} in the obvious way. Thus if β ∈ Z^{l} and γ ∈ Z^{m}, then (β, γ) is identified with an element of Z^{l+m}. The degrees of β, γ, and (β, γ) can be defined as before, so that

(35.1.6) deg(β, γ) = deg(β) + deg(γ).

In this case, we may consider

X^{β} Y^{γ} = X_{1}^{β_{1}} ⋯ X_{l}^{β_{l}} Y_{1}^{γ_{1}} ⋯ Y_{m}^{γ_{m}} (35.1.7)

as a formal Laurent monomial in X_{1}, … X_{l}, Y_{1}, …, Y_{m} with degree (35.1.6). We can define LS_{A}(X_{1}, …, X_{l}) as a module over k as before, so that

LS(LS_{A}(X_{1}, …, X_{l}))(Y_{1}, …, Y_{m}) (35.1.8)

may be defined as a module over k in the same way. There is an obvious oneto-one correspondence between (35.1.8) and

LS_{A}(X_{1}, …, X_{l}, Y_{1}, …, Y_{m}), (35.1.9)

which associates an A-valued function on Z^{l+m} with a function on Z^{m} that takes values in the space of A-valued functions on Z^{l}. This defines an isomorphism between (35.1.8) and (35.1.9), as modules over k. Similarly, we can define LP_{A}(X_{1}, …, X_{l}) as a module over k as before, so that

LP (LP_{A}(X_{1}, …, X_{l}))(Y_{1}, …, Y_{m}) (35.1.10)

may be defined as a module over k too. The one-to-one correspondence between (35.1.8) and (35.1.9) just mentioned sends (35.1.10) onto

LP_{A}(X_{1}, …, X_{l}, Y_{1}, …, Y_{m}). (35.1.11)

35.2 Multiplication and Laurent series

Let us continue with the same notation and hypotheses as in the previous section. If α, β ∈ Z^{n}, then α + β ∈ Z^{n} is defined as usual by coordinatewise addition, and (35.2.1) deg(α + β) = deg(α) + deg(β).

Similarly, multiplication of the corresponding formal Laurent monomials is defined by T^{α} T^{β} = T^{α+β}. (35.2.2)

Let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Suppose that f(T) ∈ LS_{A}(T_{1}, …, T_{n}) is as in (35.1.3), and that

g(T) = ∑_{β∈Z^{n}} g_{β} T^{β} (35.2.3)

is another element of LS_{A}(T_{1}, …, T_{n}). If γ ∈ Z^{n}, then we would like to put

h_{γ} = ∑_{α+β=γ} f_{α} g_{β}, (35.2.4)

where more precisely the sum is taken over all α, β ∈ Z^{n} with α + β = γ. This defines an element of A when all but finitely many terms in the sum on the right are equal to 0. In particular, this happens for every γ ∈ Z^{n} when f(T) and g(T) are formal power series in T_{1}, …, T_{n}, as in Section 5.8, and when at least one of f(T) and g(T) is a formal Laurent polynomial in T_{1}, …, T_{n}.

If (35.2.4) is defined as an element of A for every γ ∈ Z^{n}, then

h(T) = ∑_{γ∈Z^{n}} h_{γ} T^{γ} (35.2.5)

defines an element of LS_{A}(T_{1}, …, T_{n}). In this case, we put

f(T) g(T) = h(T). (35.2.6)

Of course, this agrees with the definition of the product in Section 5.8 when f(T), g(T) ∈ A[[T_{1}, …, T_{n}]]. If f(T) and g(T) are both formal Laurent polynomials in T_{1}, …, T_{n}, then we also get that (35.2.4) is equal to 0 for all but finitely many γ ∈ Z^{n}, so that

h(T) ∈ LP_{A}(T_{1}, …, T_{n}). (35.2.7)

This makes LP_{A}(T_{1}, …, T_{n}) an algebra over k in the strict sense, which contains A[T_{1}, …, T_{n}] as a subalgebra.

If A is an associative algebra over k, then one can check that

(35.2.8) LP_{A}(T_{1}, …, T_{n}) is an associative algebra over k too. Similarly,

(35.2.9) LS_{A}(T_{1}, …, T_{n}) is a left and right module over LP_{A}(T_{1}, …, T_{n})

under these conditions. If A has a multiplicative identity element e, then the corresponding formal Laurent polynomial in T_{1}, …, T_{n} is the multiplicative identity element in LP_{A}(T_{1}, …, T_{n}). If A is commutative, then multiplication of formal Laurent series in T_{1}, …, T_{n} with coefficients in A is commutative when it is defined. In particular,

(35.2.10) LP_{k}(T_{1}, …, T_{n}) is a commutative associative algebra over k.

Suppose that A is a module over k again, let f(T) ∈ LS_{k}(T_{1}, …, T_{n}) be as in (35.1.3), and let g(T) ∈ LS_{A}(T_{1}, …, T_{n}) be as in (35.2.3). Note that f_{α} g_{β} is defined as an element of A for every α, β ∈ Z^{n}, using scalar multiplication on A. If γ ∈ Z^{n}, then we would like to define h_{γ} as an element of A as in (35.2.4). As before, this make sense when all but finitely many terms in the sum on the right side of (35.2.4) are equal to 0. This happens for every γ ∈ Z^{n} when f(T) and g(T) are formal power series in T_{1}, …, T_{n}, and when at least one of f(T) and g(T) is a formal Laurent polynomial in T_{1}, …, T_{n}.

If (35.2.4) is defined as an element of A for every γ ∈ Z^{n}, then (35.2.5) defines an element of LS_{A}(T_{1}, …, T_{n}), which we use to define f(T) g(T) as in (35.2.6). In particular, one can check that

(35.2.11) LS_{A}(T_{1}, …, T_{n}) is a module over LP_{k}(T_{1}, …, T_{n})

in this way. If f(T) and g(T) are both formal Laurent polynomials in T_{1}, …, T_{n}, then (35.2.4) is equal to 0 for all but finitely many γ ∈ Z^{n}, so that (35.2.7) holds. This means that

(35.2.12) LP_{A}(T_{1}, …, T_{n}) is a submodule of LS_{A}(T_{1}, …, T_{n}), as a module over LP_{k}(T_{1}, …, T_{n}).

Let l, m ∈ Z_{+} and X_{1}, …, X_{l}, Y_{1}, …, Y_{m} be as in the previous section. If A is an algebra over k in the strict sense, then LP_{A}(X_{1}, …, X_{l}) is an algebra over k in the strict sense with respect to multiplication of formal Laurent polynomials, as before. One can use this to define multiplication on (35.1.10), to get an algebra over k in the strict sense. Of course, (35.1.11) is an agebra over k in the strict sense with respect to multiplication of formal Laurent polynomials as well. One can check that the bijection between (35.1.10) and (35.1.11) mentioned in the previous section is an algebra isomorphism under these conditions. In particular, LP (LP_{k}(X_{1}, …, X_{l}))(Y_{1}, …, Y_{m}) (35.2.13)

is a commutative associative algebra over k. This algebra is isomorphic to

LP_{k}(X_{1}, …, X_{l}, Y_{1}, …, Y_{m}), (35.2.14)

using the bijection just mentioned.

Similarly, if A is a module over k, then LS_{A}(X_{1}, …, X_{l}) may be considered as a module over LP_{k}(X_{1}, …, X_{l}). This permits (35.1.8) to be considered as a module over (35.2.13). We may also consider (35.1.9) as a module over (35.2.14). One can check that scalar multiplication on (35.1.8) by elements of (35.2.13) corresponds to scalar multiplication on (35.1.9) by elements of (35.2.14). More precisely, this uses the bijection between (35.1.8) and (35.1.9) mentioned in the previous section, and the analogous bijection between (35.2.13) and (35.2.14).

35.3 Laurent polynomial functions

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. Also let t = (t_{1}, …, t_{n}) be an n-tuple of elements of k, and suppose that t_{j} has a multiplicative inverse in k for each j = 1, …, n. If α ∈ Z^{n}, then t^{α} = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}} (35.3.1)

is defined as an element of k. Note that

t^{α+β} = t^{α} t^{β} (35.3.2)

for every α, β ∈ Z^{n}.

Let A be a module over k, and let T_{1}, …, T_{n} be commuting indeterminates. Suppose that f(T) ∈ LP_{A}(T_{1}, …, T_{n}) is as in (35.1.3). If t ∈ k^{n} is as in the preceding paragraph, then

f(t) = ∑_{α∈Z^{n}} f_{α} t^{α} (35.3.3)

defines an element of A. More precisely, f_{α} t^{α} is defined as an element of A for each α ∈ Z^{n}, using scalar multiplication on A. We also have that f_{α} t^{α} = 0 for all but finitely many α ∈ Z^{n}, by hypothesis. The mapping from LP_{A}(T_{1}, …, T_{n}) into k defined by f(T) ↦ f(t) is clearly linear over k. If f(T) ∈ k[T_{1}, …, T_{n}], then this is the same as the definition of f(t) in Section 5.9.

Suppose for the moment that A is an algebra over k in the strict sense, where multiplication of a b ∈ A is expressed as a b. If f(T), g(T) ∈ LP_{A}(T_{1}, …, T_{n}), then h(T) = f(T) g(T) is defined as an element of LP_{A}(T_{1}, …, T_{n}), as in the previous section. If t ∈ k^{n} is as before, then f(t), g(t), and h(t) are defined as elements of A, as in the preceding paragraph. Under these conditions, one can verify that h(t) = f(t) g(t) (35.3.4)

Thus f(T) ↦ f(t) is a homomorphism from LP_{A}(T_{1}, …, T_{n}) into A, as algebras over k.

Let A be a module over k again, and let f(T) ∈ LP_{k}(T_{1}, …, T_{n}) and g(T) in LP_{A}(T_{1}, …, T_{n}) be given. Remember that h(T) = f(T) g(T) is defined as an element of LP_{A}(T_{1}, …, T_{n}), as in the previous section. If t ∈ k^{n} is as before, then f(t) ∈ k, and g(t), h(t) are defined as elements of A. One can verify that (35.3.4) holds in this case too.

Now let A be an associative algebra over k with a multiplicative identity element e, and let A^{n} be the space of n-tuples of elements of A, as usual. Suppose that a = (a_{1}, …, a_{n}) ∈ A^{n} has commuting coordinates, so that a_{j} a_{l} = a_{l} a_{j} for every j, l = 1, …, n. Of course, this holds trivially when n = 1. Suppose also that a_{j} is invertible in A for each j = 1, …, n. It follows that a_{j}^{−1} commutes with a_{l} and with a_{l}^{−1} for every j, l = 1, …, n. If α ∈ Z^{n}, then

a^{α} = a_{1}^{α_{1}} ⋯ a_{n}^{α_{n}} (35.3.5)

defines an element of A. Observe that

a^{α+β} = a^{α} a^{β} (35.3.6)

for every α, β ∈ Z^{n}. Let f(T) ∈ LP_{k}(T_{1}, …, T_{n}) be as in (35.1.3) again. One can define f(a) as an element of A by

f(a) = ∑_{α∈Z^{n}} f_{α} a^{α}, (35.3.7)

as before. One can check that f(T) ↦ f(a) defines a homomorphism from LP_{k}(T_{1}, …, T_{n}) into A, as algebras over k.

35.4 Differentiating Laurent series

Let n be a positive integer, let α ∈ Z^{n} be given, and let l be a positive integer with l ≤ n. As in Section 5.10, we define α^{+}(l) ∈ Z^{n} by

(35.4.1)  α_{j}^{+}(l) = α_{j}  when j ≠ l
= α_{l} + 1 when j = l.

Similarly, let us define α^{−}(l) ∈ Z^{n} by

(35.4.2)  α_{j}^{−}(l) = α_{j}  when j ≠ l
= α_{l} − 1 when j = l.

This is a bit different from the way that α(l) was defined before, because we are using n-tuples of arbitrary integers here.

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, and let T_{1}, …, T_{n} be commuting indeterminates. Also let f(T) be a formal Laurent series in T_{1}, …, T_{n} with coefficients in A, as in (35.1.3). The formal partial derivative of f(T) in T_{l} can be defined as a formal Laurent series in T_{1}, …, T_{n} with coefficients in A by

∂_{l}f(T) = \frac{∂}{∂T_{l}} f(T) = ∑_{α∈Z^{n}}(α_{l} + 1) · f_{α^{+}(l)} T^{α}. (35.4.3)

This is essentially the same as

∑_{α∈Z^{n}} α_{l} · f_{α} T^{α^{−}(l)}. (35.4.4)

This agrees with the analogous definition for partial derivatives of formal power series in T_{1}, …, T_{n}, in Section 5.10. If f(T) ∈ LP_{A}(T_{1}, …, T_{n}), then these sums reduce to finite sums, and ∂_{l}f(T) ∈ LP_{A}(T_{1}, …, T_{n}) too. Note that the mapping f(T) ↦ ∂_{l}f(T) (35.4.5)

is a homomorphism from LS_{A}(T_{1}, …, T_{n}) into itself, as a module over k. One can verify that ∂_{l}(∂_{m}f(T)) = ∂_{m}(∂_{l}f(T)) (35.4.6)

for every l, m = 1, …, n and f(T) ∈ LS_{A}(T_{1}, …, T_{n}).

Suppose for the moment that A is an algebra in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Let f(T) and g(T) be formal Laurent series in T_{1}, …, T_{n} with coefficients in A, where at least one of f(T) and g(T) is a formal Laurent polynomial in T_{1}, …, T_{n}. Thus f(T) g(T) is defined as an element of LS_{A}(T_{1}, …, T_{n}), as in Section 35.1. Under these conditions, one can check that

∂_{l}(f(T) g(T)) = (∂_{l}f(T)) g(T) + f(T) (∂_{l}g(T)). (35.4.7)

Note that both terms on the right are defined as elements of LS_{A}(T_{1}, …, T_{n}). Of course, one can reduce to the case where one of f(T) and g(T) is a multiple of a single Laurent monomial, by linearity. In particular, the partial derivative defines a derivation on LP_{A}(T_{1}, …, T_{n}), as an algebra over k.

Let A be a module over k again, and let f(T) ∈ LS_{k}(T_{1}, …, T_{n}) and g(T) in LS_{A}(T_{1}, …, T_{n}) be given, where at least one of f(T) and g(T) is a formal Laurent polynomial in T_{1}, …, T_{n}. This implies that f(T) g(T) is defined as an element of LS_{A}(T_{1}, …, T_{n}), as in Section 35.1. One can verify that (35.4.7) holds in this case too. As before, both terms on the right side of (35.4.7) are defined as elements of LS_{A}(T_{1}, …, T_{n}), and one may as well suppose that one of f(T) and g(T) is a multiple of a single Laurent monomial.

35.5 Derivatives and functions

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e. Also let a be an invertible element of A, and let u be another element of A that commutes with a and satisfies u^{2} = 0. (35.5.1)

Of course, u commutes with a^{−1} too. One can check that a + u is invertible in A as well, with (a + u)^{−1} = a^{−1} − a^{−2} u. (35.5.2)

If j is any integer, then

(a + u)^{j} = a^{j} + j · a^{j−1} u. (35.5.3)

More precisely, this can be verified directly when j ≥ 0, as in Section 5.7. Similarly,

(35.5.4) (a + u)^{−j} = (a^{−1} − a^{−2} u)^{j} = a^{−j} − j · (a^{−1})^{j−1} a^{−2} u
= a^{−j} − j · a^{−j−1} u

for every j ≥ 1.

Now let n be a positive integer, and let a = (a_{1}, …, a_{n}) be an n-tuple of commuting invertible elements of A. Let u = (u_{1}, …, u_{n}) be another n-tuple of elements of A, with a_{j} u_{l} = u_{l} a_{j} (35.5.5)

and u_{j} u_{l} = 0 (35.5.6)

for every j, l = 1, …, n. It follows that a_{j}^{−1} commutes with u_{l} for every j, l = 1, …, n, as before. Note that a_{l} + u_{l} is invertible in A for each l = 1, …, n, with inverse as in (35.5.2). If α_{l} is any integer, then

(a_{l} + u_{l})^{α_{l}} = a_{l}^{α_{l}} + α_{l} · a_{l}^{α_{l}−1} u_{l}, (35.5.7)

as in (35.5.3).

Observe that a_{j} +u_{j} commutes with a_{l}+u_{l} for each j, l = 1, …, n. If α ∈ Z^{n}, then a^{α} and (a + u)^{α} can be defined as elements of A as in Section 35.3. One can check that

(a + u)^{α} = a^{α} + ∑_{l=1}^{n} α_{l} · a^{α^{−}(l)} u_{l}, (35.5.8)

where α^{−}(l) ∈ Z^{n} is as in the previous section, using (35.5.7). Let T_{1}, …, T_{n} be n commuting indeterminates, and let f(T) ∈ LP_{k}(T_{1}, …, T_{n}) be given, so that ∂_{l}f(T) ∈ LP_{k}(T_{1}, …, T_{n}) is defined as in the previous section. One can verify that

f(a + u) = f(a) + ∑_{l=1}^{n}(∂_{l}f)(a) u_{l}, (35.5.9)

using (35.5.8), where f(a + u), f(a), and (∂_{l}f)(a) are defined as elements of A as in Section 35.3.

Let t = (t_{1}, …, t_{n}) be an n-tuple of invertible elements of k, and let u = (u_{1}, …, u_{n}) be an n-tuple of elements of k such that (35.5.6) holds for every j, l = 1, …, n. Thus t_{l} + u_{l} is invertible in k for each l = 1, …, n, with inverse as in (35.5.2). If α ∈ Z^{n}, then t^{α} and (t + u)^{α} can be defined as elements of k as in Section 35.3, and (t + u)^{α} can be expressed as in (35.5.8). Let A be a module over k, and let f(T) ∈ LP_{A}(T_{1}, …, T_{n}) be given, so that ∂_{l}f(T) can be defined as an element of LP_{A}(T_{1}, …, T_{n}) as in the previous section. As before, one can check that

f(t + u) = f(t) + ∑_{l=1}^{n}(∂_{l}f)(t) u_{l}, (35.5.10)

where f(t + u), f(t), and (∂_{l}f)(t) are defined as elements of A as in Section 35.3.

35.6 Differential operators and Laurent series

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let ∂_{1}, …, ∂_{n} be commuting formal symbols, which may be used to represent partial derivatives. Also let α = (α_{1}, …, α_{n}) be a multi-index of length n, which is to say an n-tuple of nonnegative integers, and let

∂^{α} = ∂_{1}^{α_{1}} ⋯ ∂_{n}^{α_{n}} (35.6.1)

be the corresponding formal product of ∂_{l}’s. Remember that |α| is the same as the degree of α, as an n-tuple of integers, as in Section 35.1. A formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) can be expressed as

∑_{|α|≤N} a^{α}(T) ∂^{α}, (35.6.2)

where N is a nonnegative integer, the sum is taken over all multi-indices α of length n with |α| ≤ N, and a^{α}(T) ∈ LP_{k}(T_{1}, …, T_{n}) for each such α. Remember that the analogous notions with coefficients in k[T_{1}, …, T_{n}] or k[[T_{1}, …, T_{n}]] were discussed in Section 5.11.

As before, we can take a^{α}(T) = 0 when |α| > N, so that a^{α}(T) is defined for every multi-index α. The space of these formal differential operators can be defined as the space of functions α ↦ a^{α}(T) from (Z_{+} ∪ {0})^{n} into LP_{k}(T_{1}, …, T_{n}) with a^{α}(T) = 0 for all but finitely many α. This is a module over k with respect to pointwise addition and scalar multiplication, which corresponds to termwise addition and scalar multiplication of sums as in (35.6.2). The space of these formal differential operators corresponds to the direct sum of copies of LP_{k}(T_{1}, …, T_{n}) indexed by (Z_{+} ∪ {0})^{n}, as a module over k.

We can identify elements of LP_{k}(T_{1}, …, T_{n}) with sums of the form (35.6.2), with N = 0. Multiplication on LP_{k}(T_{1}, …, T_{n}) can be extended to formal differential operators, with

∂_{l}(b^{β}(T)∂^{β}) = (∂_{l}b^{β}(T)) ∂^{β} + b^{β}(T) ∂_{l} ∂^{β} (35.6.3)

for every l = 1, …, n, multi-index β, and b^{β}(T) ∈ LP_{k}(T_{1}, …, T_{n}), as before. Of course, ∂_{l} ∂^{β} = ∂^{β^{+}(l)}, (35.6.4)

in the notation of Section 35.4. The space of these formal partial differential operators is an associative algebra over k, which contains LP_{k}(T_{1}, …, T_{n}) as a subalgebra. The multiplicative identity element in k corresponds to an element of LP_{k}(T_{1}, …, T_{n}), and thus a formal differential operator, and this is the identity element in the space of these formal differential operators.

Let A be a module over k. If α is a multi-index of length n and f(T) is a formal Laurent series in T_{1}, …, T_{n} with coefficients in A, then

∂^{α}f(T) = ∂_{1}^{α_{1}} ⋯ ∂_{n}^{α_{n}}f(T) (35.6.5) defines an element of LS_{A}(T_{1}, …, T_{n}) too, using the definition of partial derivatives on LS_{A}(T_{1}, …, T_{n}) in Section 35.4. This is interpreted as being f(T) when α = 0. If (35.6.2) is a formal differential operator in the symbols ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}), then

( ∑_{|α|≤N} a^{α}(T) ∂^{α})f(T) = ∑_{|α|≤N} a^{α}(T) ∂^{α}f(T) (35.6.6)

defines an element of LS_{A}(T_{1}, …, T_{n}), where the products on the right are as in Section 35.1. Similarly, if f is a formal Laurent polynomial in T_{1}, …, T_{n} with coefficients in A, then (35.6.5) is in LP_{A}(T_{1}, …, T_{n}) for each α, so that (35.6.6) is in LP_{A}(T_{1}, …, T_{n}) as well.

Thus (35.6.2) determines a mapping from LS_{A}(T_{1}, …, T_{n}) into itself that is linear over k. Of course, the space

Hom_{k}(LS_{A}(T_{1}, …, T_{n}), LS_{A}(T_{1}, …, T_{n})) (35.6.7)

of homomorphisms from LS_{A}(T_{1}, …, T_{n}) into itself, as a module over k, is an associative algebra over k with respect to composition of mappings. One can verify that the mapping from the space of formal differential operators in the symbols ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) into (35.6.7) just defined is an algebra homomorphism, with respect to multiplication of formal differential operators, as described before.

The space of formal differential operators in the symbols ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] may be considered as a subalgebra of the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}). In particular, formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] determine mappings on LS_{A}(T_{1}, …, T_{n}) as before. The restrictions of these mappings to A[[T_{1}, …, T_{n}]] are the same as the analogous mappings discussed in Section 5.11.

Let us now take A = k, as a module over itself, and suppose that for every m ∈ Z_{+} and t ∈ k with m · t = 0 in k, we have that t = 0. Of course, this holds when k = Z, or k is a field of characteristic 0, or at least an algebra over Q. Under these conditions, one can check that a formal differential operator (35.6.2) in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) is uniquely determined by the corresponding mapping on LS_{k}(T_{1}, …, T_{n}). In fact, (35.6.2) is uniquely determined by the restriction of this mapping to k[T_{1}, …, T_{n}], as in Section 5.11.

As in Section 32.10, the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) is essentially the same, as a module over k, as the space (LP_{k}(T_{1}, …, T_{n}))[∂_{1}, …, ∂_{n}] (35.6.8)

of formal polynomials in ∂_{1}, …, ∂_{n}, as n commuting indeterminates, with coefficients in LP_{k}(T_{1}, …, T_{n}), as a module over k. We can also use multiplication on LP_{k}(T_{1}, …, T_{n}) to define multiplication on (35.6.8) as in Section 5.8, in which case ∂_{1}, …, ∂_{n} are not considered as being related to differentiation in T_{1}, …, T_{n}. This makes (35.6.8) a commutative algebra over k, and over LP_{k}(T_{1}, …, T_{n}), as before. If L_{1}, L_{2} are formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) of order N_{1}, N_{2} ≥ 0, respectively, then their product as formal differential operators corresponds to their product as formal polynomials in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}), plus a formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) of order N_{1} + N_{2} − 1, at least when N_{1} + N_{2} ≥ 1. In particular, the commutator of L_{1} and L_{2} with respect to multiplication of formal differential operators is a formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) of order N_{1} + N_{2} − 1, as before.

35.7 First-order operators

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let T_{1}, …, T_{n} be n commuting indeterminates. Also let ∂_{1}, …, ∂_{n} be n commuting formal symbols, as in the previous section, and let

a(T) = (a^{1}(T), …, a^{n}(T)) (35.7.1)

be an n-tuple of formal Laurent polynomials in T_{1}, …, T_{n} with coefficients in k. Thus

D_{a(T)} = ∑_{j=1}^{n} a^{j}(T) ∂_{j} (35.7.2)

defines a formal differential operator in the symbols ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}), as in the previous section.

Let b(T) = (b^{1}(T), …, b^{n}(T)) be another n-tuple of formal Laurent polynomials in T_{1}, …, T_{n} with coefficients in k, so that D_{b(T)} can be defined as in the preceding paragraph. Under these conditions, D_{a(T)} D_{b(T)} and D_{b(T)} D_{a(T)} can be defined as formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) too, as before. One can check that

D_{a(T)} D_{b(T)} − D_{b(T)} D_{a(T)} (35.7.3)

= ∑_{j=1}^{n} ∑_{l=1}^{n}(a^{j}(T) ∂_{j}b^{l}(T) − b^{j}(T) ∂_{j}a^{l}(T)) ∂_{l},

as in Section 5.12. Put

c^{l}(T) = ∑_{j=1}^{n}(a^{j}(T)∂_{j}b^{l}(T) − b^{j}(T) ∂_{j}a^{l}(T)) (35.7.4)

for each l = 1, …, n, which is a formal Laurent polynomial in T_{1}, …, T_{n} with coefficients in k. It follows that

D_{a(T)} D_{b(T)} − D_{b(T)} D_{a(T)} = D_{c(T)}, (35.7.5)

where c(T) = (c^{1}(T), …, c^{n}(T)).

If f(T) ∈ LS_{k}(T_{1}, …, T_{n}), then

D_{a(T)}f(T) = ∑_{j=1}^{n} a^{j}(T) ∂_{j}f(T) (35.7.6)

defines an element of LS_{k}(T_{1}, …, T_{n}) as well, as in the previous section. Similarly, if f(T) ∈ LP_{k}(T_{1}, …, T_{n}), then this is an element of LP_{k}(T_{1}, …, T_{n}), as before. If f(T), g(T) ∈ LS_{k}(T_{1}, …, T_{n}), and at least one of f(T) and g(T) is in LP_{k}(T_{1}, …, T_{n}), then f(T) g(T) is defined as an element of LS_{k}(T_{1}, …, T_{n}), and we have that

D_{a(T)}(f(T) g(T)) = (D_{a(T)}f(T)) g(T) + f(T) (D_{a(T)}g(T)). (35.7.7)

In particular, D_{a(T)} defines a derivation on LP_{k}(T_{1}, …, T_{n}).

Now let δ be any derivation on LP_{k}(T_{1}, …, T_{n}), as an algebra over k. Observe that δ(1) = 0, (35.7.8)

because δ(1) = δ(1 · 1) = δ(1) 1 + 1 δ(1) = δ(1) + δ(1). It follows that

δ(T_{j}^{−1}) = −T_{j}^{−2} δ(T_{j}) (35.7.9)

for every j = 1, …, n, because 0 = δ(T_{j} T_{j}^{−1}) = δ(T_{j}) T_{j}^{−1} + T_{j} δ(T_{j}^{−1}). Put

a^{j}(T) = δ(T_{j}) (35.7.10)

for every j = 1, …, n, which is an element of LP_{k}(T_{1}, …, T_{n}), and let a(T) be as in (35.7.1). One can check that

δ(f(T)) = D_{a(T)}f(T) (35.7.11)

for every f(T) ∈ LP_{k}(T_{1}, …, T_{n}) under these conditions.

Put e^{j}(T) = T_{j} for each j = 1, …, n, so that

e(T) = (e^{1}(T), …, e^{n}(T)) = (T_{1}, …, T_{n}). (35.7.12)

The corresponding formal differential operator

D_{e(T)} = ∑_{j=1}^{n} e^{j}(T) ∂_{j} = ∑_{j=1}^{n} T_{j} ∂_{j} (35.7.13)

is the analogue of the classical Euler operator. It is easy to see that

D_{e(T)}(T^{α}) = (deg(α) · 1) T^{α} (35.7.14)

for every α ∈ Z^{n}. Let A be a module over k, and let f(T) = ∑_{α∈Z^{n}} f_{α} T^{α} be a formal Laurent series in T_{1}, …, T_{n} with coefficients in A. Observe that

D_{e(T)}f(T) = ∑_{α∈Z^{n}} deg(α) · f_{α} T^{α}. (35.7.15)

35.8 Homogeneous Laurent series

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, and let n be a positive integer. Also let T_{1}, …, T_{n} be n commuting indeterminates for some positive integer n, and let f(T) = ∑_{α∈Z^{n}} f_{α} T^{α} be a formal Laurent series in T_{1}, …, T_{n} with coefficients in A. We say that f(T) is homogeneous of degree d for some integer d if

(35.8.1) f_{α} = 0 for every α ∈ Z^{n} with deg(α) ≠ d.

If f(T) is a formal polynomial in T_{1}, …, T_{n}, and d ≥ 0, then this is equivalent to the definition of homogeneity of degree d in Section 5.13.

The space LS_{A,d}(T_{1}, …, T_{n}) of formal Laurent series in T_{1}, …, T_{n} with coefficients in A that are homogeneous of degree d ∈ Z is a submodule of the space LS_{A}(T_{1}, …, T_{n}) of all formal Laurent series in T_{1}, …, T_{n} with coefficients in A, as a module over k. Observe that LS_{A,d}(T_{1}, …, T_{n}) corresponds to the direct product of copies of A indexed by α ∈ Z^{n} with deg(α) = d, and that LS_{A}(T_{1}, …, T_{n}) corresponds to the direct product of LS_{A,d}(T_{1}, …, T_{n}) over d ∈ Z, as modules over k.

Similarly, let

LP_{A,d}(T_{1}, …, T_{n}) = LP_{A}(T_{1}, …, T_{n}) ∩ LS_{A,d}(T_{1}, …, T_{n}) (35.8.2)

be the space of formal Laurent polynomials in T_{1}, …, T_{n} with coefficients in A that are homogeneous of degree d ∈ Z, as formal Laurent series in T_{1}, …, T_{n}. This corresponds to the direct sum of copies of A indexed by α ∈ Z^{n} with deg(α) = d as a module over k, and LP_{A}(T_{1}, …, T_{n}) corresponds to the direct sum of LP_{A,d}(T_{1}, …, T_{n}) over d ∈ Z.

Let A be an algebra over k in the strict sense, where multiplication of a, b ∈ A is expressed as a b. Suppose that f(T) ∈ LS_{A,d_{1}}(T_{1}, …, T_{n}) and g(T) is in LS_{A,d_{2}}(T_{1}, …, T_{n}) for some d_{1}, d_{2} ∈ Z. If f(T) or g(T) is a formal Laurent polynomial in T_{1}, …, T_{n}, then f(T) g(T) can be defined as a formal Laurent series in T_{1}, …, T_{n} with coefficients in A, as in Section 35.1. Under these conditions, it is easy to see that

f(T) g(T) ∈ LS_{A,d_{1}+d_{2}}(T_{1}, …, T_{n}). (35.8.3)

If f(T) and g(T) are both formal Laurent polynomials in T_{1}, …, T_{n}, then

f(T) g(T) ∈ LP_{A,d_{1}+d_{2}}(T_{1}, …, T_{n}). (35.8.4)

Suppose that A is an associative algebra over k, so that LP_{A}(T_{1}, …, T_{n}) is an associative algebra over k as well, as in Section 35.1. Under these conditions, LS_{A}(T_{1}, …, T_{n}) may be considered as a left and right module over LP_{A}(T_{1}, …, T_{n}), as before. Note that

(35.8.5) LP_{A,0}(T_{1}, …, T_{n}) is a subalgebra of LP_{A}(T_{1}, …, T_{n}), by (35.8.4). Thus LS_{A}(T_{1}, …, T_{n}) may be considered as a left and right module over LP_{A,0}(T_{1}, …, T_{n}) too. If d is any integer, then

(35.8.6) LS_{A,d}(T_{1}, …, T_{n}) is a submodule of LS_{A}(T_{1}, …, T_{n}), as a left or right module over LP_{A,0}(T_{1}, …, T_{n}),

by (35.8.3).

Let A be a module over k again, and suppose that f(T) ∈ LS_{k,d_{1}}(T_{1}, …, T_{n}) and g(T) ∈ LS_{A,d_{2}}(T_{1}, …, T_{n}) for some d_{1}, d_{2} ∈ Z. If f(T) or g(T) is a formal Laurent polynomial in T_{1}, …, T_{n}, then f(T) g(T) can be defined as a formal Laurent series in T_{1}, …, T_{n} with coefficients in A, as in Section 35.1. It is easy to see that (35.8.3) holds in this case. If f(T) and g(T) are both formal Laurent polynomials in T_{1}, …, T_{n}, then (35.8.4) holds.

Remember that LS_{A}(T_{1}, …, T_{n}) is a module over LP_{k}(T_{1}, …, T_{n}), as in Section 35.1. We may also consider LS_{A}(T_{1}, …, T_{n}) as a module over the subalgebra LP_{k,0}(T_{1}, …, T_{n}) of LP_{k}(T_{1}, …, T_{n}). If d ∈ Z, then

(35.8.7) LS_{A,d}(T_{1}, …, T_{n}) is a submodule of LS_{A}(T_{1}, …, T_{n}), as a module over LP_{k,0}(T_{1}, …, T_{n}).

Similarly, LP_{A}(T_{1}, …, T_{n}) is a submodule of LS_{A}(T_{1}, …, T_{n}), as a module over LP_{k}(T_{1}, …, T_{n}), and thus as a module over LP_{k,0}(T_{1}, …, T_{n}) too. If d ∈ Z, then

(35.8.8) LP_{A,d}(T_{1}, …, T_{n}) is a submodule of LP_{A}(T_{1}, …, T_{n}), as a module over LP_{k,0}(T_{1}, …, T_{n}).

If f(T) ∈ LS_{A,d}(T_{1}, …, T_{n}) for some d ∈ Z, then it is easy to see that

∂_{l}f(T) ∈ LS_{A,d−1}(T_{1}, …, T_{n}) (35.8.9)

for every l = 1, …, n, where the partial derivative ∂_{l}f(T) is as in Section 35.4. Put e(T) = (T_{1}, …, T_{n}), and let D_{e(T)} = ∑_{j=1}^{n} T_{j} ∂_{j} be the corresponding first-order differential operator, as in Section 35.7. If f(T) ∈ LS_{A,d}(T_{1}, …, T_{n}), then D_{e(T)}f(T) = d · f(T), (35.8.10)

by (35.7.15).

35.9 Homogeneity and differential operators

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let T_{1}, …, T_{n} be n commuting indeterminates. Also let ∂_{1}, …, ∂_{n} be n commuting formal symbols, which may be used to represent partial derivatives, as in Section 35.6. Consider a formal differential operator

L = ∑_{|α|≤N} a^{α}(T) ∂^{α} (35.9.1)

in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}). We say that L is homogeneous of degree d for some integer d if

a^{α}(T) ∈ LP_{k,d+|α|}(T_{1}, …, T_{n}) (35.9.2)

for each multi-index α. This agrees with the definition in Section 5.14 when a^{α}(T) is a formal polynomial in T_{1}, …, T_{n} for each α.

Let L_{1}, L_{2} be formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}), and suppose that L_{1}, L_{2} are homogeneous of degree d_{1}, d_{2} ∈ Z, respectively. Under these conditions, one can check that their product

(35.9.3) L_{1} L_{2} is homogeneous of degree d_{1} + d_{2}.

More precisely, one can first verify this when L_{1} is of the form ∂_{j} for some j = 1, …, n, as in Section 5.14. One can use this to get that (35.9.3) holds when L_{1} = ∂^{α} for some multi-index α, as before. The analogous statement for arbitrary L_{1} can be obtained from this and (35.8.4).

Let L_{1} be a formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) that is homogeneous of degree d_{1} ∈ Z again, and let A be a module over k. If f(T) is a formal Laurent series in T_{1}, …, T_{n} with coefficients in A that is homogeneous of degree d ∈ Z, then

L_{1}f(T) ∈ LS_{A,d_{1}+d}(T_{1}, …, T_{n}). (35.9.4)

This is the same as (35.8.9) when L_{1} = ∂_{l} for some l = 1, …, n. This implies that (35.9.4) holds when L_{1} = ∂^{α} for some multi-index α. One can use this and (35.8.3) to get that (35.9.4) holds for any L_{1}.

Remember that

(35.9.5) the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n})

is an associative algebra over k, as in Section 35.6. The space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] is a subalgebra of (35.9.5), as before.

The space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) that are homogeneous of degree d ∈ Z is a submodule of (35.9.5), as a module over k. In fact,

(35.9.6) the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) that are homogeneous of degree 0

is a subalgebra of (35.9.5). Of course, the space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in k[T_{1}, …, T_{n}] that are homogeneous of degree 0 is a subalgebra of (35.9.6).

If a(T) = (a^{1}(T), …, a^{n}(T)) is an n-tuple of formal Laurent polynomials in T_{1}, …, T_{n} with coefficients in k, then put D_{a(T)} = ∑_{j=1}^{n} a^{j}(T) ∂_{j}, as in Section 35.7. Observe that

{D_{a(T)} : a(T) ∈ (LP_{k}(T_{1}, …, T_{n}))^{n}} (35.9.7)

is a submodule of (35.9.5), as a module over k. More precisely, (35.9.7) is a Lie subalgebra of (35.9.5), with respect to the commutator bracket, as in Section 35.7. The space of D_{a(T)} with a(T) ∈ (k[T_{1}, …, T_{n}])^{n} is a Lie subalgebra of (35.9.7).

Similarly, {D_{a(T)} : a(T) ∈ (LP_{k,1}(T_{1}, …, T_{n}))^{n}} (35.9.8)

is a Lie subalgebra of (35.9.6), with respect to the commutator bracket. This is also a subalgebra of (35.9.7), as a Lie algebra over k. Remember that g_{n}(k) is the space of D_{a(T)}, where a^{j}(T) is in the space k_{1}[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in k that are homogeneous of degree one for each j = 1, …, n, as in Section 5.15. This is a subalgebra of (35.9.8), as a Lie algebra over k.

If a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n}, then a^{j}(T) can be expressed as ∑_{l=1}^{n} a_{l}^{j} T_{l} for each j = 1, …, n, where a_{l}^{j} ∈ k for every j, l = 1, …, n. Remember that s_{n}(k) is the space of D_{a(T)} with a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n} and ∑_{j=1}^{n} a_{j}^{j} = 0, as in Section 5.15. This is a subalgebra of g_{n}(k), as a Lie algebra over k, as before. In particular, s_{n}(k) may be considered as a subalgebra of (35.9.8), as a Lie algebra over k.

35.10 Representations and Laurent series

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, and let n be a positive integer. Also let T_{1}, …, T_{n} be n commuting indeterminates, and let ∂_{1}, …, ∂_{n} be n commuting formal symbols, which may be used to represent partial derivatives, as before. Remember that the space LS_{A}(T_{1}, …, T_{n}) of formal Laurent series in T_{1}, …, T_{n} with coefficients in A is a module over k, with respect to termwise addition and scalar multiplication, as in Section 35.1.

The space of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) is an associative algebra over k, as in Section 35.6. These formal differential operators determine homomorphisms from LS_{A}(T_{1}, …, T_{n}) into itself, as a module over k, as before. This defines a representation of (35.9.5), as an associative algebra over k, on LS_{A}(T_{1}, …, T_{n}).

Remember that the mappings from LS_{A}(T_{1}, …, T_{n}) into itself associated to formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) send LP_{A}(T_{1}, …, T_{n}) into itself, as in Section 35.6. Thus

(35.10.1) LP_{A}(T_{1}, …, T_{n}) is a submodule of LS_{A}(T_{1}, …, T_{n}),

as a left module over (35.9.5).

In particular, LS_{A}(T_{1}, …, T_{n}) may be considered as a left module over (35.9.6), as an associative algebra over k. Similarly, LP_{A}(T_{1}, …, T_{n}) is a submodule of LS_{A}(T_{1}, …, T_{n}), as a left module over (35.9.6).

If d ∈ Z, then

(35.10.2) LS_{A,d}(T_{1}, …, T_{n}) is a submodule of LS_{A}(T_{1}, …, T_{n}),

as a left module over (35.9.6), as in the previous section. We also have that

(35.10.3) LP_{A,d}(T_{1}, …, T_{n}) is a submodule of LP_{A}(T_{1}, …, T_{n}),

as a left module over (35.9.6).

Remember that (35.9.7) is a Lie algebra over k, with respect to the commutator bracket. We may consider LS_{A}(T_{1}, …, T_{n}) as a module over (35.9.7), as a Lie algebra over k. Similarly, we may consider LP_{A}(T_{1}, …, T_{n}) as a submodule of LS_{A}(T_{1}, …, T_{n}), as a module over (35.9.7).

Remember too that (35.9.8) is a subalgebra of (35.9.7), as a Lie algebra over k. This permits us to consider LS_{A}(T_{1}, …, T_{n}) as a module over (35.9.8), as a Lie algebra over k. We may consider LP_{A}(T_{1}, …, T_{n}) as a submodule of LS_{A}(T_{1}, …, T_{n}), as a module over (35.9.8) as well.

Similarly, g_{n}(k) is a subalgebra of (35.9.8), as a Lie algebra over k, and s_{n}(k) is a subalgebra of g_{n}(k). Thus

(35.10.4) we may consider LS_{A}(T_{1}, …, T_{n}) as a module over g_{n}(k) or s_{n}(k),

as Lie algebras over k. As usual, LP_{A}(T_{1}, …, T_{n}) may be considered as a submodule of LS_{A}(T_{1}, …, T_{n}), as a module over g_{n}(k) or s_{n}(k).

If d ∈ Z, then (35.10.2) holds, with LS_{A}(T_{1}, …, T_{n}) considered as a module over (35.9.8), g_{n}(k), or s_{n}(k). Similarly, (35.10.3) holds, with LP_{A}(T_{1}, …, T_{n}) considered as a module over (35.9.8), g_{n}(k), or s_{n}(k).

Note that

(35.10.5) A[[T_{1}, …, T_{n}]] is a submodule of LS_{A}(T_{1}, …, T_{n}),

as a module over g_{n}(k) or s_{n}(k). Similarly,

(35.10.6) A[T_{1}, …, T_{n}] is a submodule of LP_{A}(T_{1}, …, T_{n}),

as a module over g_{n}(k) or s_{n}(k). If d is a nonnegative integer, then the space A_{d}[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in A that are homogeneous of degree d is a submodule of A[T_{1}, …, T_{n}], as a module over g_{n}(k) or s_{n}(k), as in Section 6.15. In this case,

(35.10.7) A_{d}[T_{1}, …, T_{n}] is a submodule of LP_{A,d}(T_{1}, …, T_{n}),

as a module over g_{n}(k) or s_{n}(k).

35.11 A linear mapping onto A

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. Also let n be a positive integer, and let T_{1}, …, T_{n} be n commuting indeterminates, as before. If f(T) = ∑_{α∈Z^{n}} f_{α} T^{α} is a formal Laurent series in T_{1}, …, T_{n} with coefficients in A, then put

I_{0}(f(T)) = f_{0}. (35.11.1)

This defines a homomorphism from LS_{A}(T_{1}, …, T_{n}) onto A, as modules over k. If β ∈ Z^{n}, then f(T) T^{β} ∈ LS_{A}(T_{1}, …, T_{n}), and

I_{0}(f(T) T^{β}) = f_{−β}. (35.11.2)

Note that I_{0}(f) = 0 (35.11.3)

when f(T) is homogeneous of degree d ≠ 0.

Suppose for the moment that A is an algebra over k in the strict sense. In this case, it is easy to see that the restriction of I_{0} to A[[T_{1}, …, T_{n}]] defines an algebra homomorphism from A[[T_{1}, …, T_{n}]] onto A.

If l is a positive integer with l ≤ n, then

T_{l} ∂_{l}f(T) = ∑_{α∈Z^{n}} α_{l} · f_{α} T^{α}. (35.11.4)

This implies that I_{0}(T_{l}∂_{l}f(T)) = 0. (35.11.5)

Let ∂_{1}, …, ∂_{n} be n commuting formal symbols, which may be used to represent partial derivatives, as usual. Put e(T) = (T_{1}, …, T_{n}), so that the corresponding formal differential operator D_{e(T)} = ∑_{j=1}^{n} T_{j} ∂_{j} is the analogue of the classical Euler operator, as in Section 35.7. Thus

I_{0}(D_{e(T)}f(T)) = 0, (35.11.6)

by (35.11.5). If k is a field of characteristic 0, then it is easy to see that

(35.11.7) D_{e(T)} maps LS_{A}(T_{1}, …, T_{n}) onto the kernel of I_{0}.

Similarly, D_{e(T)} maps LP (T_{1}, …, T_{n}) onto the kernel of the restriction of I_{0} to LP_{A}(T_{1}, …, T_{n}) under these conditions.

35.12 Some related bilinear functionals

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let T_{1}, …, T_{n} be n commuting indeterminates. Thus I_{0} may be defined as a homomorphism from LS_{k}(T_{1}, …, T_{n}) onto k, as modules over k, as in the previous section. If f(T) = ∑_{α∈Z^{n}} f_{α} T^{α} and g(T) = 35.12. SOME RELATED BILINEAR FUNCTIONALS 741

∑_{β∈Z^{n}} g_{β} T^{β} are formal Laurent series in T_{1}, …, T_{n} with coefficients in k, then we would like to put B_{0}(f(T), g(T)) = I_{0}(f(T) g(T)), (35.12.1) when f(T) g(T) is defined as an element of LS_{k}(T_{1}, …, T_{n}). In particular, we can do this when at least one of f(T) and g(T) is a Laurent polynomial.

Equivalently,

B_{0}(f(T), g(T)) = ∑_{α∈Z^{n}} f_{α} g_{−α}, (35.12.2)

which is defined as an element of k when f_{α} g_{−α} = 0 for all but finitely many α ∈ Z^{n}. This holds when at least one of f(T) and g(T) is a Laurent polynomial, as before. We may consider B_{0} as a k-valued function on LP_{k}(T_{1}, …, T_{n}) × LS_{k}(T_{1}, …, T_{n}) (35.12.3) or LS_{k}(T_{1}, …, T_{n}) × LP_{k}(T_{1}, …, T_{n}). (35.12.4) We may also simply consider B_{0} as a k-valued function on LP_{k}(T_{1}, …, T_{n}) × LP_{k}(T_{1}, …, T_{n}). (35.12.5) It is easy to see that B_{0} is bilinear over k in each case.

If at least one of f(T) and g(T) is a Laurent polynomial, then B_{0}(f(T), g(T)) = B_{0}(g(T), f(T)). (35.12.6) This can be obtained from the fact that f(T) g(T) = g(T) f(T), as in Section 35.2, or from (35.12.2). Similarly, if a(T) is a Laurent polynomial in T_{1}, …, T_{n} with coefficients in k, then B_{0}(a(T) f(T), g(T)) = B_{0}(f(T), a(T) g(T)). (35.12.7) If l is a positive integer with l ≤ n, then I_{0}(T_{l} ∂_{l}(f(T) g(T))) = 0, (35.12.8) as in (35.11.5). This implies that B_{0}(T_{l} ∂_{l}f(T), g(T)) = −B_{0}(f(T), T_{l} ∂_{l}g(T)). (35.12.9)

Observe that B_{0}(f(T), T^{β}) = f_{−β} (35.12.10) for every β ∈ Z^{n}. If f(T), g(T) are homogeneous of degrees d_{1}, d_{2}, respectively, and d_{1} + d_{2} ≠ 0, then B_{0}(f(T), g(T)) = 0, (35.12.11) when the left side is defined.

35.13 Adjoints of differential operators

Let k be a commutative ring with a multiplicative identity element, and suppose that for every m ∈ Z_{+} and t ∈ k with m · t = 0, we have that t = 0. In particular, this holds when k = Z, or k is a field of characteristic 0, or at least an algebra over Q. Also let n be a positive integer, let T_{1}, …, T_{n} be n commuting indeterminates, and let ∂_{1}, …, ∂_{n} be n commuting formal symbols, which may be used to represent partial derivatives, as before. Suppose that L is a formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}). Let us say that a formal differential operator L^{∗} in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) is the adjoint of L if

B_{0}(L(f(T)), g(T)) = B_{0}(f(T), L^{∗}(g(T))) (35.13.1)

for every f(T), g(T) ∈ LS_{k}(T_{1}, …, T_{n}) with at least one of f(T), g(T) in LP_{k}(T_{1}, …, T_{n}).

If (35.13.1) holds for every f(T), g(T) ∈ LP_{k}(T_{1}, …, T_{n}), then L^{∗}(g(T)) is uniquely determined by L and g(T) for every g(T) ∈ LP_{k}(T_{1}, …, T_{n}). This means that the mapping on LP_{k}(T_{1}, …, T_{n}) corresponding to L^{∗} is uniquely determined by L. It follows that L^{∗} is uniquely determined by L under these conditions, as in Section 35.6.

If L is of order 0, then L corresponds to multiplication by an element of LP_{k}(T_{1}, …, T_{n}), and one can take L^{∗} = L, as in (35.12.7). If L = T_{l} ∂_{l} for some l = 1, …, n, then one can take L^{∗} = −T_{l} ∂_{l}, as in (35.12.9).

Let L_{1}, L_{2} be formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}), and suppose that L_{1}^{∗}, L_{2}^{∗} are formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) that are adjoints of L_{1}, L_{2}, respectively. It is easy to see that L_{1}^{∗} + L_{2}^{∗} is the adjoint of L_{1} + L_{2} in this case. Similarly, if a ∈ k, then a L_{1}^{∗} is the adjoint of a L_{1}. If f(T), g(T) are elements of LS_{k}(T_{1}, …, T_{n}), at least one of which is in LP_{k}(T_{1}, …, T_{n}), then

(35.13.2) B_{0}(L_{2}(L_{1}(f(T))), g(T)) = B_{0}(L_{1}(f(T)), L_{2}^{∗}(g(T)))
= B_{0}(f(T), L_{1}^{∗}(L_{2}^{∗}(g(T)))).

This means that the product of L_{1}^{∗} and L_{2}^{∗} is the adjoint of the product of L_{1} and L_{2}.

If L is any formal differential operator in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}), then one can use the remarks in the previous two paragraphs to get that there is a formal differential operator L^{∗} in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}) that is the adjoint of L. More precisely, L can be expressed as a sum of products of formal differential operators of order 0, and formal differential operators of the form T_{l} ∂_{l}, 1 ≤ l ≤ n. The adjoints of each of these can be obtained as before, which can be used to get the adjoint of L. Note that (L^{∗})^{∗} = L, (35.13.3) so that L ↦ L^{∗} is an involution on the algebra of formal differential operators in ∂_{1}, …, ∂_{n} with coefficients in LP_{k}(T_{1}, …, T_{n}). If L is homogeneous of degree d, then one can verify that L^{∗} is homogeneous of degree d as well.

Formal Laurent series, 2

36.1 Laurent series and sl_{2}(k)

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. We would like to consider formal Laurent polynomials and series with coefficients in A, as in the previous chapter, with n = 2. Thus we let T_{1}, T_{2} be commuting indeterminates, and we let ∂_{1}, ∂_{2} be commuting formal symbols, which may be used to represent partial derivatives, as before.

Put

(36.1.1)  H = T_{1} ∂_{1} − T_{2} ∂_{2},
(36.1.2)  X = T_{1} ∂_{2},
(36.1.3)  Y = T_{2} ∂_{1},

which are formal differential operators in ∂_{1}, ∂_{2} with coefficients in k_{1}[T_{1}, T_{2}]. Note that H, X, and Y are homogeneous of degree 0, as in Sections 5.14 and 35.9. In fact, H, X, Y ∈ s_{2}(k), (36.1.4)

where s_{2}(k) is as in Sections 5.15 and 35.9. More precisely, s_{2}(k) is freely generated by H, X, and Y, as a module over k.

Remember that s_{2}(k) is a Lie algebra over k, with respect to the commutator bracket corresponding to products of formal differential operators. It is easy to see that

(36.1.5)  [X, Y] = H,
(36.1.6)  [H, X] = 2 · X,
(36.1.7)  [H, Y] = −2 · Y.

Thus s_{2}(k) is isomorphic to sl_{2}(k), as a Lie algebra over k, where the usual elements h, x, and y of sl_{2}(k) as in Section 34.1 correspond to H, X, and Y, repsectively. This is related to some of the remarks in Section 5.15.

We may consider the space LS_{A}(T_{1}, T_{2}) of formal Laurent series in T_{1}, T_{2} with coefficients in A as a module over s_{2}(k), as a Lie algebra over k, as in Section 35.10. This may be considered as a representation of sl_{2}(k), as a Lie algebra over k, on LS_{A}(T_{1}, T_{2}), using the isomorphism mentioned in the preceding paragraph. That is to say, the actions of h, x, and y on LS_{A}(T_{1}, T_{2}) under this representation are given by H, X, and Y, respectively.

The space LP_{A}(T_{1}, T_{2}) of formal Laurent polynomials in T_{1}, T_{2} with coefficients in A is a submodule of LS_{A}(T_{1}, T_{2}), as a module over sl_{2}(k). Similarly, A[[T_{1}, T_{2}]] is a submodule of LS_{A}(T_{1}, T_{2}), and A[T_{1}, T_{2}] is a submodule of each of A[[T_{1}, T_{2}]] and LP_{A}(T_{1}, T_{2}), as modules over sl_{2}(k).

If d ∈ Z, then LS_{A,d}(T_{1}, T_{2}), LP_{A,d}(T_{1}, T_{2}) are submodules of LS_{A}(T_{1}, T_{2}), LP_{A}(T_{1}, T_{2}), respectively, as modules over s_{2}(k), as in Section 35.10. If d ≥ 0, then A_{d}[T_{1}, T_{2}] is a submodule of A[T_{1}, T_{2}] and LP_{A,d}(T_{1}, T_{2}), as modules over s_{2}(k), as before.

Let d ∈ Z and f(T) ∈ LS_{A,d}(T_{1}, T_{2}) be given. Thus

f(T) = ∑_{j=−∞}^{∞} f_{j} T_{1}^{j} T_{2}^{d−j}, (36.1.8)

where f_{j} ∈ A for every j. Of course, f(T) ∈ LP_{A,d}(T_{1}, T_{2}) exactly when f_{j} = 0 for all but finitely many integers j.

Observe that

(36.1.9)  (Hf)(T) = ∑_{j=−∞}^{∞}(j − (d − j)) · f_{j} T_{1}^{j} T_{2}^{d−j}

= ∑_{j=−∞}^{∞}(2 j − d) · f_{j} T_{1}^{j} T_{2}^{d−j}.

Similarly,

(36.1.10)  (Xf)(T) = ∑_{j=−∞}^{∞}(d − j) · f_{j} T_{1}^{j+1} T_{2}^{d−j−1}

= ∑_{j=−∞}^{∞}(d − j + 1) · f_{j−1} T_{1}^{j} T_{2}^{d−j}

and

(36.1.11)  (Y f)(T) = ∑_{j=−∞}^{∞} j · f_{j} T_{1}^{j−1} T_{2}^{d−j+1}

= ∑_{j=−∞}^{∞}(j + 1) · f_{j+1} T_{1}^{j} T_{2}^{d−j}.

36.2. SOME SUBMODULES OF LS_{A,D}(T_{1}, T_{2}) 745

36.2 Some submodules of LS_{A,d}(T_{1}, T_{2})

Let us continue with the same notation and hypotheses as in the previous section. In particular, if f(T) ∈ LS_{A,d}(T_{1}, T_{2}) and j ∈ Z, then we let f_{j} be the coefficient in A of T_{1}^{j} T_{2}^{d−j} in f(T), as in (36.1.8).

If l ∈ Z, then put (36.2.1) LS_{A,d}^{l,+}(T_{1}, T_{2}) = {f(T) ∈ LS_{A,d}(T_{1}, T_{2}) : f_{j} = 0 for every j < l}. Similarly, put (36.2.2) LS_{A,d}^{l,−}(T_{1}, T_{2}) = {f(T) ∈ LS_{A,d}(T_{1}, T_{2}) : f_{j} = 0 for every j > l}. Note that these are submodules of LS_{A,d}(T_{1}, T_{2}), as a module over k. If r ∈ Z and l ≤ r, then

(36.2.3)  LS_{A,d}^{r,+}(T_{1}, T_{2}) ⊆ LS_{A,d}^{l,+}(T_{1}, T_{2}),
(36.2.4)  LS_{A,d}^{l,−}(T_{1}, T_{2}) ⊆ LS_{A,d}^{r,−}(T_{1}, T_{2}).

Let us also put

LP_{A,d}^{l,+}(T_{1}, T_{2}) = LS_{A,d}^{l,+}(T_{1}, T_{2}) ∩ LP_{A,d}(T_{1}, T_{2})
(36.2.5)  = {f(T) ∈ LP_{A,d}(T_{1}, T_{2}) : f_{j} = 0 for every j < l}

and

LP_{A,d}^{l,−}(T_{1}, T_{2}) = LS_{A,d}^{l,−}(T_{1}, T_{2}) ∩ LP_{A,d}(T_{1}, T_{2})
(36.2.6)  = {f(T) ∈ LP_{A,d}(T_{1}, T_{2}) : f_{j} = 0 for every j > l}.

These are submodules of LP_{A,d}(T_{1}, T_{2}), as a module over k. If l ≤ r, then

(36.2.7)  LP_{A,d}^{r,+}(T_{1}, T_{2}) ⊆ LP_{A,d}^{l,+}(T_{1}, T_{2}),
(36.2.8)  LP_{A,d}^{l,−}(T_{1}, T_{2}) ⊆ LP_{A,d}^{r,−}(T_{1}, T_{2}),

as before.

Observe that

(36.2.9)  H(LS_{A,d}^{l,+}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l,+}(T_{1}, T_{2}),
(36.2.10)  H(LS_{A,d}^{l,−}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l,−}(T_{1}, T_{2})

for every l ∈ Z, by (36.1.9). Similarly,

(36.2.11)  H(LP_{A,d}^{l,+}(T_{1}, T_{2})) ⊆ LP_{A,d}^{l,+}(T_{1}, T_{2}),
(36.2.12)  H(LP_{A,d}^{l,−}(T_{1}, T_{2})) ⊆ LP_{A,d}^{l,−}(T_{1}, T_{2})

for every l ∈ Z.

One can check that

(36.2.13)  X(LS_{A,d}^{l,+}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l+1,+}(T_{1}, T_{2}),
(36.2.14)  X(LS_{A,d}^{l,−}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l+1,−}(T_{1}, T_{2})

for every l ∈ Z, using (36.1.10). Similarly,

(36.2.15)  X(LP_{A,d}^{l,+}(T_{1}, T_{2})) ⊆ LP_{A,d}^{l+1,+}(T_{1}, T_{2}),
(36.2.16)  X(LP_{A,d}^{l,−}(T_{1}, T_{2})) ⊆ LP_{A,d}^{l+1,−}(T_{1}, T_{2})

for each l ∈ Z.

If l ∈ Z, then one can check that

(36.2.17)  Y (LS_{A,d}^{l,+}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l−1,+}(T_{1}, T_{2}),
(36.2.18)  Y (LS_{A,d}^{l,−}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l−1,−}(T_{1}, T_{2}),

using (36.1.11). Similarly,

(36.2.19)  Y (LP_{A,d}^{l,+}(T_{1}, T_{2})) ⊆ LP_{A,d}^{l−1,+}(T_{1}, T_{2}),
(36.2.20)  Y (LP_{A,d}^{l,−}(T_{1}, T_{2})) ⊆ LP_{A,d}^{l−1,−}(T_{1}, T_{2})

for every l ∈ Z.

If l ∈ Z, then LS_{A,d}(T_{1}, T_{2}) corresponds to the direct sum of LS_{A,d}^{l,+}(T_{1}, T_{2}) and LS_{A,d}^{l−1,−}(T_{1}, T_{2}), as a module over k. Similarly, LP_{A,d}(T_{1}, T_{2}) corresponds to the direct sum of LP_{A,d}^{l,+}(T_{1}, T_{2}) and LP_{A,d}^{l−1,−}(T_{1}, T_{2}), as a module over k.

36.3 Submodules over s_{2}(k)

Let us continue with the same notation and hypotheses as in the previous two sections. If l = d, then we can improve (36.2.14) and (36.2.16). More precisely, one can verify that

(36.3.1)  X(LS_{A,d}^{d,−}(T_{1}, T_{2})) ⊆ LS_{A,d}^{d,−}(T_{1}, T_{2}),
(36.3.2)  X(LP_{A,d}^{d,−}(T_{1}, T_{2})) ⊆ LP_{A,d}^{d,−}(T_{1}, T_{2}),

using (36.1.10). Similarly, if l = 0, then we can improve (36.2.17) and (36.2.19). That is to say, one can verify that

(36.3.3)  Y (LS_{A,d}^{0,+}(T_{1}, T_{2})) ⊆ LS_{A,d}^{0,+}(T_{1}, T_{2}),
(36.3.4)  Y (LP_{A,d}^{0,+}(T_{1}, T_{2})) ⊆ LP_{A,d}^{0,+}(T_{1}, T_{2}),

using (36.1.11).

It follows that

(36.3.5) LS_{A,d}^{d,−}(T_{1}, T_{2}) is a submodule of LS_{A,d}(T_{1}, T_{2}), as a module over s_{2}(k), 36.4. CHARACTERISTIC 0, A = K 747

by (36.2.10), (36.3.1), and (36.2.18). Similarly,

(36.3.6) LP_{A,d}^{d,−}(T_{1}, T_{2}) is a submodule of LP_{A,d}(T_{1}, T_{2}), as a module over s_{2}(k),

by (36.2.12), (36.3.2), and (36.2.20). We also get that

(36.3.7) LS_{A,d}^{0,+}(T_{1}, T_{2}) is a submodule of LS_{A,d}(T_{1}, T_{2}), as a module over s_{2}(k),

by (36.2.9), (36.2.13), and (36.3.3). Similarly,

(36.3.8) LP_{A,d}^{0,+}(T_{1}, T_{2}) is a submodule of LP_{A,d}(T_{1}, T_{2}), as a module over s_{2}(k),

by (36.2.11), (36.2.15), and (36.3.4).

Note that

LS_{A,d}^{d,−}(T_{1}, T_{2}) ∩ LS_{A,d}^{0,+}(T_{1}, T_{2}) (36.3.9)

= LP_{A,d}^{d,−}(T_{1}, T_{2}) ∩ LP_{A,d}^{0,+}(T_{1}, T_{2})
= A_{d}[T_{1}, T_{2}] when d ≥ 0
= {0}  when d < 0.

Let us now take d = −1. Observe that

(36.3.10) LS_{A,−1}(T_{1}, T_{2}) corresponds to the direct sum of

LS_{A,−1}^{0,+}(T_{1}, T_{2}) and LS_{A,−1}^{−1,−}(T_{1}, T_{2}), as a module over s_{2}(k).

Similarly,

(36.3.11) LP_{A,−1}(T_{1}, T_{2}) corresponds to the direct sum of

LP_{A,−1}^{0,+}(T_{1}, T_{2}) and LP_{A,−1}^{−1,−}(T_{1}, T_{2}), as a module over s_{2}(k).

36.4 Characteristic 0, A = k

Let us continue with the same notation and hypotheses as in the previous three sections, now with k a field of characterisctic 0, and A = k, as a module over itself. If j ∈ Z, then T_{1}^{j} T_{2}^{d−j} ∈ LP_{k,d}(T_{1}, T_{2}), and

H(T_{1}^{j} T_{2}^{d−j}) = (2 j − d) T_{1}^{j} T_{2}^{d−j}. (36.4.1)

This means that T_{1}^{j} T_{2}^{d−j} has weight 2 j − d in LP_{k,d}(T_{1}, T_{2}), as a module over s_{2}(k), or equivalently sl_{2}(k), as in Section 15.1.

Observe that X(T_{1}^{d}) = 0. (36.4.2)

Thus T_{1}^{d} is a maximal or primitive vector of weight d in LP_{k,d}(T_{1}, T_{2}), as in Section 15.2. More precisely, T_{1}^{d} may be considered as a maximal or primitive vector of weight d in the submodule LP_{k,d}^{d,−}(T_{1}, T_{2}) of LP_{k,d}(T_{1}, T_{2}).

If d ≥ 0, then T_{1}^{d} may be considered as a maximal or primitive vector of weight d in the submodule k_{d}[T_{1}, T_{2}] of LP_{k,d}^{d,−}(T_{1}, T_{2}). In this case, it is easy to see that Y^{l}(T_{1}^{d}), l = 0, …, d, forms a basis for k_{d}[T_{1}, T_{2}], as a vector space over k, while Y^{d+1}(T_{1}^{d}) = 0. As in Section 15.5, this implies that k_{d}[T_{1}, T_{2}] is isomorphic as a module over sl_{2}(k) to the module discussed in Section 15.4, with m = d.

If d < 0, then Y^{l}(T_{1}^{d}), l ≥ 0, forms a basis for LP_{k,d}^{d,−}(T_{1}, T_{2}), as a vector space over k. Let Z_{0}(μ_{0}) be as in Section 34.2, and let us take

μ_{0} = d · 1 (36.4.3)

in k. Under these conditions, we get that LP_{k,d}^{d,−}(T_{1}, T_{2}) is isomorphic to Z_{0}(μ_{0}), as a module over sl_{2}(k), as in Section 34.3.

36.5 Some helpful projections

Let us return to the notation and hypotheses in Sections 36.1 and 36.2. Thus k is a commutative ring with a multiplicative identity element again, and A is a module over k. Let f(T) = ∑_{j=−∞}^{∞} f_{j} T_{1}^{j} T_{2}^{d−j} ∈ LS_{A,d}(T_{1}, T_{2}) be given. If l ∈ Z, then put

(36.5.1)  P_{l,+}(f(T)) = ∑_{j=l}^{∞} f_{j} T_{1}^{j} T_{2}^{d−j},

(36.5.2)  P_{l,−}(f(T)) = ∑_{j=−∞}^{l} f_{j} T_{1}^{j} T_{2}^{d−j}.

These define homomorphisms from LS_{A,d}(T_{1}, T_{2}) into itself, as a module over k.

Note that

(36.5.3)  P_{l,+}(LS_{A,d}(T_{1}, T_{2})) = LS_{A,d}^{l,+}(T_{1}, T_{2}),
(36.5.4)  P_{l,−}(LS_{A,d}(T_{1}, T_{2})) = LS_{A,d}^{l,−}(T_{1}, T_{2})

for each l ∈ Z. Similarly,

(36.5.5)  P_{l,+}(LP_{A,d}(T_{1}, T_{2})) = LP_{A,d}^{l,+}(T_{1}, T_{2}),
(36.5.6)  P_{l,−}(LP_{A,d}(T_{1}, T_{2})) = LP_{A,d}^{l,−}(T_{1}, T_{2})

for every l ∈ Z. More precisely, P_{l,+} is equal to the identity mapping on LS_{A,d}^{l,+}(T_{1}, T_{2}), and P_{l,−} is equal to the identity mapping on LS_{A,d}^{l,−}(T_{1}, T_{2}).

36.5. SOME HELPFUL PROJECTIONS 749

We also have that (36.5.7) the kernel of P_{l,+} on LS_{A,d}(T_{1}, T_{2})

is equal to LS_{A,d}^{l−1,−}(T_{1}, T_{2})

and (36.5.8) the kernel of P_{l,−} on LS_{A,d}(T_{1}, T_{2})

is equal to LS_{A,d}^{l+1,+}(T_{1}, T_{2})

for every l ∈ Z. It follows that (36.5.9) the kernel of P_{l,+} on LP_{A,d}(T_{1}, T_{2})

is equal to LP_{A,d}^{l−1,−}(T_{1}, T_{2})

and (36.5.10) the kernel of P_{l,−} on LP_{A,d}(T_{1}, T_{2})

is equal to LP_{A,d}^{l+1,+}(T_{1}, T_{2})

for every l ∈ Z.

If l ∈ Z, then we may consider the quotients LS_{A,d}(T_{1}, T_{2})/LS_{A,d}^{l,+}(T_{1}, T_{2}) (36.5.11) and LS_{A,d}(T_{1}, T_{2})/LS_{A,d}^{l,−}(T_{1}, T_{2}) (36.5.12) as modules over k. We may consider the quotients LP_{A,d}(T_{1}, T_{2})/LP_{A,d}^{l,+}(T_{1}, T_{2}) (36.5.13) and LP_{A,d}(T_{1}, T_{2})/LP_{A,d}^{l,−}(T_{1}, T_{2}) (36.5.14) as modules over k too. The natural inclusion mapping from LP_{A,d}(T_{1}, T_{2}) into LS_{A,d}(T_{1}, T_{2}) leads to injective homomorphisms from (36.5.13) into (36.5.11), and from (36.5.14) into (36.5.12), as modules over k.

We can use P_{l−1,−} to identify (36.5.11) with LS_{A,d}^{l−1,−}(T_{1}, T_{2}), as a module over k, in which case (36.5.13) corresponds to LP_{A,d}^{l−1,−}(T_{1}, T_{2}). Similarly, we can use P_{l+1,+} to identify (36.5.12) with LS_{A,d}^{l+1,+}(T_{1}, T_{2}), as a module over k, so that (36.5.14) corresponds to LP_{A,d}^{l+1,+}(T_{1}, T_{2}).

36.6 Projections and H, X, Y

Let us continue with the same notation and hypotheses as in the previous section. It is easy to see that

(36.6.1)  H ∘ P_{l,+} = P_{l,+} ∘ H,
(36.6.2)  H ∘ P_{l,−} = P_{l,−} ∘ H

on LS_{A,d}(T_{1}, T_{2}) for each l ∈ Z, using (36.1.9). If l ∈ Z, then H maps each of LS_{A,d}^{l,+}(T_{1}, T_{2}), LS_{A,d}^{l,−}(T_{1}, T_{2}), LP_{A,d}^{l,+}(T_{1}, T_{2}), and LP_{A,d}^{l,−}(T_{1}, T_{2}) into itself, as in Section 36.2. This means that H induces a mapping on each of the quotients (36.5.11), (36.5.12), (36.5.13), and (36.5.14), that is linear over k. These induced mappings correspond to H on LS_{A,d}^{l−1,−}(T_{1}, T_{2}), LS_{A,d}^{l+1,+}(T_{1}, T_{2}), LP_{A,d}^{l−1,−}(T_{1}, T_{2}), and LP_{A,d}^{l+1,+}(T_{1}, T_{2}), respectively, with respect to the identifications mentioned before, by (36.6.1) and (36.6.2).

Remember that

X(LS_{A,d}^{l,+}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l+1,+}(T_{1}, T_{2}) ⊆ LS_{A,d}^{l,+}(T_{1}, T_{2}) (36.6.3)

for each l ∈ Z, as in Section 36.2. Of course, there are analogous inclusions for Laurent polynomials, so that X induces a mapping on each of the quotients (36.5.11) and (36.5.13) that is linear over k. If l ∈ Z, then

P_{l−1,−} ∘ X = 0 on LS_{A,d}^{l−1,+}(T_{1}, T_{2}), (36.6.4)

because of the first inclusion in (36.6.3), with l replaced by l − 1. We also have that P_{l−1,−} ∘ X = X on LS_{A,d}^{l−2,−}(T_{1}, T_{2}), (36.6.5)

because X(LS_{A,d}^{l−2,−}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l−1,−}(T_{1}, T_{2}), as in (36.2.14). This determines P_{l−1,−} ∘ X on LS_{A,d}(T_{1}, T_{2}).

As before, we can use P_{l−1,−} to identify the quotients (36.5.11) and (36.5.13) with LS_{A,d}^{l−1,−}(T_{1}, T_{2}) and LP_{A,d}^{l−1,−}(T_{1}, T_{2}), respectively, as modules over k. Using this, the mapping induced by X on each of these quotients corresponds to the restriction of P_{l−1,−} ∘ X to LS_{A,d}^{l−1,−}(T_{1}, T_{2}) and LP_{A,d}^{l−1,−}(T_{1}, T_{2}), respectively.

Similarly,

Y (LS_{A,d}^{l,−}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l−1,−}(T_{1}, T_{2}) ⊆ LS_{A,d}^{l,−}(T_{1}, T_{2}) (36.6.6)

for each l ∈ Z, as in Section 36.2 again. It follows that Y induces a mapping on each of the quotients (36.5.12) and (36.5.14) that is linear over k, using also the analogue of (36.6.6) for Laurent polynomials. If l ∈ Z, then

P_{l+1,+} ∘ Y = 0 on LS_{A,d}^{l+1,−}(T_{1}, T_{2}), (36.6.7)

because of the first inclusion in (36.6.6), with l replaced with l + 1. In addition,

P_{l+1,+} ∘ Y = Y on LS_{A,d}^{l+2,+}(T_{1}, T_{2}), (36.6.8) because Y (LS_{A,d}^{l+2,+}(T_{1}, T_{2})) ⊆ LS_{A,d}^{l+1,+}(T_{1}, T_{2}), as in (36.2.17). This determines P_{l+1,+} ∘ Y on LS_{A,d}(T_{1}, T_{2}).

As usual, we can use P_{l+1,+} to identify the quotients (36.5.12) and (36.5.14) with LS_{A,d}^{l+1,+}(T_{1}, T_{2}) and LP_{A,d}^{l+1,+}(T_{1}, T_{2}), respectively, as modules over k. The mapping induced by Y on each of these quotients corresponds to the restriction of P_{l+1,+} ∘ Y to LS_{A,d}^{l+1,+}(T_{1}, T_{2}) and LP_{A,d}^{l+1,+}(T_{1}, T_{2}), respectively.

One can check that

X ∘ P_{d+1,+} = P_{d+1,+} ∘ X (36.6.9)

on LS_{A,d}(T_{1}, T_{2}), using (36.1.10). Alternatively, both sides are equal to 0 on LS_{A,d}^{d,−}(T_{1}, T_{2}), by definition of P_{d+1,+}, and because X maps LS_{A,d}^{d,−}(T_{1}, T_{2}) into itself, as in Section 36.3. Both sides are equal to X on LS_{A,d}^{d+1,+}(T_{1}, T_{2}), by definition of P_{l+1,+}, and because X maps LS_{A,d}^{d+1,+}(T_{1}, T_{2}) into itself, as in Section 36.2.

Similarly, one can verify that

Y ∘ P_{−1,−} = P_{−1,−} ∘ Y (36.6.10)

on LS_{A,d}(T_{1}, T_{2}), using (36.1.11). Alternatively, both sides are equal to 0 on LS_{A,d}^{0,+}(T_{1}, T_{2}), by definition of P_{−1,−}, and because Y maps LS_{A,d}^{0,+}(T_{1}, T_{2}) into itself, as in Section 36.3 again. Both sides are equal to Y on LS_{A,d}^{−1,−}(T_{1}, T_{2}), by definition of P_{−1,−}, and because Y maps LS_{A,d}^{−1,−}(T_{1}, T_{2}) into itself, as in Section 36.2.

36.7 Some related quotient modules

Let us continue with the same notation and hypotheses as in the previous two sections. Remember that s_{2}(k) is a Lie algebra over k, as in Section 36.1, and that LS_{A,d}^{d,−}(T_{1}, T_{2}) and LS_{A,d}^{0,+}(T_{1}, T_{2}) are submodules of LS_{A,d}(T_{1}, T_{2}), as a module over s_{2}(k), as in Section 36.3. Thus the quotients

LS_{A,d}(T_{1}, T_{2})/LS_{A,d}^{d,−}(T_{1}, T_{2}) (36.7.1)

and LS_{A,d}(T_{1}, T_{2})/LS_{A,d}^{0,+}(T_{1}, T_{2}) (36.7.2)

may be considered as modules over s_{2}(k) as well. We may identify these quotients with LS_{A,d}^{d+1,+}(T_{1}, T_{2}) and LS_{A,d}^{−1,−}(T_{1}, T_{2}), respectively, as modules over k, using P_{d+1,+} and P_{−1,−}, respectively, as in Section 36.5.

Remember that the actions of H induced on these quotients by the action of H on LS_{A,d}(T_{1}, T_{2}) correspond to the usual actions of H on LS_{A,d}^{d+1,+}(T_{1}, T_{2}) and LS_{A,d}^{−1,−}(T_{1}, T_{2}), as in (36.6.1) and (36.6.2). Similarly, the action of X induced on (36.7.1) by the action of X on LS_{A,d}(T_{1}, T_{2}) corresponds to the usual action of X on LS_{A,d}^{d+1,+}(T_{1}, T_{2}), because of (36.6.9). We also have that the action of Y induced on (36.7.2) by the action of Y on LS_{A,d}(T_{1}, T_{2}) corresponds to the usual action of Y on LS_{A,d}^{−1,−}(T_{1}, T_{2}), because of (36.6.10).

The action of Y induced on (36.7.1) by the action of Y on LS_{A,d}(T_{1}, T_{2}) corresponds to

(36.7.3) the restriction of P_{d+1,+} ∘ Y to LS_{A,d}^{d+1,+}(T_{1}, T_{2}),

as in the previous section. This mapping is described by (36.6.7) and (36.6.8), with l = d. Similarly, the action of X induced on (36.7.2) by the action of X on LS_{A,d}(T_{1}, T_{2}) corresponds to

(36.7.4) the restriction of P_{−1,−} ∘ X to LS_{A,d}^{−1,−}(T_{1}, T_{2}),

as before. This mapping is described by (36.6.4) and (36.6.5), with l = 0.

Of course, LP_{A,d}(T_{1}, T_{2}) is a submodule of LS_{A,d}(T_{1}, T_{2}), as a module over s_{2}(k). Remember that LP_{A,d}^{d,−}(T_{1}, T_{2}) and LP_{A,d}^{0,+}(T_{1}, T_{2}) are submodules of LP_{A,d}(T_{1}, T_{2}), as a module over s_{2}(k), as in Section 36.3. This means that the quotients LP_{A,d}(T_{1}, T_{2})/LP_{A,d}^{d,−}(T_{1}, T_{2}) (36.7.5)

and LP_{A,d}(T_{1}, T_{2})/LP_{A,d}^{0,+}(T_{1}, T_{2}) (36.7.6)

may be considered as modules over s_{2}(k), as before. Note that (36.7.5) and (36.7.6) may be considered as submodules of (36.7.1) and (36.7.2), respectively, as modules over s_{2}(k). We may also identify (36.7.5) and (36.7.6) with LP_{A,d}^{d+1,+}(T_{1}, T_{2}) and LP_{A,d}^{−1,−}(T_{1}, T_{2}), respectively, as modules over k, using P_{d+1,+} and P_{−1,−}, respectively, as before.

Suppose for the moment that d ≥ 0, so that

LS_{A,d}^{d,−}(T_{1}, T_{2}) ∩ LS_{A,d}^{0,+}(T_{1}, T_{2}) (36.7.7)

= LP_{A,d}^{d,−}(T_{1}, T_{2}) ∩ LP_{A,d}^{0,+}(T_{1}, T_{2}) = A_{d}[T_{1}, T_{2}],

as in Section 36.3. In particular, A_{d}[T_{1}, T_{2}] is a submodule of LS_{A,d}^{d,−}(T_{1}, T_{2}), LS_{A,d}^{0,+}(T_{1}, T_{2}), LP_{A,d}^{d,−}(T_{1}, T_{2}), and LP_{A,d}^{0,+}(T_{1}, T_{2}), as modules over s_{2}(k). This means that the quotients

LS_{A,d}^{0,+}(T_{1}, T_{2})/A_{d}[T_{1}, T_{2}], (36.7.8)

LS_{A,d}^{d,−}(T_{1}, T_{2})/A_{d}[T_{1}, T_{2}], (36.7.9)

LP_{A,d}^{0,+}(T_{1}, T_{2})/A_{d}[T_{1}, T_{2}], (36.7.10)

LP_{A,d}^{d,−}(T_{1}, T_{2})/A_{d}[T_{1}, T_{2}], (36.7.11)

may be considered as modules over s_{2}(k) too. Of course, (36.7.10) and (36.7.11) correspond to submodules of (36.7.8) and (36.7.9), respectively, as modules over s_{2}(k).

36.8. QUOTIENTS AND CHARACTERISTIC 0, A = K 753

It is easy to see that these modules are isomorphic to (36.7.1), (36.7.2), (36.7.5), and (36.7.6), respectively, in a natural way, as modules over s_{2}(k). Indeed, the natural quotient mapping from LS_{A,d}(T_{1}, T_{2}) onto (36.7.1) maps LS_{A,d}^{0,+}(T_{1}, T_{2}) onto (36.7.1) under these conditions, because LS_{A,d}(T_{1}, T_{2}) is generated by LS_{A,d}^{0,+}(T_{1}, T_{2}) and LS_{A,d}^{d,−}(T_{1}, T_{2}), as a module over k. The kernel of the restriction to LS_{A,d}^{0,+}(T_{1}, T_{2}) of the natural quotient mapping from LS_{A,d}(T_{1}, T_{2}) onto (36.7.1) is equal to A_{d}[T_{1}, T_{2}], by (36.7.7). This leads to the desired module isomorphism, and the other cases are analogous.

Suppose now that d < 0, so that

LS_{A,d}^{d,−}(T_{1}, T_{2}) ∩ LS_{A,d}^{0,+}(T_{1}, T_{2}) (36.7.12)

= LP_{A,d}^{d,−}(T_{1}, T_{2}) ∩ LP_{A,d}^{0,+}(T_{1}, T_{2}) = {0},

as in Section 36.3 again. This implies that the restriction to LS_{A,d}^{0,+}(T_{1}, T_{2}) of the natural quotient mapping from LS_{A,d}(T_{1}, T_{2}) onto (36.7.1) is injective. Thus LS_{A,d}^{0,+}(T_{1}, T_{2}) corresponds to a submodule of the quotient (36.7.1), as a module over s_{2}(k). Similarly, LS_{A,d}^{d,−}(T_{1}, T_{2}), LP_{A,d}^{0,+}(T_{1}, T_{2}), and LP_{A,d}^{d,−}(T_{1}, T_{2}) correspond to submodules of (36.7.2), (36.7.5), and (36.7.6), respectively, as modules over s_{2}(k), in this case. If d = −1, then each of these submodules is the same as the corresponding quotient module.

36.8 Quotients and characteristic 0, A = k

Let us continue with the same notation and hypotheses as in the previous three sections, and take k to be a field of characteristic 0, and A = k, as a module over itself. Observe that T_{1}^{−1} T_{2}^{d+1} is an element of LP_{k,d}^{−1,−}(T_{1}, T_{2}), with

H(T_{1}^{−1} T_{2}^{d+1}) = (−d − 2) T_{1}^{−1} T_{2}^{d+1}, (36.8.1)

as in (36.4.1). We also have that

X(T_{1}^{−1} T_{2}^{d+1}) = (d + 1) T_{2}^{d} ∈ LP_{k,d}^{0,+}(T_{1}, T_{2}). (36.8.2)

The image of T_{1}^{−1} T_{2}^{d+1} in

LP_{k,d}(T_{1}, T_{2})/LP_{k,d}^{0,+}(T_{1}, T_{2}) (36.8.3)

under the natural quotient mapping from LP_{k,d}(T_{1}, T_{2}) is clearly nonzero. It follows that the image of T_{1}^{−1} T_{2}^{d+1} in (36.8.3) is a maximal or primitive vector of weight −d − 2, as in Section 15.2.

One can check that Y^{l}(T_{1}^{−1} T_{2}^{d+1}), l ≥ 0, form a basis for LP_{k,d}^{−1,−}(T_{1}, T_{2}), as a vector space over k. This means that the images of these vectors in the quotient (36.8.3) form a basis there. Let Z_{0}(μ_{0}) be as in Section 34.2, and let us take μ_{0} = (−d − 2) · 1 (36.8.4) in k. Thus (36.8.3) is isomorphic to Z_{0}(μ_{0}), as modules over sl_{2}(k), as in Section 34.3.

If d ≥ 0, then k_{d}[T_{1}, T_{2}] is a submodule of LP_{k,d}^{d,−}(T_{1}, T_{2}), as a module over s_{2}(k). In this case, (36.8.3) is isomorphic to

LP_{k,d}^{d,−}(T_{1}, T_{2})/k_{d}[T_{1}, T_{2}] (36.8.5)

as modules over s_{2}(k), as in the previous section. If d < 0, then LP_{k,d}^{d,−}(T_{1}, T_{2}) corresponds to a submodule of (36.8.3), as a module over s_{2}(k), as before. In particular, if d = −1, then (36.8.3) is isomorphic to LP_{k,−1}^{−1,−}(T_{1}, T_{2}), as a module over s_{2}(k). Note that the remarks in the preceding paragraph correspond to some of those in Section 36.4 in this case.

If d ≤ −2, so that m = −d−2 ≥ 0, then we get a proper submodule Z_{0,m}(μ_{0}) of Z_{0}(μ_{0}), as a module over sl_{2}(k), as in Section 34.2. One can verify that this corresponds to LP_{k,d}^{d,−}(T_{1}, T_{2}) as a submodule of (36.8.3) under these conditions.

36.9 Some more submodules and quotients

Let k be a commutative ring with a multiplicative identity element, and let B be a Lie algebra over k. Also let V be a module over k that is a module over B, as a Lie algebra over k, and let W, Z be submodules of V, as a module over k and B. Thus W + Z and W ∩ Z are submodules of V as well, as a module over k and B.

The quotients V/W, V/Z may be considered as modules over k and B too, as usual. The natural quotient mapping from V onto V/W maps Z and W + Z onto (W +Z)/W, and the kernel of the restriction to Z of the quotient mapping is equal to W ∩Z. Thus (W +Z)/W is isomorphic in a natural way to Z/(W ∩Z), as a module over k and B. Similarly, (W + Z)/Z is isomorphic to W/(W ∩ Z) in a natural way, as a module over k and B.

Of course, (W/(W ∩ Z)) ∩ (Z/(W ∩ Z)) = {0} (36.9.1)

in V/(W ∩ Z). This implies that

(W + Z)/(W ∩ Z) = (W/(W ∩ Z)) + (Z/(W ∩ Z)) (36.9.2)

corresponds to the direct sum of W/(W ∩ Z) and Z/(W ∩ Z) in V/(W ∩ Z), as a module over k and B.

The natural quotient mapping from V onto V/(W + Z) has kernel W + Z, which leads to natural homomorphisms from V/W and V/Z onto V/(W + Z), as modules over k and B. The kernels of these homomorphisms are (W +Z)/W and (W + Z)/Z, respectively. This leads to natural isomorphisms from

(V/W)/((W + Z)/W), (V/Z)/((W + Z)/Z) (36.9.3)

onto V/(W + Z), as modules over k and B.

Let us now return to the same notation and hypotheses as in Section 36.7. Thus k is still a commutative ring with a multiplicative identity element, and A is a module over k. Suppose for the moment that d ≥ 0, so that (36.7.7) holds. We may consider LP_{A,d}(T_{1}, T_{2})/A_{d}[T_{1}, T_{2}] (36.9.4)

and LS_{A,d}[T_{1}, T_{2}]/A_{d}[T_{1}, T_{2}] (36.9.5)

as modules over s_{2}(k), as a Lie algebra over k, because A_{d}[T_{1}, T_{2}] is a submodule of LP_{A,d}[T_{1}, T_{2}] and LS_{A,d}(T_{1}, T_{2}), as modules over s_{2}(k). Note that (36.9.4) may be considered as a submodule of (36.9.5), as a module over s_{2}(k).

It is easy to see that

LP_{A,d}(T_{1}, T_{2}) = LP_{A,d}^{0,+}(T_{1}, T_{2}) + LP_{A,d}^{d,−}(T_{1}, T_{2}) (36.9.6)

and LS_{A,d}(T_{1}, T_{2}) = LS_{A,d}^{0,+}(T_{1}, T_{2}) + LS_{A,d}^{d,−}(T_{1}, T_{2}), (36.9.7)

because d ≥ 0. In fact, (36.9.4) corresponds to the direct sum of (36.7.10) and (36.7.11), as a module over s_{2}(k). Similarly, (36.9.5) corresponds to the direct sum of (36.7.8) and (36.7.9), as a module over s_{2}(k). This follows from the analogous statement for (36.9.2) mentioned earlier.

Suppose now that d < 0, so that (36.7.12) holds. We may consider

LP_{A,d}(T_{1}, T_{2})/(LP_{A,d}^{0,+}(T_{1}, T_{2}) + LP_{A,d}^{d,−}(T_{1}, T_{2})) (36.9.8)

and LS_{A,d}(T_{1}, T_{2})/(LS_{A,d}^{0,+}(T_{1}, T_{2}) + LS_{A,d}^{d,−}(T_{1}, T_{2})) (36.9.9)

as modules over s_{2}(k), as a Lie algebra over k. If d = −1, then (36.9.6) and (36.9.7) hold, and so (36.9.8) and (36.9.9) are equal to 0. Otherwise, we have that

LS_{A,d}(T_{1}, T_{2}) = LS_{A,d}^{0,+}(T_{1}, T_{2}) + LS_{A,d}^{d,−}(T_{1}, T_{2}) + LP_{A,d}(T_{1}, T_{2}). (36.9.10)

This implies that (36.9.8) is isomorphic to (36.9.9), as modules over s_{2}(k).

Under these conditions, (36.9.8) can be described as in (36.9.3). This means that (36.9.8) is isomorphic in a natural way to quotients of (36.7.5) and (36.7.6) by submodules corresponding to LP_{A,d}^{0,+}(T_{1}, T_{2}) and LP_{A,d}^{d,−}(T_{1}, T_{2}), respectively, as a module over s_{2}(k).

36.10 Some additional submodules

Let k be a field of characteristic 0, and let us continue with the same notation and hypotheses as in Sections 36.1 and 36.2, with A = k. Put

f_{0}(T) = exp(T_{1}^{−1} T_{2}) = ∑_{j=0}^{∞}(1/j!) T_{1}^{−j} T_{2}^{j}, (36.10.1)

which is an element of LS_{k,0}(T_{1}, T_{2}), using the natural embedding of Q into k. More precisely, f_{0}(T) ∈ LS_{k,0}^{0,−}(T_{1}, T_{2}), in the notation of Section 36.2.

It is easy to see that

∂_{1}f_{0}(T) = −T_{1}^{−2} T_{2} f_{0}(T) (36.10.2)

and ∂_{2}f_{0}(T) = T_{1}^{−1} f_{0}(T). (36.10.3)

This implies that

(36.10.4)  H(f_{0}(T)) = −2 T_{1}^{−1} T_{2} f_{0}(T),
(36.10.5)  X(f_{0}(T)) = f_{0}(T),
(36.10.6)  Y (f_{0}(T)) = −T_{1}^{−2} T_{2}^{2} f_{0}(T).

Note that

(36.10.7)  H(T_{1}^{−m} T_{2}^{m}) = −2 m T_{1}^{−m} T_{2}^{m},
(36.10.8)  X(T_{1}^{−m} T_{2}^{m}) = m T_{1}^{1−m} T_{2}^{m−1},
(36.10.9)  Y (T_{1}^{−m} T_{2}^{m}) = −m T_{1}^{−m−1} T_{2}^{m+1}

for every m ∈ Z.

Put f_{m}(T) = T_{1}^{−m} T_{2}^{m} f_{0}(T) (36.10.10)

for every m ∈ Z, which is the same as f_{0}(T) when m = 0. By construction,

f_{m}(T) ∈ LS_{k,0}^{−m,−}(T_{1}, T_{2}) (36.10.11)

for every m ∈ Z, using the notation in Section 36.2 again. We also have that the coefficient of T_{1}^{−m} T_{2}^{m} in f_{m}(T) is equal to 1. One can check that the f_{m}(T)’s are linearly independent in LS_{k,0}(T_{1}, T_{2}), using these two properties.

If m ∈ Z, then

H(f_{m}(T)) = H(T_{1}^{−m} T_{2}^{m}) f_{0}(T) + T_{1}^{−m} T_{2}^{m} H(f_{0}(T))
(36.10.12)  = −2 m T_{1}^{−m} T_{2}^{m}f_{0}(T) + T_{1}^{−m} T_{2}^{m} (−2T_{1}^{−1} T_{2} f_{0}(T))
= −2 m f_{m}(T) − 2 f_{m+1}(T).

Similarly,

X(f_{m}(T)) = X(T_{1}^{−m} T_{2}^{m}) f_{0}(T) + T_{1}^{−m} T_{2}^{m} X(f_{0}(T))
(36.10.13)  = m T_{1}^{1−m} T_{2}^{m−1} f_{0}(T) + T_{1}^{−m} T_{2}^{m} f_{0}(T)
= m f_{m−1}(T) + f_{m}(T).

In the same way,

Y (f_{m}(T)) = Y (T_{1}^{−m} T_{2}^{m}) f_{0}(T) + T_{1}^{−m} T_{2}^{m} Y (f_{0}(T))
(36.10.14)  = −m T_{1}^{−m−1} T_{2}^{m+1} f_{0}(T) + T_{1}^{−m} T_{2}^{m} (−T_{1}^{−2} T_{2}^{2} f_{0}(T))
= −m f_{m+1}(T) − f_{m+2}(T).

Let E_{k}(T_{1}, T_{2}) be the linear span of the f_{m}(T)’s, m ∈ Z, in LS_{k,0}(T_{1}, T_{2}), as a vector space over k. This is a submodule of LS_{k,0}(T_{1}, T_{2}), as a module over s_{2}(k), as in the preceding paragraph.

Similarly, let E_{k}^{0}(T_{1}, T_{2}) be the linear span of the f_{m}(T)’s, m ≥ 0. It is easy to see that this is a submodule of E_{k}(T_{1}, T_{2}), as a module over s_{2}(k).

One can check that there is no nonzero eigenvector of H in E_{k}(T_{1}, T_{2}), using (36.10.12) and the linear independence of the f_{m}(T)’s.

One can verify that E_{k}^{0}(T_{1}, T_{2}) is irreducible as a module over s_{2}(k). To see this, observe that (X − I)(f_{m}(T)) = m f_{m−1}(T) (36.10.15)

for every m ∈ Z, where I is the identity mapping. This implies that every nonzero element of E_{k}^{0}(T_{1}, T_{2}) is mapped to a nonzero multiple of f_{0}(T) by a suitable power of X − I. It follows that any nonzero submodule of E_{k}^{0}(T_{1}, T_{2}) contains f_{m}(T) for every m ≥ 0, because of (36.10.12).

This is related to part (c) of Exercise 2 on p111 of [14].

Formal Laurent series, 3

37.1 Some commutativity conditions

Let k be a commutative ring with a multiplicative identity element, let m, n be positive integers, and let T_{1}, …, T_{n}, R_{1}, …, R_{m}, be n + m commuting indeterminates. Also let A be a module over k, and let us use

f(T, R) = f(T_{1}, …, T_{n}, R_{1}, …, R_{m}) (37.1.1)

to denote a formal Laurent series in T_{1}, …, T_{n}, R_{1}, …, R_{m} with coefficients in A. The corresponding formal derivatives of f(T, R) in T_{j} and R_{l} may be expressed as

∂_{T_{j}}f(T, R) = \frac{∂}{∂T_{j}} f(T, R), ∂_{R_{l}}f(T, R) = \frac{∂}{∂R_{l}} f(T, R), (37.1.2)

respectively, for each j = 1, …, n, l = 1, …, m.

Let ∂_{T_{1}}, …, ∂_{T_{n}}, ∂_{R_{1}}, …, ∂_{R_{m}} be n + m commuting formal symbols, which may be used to represent partial derivatives in the T_{j}’s and R_{l}’s. Suppose that for each j = 1, …, n and l = 1, …, m,

ζ^{j,l}(T, R) = ζ^{j,l}(T_{1}, …, T_{n}, R_{1}, …, R_{m}) (37.1.3)

is a formal Laurent polynomial in T_{1}, …, T_{n}, R_{1}, …, R_{m} with coefficients in k. Put

δ_{j} = ∂_{T_{j}} + ∑_{l=1}^{m} ζ^{j,l}(T, R) ∂_{R_{l}} (37.1.4)

for each j = 1, …, n. This is a formal first-order differential operator in ∂_{T_{1}}, …, ∂_{T_{n}}, ∂_{R_{1}}, …, ∂_{R_{m}} with coefficients in

LS_{k}(T, R) = LP_{k}(T_{1}, …, T_{n}, R_{1}, …, R_{m}). (37.1.5)

Let 1 ≤ j_{1}, j_{2} ≤ n be given, and observe that

δ_{j_{1}} δ_{j_{2}} − δ_{j_{2}} δ_{j_{1}} = ∑_{l=1}^{m}(δ_{j_{1}}(ζ^{j_{2},l}(T, R)) − δ_{j_{2}}(ζ^{j_{1},l}(T, R))) ∂_{R_{l}}. (37.1.6)

Thus δ_{j_{1}} δ_{j_{2}} = δ_{j_{2}} δ_{j_{1}} (37.1.7)

if and only if δ_{j_{1}}(ζ^{j_{2},l}(T, R)) = δ_{j_{2}}(ζ^{j_{1},l}(T, R)) (37.1.8)

for every l = 1, …, m.

Let f(T, R) be a formal Laurent series in T_{1}, …, T_{n}, R_{1}, …, R_{m} with coefficients in A, and suppose that we were interested in substituting R_{l} with something that depends on T_{1}, …, T_{n}, for each l = 1, …, m. In order to differentiate the result in T_{j} for some j = 1, …, n, we should differentiate f in T_{j}, and add to that the derivative of f in R_{l} times the derivative in T_{j} of the expression being substituted for R_{l} for each l = 1, …, m. This corresponds to δ_{j}(f(T, R)), where the derivative in T_{j} of the expression being substituted for R_{l} is given by ζ^{j,l}(T, R). The commutativity condition (37.1.7) says exactly that these total derivatives in T_{j_{1}} and T_{j_{2}} should commute. This holds exactly when the total derivatives in T_{j_{1}} and T_{j_{2}} of the expression being substituted for R_{l} commute for each l = 1, …, m, as in (37.1.8).

Let us suppose for the rest of the section that (37.1.8) holds for every j_{1}, j_{2} = 1, …, n and l = 1, …, m. Of course, we may identify LS_{A}(T_{1}, …, T_{n}) with a submodule of LS_{A}(T, R) = LS_{A}(T_{1}, …, T_{n}, R_{1}, …, R_{m}), (37.1.9)

as a module over k, in the obvious way. The restriction of the mapping on LS_{A}(T, R) determined by δ_{j} to LS_{A}(T_{1}, …, T_{n}) is the same as ∂_{T_{j}} for each j = 1, …, n, by construction. If α = (α_{1}, …, α_{n}) is a multi-index of length n, then let δ^{α} = δ_{1}^{α_{1}} ⋯ δ_{n}^{α_{n}} (37.1.10)

be the corresponding product of the δ_{j}’s. This is a formal differential operator in ∂_{T_{1}}, …, ∂_{T_{n}}, ∂_{R_{1}}, …, ∂_{R_{m}} with coefficients in LP_{k}(T, R).

Let β be a multi-index of length n, and let b^{β}(T, R) be a formal Laurent polynomial in T_{1}, …, T_{n}, R_{1}, …, R_{m} with coefficients in k. If 1 ≤ j ≤ n, then

δ_{j} (b^{β}(T, R) δ^{β}) = (δ_{j}(b^{β}(T, R))) δ^{β} + b^{β}(T, R) δ_{j} δ^{β}, (37.1.11)

as a formal differential operator in ∂_{T_{1}}, …, ∂_{T_{n}}, ∂_{R_{1}}, …, ∂_{R_{n}} with coefficients in LP_{k}(T, R). If b^{β}(T) is a formal Laurent polynomial in T_{1}, …, T_{n} with coefficients in k, then we get that

δ_{j} (b^{β}(T) δ^{β}) = (∂_{T_{j}}(b^{β}(T))) δ^{β} + b^{β}(T) δ_{j} δ^{β}. (37.1.12)

Let a(T) = (a^{1}(T), …, a^{n}(T)) be an n-tuple of formal Laurent polynomials in T_{1}, …, T_{n} with coefficients in k, and put

D̃_{a(T)} = ∑_{j=1}^{n} a^{j}(T) δ_{j}. (37.1.13)

This is a formal first-order differential operator in ∂_{T_{1}}, …, ∂_{T_{n}}, ∂_{R_{1}}, …, ∂_{R_{n}} with coefficients in LP_{k}(T, R). If b(T) = (b^{1}(T), …, b^{n}(T)) is another n-tuple of formal Laurent polynomials in T_{1}, …, T_{n} with coefficients in k, then

D̃_{a(T)} D̃_{b(T)} − D̃_{b(T)} D̃_{a(T)} (37.1.14)

= ∑_{j=1}^{n} ∑_{r=1}^{n}(a^{j}(T) ∂_{T_{j}}b^{r}(T) − b^{j}(T) ∂_{T_{j}}a^{r}(T)) δ_{r},

as in Section 35.7. Put

c^{r}(T) = ∑_{j=1}^{n}(a^{j}(T) ∂_{T_{j}}b^{r}(T) − b^{j}(T) ∂_{T_{j}}a^{r}(T)) (37.1.15)

for each r = 1, …, n, which defines a formal Laurent polynomial in T_{1}, …, T_{n} with coefficients in k, as before. Thus

D̃_{a(T)} D̃_{b(T)} − D̃_{b(T)} D̃_{a(T)} = D̃_{c(T)}, (37.1.16)

where c(T) = (c^{1}(T), …, c^{n}(T)).

Put

g̃_{n}(k) = {D̃_{a(T)} : a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n}}, (37.1.17)

which is a Lie subalgebra of the algebra of formal differential operators in ∂_{T_{1}}, …, ∂_{T_{n}}, ∂_{R_{1}}, …, ∂_{R_{n}} with coefficients in LP_{k}(T, R). Remember that if a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n}, then a^{j}(T) can be expressed as ∑_{r=1}^{n} a_{r}^{j} T_{r} for each j = 1, …, n, where a_{r}^{j} ∈ k for every j, r = 1, …, n. Put

s̃_{n}(k) = {D̃_{a(T)} : a(T) ∈ (k_{1}[T_{1}, …, T_{n}])^{n}, ∑_{j=1}^{n} a_{j}^{j} = 0}, (37.1.18)

which is a Lie subalgebra of g̃_{n}(k). As in Section 5.15, g̃_{n}(k) is isomorphic to gl_{n}(k) as a Lie algebra over k, and s̃_{n}(k) corresponds to sl_{n}(k) under this isomorphism. In particular,

[g̃_{n}(k), g̃_{n}(k)] ⊆ s̃_{n}(k), (37.1.19)

as before.

37.2. SIMPLER COEFFICIENTS 761

37.2 Simpler coefficients

Let us continue with the same notation and hypotheses as in the previous section. Suppose now that θ^{j,l}(T) is a formal Laurent polynomial in T_{1}, …, T_{n} with coefficients in k for each j = 1, …, n and l = 1, …, m. We would like to consider the remarks in the previous section with ζ^{j,l}(T, R) = θ^{j,l}(T) R_{l} (37.2.1) for every j = 1, …, n and l = 1, …, m. Thus we take

δ_{j} = ∂_{T_{j}} + ∑_{l=1}^{m} θ^{j,l}(T) R_{l} ∂_{R_{l}} (37.2.2)

for each j = 1, …, n. We also ask that δ_{j_{1}}(θ^{j_{2},l}(T) R_{l}) = δ_{j_{2}}(θ^{j_{1},l}(T) R_{l}) (37.2.3) for every j_{1}, j_{2} = 1, …, n and l = 1, …, m, as in (37.1.8). In this case, this reduces to asking that ∂_{T_{j_{1}}} θ^{j_{2},l}(T) = ∂_{T_{j_{2}}} θ^{j_{1},l}(T) (37.2.4) for every j_{1}, j_{2} = 1, …, n and l = 1, …, m. This means that δ_{j_{1}} commutes with δ_{j_{2}}, as before.

Let γ ∈ Z^{m} be given, so that R^{γ} is defined as a formal Laurent monomial in R_{1}, …, R_{m}. Note that LS_{A}(T_{1}, …, T_{n}) R^{γ} = {f(T) R^{γ} : f(T) ∈ LS_{A}(T_{1}, …, T_{n})} (37.2.5) is a submodule of LS_{A}(T, R), as a module over k. More precisely, this is a submodule of LS_{A}(T, R), as a module over LP_{k}(T_{1}, …, T_{n}). It is easy to see that δ_{j}(LS_{A}(T_{1}, …, T_{n}) R^{γ}) ⊆ LS_{A}(T_{1}, …, T_{n}) R^{γ} (37.2.6) for every j = 1, …, n. Similarly, LP_{A}(T_{1}, …, T_{n}) R^{γ} = {f(T) R^{γ} : f(T) ∈ LP_{A}(T_{1}, …, T_{n})} (37.2.7) is a submodule of LP_{A}(T, R), as a module over k, and over LP_{k}(T_{1}, …, T_{n}). We also have that δ_{j}(LP_{A}(T_{1}, …, T_{n}) R^{γ}) ⊆ LP_{A}(T_{1}, …, T_{n}) R^{γ} (37.2.8) for every j = 1, …, n. If a(T) = (a^{1}(T), …, a^{n}(T)) is an n-tuple of elements of LP_{k}(T_{1}, …, T_{n}), and D̃_{a(T)} is as in (37.1.13), then we get that D̃_{a(T)}(LS_{A}(T_{1}, …, T_{n}) R^{γ}) ⊆ LS_{A}(T_{1}, …, T_{n}) R^{γ} (37.2.9)

762 CHAPTER 37. FORMAL LAURENT SERIES, 3 and D̃_{a(T)}(LP_{A}(T_{1}, …, T_{n}) R^{γ}) ⊆ LP_{A}(T_{1}, …, T_{n}) R^{γ}. (37.2.10)

Suppose that θ^{j,l}(T) ∈ LP_{k,−1}(T_{1}, …, T_{n}) (37.2.11) for every j = 1, …, n and l = 1, …, m, in the notation of Section 35.8. One can check that δ_{j}(LS_{A,d}(T_{1}, …, T_{n}) R^{γ}) ⊆ LS_{A,d−1}(T_{1}, …, T_{n}) R^{γ} (37.2.12) for every j = 1, …, n and d ∈ Z. Similarly, δ_{j}(LP_{A,d}(T_{1}, …, T_{n}) R^{γ}) ⊆ LP_{A,d−1}(T_{1}, …, T_{n}) R^{γ} (37.2.13) for every j = 1, …, n and d ∈ Z. If d(a(T)) ∈ Z and a^{j}(T) ∈ LP_{k,d(a(T))}(T_{1}, …, T_{n}) (37.2.14) for every j = 1, …, n, then it follows that D̃_{a(T)}(LS_{A,d}(T_{1}, …, T_{n}) R^{γ}) ⊆ LS_{A,d(a(T))+d−1}(T_{1}, …, T_{n}) R^{γ} (37.2.15) and D̃_{a(T)}(LP_{A,d}(T_{1}, …, T_{n}) R^{γ}) ⊆ LP_{A,d(a(T))+d−1}(T_{1}, …, T_{n}) R^{γ} (37.2.16) for every d ∈ Z. In particular, if d(a(T)) = 1, then D̃_{a(T)}(LS_{A,d}(T_{1}, …, T_{n}) R^{γ}) ⊆ LS_{A,d}(T_{1}, …, T_{n}) R^{γ} (37.2.17) and D̃_{a(T)}(LP_{A,d}(T_{1}, …, T_{n}) R^{γ}) ⊆ LP_{A,d}(T_{1}, …, T_{n}) R^{γ} (37.2.18) for every d ∈ Z.

Suppose for the moment that δ_{j} is as in the previous section again, where the coefficients satisfy (37.1.8), as before. Under these conditions, (37.2.19) we may consider LS_{A}(T, R) as a module over g̃_{n}(k) or s̃_{n}(k), as Lie algebras over k, and (37.2.20) LP_{A}(T, R) is a submodule of LS_{A}(T, R), as a module over g̃_{n}(k) or s̃_{n}(k). Note that (37.2.21) LS_{A}(T_{1}, …, T_{n}) is a submodule of LS_{A}(T, R) and (37.2.22) LP_{A}(T_{1}, …, T_{n}) is a submodule of LP_{A}(T, R), 37.3. SOME ACTIONS ON LS_{A}(T_{1}, T_{2}, R) 763

as modules over g̃_{n}(k) or s̃_{n}(k). The actions of g̃_{n}(k), s̃_{n}(k) on LS_{A}(T_{1}, …, T_{n}) correspond exactly to the actions of g_{n}(k), s_{n}(k), respectively, as in Section 35.10, in a natural way, by construction.

With δ_{j} as in (37.2.2), we get that

(37.2.23) LS_{A}(T_{1}, …, T_{n}) R^{γ} is a submodule of LS_{A}(T, R)

and (37.2.24) LP_{A}(T_{1}, …, T_{n}) R^{γ} is a submodule of LP_{A}(T, R),

as modules over g̃_{n}(k) or s̃_{n}(k). Of course, these reduce to (37.2.21) and (37.2.22), respectively, when γ = 0. If (37.2.11) holds, then we obtain that

(37.2.25) LS_{A,d}(T_{1}, …, T_{n}) R^{γ} is a submodule of LS_{A}(T_{1}, …, T_{n}) R^{γ}

and (37.2.26) LP_{A,d}(T_{1}, …, T_{n}) R^{γ} is a submodule of LP_{A}(T_{1}, …, T_{n}) R^{γ}

for every d ∈ Z, as modules over g̃_{n}(k) or s̃_{n}(k).

37.3 Some actions on LS_{A}(T_{1}, T_{2}, R)

Let k be a commutative ring with a multiplicative identity element, let A be a module over k, and let T_{1}, T_{2}, and R be commuting indeterminates. We would like to consider formal Laurent polynomials and series in T_{1}, T_{2}, and R with coefficients in k or A, as in the previous two sections, with n = 2 and m = 1. If f(T_{1}, T_{2}, R) is a formal Laurent series in T_{1}, T_{2}, R with coefficients in A, then its formal derivatives in T_{1}, T_{2}, and R may be expressed as ∂_{T_{j}}f(T_{1}, T_{2}, R) = \frac{∂}{∂T_{j}} f(T_{1}, T_{2}, R), j = 1, 2, and ∂_{R}f(T_{1}, T_{2}, R) = \frac{∂}{∂R}f(T_{1}, T_{2}, R), as before.

Let ∂_{T_{1}}, ∂_{T_{2}}, and ∂_{R} be commuting formal symbols, which may be used to represent partial derivatives in T_{1}, T_{2}, and R, respectively, as usual. Let ν_{0} ∈ k be given, and put

δ_{1} = ∂_{T_{1}} + ν_{0} T_{1}^{−1} R ∂_{R}, δ_{2} = ∂_{T_{2}}. (37.3.1)

Note that δ_{1} and δ_{2} commute, as formal differential operators in ∂_{T_{1}}, ∂_{T_{2}}, and ∂_{R} with coefficients in LP_{k}(T_{1}, T_{2}, R).

If f(T_{1}, T_{2}, R) ∈ LS_{A}(T_{1}, T_{2}, R), then we may be interested in trying to substitute R with something that depends on T_{1} and T_{2}, as in Section 37.1. The derivative of the result in T_{j}, j = 1, 2, should include the derivative of f in R times the derivitive in T_{j} of the expression being substituted for R, as before. Here we think of substituting R with an expression whose derivative in T_{1} is ν_{0} T_{1}^{−1} R, and whose derivative in T_{2} is equal to 0. In effect, we may think of substituting R with T_{1} to the power ν_{0}, formally.

Note that δ_{1}, δ_{2} satisfy the conditions mentioned at the beginning of the previous section. As in Section 37.1, we may identify LS_{A}(T_{1}, T_{2}) with a submodule of LS_{A}(T_{1}, T_{2}, R), as a module over k, in the obvious way. If γ ∈ Z, then LS_{A}(T_{1}, T_{2}) R^{γ} = {f(T_{1}, T_{2}) R^{γ} : f(T_{1}, T_{2}) ∈ LS_{A}(T_{1}, T_{2})} (37.3.2)

764 CHAPTER 37. FORMAL LAURENT SERIES, 3 is a submodule of LS_{A}(T_{1}, T_{2}, R) too, as a module over k. In fact, this is a submodule of LS_{A}(T_{1}, T_{2}, R), as a module over LP_{k}(T_{1}, T_{2}), as before. Similarly, LP_{A}(T_{1}, T_{2}) R^{γ} = {f(T_{1}, T_{2}) R^{γ} : f(T_{1}, T_{2}) ∈ LP_{A}(T_{1}, T_{2})} (37.3.3) is a submodule of LP_{A}(T_{1}, T_{2}, R), as a module over k, and over LP_{k}(T_{1}, T_{2}).

It is easy to see that δ_{j}(LS_{A}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A}(T_{1}, T_{2}) R^{γ} (37.3.4) for j = 1, 2, as in the previous section. Similarly, δ_{j}(LP_{A}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A}(T_{1}, T_{2}) R^{γ} (37.3.5) for j = 1, 2. If a(T) = (a^{1}(T), a^{2}(T)) is an ordered pair of elements of formal Laurent polynomials in T_{1}, T_{2} with coefficients in k, then put D̃_{a(T)} = a^{1}(T) δ_{1} + a^{2}(T) δ_{2}, (37.3.6) as in Section 37.1. Under these conditions, D̃_{a(T)}(LS_{A}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A}(T_{1}, T_{2}) R^{γ} (37.3.7) and D̃_{a(T)}(LP_{A}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A}(T_{1}, T_{2}) R^{γ}. (37.3.8)

More precisely, if d ∈ Z, then δ_{j}(LS_{A,d}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d−1}(T_{1}, T_{2}) R^{γ} (37.3.9) for j = 1, 2, using the notation in Section 35.8. Similarly, δ_{j}(LP_{A,d}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d−1}(T_{1}, T_{2}) R^{γ} (37.3.10) for j = 1, 2, as in the previous section. In these inclusions, we use the same type of notation as in (37.3.2) and (37.3.3). If d(a(T)) ∈ Z and a^{j}(T) ∈ LP_{k,d(a(T))}(T_{1}, T_{2}) (37.3.11) for j = 1, 2, then we get that D̃_{a(T)}(LS_{A,d}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d(a(T))+d−1}(T_{1}, T_{2}) R^{γ} (37.3.12) and D̃_{a(T)}(LP_{A,d}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d(a(T))+d−1}(T_{1}, T_{2}) R^{γ}, (37.3.13) as before.

Remember that g̃_{2}(k) = {D̃_{a(T)} : a(T) ∈ (k_{1}[T_{1}, T_{2}])^{2}}, (37.3.14)

37.4. THE ACTION OF S̃_{2}(K) 765

as in Section 37.1. If a(T) ∈ (k_{1}[T_{1}, T_{2}])^{2}, then a^{j}(T) can be expressed as a_{1}^{j} T_{1} + a_{2}^{j} T_{2} for j = 1, 2, where a_{l}^{j} ∈ k for j, l = 1, 2, and s̃_{2}(k) = {D̃_{a(T)} : a(T) ∈ (k_{1}[T_{1}, T_{2}])^{2}, a_{1}^{1} + a_{2}^{2} = 0}, (37.3.15) as before. As in the previous section, (37.3.16) we may consider LS_{A}(T_{1}, T_{2}, R) as a module over g̃_{2}(k) or s̃_{2}(k), as Lie algebras over k, and (37.3.17) LP_{A}(T_{1}, T_{2}, R) is a submodule of LS_{A}(T_{1}, T_{2}, R), as a module over g̃_{2}(k) or s̃_{2}(k). We also have that (37.3.18) LS_{A}(T_{1}, T_{2}) R^{γ} is a submodule of LS_{A}(T_{1}, T_{2}, R) and (37.3.19) LP_{A}(T_{1}, T_{2}) R^{γ} is a submodule of LP_{A}(T_{1}, T_{2}, R), as modules over g̃_{2}(k) or s̃_{2}(k). If d ∈ Z, then (37.3.20) LS_{A,d}(T_{1}, T_{2}) R^{γ} is a submodule of LS_{A}(T_{1}, T_{2}) R^{γ} and (37.3.21) LP_{A,d}(T_{1}, T_{2}) R^{γ} is a submodule of LP_{A}(T_{1}, T_{2}) R^{γ}, as modules over g̃_{2}(k) and s̃_{2}(k).

37.4 The action of s̃_{2}(k)

Let us continue with the same notation and hypotheses as in the previous section. Put

(37.4.1)  H̃ = T_{1} δ_{1} − T_{2} δ_{2} = T_{1}∂_{T_{1}} + ν_{0} R ∂_{R} − T_{2} ∂_{T_{2}},
(37.4.2)  X̃ = T_{1} δ_{2} = T_{1} ∂_{T_{2}},
(37.4.3)  Ỹ = T_{2} δ_{1} = T_{2} ∂_{T_{1}} + ν_{0} T_{1}^{−1} T_{2} R ∂_{R},

which are formal differential operators in ∂_{T_{1}}, ∂_{T_{2}}, and ∂_{R} with coefficients in LP_{k,1}(T_{1}, T_{2}, R), in the notation of Section 35.8. Note that H̃, X̃, Ỹ ∈ s̃_{2}(k), (37.4.4) and that s̃_{2}(k) is freely generated by H̃, X̃, and Ỹ, as a module over k.

Remember that s̃_{2}(k) is a Lie subalgebra of g̃_{2}(k), as a Lie algebra over k, as in Section 37.1. One can check that

(37.4.5)  [X̃, Ỹ] = H̃,
(37.4.6)  [H̃, X̃] = 2 · X̃,
(37.4.7)  [H̃, Ỹ] = −2 · Ỹ.

This shows that s̃_{2}(k) is isomorphic to sl_{2}(k) as a Lie algebra over k, where H̃, X̃, and Ỹ correspond to the usual elements h, x, and y of sl_{2}(k) as in Section 34.1, respectively. This is basically the same as in Section 36.1 when ν_{0} = 0.

As in the previous section, we may consider LS_{A}(T_{1}, T_{2}, R) as a module over s̃_{2}(k), as a Lie algebra over k. Thus LS_{A}(T_{1}, T_{2}, R) may be considered as a module over sl_{2}(k), as a Lie algebra over k, using the isomorphism mentioned in the preceding paragraph. This means that the actions of h, x, and y on LS_{A}(T_{1}, T_{2}, R) are given by H̃, X̃, and Ỹ, respectively. Note that LS_{A}(T_{1}, T_{2}) is a submodule of LS_{A}(T_{1}, T_{2}, R), as a module over s̃_{2}(k) or sl_{2}(k). More precisely, the actions of H̃, X̃, and Ỹ on LS_{A}(T_{1}, T_{2}), as a submodule of LS_{A}(T_{1}, T_{2}, R), as a module over k, are the same as the actions of H, X, and Y from Section 36.1, respectively.

Remember that LP_{A}(T_{1}, T_{2}, R) is a submodule of LS_{A}(T_{1}, T_{2}, R), as a module over s̃_{2}(k), or equivalently sl_{2}(k). Let γ and d be integers, as in the previous section, so that LS_{A,d}(T_{1}, T_{2}) R^{γ} is a submodule of LS_{A}(T_{1}, T_{2}, R), as a module over s̃_{2}(k) or sl_{2}(k). Similarly, LP_{A,d}(T_{1}, T_{2}) R^{γ} is a submodule of LP_{A}(T_{1}, T_{2}, R), as a module over s̃_{2}(k) or sl_{2}(k).

Let f(T_{1}, T_{2}) = ∑_{j=−∞}^{∞} f_{j} T_{1}^{j} T_{2}^{d−j} ∈ LS_{A}(T_{1}, T_{2}) be given, so that

f(T_{1}, T_{2}) R^{γ} = ∑_{j=−∞}^{∞} f_{j} T_{1}^{j} T_{2}^{d−j} R^{γ} (37.4.8)

is an element of LS_{A,d}(T_{1}, T_{2}) R^{γ}. Thus f_{j} ∈ A for every j, and f(T_{1}, T_{2}) is an element of LP_{A}(T_{1}, T_{2}) exactly when f_{j} = 0 for all but finitely many j. It is easy to see that

H̃(f(T_{1}, T_{2}) R^{γ}) = ∑_{j=−∞}^{∞}(j + ν_{0} γ − (d − j)) · f_{j} T_{1}^{j} T_{2}^{d−j} R^{γ}

(37.4.9)  = ∑_{j=−∞}^{∞}(2 j − d + ν_{0} γ) · f_{j} T_{1}^{j} T_{2}^{d−j} R^{γ}.

Here we are implicitly using the natural ring homomorphism from Z into k, which sends an integer to the corresponding multiple of the multiplicative identity element in k. Similarly,

X̃(f(T_{1}, T_{2}) R^{γ}) = ∑_{j=−∞}^{∞}(d − j) · f_{j} T_{1}^{j+1} T_{2}^{d−j−1} R^{γ}

(37.4.10)  = ∑_{j=−∞}^{∞}(d − j + 1) · f_{j−1} T_{1}^{j} T_{2}^{d−j} R^{γ}

and

Ỹ(f(T_{1}, T_{2}) R^{γ}) = ∑_{j=−∞}^{∞}(j + ν_{0} γ) · f_{j} T_{1}^{j−1} T_{2}^{d−j+1} R^{γ}

37.5. SOME SUBMODULES OF LS_{A,D}(T_{1}, T_{2}) R^{γ} 767

(37.4.11)  = ∑_{j=−∞}^{∞}(j + 1 + ν_{0} γ) · f_{j+1} T_{1}^{j} T_{2}^{d−j} R^{γ}.

Note that these expressions correspond to those in Section 36.1 when γ = 0, and when ν_{0} = 0.

37.5 Some submodules of LS_{A,d}(T_{1}, T_{2}) R^{γ}

We continue with the same notation and hypotheses as in the previous two sections. In particular, if f(T_{1}, T_{2}) ∈ LS_{A,d}(T_{1}, T_{2}) and j ∈ Z, then

(37.5.1) f_{j} denotes the coefficient in A of T_{1}^{j} T_{2}^{d−j} in f(T_{1}, T_{2}),

as before.

If l ∈ Z, then LS_{A,d}^{l,+}(T_{1}, T_{2}) is the space of f(T_{1}, T_{2}) ∈ LS_{A,d}(T_{1}, T_{2}) such that f_{j} = 0 for every j > l, as in Section 36.2. Similarly, LS_{A,d}^{l,−}(T_{1}, T_{2}) is the space of f(T_{1}, T_{2}) ∈ LS_{A,d}(T_{1}, T_{2}) such that f_{j} = 0 for every j > l, as before. Of course, these ars submodules of LS_{A,d}(T_{1}, T_{2}), as a module over k. Remember that LP_{A,d}^{l,+}(T_{1}, T_{2}) and LP_{A,d}^{l,−}(T_{1}, T_{2}) are defined to be the intersections of LS_{A,d}^{l,+}(T_{1}, T_{2}) and LS_{A,d}^{l,−}(T_{1}, T_{2}) with LP_{A,d}(T_{1}, T_{2}), respectively. These are submodules of LP_{A,d}(T_{1}, T_{2}), as a module over k.

Thus LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}, LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ} (37.5.2)

are submodules of LS_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over k, using the same type of notation for the former as before. More precisely, LS_{A,d}(T_{1}, T_{2}) R^{γ} corresponds to the direct sum of these two submodules, as a module over k. Similarly,

LP_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}, LP_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ} (37.5.3)

are submodules of LP_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over k, and LP_{A,d}(T_{1}, T_{2}) R^{γ} corresponds to the direct sum of these two submodules.

If r ∈ Z and l ≤ r, then

(37.5.4)  LS_{A,d}^{r,+}(T_{1}, T_{2}) R^{γ} ⊆ LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ},
(37.5.5)  LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ} ⊆ LS_{A,d}^{r,−}(T_{1}, T_{2}) R^{γ},

as in Section 36.2. Similarly,

(37.5.6)  LP_{A,d}^{r,+}(T_{1}, T_{2}) R^{γ} ⊆ LP_{A,d}^{l,+}(T_{1}, T_{2}) T^{γ},
(37.5.7)  LP_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ} ⊆ LP_{A,d}^{r,−}(T_{1}, T_{2}) R^{γ}.

It is easy to see that

(37.5.8)  H̃(LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ},
(37.5.9)  H̃(LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}

for every l ∈ Z, using (37.4.9). Similarly,

(37.5.10)  H̃(LP_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ},
(37.5.11)  H̃(LP_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}

for every l ∈ Z.

One can verify that

(37.5.12)  X̃(LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ},
(37.5.13)  X̃(LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{l+1,−}(T_{1}, T_{2}) R^{γ}

for every l ∈ Z, using (37.4.10). Similarly,

(37.5.14)  X̃(LP_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ},
(37.5.15)  X̃(LP_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d}^{l+1,−}(T_{1}, T_{2}) R^{γ}

for each l ∈ Z.

One can also verify that

(37.5.16)  Ỹ(LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{l−1,+}(T_{1}, T_{2}) R^{γ},
(37.5.17)  Ỹ(LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ}

for every l ∈ Z, using (37.4.11). Similarly,

(37.5.18)  Ỹ(LP_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d}^{l−1,+}(T_{1}, T_{2}) R^{γ},
(37.5.19)  Ỹ(LP_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ}

for each l ∈ Z. Of course, these inclusions are analogous to those in Section 36.2.

37.6 Submodules over s̃_{2}(k)

Let us continue with the same notation and hypotheses as in the previous three sections. If l = d, then we can improve (37.5.13) and (37.5.15), as in Section 36.3. Indeed, one can check that

(37.6.1)  X̃(LS_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ},
(37.6.2)  X̃(LP_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ},

using (37.4.10). It follows that

(37.6.3) LS_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ} is a submodule of LS_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over s̃_{2}(k), 37.6. SUBMODULES OVER S̃_{2}(K) 769

by (37.5.9), (37.6.1), and (37.5.17). Similarly,

(37.6.4) LP_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ} is a submodule of LP_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over s̃_{2}(k),

by (37.5.11), (37.6.2), and (37.5.19).

Suppose that there is an integer m such that

m · 1 + γ · ν_{0} = 0 (37.6.5)

in k. In particular, this holds with m = 0 when γ = 0 or ν_{0} = 0. Under these conditions, we can improve (37.5.16) and (37.5.18) when l = m, as in Section 36.3. More precisely, one can check that

(37.6.6)  Ỹ(LS_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ},
(37.6.7)  Ỹ(LP_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ}) ⊆ LP_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ},

using (37.4.11).

In this case, we get that

(37.6.8) LS_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ} is a submodule of LS_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over s̃_{2}(k),

by (37.5.8), (37.5.12), and (37.6.6). Similarly,

(37.6.9) LP_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ} is a submodule of LP_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over s̃_{2}(k),

by (37.5.10), (37.5.14), and (37.6.7).

Consider the mapping from LS_{A,d}(T_{1}, T_{2}) R^{γ} into LS_{A,d−m}(T_{1}, T_{2}) with

f(T_{1}, T_{2}) R^{γ} ↦ f(T_{1}, T_{2}) T_{1}^{−m} (37.6.10)

for every f(T_{1}, T_{2}) ∈ LS_{A}(T_{1}, T_{2}). Equivalently, this mapping is defined by multiplication by T_{1}^{−m} R^{−γ}. This is an isomorphism from LS_{A,d}(T_{1}, T_{2}) R^{γ} onto LS_{A,d−m}(T_{1}, T_{2}), as modules over k. This mapping also sends LP_{A,d}(T_{1}, T_{2}) R^{γ} onto LP_{A,d−m}(T_{1}, T_{2}).

Let f(T_{1}, T_{2}) = ∑_{j=−∞}^{∞} f_{j} T_{1}^{j} T_{2}^{d−j} ∈ LS_{A,d}(T_{1}, T_{2}) be given, so that

f(T_{1}, T_{2}) T_{1}^{−m} = ∑_{j=−∞}^{∞} f_{j} T_{1}^{j−m} T_{2}^{d−j} = ∑_{j=−∞}^{∞} f_{j−m} T_{1}^{j} T_{2}^{d−m−j}. (37.6.11)

Remember that H, X, and Y are as defined in Section 36.1. Observe that

H(f(T_{1}, T_{2}) T_{1}^{−m}) = ∑_{j=−∞}^{∞}((j − m) − (d − j)) · f_{j} T_{1}^{j−m} T_{2}^{d−j}

(37.6.12)  = ∑_{j=−∞}^{∞}(2 j − m − d) · f_{j} T_{1}^{j−m} T_{2}^{d−j}.

Similarly,

(37.6.13) X(f(T_{1}, T_{2}) T_{1}^{−m}) = ∑_{j=−∞}^{∞}(d − j) · f_{j} T_{1}^{j−m+1} T_{2}^{d−j−1}

= ∑_{j=−∞}^{∞}(d − j + 1) · f_{j−1} T_{1}^{j−m} T_{2}^{d−j}

and

(37.6.14) Y (f(T_{1}, T_{2}) T_{1}^{−m}) = ∑_{j=−∞}^{∞}(j − m) · f_{j} T_{1}^{j−m−1} T_{2}^{d−j+1}

= ∑_{j=−∞}^{∞}(j + 1 − m) · f_{j+1} T_{1}^{j−m} T_{2}^{d−j}.

Let us compare these expressions with (37.4.9), (37.4.10), and (37.4.11), using (37.6.5). It is easy to see that

(37.6.15)  H̃(f(T_{1}, T_{2}) R^{γ}) T_{1}^{−m} R^{−γ} = H(f(T_{1}, T_{2}) T_{1}^{−m}),
(37.6.16)  X̃(f(T_{1}, T_{2}) R^{γ}) T_{1}^{−m} R^{−γ} = X(f(T_{1}, T_{2}) T_{1}^{−m}),
(37.6.17)  Ỹ(f(T_{1}, T_{2}) R^{γ}) T_{1}^{−m} R^{−γ} = Y (f(T_{1}, T_{2}) T_{1}^{−m}).

This shows that (37.6.10) defines an isomorphism from LS_{A,d}(T_{1}, T_{2}) R^{γ} onto LS_{A,d−m}(T_{1}, T_{2}), as modules over sl_{2}(k). More precisely, this uses the isomorphisms between sl_{2}(k) and s_{2}(k), s̃_{2}(k), as Lie algebras over k, mentioned in Sections 36.1, 37.4, respectively. Similarly, (37.6.10) defines an isomorphism from LP_{A,d}(T_{1}, T_{2}) R^{γ} onto LP_{A,d−m}(T_{1}, T_{2}), as modules over sl_{2}(k).

37.7 Characteristic 0 and LP_{k,d}(T_{1}, T_{2}) R^{γ}

Let us continue with the same notation and hypotheses as in the previous four sections, and suppose now that k is a field of characteristic 0, and that A = k, as a module over itself. Observe that

H̃(T_{1}^{j} T_{2}^{d−j} R^{γ}) = (2 j − d + γ ν_{0}) T_{1}^{j} T_{2}^{d−j} R^{γ} (37.7.1)

for each j ∈ Z, by the definition (37.4.1) of H̃. This can be obtained from (37.4.9) as well. Thus T_{1}^{j} T_{2}^{d−j} R^{γ} has weight

2 j − d + γ ν_{0} (37.7.2)

in LP_{k,d}(T_{1}, T_{2}) R^{γ}, as a module over sl_{2}(k), as in Section 15.1. Of course, this uses the isomorphism between sl_{2}(k) and s̃_{2}(k), as Lie algebras over k, mentioned in Section 37.4.

37.8. HELPFUL PROJECTIONS ON LS_{A,D}(T_{1}, T_{2}) R^{γ} 771

Using the definition (37.4.2) of X̃, we get that

X̃(T_{1}^{d} R^{γ}) = 0. (37.7.3)

This means that T_{1}^{d} R^{γ} is a maximal or primitive vector of weight

d + γ ν_{0} (37.7.4)

in LP_{k,d}(T_{1}, T_{2}) R^{γ}, as a module over sl_{2}(k), as in Section 15.2. Remember that LP_{k,d}^{d,−}(T_{1}, T_{2}) R^{γ} is a submodule of LP_{k,d}(T_{1}, T_{2}) R^{γ}, as a module over s̃_{2}(k), or equivalently sl_{2}(k), as in the previous section. Clearly

T_{1}^{d} R^{γ} ∈ LP_{k,d}^{d,−}(T_{1}, T_{2}) R^{γ}. (37.7.5)

Thus T_{1}^{d} R^{γ} may be considered as a maximal or primitive vector of weight (37.7.4) in LP_{k,d}^{d,−}(T_{1}, T_{2}) R^{γ}, as a module over sl_{2}(k).

Note that

Ỹ(T_{1}^{j} T_{2}^{d−j} R^{γ}) = (j + γ ν_{0}) T_{1}^{j−1} T_{2}^{d−j+1} R^{γ} (37.7.6)

for every integer j, by the definition (37.4.3) of Ỹ. If l is a nonnegative integer, then Ỹ^{l}(T_{1}^{d} R^{γ}) is a multiple of T_{1}^{d−l} T_{2}^{l} R^{γ} by an element of k. Suppose that

(37.7.7) d + γ ν_{0} does not correspond to a nonnegative integer,

under the natural embedding of Q into k. Under these conditions, we get that Ỹ^{l}(T_{1}^{d} R^{γ}) is a nonzero multiple of T_{1}^{d−l} T_{2}^{l} R^{γ} for each l ≥ 0. This implies that Ỹ^{l}(T_{1}^{d} R^{γ}), l ≥ 0, is a basis for LP_{k,d}^{d,−}(T_{1}, T_{2}) R^{γ}, as a vector space over k.

Let Z_{0}(μ_{0}) be as in Section 34.2, and let us take

μ_{0} = d · 1 + γ · ν_{0} (37.7.8)

in k. If (37.7.7) holds, then we get that LP_{k,d}^{d,−}(T_{1}, T_{2}) R^{γ} is isomorphic to Z_{0}(μ_{0}), as a module over sl_{2}(k), as in Section 34.3.

Of course, d + γ ν_{0} corresponds to an integer under the natural embedding of Q into k if and only if γ ν_{0} corresponds to an integer. In this case, if m is an integer that satisfies (37.6.5), then LP_{k,d}(T_{1}, T_{2}) R^{γ} is isomorphic to LP_{k,d−m}(T_{1}, T_{2}) as modules over sl_{2}(k), as in the previous section. This permits us to use the remarks in Section 36.4, with d replaced with d − m.

37.8 Helpful projections on LS_{A,d}(T_{1}, T_{2}) R^{γ}

Let us return to the notation and hypotheses in Section 37.6, so that k is a commutative ring with a multiplicative identity element again, and A is a 772 CHAPTER 37. FORMAL LAURENT SERIES, 3

module over k. Let f(T_{1}, T_{2}) = ∑_{j=−∞}^{∞} f_{j} T_{1}^{j} T_{2}^{d−j} ∈ LS_{A,d}(T_{1}, T_{2}) and l ∈ Z be given, and put

(37.8.1)  P_{l,+}^{γ}(f(T_{1}, T_{2}) R^{γ}) = ∑_{j=l}^{∞} f_{j} T_{1}^{j} T_{2}^{d−j} R^{γ},
(37.8.2)  P_{l,−}^{γ}(f(T_{1}, T_{2}) R^{γ}) = ∑_{j=−∞}^{l} f_{j} T_{1}^{j} T_{2}^{d−j} R^{γ}.

Equivalently,

(37.8.3)  P_{l,+}^{γ}(f(T_{1}, T_{2}) R^{γ}) = P_{l,+}(f(T_{1}, T_{2})) R^{γ},
(37.8.4)  P_{l,−}^{γ}(f(T_{1}, T_{2}) R^{γ}) = P_{l,−}(f(T_{1}, T_{2})) R^{γ},

in the notation of Section 36.5. Note that these define homomorphisms from LS_{A,d}(T_{1}, T_{2}) R^{γ} into itself, as a module over k.

Clearly

(37.8.5)  P_{l,+}^{γ}(LS_{A,d}(T_{1}, T_{2}) R^{γ}) = LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ},
(37.8.6)  P_{l,−}^{γ}(LS_{A,d}(T_{1}, T_{2}) R^{γ}) = LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}.

In fact, P_{l,+}^{γ} is the same as the identity mapping on LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}, and P_{l,−}^{γ} is the identity mapping on LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}. Similarly,

(37.8.7)  P_{l,+}^{γ}(LP_{A,d}(T_{1}, T_{2}) R^{γ}) = LP_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ},
(37.8.8)  P_{l,−}^{γ}(LP_{A,d}(T_{1}, T_{2}) R^{γ}) = LP_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}.

By construction, (37.8.9) the kernel of P_{l,+}^{γ} on LS_{A,d}(T_{1}, T_{2}) R^{γ}

is equal to LS_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ},

and (37.8.10) the kernel of P_{l,−}^{γ} on LS_{A,d}(T_{1}, T_{2}) R^{γ}

is equal to LS_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ}.

Similarly, (37.8.11) the kernel of P_{l,+}^{γ} on LP_{A,d}(T_{1}, T_{2}) R^{γ}

is equal to LP_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ},

and (37.8.12) the kernel of P_{l,−}^{γ} on LP_{A,d}(T_{1}, T_{2}) R^{γ}

is equal to LP_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ}.

37.9. PROJECTIONS AND H̃, X̃, Ỹ 773

The quotients (LS_{A,d}(T_{1}, T_{2}) R^{γ})/(LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}) (37.8.13) and (LS_{A,d}(T_{1}, T_{2}) R^{γ})/(LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}) (37.8.14) can be defined as modules over k in the usual way. Similarly, the quotients (LP_{A,d}(T_{1}, T_{2}) R^{γ})/(LP_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}) (37.8.15) and (LP_{A,d}(T_{1}, T_{2}) R^{γ})/(LP_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}) (37.8.16) are defined as modules over k too. The natural inclusion mapping (37.8.17) from LP_{A,d}(T_{1}, T_{2}) R^{γ} into LS_{A,d}(T_{1}, T_{2}) R^{γ} leads to natural injective homomorphisms from (37.8.15) into (37.8.13), and from (37.8.16) into (37.8.14), as modules over k.

We can use P_{l−1,−}^{γ} to identify (37.8.13) with LS_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ}, as a module over k, as in Section 36.5. Of course, (37.8.15) corresponds to LP_{A,d}^{l−1,−}(T_{1}, T_{2}) with respect to this identification. Similarly, we can use P_{l+1,+}^{γ} to identify (37.8.14) with LS_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ}, as a module over k, in which case (37.8.16) corresponds to LP_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ}.

37.9 Projections and H̃, X̃, Ỹ

We continue with the same notation and hypotheses as in the previous section. Observe that

(37.9.1)  H̃ ∘ P_{l,+}^{γ} = P_{l,+}^{γ} ∘ H̃,
(37.9.2)  H̃ ∘ P_{l,−}^{γ} = P_{l,−}^{γ} ∘ H̃

on LS_{A,d}(T_{1}, T_{2}) R^{γ}, by (37.4.9). Remember that H̃ maps each of LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}, LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}, LP_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}, (37.9.3)

and LP_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}

into itself, as in Section 37.5. This implies that H̃ induces a mapping on each of the quotients (37.8.13), (37.8.14), (37.8.15), and (37.8.16) that is linear over k. These induced mappings correspond to H̃ on LS_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ}, LS_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ}, LP_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ}, and LP_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ}, respectively, with respect to the identifications mentioned in the prevous section, by (37.9.1) and (37.9.2).

774 CHAPTER 37. FORMAL LAURENT SERIES, 3

As in Section 37.5, X̃(LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ} ⊆ LS_{A,d}^{l,+}(T_{1}, T_{2}) R^{γ}, (37.9.4) and similarly for the corresponding spaces of formal Laurent polynomials. Thus X̃ induces mappings on the quotients (37.8.13) and (37.8.15) that are linear over k. Note that P_{l−1,−}^{γ} ∘ X̃ = 0 on LS_{A,d}^{l−1,+}(T_{1}, T_{2}) R^{γ}, (37.9.5) because of the first inclusion in (37.9.4), with l replaced with l − 1. It is easy to see that P_{l−1,−}^{γ} ∘ X̃ = X̃ on LS_{A,d}^{l−2,−}(T_{1}, T_{2}) R^{γ}, (37.9.6) because X̃ maps LS_{A,d}^{l−2,−}(T_{1}, T_{2}) R^{γ} into LS_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ}, as in (37.5.13). Of course, P_{l−1,−}^{γ} ∘ X̃ is determined on LS_{A,d}(T_{1}, T_{2}) R^{γ} by (37.9.5) and (37.9.6).

We can use P_{l−1,−}^{γ} to identify the quotients (37.8.13) and (37.8.15) with LS_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ} and LP_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ}, respectively, as modules over k, as in the previous section. It is easy to see that the mappings induced by X̃ on these quotients correspond to the restrictions of P_{l−1,−}^{γ}∘X̃ to LS_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ} and LP_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ}, respectively, with respect to these identifications.

Similarly, Ỹ(LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}) ⊆ LS_{A,d}^{l−1,−}(T_{1}, T_{2}) R^{γ} ⊆ LS_{A,d}^{l,−}(T_{1}, T_{2}) R^{γ}, (37.9.7) as in Section 37.5, and there are analogous inclusions for the corresponding spaces of formal Laurent polynomials. This implies that Ỹ induces mappings on the quotients (37.8.14) and (37.8.16) that are linear over k. Observe that P_{l+1,+}^{γ} ∘ Ỹ = 0 on LS_{A,d}^{l+1,−}(T_{1}, T_{2}) R^{γ}, (37.9.8) because of the first inclusion in (37.9.7), with l replaced with l + 1. We also have that P_{l+1,+}^{γ} ∘ Ỹ = Ỹ on LS_{A,d}^{l+2,+}(T_{1}, T_{2}) R^{γ}, (37.9.9) because Ỹ maps LS_{A,d}^{l+2,+}(T_{1}, T_{2}) R^{γ} into LS_{A,d}^{l+1,+}(T_{1}, T_{2}), as in (37.5.16). As before, P_{l+1,+}^{γ} ∘ Ỹ is determined on LS_{A,d}(T_{1}, T_{2}) R^{γ} by (37.9.8) and (37.9.9).

We can use P_{l+1,+}^{γ} to identify the quotients (37.8.14) and (37.8.16) with LS_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ} and LP_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ}, respectively, as modules over k, as in the previous section again. The mappings induced by Ỹ on these quotients correspond to the restrictions of P_{l+1,+}^{γ} ∘ Ỹ to LS_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ} and LP_{A,d}^{l+1,+}(T_{1}, T_{2}) R^{γ}, respectively, with respect to these identifications.

One can verify that X̃ ∘ P_{d+1,+}^{γ} = P_{d+1,+}^{γ} ∘ X̃ (37.9.10)

37.10. RELATED QUOTIENTS OVER S̃_{2}(K) 775

on LS_{A,d}(T_{1}, T_{2}) R^{γ}, using (37.4.10). This can also be obtained using the fact that both sides of (37.9.10) are equal to 0 on LS_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ}, by definition of P_{d+1,+}^{γ}, and because X̃ maps LS_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ} into itself, as in Section 37.6. Similarly, both sides of (37.9.10) are equal to X̃ on LS_{A,d}^{d+1,+}(T_{1}, T_{2}) R^{γ}, by definition of P_{d+1,+}^{γ}, and because X̃ maps LS_{A,d}^{d+1,+}(T_{1}, T_{2}) R^{γ} into itself, as in Section 37.5.

Suppose for the moment that there is an integer m such that m·1+γ ·ν_{0} = 0 in k, as in Section 37.6. In this case, one can check that

Ỹ ∘ P_{m−1,−}^{γ} = P_{m−1,−}^{γ} ∘ Ỹ (37.9.11)

on LS_{A,d}(T_{1}, T_{2}) R^{γ}, using (37.4.11). Alternatively, both sides of (37.9.11) are equal to 0 on LS_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ}, by definition of P_{m−1,−}^{γ}, and because Ỹ maps LS_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ} into itself, as in Section 37.6. Both sides of (37.9.11) are equal to Ỹ on LS_{A,d}^{m−1,−}(T_{1}, T_{2}) R^{γ}, by definition of P_{m−1,−}^{γ}, and because Ỹ maps LS_{A,d}^{m−1,−}(T_{1}, T_{2}) R^{γ} into itself, as in Section 37.5.

37.10 Related quotients over s̃_{2}(k)

Let us continue with the same notation and hypotheses as in the previous two sections. Remember that LS_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ} is a submodule of LS_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over s̃_{2}(k), as in Section 37.6. This means that the quotient

(LS_{A,d}(T_{1}, T_{2}) R^{γ})/(LS_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ}) (37.10.1)

may be considered as a module over s̃_{2}(k) as well. As a module over k, this quotient can be identified with LS_{A,d}^{d+1,+}(T_{1}, T_{2}) R^{γ} using P_{d+1,+}^{γ}, as in Section 37.8.

The action of H̃ induced on (37.10.1) by the action of H̃ on LS_{A,d}(T_{1}, T_{2}) R^{γ} corresponds to the usual action of H̃ on LS_{A,d}^{d+1,+}(T_{1}, T_{2}) R^{γ}, as in (37.9.1). Similarly, the action of X̃ induced on (37.10.1) corresponds to the usual action of X̃ on LS_{A,d}^{d+1,+}(T_{1}, T_{2}) R^{γ}, as in (37.9.10). The action of Ỹ induced on (37.10.1) by the action of Ỹ on LS_{A,d}(T_{1}, T_{2}) R^{γ} corresponds to

(37.10.2) the restriction of P_{d+1,+}^{γ} ∘ Ỹ to LS_{A,d}^{d+1,+}(T_{1}, T_{2}) R^{γ},

as in the previous section. This mapping is described by (37.9.8) and (37.9.9), with l = d.

Remember that LP_{A,d}(T_{1}, T_{2}) R^{γ} is a submodule of LS_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over s̃_{2}(k). We also have that LP_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ} is a submodule of LP_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over s̃_{2}(k), as in Section 37.6. It follows that the quotient (LP_{A,d}(T_{1}, T_{2}) R^{γ})/(LP_{A,d}^{d,−}(T_{1}, T_{2}) R^{γ}) (37.10.3) may be considered as a module over s̃_{2}(k) as well. This may be considered as a submodule of (37.10.1), as a module over s̃_{2}(k). This can also be identified with LP_{A,d}^{d+1,+}(T_{1}, T_{2}) R^{γ}, as a module over k, using P_{d+1,+}^{γ}, as before.

If there is an integer m such that m·1+γ·ν_{0} = 0 in k, then LS_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ} is a submodule of LS_{A,d}(T_{1}, T_{2}) R^{γ}, as a module over s̃_{2}(k), as in Section 37.6. In this case, the quotient

(LS_{A,d}(T_{1}, T_{2}) R^{γ})/(LS_{A,d}^{m,+}(T_{1}, T_{2}) R^{γ}) (37.10.4)

may be considered as a module over s̃_{2}(k), which can be described more precisely as before. One can also use the isomorphism mentioned in Section 37.6 to get that this quotient is isomorphic to the analogous one in Section 36.7, with d replaced with d − m.

Representations and semisimplicity, 2

38.1 Weights and finite dimension

As before, we use the same notation and hypotheses as in Section 33.1. In particular, (A, [·, ·]_{A}) is a semisimple Lie algebra of positive finite dimension over a field k of characteristic 0, and A_{0} is a Lie subalgebra of A consisting of ad-diagonalizable elements of A. We also let Δ be a base for the root system Φ, and B_{Δ} be the standard Borel subalgebra of A associated to A_{0} and Δ, as before.

Let V be a vector space over k that is a module over A, as a Lie algebra over k. If μ ∈ A′_{0}, then V_{μ} is the linear subspace of V consisting of v ∈ V such that w · v = μ(w) v for every w ∈ A_{0}, as in Section 33.5.

In this section, we suppose also that k is algebraically closed, and that V has finite dimension, as a vector space over k. Under these conditions, it is well known that

(38.1.1) V corresponds to the direct sum of V_{μ} for finitely many μ ∈ A′_{0}, as a vector space over k.

This is mentioned on p107 of [14], and corresponds to part (a) of Proposition 3 on p60 of [24], and to Proposition 2.2 on p57 of [25].

To see this, note that for each w ∈ A_{0},

v ↦ w · v (38.1.2)

is diagonalizable on V, as in Section 14.9. These linear mappings on V commute with each other, because A_{0} is commutative as a Lie subalgebra of A, and V is a module over A. This implies (38.1.1), which is the same as saying that the linear mappings (38.1.2) with w ∈ A_{0} can be simultaneously diagonalized on V. This is the argument indicated in [14, 24].

Alternatively, let V̂ be the linear subspace of V spanned by the V_{μ}’s, μ ∈ A′_{0}, as in Section 33.5. Remember that V̂ corresponds to the direct sum of the V_{μ}’s, and that V̂ is a submodule of V, as a module over A, as before. If V̂ ≠ V, then Weyl’s theorem implies that V corresponds to the direct sum of V̂ and a nonzero submodule Y, as a module over A.

If w ∈ A_{0}, then (38.1.2) maps Y into itself, because Y is a submodule of V, as a module over A. This implies that (38.1.2) has a nonzero eigenvector in Y, because k is algebraically closed, and Y is a vector space over k of positive finite dimension. It follows that there is a nonzero element y of Y that is a simultaneous eigenvector of (38.1.2) for every w ∈ A_{0}, because these linear mappings commute on V and thus Y, as before. This means that y ∈ V_{μ} for some μ ∈ A′_{0}, so that y ∈ V̂, which is a contradiction. This is the argument used in the proof of Proposition 2.2 on p57 of [25].

If V ≠ {0}, then V has a primitive or maximal vector, in the sense of Section 33.6. This is mentioned on p108 of [14], and corresponds to part (c) of Proposition 3 on p60 of [24], and to Proposition 2.7 on p58 of [25]. To see this, remember that B_{Δ} is solvable as a Lie algebra over k, so that there is a nonzero element v of V that is an eigenvector of the action of every element of B_{Δ} on V, by Lie’s theorem. This implies that v is a primitive or maximal vector, as before.

Another argument is mentioned on p58 of [25], as follows. Remember that μ ∈ A′_{0} is said to be a weight of V with respect to A_{0} when V_{μ} ≠ {0}. In this case, the set of weights is finite, because V has finite dimension, and the set of weights is nonempty, because V ≠ {0}. One can use this to find a weight μ of V such that for each α ∈ Δ, μ + α is not a weight of V. This implies that any nonzero element of V_{μ} is primitive or maximal, because of (33.5.3), as desired.

Remember that for each α ∈ Φ, there is a unique element h_{α} of [A_{α}, A_{−α}] such that α(h_{α}) = 2 = 1 + 1 in k, as in Section 33.1. Note that h_{−α} = −h_{α} for every α ∈ Φ. If α ∈ Φ and ν ∈ A′_{0} is a weight of V, then

ν(h_{α}) ∈ Z, (38.1.3)

using the natural embedding of Q into k. This is part (b) of Proposition 3 on p60 of [24], which corresponds to a remark after the statement of the theorem on p112 of [14]. It suffices to verify this when α is an element of the set Φ^{+} of positive roots with respect to Δ.

Let α ∈ Φ^{+} be given, let x_{α} be a nonzero element of A_{α}, and let y_{α} be an element of A_{−α} such that [x_{α}, y_{α}]_{A} = h_{α}. The linear span of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k). We may consider V as a module over this subalgebra of A, as a Lie algebra over k. The eigenvalues of the action of h_{α} on V correspond to integers, as in Section 15.9. These eigenvalues are the same as the values of the weights of V at h_{α}, as desired.

The condition (38.1.3) is also mentioned in Exercise 2 on p62 of [25], and could be obtained from Theorem 3.1 on p58 and Theorem 4.1 on p59 of [25].

38.2 Weights and a larger field

Let us continue with the same notation and hypotheses as in the previous section, except that we no longer ask k to be algebraically closed. Let k_{1} be an algebraically closed field with k ⊆ k_{1}.

Remember that we can get a vector space Ṽ over k_{1} from V as in Section 33.2. In particular, the dimension of Ṽ as a vector space over k_{1} is the same as the dimension of V as a vector space over k, which is finite, by hypothesis. As before, we may consider V as a linear subspace of Ṽ, as a vector space over k.

If T is a linear mapping from V into itself, then T may be considered as a linear mapping from V into Ṽ, as a vector space over k. This has a unique extension to a linear mapping T̃ from Ṽ into itself, as a vector space over k_{1}.

Suppose for the moment that T̃ is diagonalizable on Ṽ, with distinct eigenvalues λ_{1}, …, λ_{l} ∈ k_{1}. This implies that

∏_{j=1}^{l}(T̃ − λ_{j} I_{V}) = 0 (38.2.1)

as a linear mapping on Ṽ, where I_{V} is the identity mapping on Ṽ. If

λ_{1}, …, λ_{l} ∈ k, (38.2.2)

then it follows that

∏_{j=1}^{l}(T − λ_{j} I_{V}) = 0 (38.2.3)

on V, where I_{V} is the identity mapping on V. This means that T is diagonalizable on V, by standard results.

We also get a Lie algebra (Ã, [·, ·]_{A}) over k_{1} from A, as in Section 33.2. The action of A on V has a unique extension to an action of Ã on Ṽ, that makes Ṽ a module over Ã, as a Lie algebra over k_{1}, as before.

The conditions on A discussed in Section 33.1 lead to analogous properties of Ã, as in Section 33.4. In particular, Ã is semisimple as a Lie algebra over k_{1}, which could also be obtained from the semisimplicity of A as in Section 11.5.

Let ˜(A_{0}) be the linear subspace of Ã corresponding to A_{0}, which is a Lie subalgebra of Ã that is commutative as a Lie algebra over k_{1}, as in Section 33.4. Remember that the elements of ˜(A_{0}) are ad-diagonalizable in Ã.

Let ˜(A_{0})′ be the dual of ˜(A_{0}), as a vector space over k_{1}, as before. If

μ ∈ ˜(A_{0})′, then we take Ṽ_{μ} to be the set of v ∈ Ṽ such that w · v = μ(w) v for every w ∈ ˜(A_{0}), as in Section 33.5. Because k_{1} is algebraically closed, we have

that Ṽ corresponds to the direct sum of Ṽ_{μ} for finitely many μ ∈ ˜(A_{0})′, as in the previous section.

If α ∈ A′_{0}, then let α̃ be the unique extension of α to an element of ˜(A_{0})′, as in Section 33.4. Remember that the analogue Φ̃ of Φ for Ã and ˜(A_{0}) consists of α̃, α ∈ Φ.

If α ∈ Φ, then h_{α} is the unique element of [A_{α}, A_{−α}] with α(h_{α}) = 2, as in Section 33.1. We may also consider h_{α} as an element of Ã, and indeed it is the unique element of [Ã_{α}, Ã_{−_{α}}] = [ ˜(A_{α}), ˜ (A_{−α})] with α̃(h_{α}) = 2.

If α ∈ Φ and ν ∈ ˜(A_{0})′ is a weight of Ṽ, then ν(h_{α}) corresponds to an integer, as in the previous section. This means that the action of h_{α} on Ṽ is diagonalizable, and that the eigenvalues of this action correspond to integers, and thus elements of k.

The action of h_{α} on Ṽ is the same as the unique extension of the action of h_{α} on V to a linear mapping from Ṽ into itself, by construction. It follows that the action of h_{α} on V is diagonalizable too, as before.

Remember that A_{0} is spanned by the h_{α}’s, α ∈ Φ, as in Section 33.1. This means that the action of any element of A_{0} on V is diagonalizable, because the action of h_{α} is diagonalizable for every α ∈ Φ.

The actions of the elements of A_{0} on V commute with each other, because A_{0} is commutative as a Lie subalgebra of A, and V is a module over A, as a Lie algebra over k. This implies that the actions of the elements of A_{0} on V can be simultaneously diagonalized, as in the previous section.

This shows that (38.1.1) holds, even if k is not necessarily algebraically closed. This corresponds to part of Exercise 2 on p62 of [25].

If V ≠ {0}, then one can check that V has a primitive or maximal vector, using the statement in the preceding paragraph, and the second argument for the analogous assertion in the previous section. This corresponds to another part of Exercise 2 on p62 of [25].

If V ≠ {0} and V is irreducible as a module over A, then it follows that V is standard cyclic as a module over A, as in the remarks at the beginning of Section 33.10. This corresponds to the corollary to Proposition 3 on p60 of [24], and is mentioned in the proof of part (1) of Theorem 3.2 on p59 of [25]. This also corresponds to some remarks on p108f of [14].

38.3 Conditions for finite dimension

We continue to use the same notation and hypotheses as in Section 33.1. Let V be a vector space over k that is a module over A, as a Lie algebra over k. In this section, we suppose that V is irreducible as a module over A, and that V is standard cyclic of weight μ ∈ A′_{0}.

Suppose for the moment that V has finite dimension, as a vector space over k. We would like to show that

(38.3.1) for every α ∈ Φ^{+}, μ(h_{α}) is a nonnegative integer.

This corresponds to the theorem on p112 of [14], to part of Theorem 3 on p60 of [24], and part of Theorem 4.1 on p59 of [25].

Let α ∈ Φ^{+} be given. Remember that μ(h_{α}) corresponds to an integer, using the natural embedding of Q into k, as in (38.1.3). This also uses the remarks in the previous section, when k is not necessarily algebraically closed.

As before, let x_{α} be a nonzero element of A_{α}, and let y_{α} be an element of A_{−α} such that [x_{α}, y_{α}]_{A} = h_{α}, so that the linear span of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A isomorphic to sl_{2}(k). Thus V may be considered as a module over sl_{2}(k) as well.

By hypothesis, there is a v ∈ V that is primitive or maximal of weight μ in V, as a module over A. This implies that v is primitive or maximal of weight μ(h_{α}) in V as a module over the linear span of x_{α}, y_{α}, and h_{α}, as in Section 15.2. It follows that μ(h_{α}) ≥ 0, as in Section 15.3.

Conversely, it is well known that (38.3.1) implies that the dimension of V is finite under these conditions. This corresponds to part of the theorem on p113 of [14], to part of Theorem 3 on p60 of [24], and part of Theorem 4.1 on p59 of [25] when A = sl_{n}(k). Before we begin the proof of this, let us review some facts related to inverse roots.

Remember from Section 33.1 that Φ is a reduced root system in the finitedimensional vector space E_{R} over the real numbers. If α ∈ Φ, then λ_{α} is the linear functional on E_{R} associated to the symmetry on E_{R} with vector α that maps Φ onto itself in the usual way. We have seen that the collection Φ′ of λ_{α}, α ∈ Φ, is a root system in the dual E′_{R} of E_{R}, as in Section 19.8. Remember that Δ is a base for Φ, and let Δ′ be the set of λ_{α}, α ∈ Δ. We have also seen that Δ′ is a base for Φ′, because Φ is reduced, as in Section 19.13.

Using Δ′, we can define (Φ′)^{+} = (Φ′)^{+,Δ′} to be the set of dual roots in Φ′ that are positive with respect to Δ′, which is to say that they can be expressed as linear combinations of the elements of Δ′ with nonnegative coefficients. Equivalently, (Φ′)^{+} consists exactly of λ_{α}, α ∈ Φ^{+}, as in Section 30.3.

If α ∈ Φ, then

γ(h_{α}) = λ_{α}(γ) (38.3.2)

for every γ ∈ Φ, as in (33.1.2). More precisely, the right side is an integer, by the definition of a root system, which corresponds to an element of k under the natural embedding of Q into k. Remember that h_{α} is uniquely determined by (38.3.2), because A′_{0} is spanned by Φ, as in Section 33.1.

If α ∈ Φ, then λ_{α} can be expressed as a linear combination of λ_{β}, β ∈ Δ, with integer coefficients, because Δ′ is a base for Φ′. This implies that h_{α} can be expressed as a linear combination of h_{β}, β ∈ Δ, with the same integer coefficients, because h_{α} is uniquely determined by (38.3.2). Similarly, if α ∈ Φ^{+}, then λ_{α} can be expressed as a linear combination of λ_{β}, β ∈ Δ, with nonnegative integer coefficients, because λ_{α} ∈ Φ^{+}, as before. This means that h_{α} can be expressed as a linear combination of h_{β}, β ∈ Δ, with the same nonnegative integer coefficients.

It follows that (38.3.1) holds when μ(h_{β}) is a nonnegative integer for every β ∈ Δ. This corresponds to a remark just after Theorem 3 on p60 of [24]. Similarly, if ν ∈ A′_{0}, then ν(h_{α}) corresponds to an integer for every α ∈ Φ if and only if ν(h_{β}) corresponds to an integer for every β ∈ Δ.

38.4 Some initial steps

We continue to use the same notation and hypotheses as in Section 33.1. If α is in the base Δ for Φ, then let x_{α} be a nonzero element of A_{α}, and let y_{α} be an element of A_{−α} such that [x_{α}, y_{α}]_{A} = h_{α}, as before. If α, β ∈ Δ and α ≠ β, then [x_{α}, y_{β}]_{A} = 0, as in Section 23.5. Indeed, it is easy to see that α − β ∉ Φ in this case, so that A_{α−β} = {0}.

Let UA be a universal enveloping algebra of A again, and let us identify A with its image in UA under the associated Lie algebra homomorphism, as before. If α, β ∈ Δ and α ≠ β, then

[x_{α}, y_{β}^{l}] = 0 (38.4.1)

for every positive integer l. More precisely, y_{β}^{l} is defined as an element of UA, and the left side uses the commutator bracket associated to multiplication in UA. Of course, (38.4.1) holds when l = 1 as in the preceding paragraph, which implies the analogous statement for l ≥ 2. This corresponds to part (a) of the lemma on p113 of [14].

If α, β ∈ Δ, then [h_{α}, y_{β}^{l}] = −l β(h_{α}) y_{β}^{l} (38.4.2)

for every positive integer l. If l = 1, then this reduces to the fact that [h_{α}, y_{β}]_{A} = −β(h_{α}), because h_{α} ∈ A_{0} and y_{β} ∈ A_{−β}. Otherwise, one can use induction and the fact that the commutator with h_{α} defines a derivation on UA, as in Section 2.5. This corresponds to part (b) of the lemma on p113 of [14].

If α ∈ Δ, then [x_{α}, y_{α}^{l}] = −l y_{α}^{l−1} ((l − 1) · e − h_{α}) (38.4.3)

for every positive integer l, where e = e_{UA} is the multiplicative identity element in UA. If l = 1, then y_{α}^{l−1} is interpreted as being equal to e, and (38.4.3) follows from the way that y_{α} was chosen. Otherwise, if l ≥ 2, then

[x_{α}, y_{α}^{l}] = [x_{α}, y_{α}] y_{α}^{l−1} + y_{α} [x_{α}, y_{α}^{l−1}] = h_{α} y_{α}^{l−1} + y_{α} [x_{α}, y_{α}^{l−1}], (38.4.4)

because the commutator with x_{α} defines a derivation on UA, as before. Using induction, we may suppose that the analogue of (38.4.3) for l − 1 holds, to get that [x_{α}, y_{α}^{l}] = h_{α} y_{α}^{l−1} − (l − 1) y_{α}^{l−1} ((l − 2) · e − h_{α}). (38.4.5)

Observe that

h_{α} y_{α}^{l−1} = [h_{α}, y_{α}^{l−1}] + y_{α}^{l−1} h_{α} = −2 (l − 1) y_{α}^{l−1} + y_{α}^{l−1} h_{α}, (38.4.6)

by (38.4.2). One can combine this with (38.4.5) to get (38.4.3), as desired. This corresponds to part (c) of the lemma on p113 of [14].

Let V be a vector space over k that is a module over A, as a Lie algebra over k. Suppose that V is irreducible as a module over A, that V is standard cyclic of weight μ ∈ A′_{0}, and that

m_{α} = μ(h_{α}) (38.4.7) is a nonnegative integer for every α ∈ Δ. We would like to show that V has finite dimension, as in the previous section.

By hypothesis, there is a primitive or maximal vector v ∈ V of weight μ such that V = (UA) · v. Let α ∈ Δ be given, and put

v_{α} = y_{α}^{m_{α}+1} · v. (38.4.8)

If β ∈ Δ and α ≠ β, then

x_{β} · v_{α} = y_{α}^{m_{α}+1} · (x_{β} · v) = 0. (38.4.9)

This uses (38.4.1) in the first step, and the fact that v is primitive or maximal in the second step.

Observe that

(38.4.10)  x_{α} · v_{α} = ([x_{α}, y_{α}^{m_{α}+1}]) · v + y_{α}^{m_{α}+1} · (x_{α} · v)
= −(m_{α} + 1) y_{α}^{m_{α}} · ((m_{α} · e − h_{α}) · v),

using (38.4.3) and the fact that x_{α} · v = 0 in the second step. However,

(m_{α} · e − h_{α}) · v = m_{α} v − h_{α} · v = m_{α} v − μ(h_{α}) v = 0, (38.4.11)

because v has weight μ in V. Thus

x_{α} · v_{α} = 0. (38.4.12)

Note that v_{α} has weight μ − (m_{α} + 1) α with respect to A_{0}, as in Section 33.5. If v_{α} ≠ 0, then v_{α} would be a primitive or maximal vector in V, because of (38.4.9) and (38.4.12). This is not possible, because V is irreducible as a module over A and m_{α} ≠ −1, as in Section 33.10. This shows that

y_{α}^{m_{α}+1} · v = v_{α} = 0, (38.4.13)

as in step (1) on p113 of [14].

Alternatively, the linear span of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k). Thus V may be considered as a module over this subalgebra, and v is primitive or maximal of weight m_{α} in the sense of Section 15.2. One can use this to get (38.4.12) as in Section 15.3. This is the argument used on p61 of [24].

Using (38.4.13), we get that

(38.4.14) the linear subspace of V spanned by y_{α}^{j} · v, j = 0, …, m_{α}, is a submodule of V, as a module over the Lie subalgebra of A spanned by x_{α}, y_{α}, and h_{α}.

This follows from the remarks in Section 15.3, as on p61 of [24]. Alternatively, this linear subspace of V is mapped into itself by the action of y_{α}, because of (38.4.13). This linear subspace of V is also mapped into itself by the action of h_{α}, because y_{α}^{j} · v has weight μ − j α in V for each j ≥ 0, as in Section 33.5. One can check that this linear subspace of V is mapped into itself by the action of x_{α}, using (38.4.3) and the fact that x_{α} · v = 0. This is the argument used in step (2) on p113 of [14]. Note that this linear subspace of V is nontrivial, because v ≠ 0.

38.5 Submodules over subalgebras

Let k be a commutative ring with a multiplicative identity element, and let (A, [·, ·]_{A}) be a Lie algebra over k. Suppose that V is a module over k that is a module over A, as a Lie algebra over k. Let C be a Lie subalgebra of A, so that V may be considered as a module over C as well. Also let Z be a submodule of V, as a module over k, that is a submodule of V, as a module over C as well.

Consider the subset A · Z of V consisting of finite sums of elements of the form a · z, where a ∈ A and z ∈ Z. This is clearly a submodule of V, as a module over k. Let us check that A · Z is a submodule of V as a module over C. This corresponds to part of the argument in step (3) on p113 of [14], and to a remark on p61 of [24].

If a ∈ A, c ∈ C, and z ∈ Z, then

c · (a · z) = ([c, a]_{A}) · z + a · (c · z). (38.5.1)

Of course, ([c, a]_{A}) · z ∈ A · Z automatically, and c · z ∈ Z, because Z is a submodule of V, as a module over C. It follows that c · (a · z) ∈ A · Z, which implies that A · Z is a submodule of V, as a module over C.

Suppose now that k is a field, and that A has finite dimension, as a vector space over k. Let V (C) be the linear subspace of V spanned by all finitedimensional linear subspaces Z of V such that Z is a submodule of V, as a module over C. We would like to verify that V (C) is a submodule of V, as a module over A. This corresponds to another part of the argument in step (3) on p113 of [14], and to another remark on p61 of [24].

Let Z be a finite-dimensional linear subspace of V that is a submodule of V, as a module over C. Thus A · Z is a submodule of V, as a module over C, as before. It is easy to see that A · Z has finite dimension as a vector space over k, because A and Z have finite dimension. This implies that

A · Z ⊆ V (C). (38.5.2)

One can use this to check that V (C) is a submodule of V, as a module over A.

If Z_{1}, …, Z_{n} are finitely many finite-dimensional linear subspaces of V that are submodules of Z, as a module over C, then it is easy to see that the linear subspace of V spanned by Z_{1}, …, Z_{n} has finite dimension and is a submodule of V, as a module over C. This implies that every element of V (C) is contained in a finite-dimensional linear subspace of V that is a submodule of V, as a module over C. Similarly, every finite-dimensional linear subspace of V (C) is contained in a finite-dimensional linear subspace of V that is a submodule of V, as a module over C.

Let us return now to the same notation and hypotheses as in Section 33.1, so that A is a finite-dimensional Lie algebra over a field k of characteristic 0 in particular. Let V be a vector space over k that is a module over A again, as a Lie algebra over k, and suppose that V is irreducible as a module over A, that V is standard cyclic of weight μ ∈ A′_{0}, and that μ(h_{α}) corresponds to a nonnegative integer for each α ∈ Δ, with respect to the standard embedding 38.6. SOME RELATED EXPONENTIALS ON V 785

of Q into k, as in the previous section. If α ∈ Δ, then let x_{α} and y_{α} be as before, and let C_{α} be the linear span of x_{α}, y_{α}, and h_{α} in A. Thus C_{α} is a Lie subalgebra of A that is isomorphic to sl_{2}(k).

Under these conditions, there is a nontrivial finite-dimensional linear subspace of V that is a submodule of V, as a module over C_{α}, as in the previous section. This means that V (C_{α}) ≠ {0}, where V (C_{α}) is as defined earlier. It follows that V (C_{α}) = V, (38.5.3)

because V (C_{α}) is a submodule of V, as a module over A, as before. This corresponds to step (3) on p113 of [14], and to an argument on p61 of [24].

38.6 Some related exponentials on V

Let us continue with the same notation and hypotheses as in the previous section, including those from Section 33.1. Thus V is a vector space over k that is a module over A, as a Lie algebra over k, V is irreducible as a module over A, V is standard cyclic of weight μ ∈ A′_{0}, and μ(h_{α}) corresponds to a nonnegative integer for every α ∈ Δ, with respect to the standard embedding of Q into k, as before. If α ∈ Δ, then we let x_{α}, y_{α} ∈ A be as before, so that the linear span C_{α} of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k).

Let ρ be the representation of A on V, so that ρ_{a}(v) = a · v for every a ∈ A and v ∈ V. If Z is a finite-dimensional linear subspace of V that is a submodule of V, as a module over C_{α}, then ρ_{x_{α}} and ρ_{y_{α}} are nilpotent on Z. Indeed, Z is isomorphic as a module over C_{α} to a direct sum of finitely many irreducible finite-dimensional modules over C_{α}, by Weyl’s theorem. Every irreducible finitedimensional module over C_{α} is isomorphic to the module W (m) described in Section 15.4 for some nonnegative integer m, as in Sections 15.5 and 15.9. It follows that ρ_{x_{α}} and ρ_{y_{α}} are nilpotent on Z, because of the analogous statement for W (m), as in Section 15.8. This implies that ρ_{x_{α}} and ρ_{y_{α}} are locally nilpotent on V, as in Section 27.12, because of (38.5.3). This corresponds to step (4) on p114 of [14].

Of course, ρ_{−y_{α}} = −ρ_{y_{α}} is locally nilpotent on V as well. Thus exp ρ_{x_{α}} and exp ρ_{−y_{α}} are defined as linear mappings from V into itself, as in Section 27.12. More precisely, these are one-to-one linear mappings from V onto itself, as before. It follows that

(38.6.1) θ_{α} = (exp ρ_{x_{α}}) ∘ (exp ρ_{−y_{α}}) ∘ (exp ρ_{x_{α}})

defines a one-to-one linear mapping from V onto itself, as in step (5) on p114 of [14].

Let us check that ρ_{h_{α}} ∘ θ_{α} = −θ_{α} ∘ ρ_{h_{α}} (38.6.2)

on V. It suffices to verify that this holds on any finite-dimensional linear subspace Z of V that is a submodule of V, as a module over C_{α}, because of (38.5.3). Remember that Z is isomorphic as a module over C_{α} to the direct sum of finitely many irreducible finite-dimensional modules over C_{α}, each of which is isomorphic to a module of the form W (m) as in Section 15.4. Thus (38.6.2) follows from the analogous statement for W (m), as in Section 15.8.

If h ∈ A_{0} and α(h) = 0, then [h, x_{α}]_{A} = [h, y_{α}]_{A} = 0. This means that ρ_{h} commutes with ρ_{x_{α}} and ρ_{y_{α}} on V, and thus with exp ρ_{x_{α}}, exp ρ_{−y_{α}}. It follows that ρ_{h} ∘ θ_{α} = θ_{α} ∘ ρ_{h} (38.6.3)

on V under these conditions.

Let ν ∈ A′_{0} be given, and let V_{ν} be the linear subspace of V consisting of vectors with weight ν with respect to A_{0}, as in Section 33.5. Thus V_{ν} consists of z ∈ V such that ρ_{h}(z) = ν(h) z for every h ∈ A_{0}. More precisely, it suffices to check this when h = h_{α}, and when α(h) = 0, because A_{0} is spanned by h_{α} and the kernel of α, as a vector space over k. This uses the fact that α(h_{α}) = 2 = 1 + 1 ≠ 0 in k, because k has characteristic 0, by hypothesis.

Put τ_{α}(ν) = ν − ν(h_{α}) α, (38.6.4)

which is also an element of A′_{0}. Note that τ_{α}(ν) = ν on the kernel of α, that (τ_{α}(ν))(h_{α}) = −ν(h_{α}), and that τ_{α}(ν) is uniquely determined in A′_{0} by these conditions. One can check that

θ_{α}(V_{ν}) = V_{τ_{α}(ν)}, (38.6.5)

using (38.6.2) and (38.6.3). This corresponds to step (6) on p114 of [14], and to part of Remark (1) on p62 of [24].

It follows that dim V_{ν} = dim V_{τ_{α}(ν)}, (38.6.6)

as vector spaces over k. In particular, V_{ν} ≠ {0} exactly when V_{τ_{α}(ν)} ≠ {0}. This is the same as saying that ν is a weight of V with respect to A_{0} if and only if τ_{α}(ν) is a weight of V.

Alternatively, suppose that V_{ν} ≠ {0}, and let z be a nonzero element of V_{ν}. Because V is standard cyclic of weight μ as a module over A, ν can be expressed as μ minus a linear combination of roots with integer coefficients, as in Section 33.8. In particular, this implies that ν(h_{α}) corresponds to an integer, under the standard embedding of Q into k, because μ(h_{α}) corresponds to an integer, by hypothesis.

Let Z be a finite-dimensional linear subspace of V that is a submodule of V, as a module over C_{α}, such that z ∈ Z, as in the previous section. Put

(38.6.7)  u = (ρ_{y_{α}})^{ν(h_{α})}(z) when ν(h_{α}) ≥ 0,
= (ρ_{x_{α}})^{−ν(h_{α})}(z) when ν(h_{α}) ≤ 0,

which is an element of Z. One can verify that u ≠ 0, because Z is isomorphic as a module over C_{α} to the direct sum of finitely many modules over C_{α} of the form W (m) as in Section 15.4, as before.

We also have that u has weight (38.6.4) in V with respect to A_{0}, as in Section 33.5. This implies that V_{τ_{α}(ν)} ≠ {0}, as before. Conversely, if V_{τ_{α}(ν)} ≠ {0}, then the same argument shows that V_{ν} ≠ {0}, because τ_{α}(τ_{α}(ν)) = ν. This corresponds to an argument on p61 of [24].

38.7 Weights and the Weyl group

We continue to use the same notation and hypotheses as in Section 33.1. Remember that Φ is a root system in E_{R}, which is a vector space over the real numbers of positive finite dimension, and that E_{Q} is the same as the linear span of Φ in E_{R} as a vector space over Q, as in Section 32.3. In particular, every automorphism of Φ maps E_{Q} onto itself.

As before, we let V be a vector space over k that is a module over A, as a Lie algebra over k, with V irreducible as a module over A, and standard cyclic with weight μ ∈ A′_{0}, and se suppose that μ(h_{α}) corresponds to a nonnegative integer for each α ∈ Δ, with respect to the standard embedding of Q into k. If ν ∈ A′_{0} is a weight of V with respect to A_{0}, then it follows that ν(h_{α}) corresponds to an integer for every α ∈ Δ, as in the previous section. This implies that ν ∈ E_{Q}, as in Section 32.3.

If α ∈ Φ, then we let σ_{α} be the symmetry on E_{R} with vector α that maps Φ onto itself, and λ_{α} be the corresponding linear functional on E_{R}, as usual. Remember that λ_{α} maps E_{Q} into Q, because λ_{α} takes integer values on Φ. If ν ∈ E_{Q} ⊆ A′_{0}, then λ_{α}(ν) = ν(h_{α}), (38.7.1)

by (33.1.2), and using the natural embedding of Q into k. This implies that

σ_{α}(ν) = ν − λ_{α}(ν) α = ν − ν(h_{α}) α. (38.7.2)

If ν ∈ E_{Q} and σ is in the Weyl group of Φ, then we would like to check that ν is a weight of V with respect to A_{0} if and only if σ(ν) has this property. If σ = σ_{α} for some α ∈ Δ, then this follows from the remarks in the previous section, because (38.7.2) is the same as (38.6.4). Otherwise, one can use the fact that the Weyl group of Φ is generated byσ_{α}, α ∈ Δ, as in Section 19.14. Similarly, dim V_{ν} = dim V_{σ(ν)} (38.7.3)

for every σ in the Weyl group of Φ. More precisely, this is the same as (38.6.6) when σ = σ_{α} for some α ∈ Δ.

The weights of V with respect to A_{0} are elements of E_{Q}, as before. The Weyl group of Φ maps the set of weights of V with respect to A_{0} onto itself, as in the preceding paragraph. This and (38.7.3) correspond to step (7) on p114 of [14], and are included in the theorem on p113 of [14]. These statements also correspond to Remark (1) on p62 of [24].

We would like to show that there are only finitely many weights of V with respect to A_{0}. If ν is a weight of V, then ν can be expressed as μ minus a linear combination of the elements of Δ with nonnegative integer coefficients, as in Section 33.8. This means that ν ⪯ μ, in the notation of Section 30.5.

If ν is a weight of V, then ν(h_{α}) corresponds to an integer for every α ∈ Δ, so that ν ∈ Υ, as in Section 32.3. In this case, there is an element σ of the Weyl group of Φ such that σ(ν) ∈ Υ^{+}, as in Section 30.4. Note that σ(ν) is a weight of V as well, so that σ(ν) ⪯ μ, as before.

However, there are only finitely many elements ρ of Υ^{+} such that ρ ⪯ μ, as in Section 30.6. This implies that there are only finitely many weights of V, because there are only finitely many elements of the Weyl group of Φ. This corresponds to step (8) on p114 of [14].

Alternatively, if ν is a weight of V, then ν can be expressed as μ minus a linear combination of the elements of Δ with nonnegative integer coefficients, as before. In order to show that there are only finitely many weights of V, it suffices to find upper bounds for these coefficients that do not depend on ν, as on p61 of [24].

Of course, μ ∈ E_{Q}, because μ(h_{α}) corresponds to an integer for every α ∈ Δ, by hypothesis. Remember that Δ is a basis for E_{Q} as a vector space over Q, as in Section 33.1. Thus ν can be expressed in a unique way as a linear combination of elements of Δ with coefficients in Q, and we would like to find lower bounds for these coefficients that do not depend on ν.

Remember that −Δ is a base for Φ too, so that there is an element σ of the Weyl group of Φ such that σ(Δ) = −Δ, as in Section 19.14. We also have that σ(ν) is a weight of V, because ν is a weight of V, as before. This means that σ(ν) can be expressed as μ minus a linear combination of elements of Δ with nonnegative integer coefficients, as in Section 33.8 again.

It follows that ν can be expressed as σ^{−1}(μ) minus a linear combination of the elements of σ^{−1}(Δ) with nonnegative integer coefficients. Equivalently, ν can be expressed as σ^{−1}(μ) plus a linear combination of elements of Δ with nonnegative integer coefficients. In particular, the coefficients of ν with respect to Δ are greater than or equal to the coefficients of σ^{−1}(μ).

This implies that the coefficients of ν with respect to Δ have lower bounds that do not depend on ν, because there are only finitely many elements of the Weyl group. It follows that there are only finitely many weights of V, as before.

Remember that V is spanned by its weight spaces, each of which has finite dimension, as in Section 33.8. This means that V has finite dimension, because there are only finitely many weights of V. This corresponds to step (9) on p114 of [14], and to the last step in the proof on p61 of [24].

38.8 Sums of weights

Let us continue to use the same notation and hypotheses as in Section 33.1. Let V^{1}, V^{2} be vector spaces over k that are modules over A, as a Lie algebra over k. Consider the tensor product V^{1} ⨂ V^{2}, initially as a vector space over k. As in Section 7.12, V^{1} ⨂ V^{2} may be considered as a module over A, where

a · (z_{1} ⊗ z_{2}) = (a · z_{1}) ⊗ z_{2} + z_{1} ⊗ (a · z_{2}) (38.8.1) for every a ∈ A, z_{1} ∈ V^{1}, and z_{2} ∈ V^{2}.

Let ν_{1}, ν_{2} ∈ A′_{0} be given, and remember that V_{ν_{1}}^{1}, V_{ν_{2}}^{2} consist of elements of V^{1}, V^{2} with weight ν_{1}, ν_{2}, respectively, with respect to A_{0}, as in Section 33.5. If w ∈ A_{0}, z_{1} ∈ V_{ν_{1}}^{1}, and z_{2} ∈ V_{ν_{2}}^{2}, then

(38.8.2)  w · (z_{1} ⊗ z_{2}) = (w · z_{1}) ⊗ z_{2} + z_{1} ⊗ (w · z_{2})
= (ν_{1}(w) z_{1}) ⊗ z_{2} + z_{1} ⊗ (ν_{2}(w) z_{2})
= (ν_{1}(w) + ν_{2}(w)) (z_{1} ⊗ z_{2}).

This means that z_{1} ⊗ z_{2} is in the set (V^{1} ⨂ V^{2})_{ν_{1}+ν_{2}} of elements of V^{1} ⨂ V^{2} with weight ν_{1} + ν_{2} with respect to A_{0}. The tensor product V_{ν_{1}}^{1} ⨂ V_{ν_{2}}^{2} may be considered as a linear subspace of V^{1} ⨂ V^{2}, and we get that

V_{ν_{1}}^{1} ⨂ V_{ν_{2}}^{2} ⊆ (V^{1} ⨂ V^{2})_{ν_{1}+ν_{2}}. (38.8.3)

Let μ_{1}, μ_{2} ∈ A′_{0} be given, and suppose that

(38.8.4) v_{1} ∈ V^{1}, v_{2} ∈ V^{2} are primitive or maximal vectors of weight μ_{1}, μ_{2}, respectively,

as in Section 33.6. Under these conditions, it is easy to see that

(38.8.5) v_{1} ⊗ v_{2} is a primitive or maximal vector in V^{1} ⨂ V^{2}

of weight μ_{1} + μ_{2}.

If UA is a universal enveloping algebra of A, then

V = (UA) · (v_{1} ⊗ v_{2}) (38.8.6)

is the submodule of V^{1} ⨂ V^{2}, as a module over A, generated by v_{1} ⊗ v_{2}, as before. Thus (38.8.7) V is standard cyclic of weight μ_{1} + μ_{2},

as a module over A.

If V^{1}, V^{2} have finite dimension as vector spaces over k, then V^{1} ⨂ V^{2} has finite dimension as well, which implies that V has finite dimension too. This shows that

(38.8.8) if there are standard cyclic modules over A with weights μ_{1}, μ_{2} that are finite dimensional as vector spaces over k, then there is a standard cyclic module over A with weight μ_{1} + μ_{2} that is finite dimensional as a vector space over k,

as in Proposition 4.4 on p61 of [25]. Of course, this also follows from the characterization of weights of finite-dimensional standard cyclic modules stated in Section 38.3. This more direct argument for sums is used in [25] to obtain the sufficiency of the condition in Section 38.3 when A = sl_{n}(k).

38.9 Weights and tensor products

Let us continue with the same notation and hypotheses as in the previous section. If V is a vector space over k that is a module over A, then put

Π(V) = {ν ∈ A′_{0} : V_{ν} ≠ {0}}, (38.9.1)

which is the set of weights of V with respect to A_{0}. Observe that

{ν_{1} + ν_{2} : ν_{1} ∈ Π(V^{1}), ν_{2} ∈ Π(V^{2})} ⊆ Π(V^{1} ⨂ V^{2}), (38.9.2)

by (38.8.3).

Suppose that

(38.9.3) V^{1}, V^{2} correspond to the direct sums of their weight spaces, as vector spaces over k.

This implies that

(38.9.4) V^{1} ⨂ V^{2} corresponds to the direct sum of V_{ν_{1}}^{1} ⨂ V_{ν_{2}}^{2},

ν_{1} ∈ Π(V^{1}), ν_{2} ∈ Π(V^{2}), as a vector space over k,

as in Section 7.14.

Under these conditions, we get that

Π(V^{1} ⨂ V^{2}) = {ν_{1} + ν_{2} : ν_{1} ∈ Π(V^{1}), ν_{2} ∈ Π(V^{2})}, (38.9.5)

using (38.8.3). More precisely, if ν ∈ A′_{0}, then

(38.9.6) (V^{1} ⨂ V^{2})_{ν} corresponds to the direct sum of V_{ν_{1}}^{1} ⨂ V_{ν_{2}}^{2},

where ν_{1} ∈ Π(V^{1}), ν_{2} ∈ Π(V^{2}), and ν_{1} + ν_{2} = ν, as a vector space over k.

Note that

(38.9.7) V^{1} ⨂ V^{2} corresponds to the direct sum of

its weight spaces, as a vector space over k,

in this case.

If V^{1}, V^{2} have finite dimension as vector spaces over k, then Π(V^{1}), Π(V^{2}) have only finitely many elements. This implies that Π(V^{1} ⨂ V^{2}) has only finitely many elements as well, by (38.9.5). If ν ∈ Π(V^{1} ⨂ V^{2}), then we get that

(38.9.8) dim (V^{1} ⨂ V^{2})_{ν} = ∑_{ν_{1}+ν_{2}=ν}(dim V_{ν_{1}}^{1}) (dim V_{ν_{2}}^{2}),

where more precisely the sum is taken over all ν_{1} ∈ Π(V^{1}) and ν_{2} ∈ Π(V^{2}) with ν_{1} + ν_{2} = ν.

Suppose now that

(38.9.9) V^{1}, V^{2} are standard cyclic of weights μ_{1}, μ_{2} ∈ A′_{0}, respectively, as modules over A.

In particular, this implies that (38.9.3) holds, as in Section 33.8. Remember that the elements of Π(V^{1}), Π(V^{2}) can be expressed as μ_{1}, μ_{2} minus linear combinations of elements of Δ with nonnegative integer coefficients, respectively, as in Section 33.8. It follows that

(38.9.10) the elements of Π(V^{1} ⨂ V^{2}) can be expressed as

μ_{1} + μ_{2} minus linear combinations of elements of Δ with nonnegative integer coefficients.

If ν_{1} ∈ Π(V^{1}), ν_{2} ∈ Π(V^{2}), then we get that

(38.9.11) ν_{1} + ν_{2} = μ_{1} + μ_{2} only when ν_{1} = μ_{1}, ν_{2} = μ_{2}.

This implies that (V^{1} ⨂ V^{2})_{μ_{1}+μ_{2}} = V_{μ_{1}}^{1} ⨂ V_{μ_{2}}^{2}, (38.9.12)

as in (38.9.6). Remember that V_{μ_{1}}^{1}, V_{μ_{2}}^{2} have dimension one as vector spaces over k, as in Section 33.8. This means that

dim (V^{1} ⨂ V^{2})_{μ_{1}+μ_{2}} = 1, (38.9.13)

by (38.9.12).

If ν ∈ Π(V^{1} ⨂ V^{2}), then one can check that

(38.9.14) ν = ν_{1} + ν_{2} for only finitely many ν_{1} ∈ Π(V^{1}), ν_{2} ∈ Π(V^{2}),

because ν_{1}, ν_{2} can be expressed as μ_{1}, μ_{2} minus linear combinations of elements of Δ with nonnegative integer coefficients, as before. Remember that V_{ν_{1}}^{1}, V_{ν_{2}}^{2} have finite dimensions as vector spaces over k for every ν_{1}, ν_{2} ∈ A′_{0}, as in Section 33.8. Thus (38.9.15) (V^{1} ⨂ V^{2})_{ν} has finite dimension

as a vector space over k, because it corresponds to the direct sum of finitely many finite-dimensional subspaces, by (38.9.6). More precisely, the dimension of (V^{1} ⨂ V^{2})_{ν} can be expressed as in (38.9.8) again.

This corresponds to Exercise 7 on p117 of [14], and to part of part (b) of Proposition 5 on p63 of [24].

38.10 Fundamental weights and modules

Let us return to some connections with abstract weights, as in Section 32.3. Remember that the collection Φ′ of λ_{α}, α ∈ Φ, is a root system in the dual E′_{R} of E_{R}, and that the set Δ′ of λ_{α}, α ∈ Δ, is a base for Φ′, because Φ is reduced as a root system in E_{R}. In particular, Δ′ is a basis for E′_{R}, as a vector space over R, and if α ∈ Δ, then there is a unique element μ_{α} of E_{R} such that

(38.10.1)  λ_{β}(μ_{α}) = 1 when β = α
= 0 for every β ∈ Δ with β ≠ α,

as in Section 30.1. The μ_{α}’s, α ∈ Δ, form a basis for E_{R} dual to the λ_{β}’s, β ∈ Δ, as before.

Remember that Υ consists of z ∈ E_{R} such that λ_{β}(z) ∈ Z for every β ∈ Φ, or equivalently for every β ∈ Δ, because Δ′ is a base for Φ′. Thus μ_{α} ∈ Υ for every α ∈ Δ, as in Section 30.1. We also have that Υ ⊆ E_{Q}, as in Section 32.3, so that μ_{α} ∈ E_{Q} ⊆ A′_{0} for every α ∈ Δ.

Similarly, Υ^{+} consists of z ∈ Υ such that λ_{β}(z) ≥ 0 for every β ∈ Δ, or equivalently for every β ∈ Φ. Clearly μ_{α} ∈ Υ^{+} for every α ∈ Δ, as in Section 30.3. Remember that the μ_{α}’s, α ∈ Δ, are called the fundamental (dominant) weights with respect to Δ, as in Section 30.1. It is easy to see that Υ^{+} consists exactly of linear combinations of the μ_{α}’s, α ∈ Δ, whose coefficients are nonnegative integers.

As in Section 32.3, Υ consists exactly of ν ∈ A′_{0} such that ν(h_{β}) corresponds to an integer for every β ∈ Δ, or equivalently for every β ∈ Φ. Similarly, Υ^{+} consists exactly of μ ∈ A′_{0} such that μ(h_{β}) corresponds to a nonnegative integer for every β ∈ Δ, or equivalently β ∈ Φ. This is the same as the set of μ ∈ A′_{0} for which there is a finite-dimensional standard cyclic module over A with weight μ, as in Section 38.3. The modules associated to μ_{α}, α ∈ Δ, in this way are called the fundamental modules over A, or the fundamental representations of A, as in Remark (2) on p62 of [24].

If μ ∈ Υ^{+}, then μ can be expressed as a sum of μ_{α}’s, α ∈ Δ, with suitable repetitions. One can use this to get a finite-dimensional standard cyclic module over A with weight μ as a submodule of a tensor product of fundamental modules over A, with suitable repetitions, as in Exercise 8 on p117 of [14]. This is analogous to the argument discussed in Section 38.8. This type of argument is used on p61 of [25], to get the existence of finite-dimensional standard cyclic modules over A = sl_{n}(k) with any weight μ ∈ Υ^{+}.

38.11 Simple transitivity

Let us continue with the same notation and hypotheses as in Section 33.1. In particular, Φ is a reduced root system in the vector space E_{R} over R of positive finite dimension, and Δ is a base for Φ. Remember that any reduced root system can be as in Section 33.1, because of Serre’s theorem, as in Section 27.8.

If σ is an element of the Weyl group of Φ such that

σ(Δ) = Δ, (38.11.1)

then σ is the identity mapping on E_{R}. This was discussed in Section 20.1, and it can also be obtained using results about representations of the Lie algebra A, as in Proposition 4 on p62 of [24]. This corresponds to Exercise 1 on p116 of [14].

If β ∈ Φ, then λ_{σ(β)} = λ_{β} ∘ σ^{−1}, (38.11.2)

because σ is an automorphism of Φ, as in Section 19.4. Let μ_{α}, α ∈ Δ, be the fundamental weights with respect to Δ, as in the previous section. If α, β ∈ Δ, then

(38.11.3)  λ_{σ(β)}(σ(μ_{α})) = λ_{β}(μ_{α}) = 1 when β = α
= 0 when β ≠ α,

by (38.11.2) and (38.10.1). This implies that

σ(μ_{α}) = μ_{σ(α)} (38.11.4)

for every α ∈ Δ, because of (38.11.1).

Let α ∈ Δ be given, and let V^{α} be a finite-dimensional standard cyclic module over A with weight μ_{α}. In particular, μ_{α} is a weight of V^{α}, which implies that (38.11.4) is a weight of V^{α} too, as in Section 38.7. It follows that

μ_{α} − σ(μ_{α}) = μ_{α} − μ_{σ(α)} (38.11.5)

can be expressed as a linear combination of the elements of Δ, with coefficients that are nonnegative integers, as in Section 33.8.

Of course, the sum of (38.11.5) over α ∈ Δ is equal to 0, by (38.11.1). This implies that (38.11.5) is equal to 0 for every α ∈ Δ, because Δ is a basis for E_{R}, and the coefficients of (38.11.5) with respect to Δ are all nonnegative. It follows that σ is the identity mapping on E_{R}, as desired.

38.12 Weights and saturated sets

We continue with the same notation and hypotheses as in Section 33.1. Let V be a finite-dimensional vector space over k, and suppose that V is a standard cyclic module over A of weight μ ∈ A′_{0}. If ν ∈ A′_{0}, then let V_{ν} be the linear subspace of V consisting of vectors of weight ν with respect to A_{0}, as in Section 33.5. Put Π = Π(V) = {ν ∈ A′_{0} : V_{ν} ≠ {0}}. (38.12.1)

Note that Π has only finitely many elements, because V has finite dimension, by hypothesis.

If ν ∈ Π and α ∈ Φ, then ν(h_{α}) corresponds to an integer under the standard embedding of Q into k, as in Sections 38.1 and 38.2. This means that

Π ⊆ Υ, (38.12.2)

where Υ is as in Section 32.3. We would like to show that

(38.12.3) Π is saturated as a subset of Υ, in the sense of Section 30.7. This corresponds to the first part of the proposition on p114 of [14].

Let ν ∈ Π and α ∈ Φ be given, and remember that ν ∈ E_{Q} and ν(h_{α}) = λ_{α}(ν), as in Sections 32.2 and 32.3. Let x_{α} be a nonzero element of A_{α}, and let y_{α} be an element of A_{−α} such that [x_{α}, y_{α}]_{A} = h_{α}, as usual. The linear span of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k), as a Lie algebra over k.

Let Z be the linear subspace of V spanned by the subspaces of the form

V_{ν+j·α}, j ∈ Z. (38.12.4)

It is easy to see that Z is a submodule of V, as a module over the linear span of x_{α}, y_{α}, and h_{α} in A, using the remarks in Section 33.5.

Suppose that j_{1}, j_{2} ∈ Z satisfy j_{1} ≤ j_{2} and

ν + j_{1} · α, ν + j_{2} · α ∈ Π. (38.12.5)

If j ∈ Z and j_{1} ≤ j ≤ j_{2}, then we would like to verify that

ν + j · α ∈ Π. (38.12.6)

Of course, Z has finite dimension as a vector space over k, because V has finite dimension, by hypothesis. Thus we can use Weyl’s theorem to get that Z corresponds to the direct sum of irreducible modules over the linear span of x_{α}, y_{α}, and h_{α} in A. Each of these irreducible modules is isomorphic to W (m) in Section 15.4 for some nonnegative m, as a module over sl_{2}(k), as in Sections 15.5 and 15.9. One can use this to get (38.12.6), because of the analogous property of W (m).

Remember that elements of the Weyl group of Φ map Π onto itself, as in Section 38.7. In particular, σ_{α}(ν) ∈ Π, where σ_{α} is the symmetry on E_{R} with vector α that maps Φ onto itself, as usual. Equivalently,

ν − λ_{α}(ν) α ∈ Π, (38.12.7)

where λ_{α}(ν) = ν(h_{α}) corresponds to an integer under the natural embedding of Q into k, as before. If j ∈ Z is between 0 and λ_{α}(ν), then (38.12.6) holds, because ν ∈ Π, by hypothesis. This implies (38.12.3), as desired.

Let r, q be the largest integers such that

ν − r · α, ν + q · α ∈ Π, (38.12.8)

so that q, r ≥ 0, because ν ∈ Π. It follows that (38.12.6) holds for some j ∈ Z if and only if −r ≤ j ≤ q. The sequence of elements of Π of the form

ν + j · α, (38.12.9)

with j ∈ Z and −r ≤ j ≤ q, is called the α-string through ν, as on p114 of [14]. It is easy to see that σ_{α} maps this sequence onto itself, but in the opposite order. This implies that r − q = λ_{α}(ν), (38.12.10) as on p114 of [14].

Of course, μ ∈ Π, and μ(h_{α}) corresponds to a nonnegative integer for every α ∈ Φ^{+}, as in Section 38.3. This means that μ ∈ Υ^{+}, as in Section 32.3. In fact, (38.12.11) Π has highest weight μ,

in the sense of Section 30.7. Indeed, if ν ∈ Π, then ν can be expressed as μ minus a linear combination of elements of Δ with nonnegative integer coefficients, as in Section 33.8. Thus ν ⪯ μ, in the notation of Section 30.5.

If ρ ∈ Υ^{+} and ρ ⪯ μ, then ρ ∈ Π, (38.12.12)

as in Section 30.8. This implies that

σ(ρ) ∈ Π (38.12.13)

for every element σ of the Weyl group of Φ, as before.

If ν ∈ Π and σ is an element of the Weyl group of Φ, then σ(ν) ∈ Π, so that

σ(ν) ⪯ μ, (38.12.14)

as before. Conversely, if ν ∈ Υ and (38.12.14) holds for every element σ of the Weyl group of Φ, then ν ∈ Π. Indeed, if ν ∈ Υ, then there is an element σ of the Weyl group of Φ such that σ(ν) ∈ Υ^{+}, as in Section 30.4. This implies that σ(ν) ∈ Π, and thus ν ∈ Π, as in the preceding paragraph. This corresponds to the second part of the proposition on p114 of [14].

38.13 Formal characters in Z[Υ]

Let us continue with the same notation and hypotheses as in Section 33.1, and let Υ be as in Section 32.3 again. Remember that Z[Υ] is the group ring of Υ with coefficients in Z, and that e_{ν}, ν ∈ Υ, are the standard basis elements of Z[Υ], as in Section 32.4.

Let V be a finite-dimensional vector space over k that is a module over A, as a Lie algebra over k. If ν ∈ A′_{0}, then V_{ν} is the linear subspace of V consisting of vectors of weight ν with respect to A_{0}, as in Section 33.5. Remember that V_{ν} = {0} unless ν ∈ Υ, as in (38.12.2).

The formal character of V is defined as an element of Z[Υ] by

ch(V) = ∑_{ν∈Υ}(dim V_{ν}) e_{ν}, (38.13.1)

as on p124 of [14], and p63 of [24]. More precisely, the dimension of V_{ν} as a vector space over k is a nonnegative integer for every ν ∈ Υ, which is equal to zero for all but finitely many ν ∈ Υ, because V has finite dimension. Thus the right side of (38.13.1) is an element of Z[Υ].

Let V^{1}, V^{2} be finite-dimensional vector spaces over k that are modules over A, as a Lie algebra over k. Under these conditions, the direct sum V^{1} ⨁ V^{2} is also a finite-dimensional vector space over k that is a module over A, and it is easy to see that ch (V^{1} ⨁ V^{2}) = ch(V^{1}) + ch(V^{2}). (38.13.2)

Similarly, the tensor product V^{1} ⨂ V^{2} is a finite-dimensional vector space over k and a module over A, and one can check that

ch (V^{1} ⨂ V^{2}) = ch(V^{1}) · ch(V^{2}), (38.13.3)

using the remarks in Section 38.9. This corresponds to Proposition B on p125 of [14], and part (b) of Proposition 5 on p63 of [24].

Remember that any automorphism of Φ in E_{R} maps Υ onto itself, as in Section 30.4. In fact, the restriction of any automorphism of Φ to Υ is an automorphism of Υ, as a commutative group with respect to addition. In particular, elements of the Weyl group of Φ determine automorphisms of Υ. These automorphisms of Υ lead to ring automorphisms of Z[Υ], as in Section 32.4.

If V is irreducible as a module over A, then V is standard cyclic as a module over A, as in Section 38.2. In this case, we have seen that elements of the Weyl group of Φ map the set of weights of V to itself, and preserve the dimensions of the corresponding weight spaces, as in Section 38.7. It is easy to see that this holds when V is not necessarily irreducible as a module over A too, because V corresponds to the direct sum of finitely many irreducible submodules, by Weyl’s theorem. This implies that

(38.13.4) ch(V) is invariant under the action of the Weyl group on Z[Υ].

This corresponds to a remark on p125 of [14], and to part (a) of Proposition 5 on p63 of [24].

Let V, W be finite-dimensional vector spaces over k that are modules over A. If V and W are isomorphic as modules over A, then

ch(V) = ch(W). (38.13.5)

Conversely, if (38.13.5) holds, then V and W are isomorphic as modules over A. This is part (c) of Proposition 5 on p63 of [24], which corresponds to a remark about Proposition A on p125 of [14].

Let Π(V), Π(W) be the sets of weights of V, W, respectively, as in (38.12.1). Suppose that (38.13.5) holds, which implies in particular that

Π(V) = Π(W) (38.13.6)

and dim V = dim W. (38.13.7)

To show that V is isomorphic to W, one can use induction on the common dimension of V and W. Of course, this is trivial when the dimension if 0, and so we may suppose that the dimension is positive.

Observe that Π(V) ≠ ∅, because V ≠ {0}. One can find μ ∈ Π(V) such that for each α ∈ Δ, μ+α ∉ Π(V), because Π(V) has only finitely many elements, as in Section 38.1. Let v be a nonzero element of V_{μ}, and note that v is primitive or maximal of weight μ, as in Section 38.1 again.

Let UA be a universal enveloping algebra of A, and put

V^{1} = (UA) · v. (38.13.8)

This is the submodule of V, as a module over A, generated by v, as in Section 33.7. By construction, V^{1} is standard cyclic of weight μ, as a module over A, as in Section 33.8. Of course, V^{1} has finite dimension as a vector space over k, because V has finite dimension. This implies that V^{1} is irreducible as a module over A, as in Section 33.9.

Note that μ ∈ Π(W) too, by (38.13.6). Let w be a nonzero element of W_{μ}, so that w is primitive or maximal of weight μ, as before. Put

W^{1} = (UA) · w, (38.13.9)

which is the submodule of W, as a module over A, generated by w. Thus W^{1} is standard cyclic of weight μ, as a module over A. We also have that W^{1} is irreducible as a module over A, because it has finite dimension, as a vector space over k.

Under these conditions, V^{1} and W^{1} are isomorphic as modules over A, as in Section 33.10. This implies that

ch(V^{1}) = ch(W^{1}), (38.13.10)

as before. Using Weyl’s theorem, we get that there are submodules V^{2}, W^{2} of V, W, respectively, as modules over A, such that V, W correspond to V^{1} ⨁ V^{2}, W^{1} ⨁ W^{2}, respectively, as modules over W. This means that

(38.13.11) ch(V) = ch(V^{1}) + ch(V^{2}), ch(W) = ch(W^{1}) + ch(W^{2}),

as in (38.13.2). It follows that

ch(V^{2}) = ch(W^{2}), (38.13.12)

by (38.13.10).

The common dimension of V^{2} and W^{2} is strictly less than the common dimension of V and W, by construction. Thus we get that V^{2} and W^{2} are isomorphic as modules over A, by induction. This implies that V and W are isomorphic as modules over A, because of the analogous statement for V^{1} and W^{1}, as desired.

38.14 Invariant elements of Z[Υ]

Let us continue with the same notation and hypotheses as in the previous section. Remember that elements of the Weyl group W of Φ in E_{R} determine ring automorphisms on Z[Υ], as before. Let Z[Υ]^{W} be the subring of Z[Υ] consisting of elements that are invariant under the action of W. If V is a finite-dimensional vector space over k that is a module over A, as a lie algebra over k, then

ch(V) ∈ Z[Υ]^{W} , (38.14.1)

as in (38.13.4).

Let

z = ∑_{ν∈Υ} z_{ν} e_{ν} (38.14.2)

be an element of Z[Υ], so that z_{ν} ∈ Z for every ν ∈ Υ, and z_{ν} = 0 for all but finitely many ν ∈ Υ. Suppose that z ∈ Z[Υ]^{W}, which means that

z_{σ(ν)} = z_{ν} (38.14.3)

for every ν ∈ Υ. This permits us to arrange the sum in (38.14.2) into a sum over orbits of W in Υ.

Remember that Υ^{+} consists of the μ ∈ Υ such that μ(h_{α}) ≥ 0 for every α ∈ Φ^{+}, where Φ^{+} is the set of positive roots with respect to the base Δ for Φ, as in Section 32.3. If μ ∈ Υ^{+}, then put

O(μ) = {σ(μ) : σ ∈ W }, (38.14.4)

which is the orbit of W in Υ that contains ν. Every orbit of W in Υ contains exactly one element of Υ^{+}, as in Section 30.4. This means that every orbit of W in Υ is of the form (38.14.4) for exactly one ν ∈ Υ^{+}. Thus

z = ∑_{μ∈Υ^{+}} z_{μ} ( ∑_{ν∈O(μ)} e_{ν}), (38.14.5)

by (38.14.3).

If μ ∈ Υ^{+}, then there is a finite-dimensional module V (μ) over A that is standard cyclic of weight μ, as in Section 38.3. In particular, ch(V (μ)) ∈ Z[Υ]^{W}, as in (38.14.1). Note that V (λ) is irreducible as a module over A, as in Section 33.9, and unique up to isomorphism, as in Section 33.10. Thus ch(V (μ)) only depends on μ, and not the choice of V (μ).

Proposition A on p125 of [14] says that every element of Z[Υ] can be expressed in a unique way as a linear combination of finitely many ch(V (μ))’s, μ ∈ Υ^{+}, with coefficients in Z. This is related to the corollary on p64 of [24]. To see this, let z ∈ Z[Υ]^{W} be given, and let us show first that z can be expressed in this way.

Let ⪯ be the partial ordering defined on E_{R} as in Section 30.5, although we shall only use it on Υ here. If γ ∈ Υ^{+}, then there are only finitely many β ∈ Υ^{+} such that β ⪯ γ, as in Section 30.6. Of course, there are only finitely many γ ∈ Υ such that z_{γ} ≠ 0. Put

(38.14.6) M_{z} = {β ∈ Υ^{+} : β ⪯ γ for some γ ∈ Υ^{+} such that z_{γ} ≠ 0}.

Note that M_{z} has only finitely many elements, by the previous statements.

To show that z can be expressed as before, we use induction on the number of elements of M_{z}. If M_{z} = ∅, then z_{γ} = 0 for every γ ∈ Υ^{+}. This implies that z = 0, as in (38.14.5).

Suppose now that M_{z} ≠ ∅, so that z_{γ} ≠ 0 for some γ ∈ Υ^{+}. Let μ be an element of Υ^{+} such that z_{μ} ≠ 0, and μ is maximal among γ ∈ Υ^{+} with z_{γ} ≠ 0 with respect to ⪯. Also let V (μ) be as before, and put

z̃ = z − z_{μ} ch(V (μ)). (38.14.7)

Clearly z̃ ∈ Z[Υ]^{W}, and we let M_{z} ⊆ Υ^{+} be defined as before.

If ν ∈ Υ is a weight of V (μ), then ν ⪯ μ, as in Section 33.8. Equivalently, this means that e_{ν} has nonzero coefficient in ch(V (μ)) only when ν ⪯ μ. This implies that

M_{z} ⊆ M_{z}, (38.14.8)

because μ ∈ Υ^{+} and z_{μ} ≠ 0, by construction.

The dimension of the weight space in V (μ) corresponding to μ is equal to one, as in Section 33.8. This means that the coefficient of e_{μ} in ch(V (μ)) is equal to one. It follows that the coefficient of e_{μ} in z̃ is equal to zero, by construction. One can use this to check that

μ ∉ M_{z}, (38.14.9)

because of (38.14.8) and the maximality property of μ. This implies that

M_{z} ≠ M_{z}, (38.14.10)

because μ ∈ M_{z}, by construction.

Thus the number of elements of M_{z} is strictly less than than the number of elements of M_{z}. This permits us to use induction to get that z̃ can be expressed in the desired way. It follows that z can be expressed similarly, by the definition of z̃.

To get uniqueness, let μ_{1}, …, μ_{r} be finitely many distinct elements of Υ^{+}, and let c_{1}, …, c_{r} be finitely many nonzero integers. We would like to check that

∑_{j=1}^{r} c_{j} ch(V (μ_{j})) (38.14.11)

is nonzero as an element of Z[Υ]. Let l be a positive integer such that l ≤ r and μ_{l} is maximal among μ_{1}, …, μ_{r}, with respect to ⪯. Thus the coefficient of e_{μ_{l}} in ch(V (μ_{j})) is equal to one when j = l, and to 0 otherwise, as in Section 33.8. This means that the coefficient of e_{μ_{l}} in (38.14.11) is c_{l} ≠ 0, so that (38.14.11) is not zero.

Dimensions of weight spaces

39.1 Casimir elements in UA

Let k be a field, and let (A, [·, ·]_{A}) be a Lie algebra over k that has positive finite dimension, as a vector space over k. Also let b(·, ·) be a nondegenerate symmetric bilinear form on A, and suppose that b(·, ·) is associative on A, or equivalently invariant with respect to the adjoint representation on A. Thus

b([x, w]_{A}, y) = b(x, [w, y]_{A}) (39.1.1)

for every w, x, y ∈ A, as in Sections 6.10 and 7.7. Let u_{1}, …, u_{n} be a basis for A as a vector space over k, and let w_{1}, …, w_{n} be the corresponding dual basis for A with respect to b(·, ·), so that

b(u_{j}, w_{l}) = δ_{j,l} (39.1.2)

for every j, l = 1, …, n. As usual, δ_{j,l} ∈ k is equal to 1 when j = l, and to 0 otherwise.

Let UA be a universal enveloping algebra of A, with multiplicative identity element e = e_{UA} and mapping i = i_{UA} from A into UA, as in Section 25.4. Remember that i is injective, by the Poincaré–Birkhoff–Witt theorem, as in Section 25.12, so that we may as well identify A with a Lie subalgebra of UA. The Casimir element of UA associated to b is

c_{A}(b) = ∑_{j=1}^{n} u_{j} w_{j}, (39.1.3)

as on p46 of [25]. This corresponds to some remarks on p118 of [14], and was also mentioned in Section 13.4.

More precisely, remember that b leads to a one-to-one linear mapping from A onto its dual A′ as a vector space over k, as in Section 13.4. This lead to a one-to-one linear mapping from A ⨂ A onto A ⨂ A′, and thus onto the space L(A) of linear mappings from A into itself, as a vector space over k, as before.

The u_{j}’s and w_{j}’s were used to define an element of A ⨂ A, that corresponds to the identity mapping as an element of L(A). In particular, this element of A ⨂ A does not depend on the choice of u_{j}’s, from which the w_{j}’s were obtained using b, as before.

There is a natural mapping from A × A into UA that is bilinear over k, which is defined using the natural mapping from A into UA and multiplication in UA. This leads to a mapping from A ⨂ A into UA that is linear over k, as usual. By construction, (39.1.3) is the image under this mapping of the element of A ⨂ A defined in Section 13.4 using the u_{j}’s and w_{j}’s. Note that (39.1.3) does not depend on the choice of u_{j}’s, as on p46 of [25]. This corresponds to the second part of Exercise 2 on p125 of [14].

We also have that

(39.1.4) c_{A}(b) is in the center of UA,

as an algebra over k, which is to say that c_{A}(b) commutes with every element of UA. To see this, it suffices to check that c_{A}(b) commutes with every element of A, because UA is generated as an algebra over k by A and e. This property corresponds to the fact that the analogous element of A ⨂ A is invariant with respect to the representation of A on A ⨂ A obtained from the adjoint representation of A on itself, as in Section 13.4. One can get this from the fact that the identity mapping on A is invariant with respect to the representation on L(A) obtained from the adjoint representation of A on itself, as on p46 of [25]. This corresponds to the first part of Exercise 2 on p125 of [14].

Let V be a vector space over k that is a module over A, as a Lie algebra over k. Thus V may be considered as a left module over UA, as an associative algebra over k, as in Section 25.6. The action of c_{A}(b) on V is the same as the Casimir element of the space L(V) of linear mappings from V into itself associated to b as in Section 13.3, as mentioned in Section 13.4.

Using (39.1.4), we get that

(39.1.5) the action of c_{A}(b) on V commutes with the actions of all other elements of UA on V,

which is the same as saying that the action of c_{A}(b) on V commutes with the actions of all elements of A on V. This was verified directly in Section 13.3, and (39.1.4) can be obtained in the same way, as in the first part of Exercise 2 on p125 of [14].

39.2 Casimir elements and nice bases

Let us return now to the same notation and hypotheses as in Section 33.1. Also let b(·, ·) be a nondegenerate symmetric bilinear form on A that is associative on A, as in the previous section. If α, β ∈ A′_{0} and α + β ≠ 0, then

(39.2.1) b(x, y) = 0 for every x ∈ A_{α}, y ∈ A_{β}, as in Section 17.3. This implies that the restriction of b(·, ·) to A_{0} is nondegenerate, as before.

Let α ∈ Φ and x ∈ A_{α} be given. If w ∈ A_{0} and y ∈ A_{−α}, then

b(w, [x, y]_{A}) = α(w) b(x, y), (39.2.2)

as in Section 17.3 again. If x ≠ 0, then there is a y ∈ A_{−α} such that b(x, y) ≠ 0, as before. Of course, this means that b(x, y) ≠ 0 for every y ∈ A_{−α} with y ≠ 0, because A_{−α} has dimension one as a vector space over k.

Remember that Δ is a base for Φ, and that h_{α}, α ∈ Δ, is a basis for A_{0}, as a vector space over k. If β ∈ Δ, then there is a unique w_{β} ∈ A_{0} such that

(39.2.3)  b(h_{α}, w_{β}) = 1 when α = β
= 0 for every α ∈ Δ with α ≠ β,

because the restriction of b(·, ·) to A_{0} is nondegenerate. Note that w_{β}, β ∈ Δ, is a basis for A_{0}, as a vector space over k.

Let x_{α} be a nonzero element of A_{α} for each α ∈ Φ. This leads to a unique z_{α} ∈ A_{−α} such that b(x_{α}, z_{α}) = 1 (39.2.4)

for each α ∈ Φ.

Observe that {h_{α} : α ∈ Δ} ∪ {x_{α} : α ∈ Φ} (39.2.5)

is a basis for A, as a vector space over k. Similarly,

{w_{α} : α ∈ Δ} ∪ {z_{α} : α ∈ Φ} (39.2.6)

is a basis for A. These two bases are dual to each other with respect to b(·, ·), by construction.

If α ∈ A′_{0}, then there is a unique t_{b,α} ∈ A_{0} such that

α(w) = b(w, t_{b,α}) (39.2.7)

for every w ∈ A_{0}, because the restriction of b(·, ·) to A_{0} is nondegenerate, as in Section 17.5. If α ∈ Φ, then

[x_{α}, z_{α}]_{A} = t_{b,α}, (39.2.8)

by (39.2.2) and (39.2.4). Note that

α(t_{b,α}) = b(t_{b,α}, t_{b,α}) (39.2.9)

for every α ∈ A′_{0}.

If α ∈ Φ, then there is a unique y_{α} ∈ A_{−α} such that y_{α} ≠ 0 and [x_{α}, y_{α}]_{A} = h_{α}, as usual. This implies that

h_{α} = [x_{α}, y_{α}]_{A} = b(x_{α}, y_{α}) t_{b,α}, (39.2.10)

39.3. ADJUSTED BASES FOR SL_{2}(K) MODULES 803

by (39.2.2). It follows that

b(x_{α}, y_{α}) α(t_{b,α}) = α(h_{α}) = 2, (39.2.11)

and in particular that b(x_{α}, y_{α}), α(t_{b,α}) ≠ 0. We also have that

y_{α} = b(x_{α}, y_{α}) z_{α}, (39.2.12)

because A_{−α} has dimension one as a vector space over k.

Let UA be a universal enveloping algebra of A again, and let c_{A}(b) be the Casimir element of UA associated to b, as in the previous section. In this case, we can use the dual bases (39.2.5) and (39.2.6) to express c_{A}(b) as

c_{A}(b) = ∑_{α∈Δ} h_{α} w_{α} + ∑_{α∈Φ} x_{α} z_{α}, (39.2.13)

as on p118 of [14].

39.3 Adjusted bases for sl_{2}(k) modules

Let us continue with the same notation and hypotheses as in the previous section. Let α ∈ Φ be given, and remember that the linear span of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A, that is isomorphic to sl_{2}(k), as a Lie algebra over k. Of course, this is the same as the linear span of x_{α}, z_{α}, and t_{b,α} in A.

Let m be a nonnegative integer, and let W (m) be the module over the linear span of x_{α}, y_{α}, and h_{α} in A, as a Lie algebra over k, discussed in Section 15.4. Thus W (m) is a vector space over k with a basis v_{0}, v_{1}, …, v_{m}, and we put v_{−1} = v_{m+1} = 0 for convenience. The actions of x_{α}, y_{α}, and h_{α} on W (m) are defined by

(39.3.1)  h_{α} · v_{j} = (m − 2 j) v_{j},
(39.3.2)  y_{α} · v_{j} = (j + 1) v_{j+1},
(39.3.3)  x_{α} · v_{j} = (m − j + 1) v_{j−1}

for j = 0, 1, …, m, as before.

It will be convenient to use another basis for W (m), to deal with the actions of x_{α}, z_{α}, and t_{b,α} on W (m), as on p119 of [14]. Note that

(39.3.4)  z_{α} = (α(t_{b,α})/2) y_{α},
(39.3.5)  t_{b,α} = (α(t_{b,α})/2) h_{α},

by (39.2.10), (39.2.11), and (39.2.12). Put

u_{j} = j! (α(t_{b,α})^{j}/2^{j}) v_{j} (39.3.6)

for j = 0, 1, …, m, so that u_{0}, u_{1}, …, u_{m} is a basis for W (m), as a vector space over k. As before, it is convenient to put u_{−1} = u_{m+1} = 0.

804 CHAPTER 39. DIMENSIONS OF WEIGHT SPACES

It is easy to see that

(39.3.7)  t_{b,α} · u_{j} = (m − 2 j) (α(t_{b,α})/2) u_{j},
(39.3.8)  z_{α} · u_{j} = u_{j+1},
(39.3.9)  x_{α} · u_{j} = j (m − j + 1) (α(t_{b,α})/2) u_{j−1}

for j = 0, 1, …, m. It follows that x_{α} · (z_{α} · u_{j}) = (m − j) (j + 1) (α(t_{b,α})/2) u_{j} (39.3.10) for j = 0, 1, …, m, as on p119 of [14].

39.4 Some preliminary subspaces

We continue with the same notation and hypotheses as in the previous two sections. Let V be a vector space over k of positive finite dimension that is an irreducible module over A, as a Lie algebra over k. Remember that if ν ∈ A′_{0}, then V_{ν} is the linear subspace of V consisting of vectors of weight ν with respect to A_{0}, as in Section 33.5.

Let ρ be the given representation of A on V, and let α ∈ Φ be given, as in the previous section. If ν ∈ A′_{0}, then ρ_{z_{α}}(V_{ν}) ⊆ V_{ν−α}, (39.4.1) because z_{α} ∈ A_{−α}, as in Section 33.5. Similarly, ρ_{x_{α}}(V_{ν−α}) ⊆ V_{ν}, (39.4.2) because x_{α} ∈ A_{α}. Thus ρ_{x_{α}}(ρ_{z_{α}}(V_{ν})) ⊆ V_{ν}. (39.4.3) We would like to consider the trace of ρ_{x_{α}} ∘ ρ_{z_{α}} as a linear mapping from V_{ν} into itself, as on p119 of [14].

Let ν ∈ A′_{0} be a weight of V with respect to A_{0}, and suppose that (39.4.4) ν + α is not a weight of V. This implies that the α-string of weights through ν is of the form ν, ν − α, …, ν − m α, (39.4.5) where m is a nonnegative integer given by m = λ_{α}(ν), (39.4.6) as in Section 38.12.

Let W be the linear subspace of V spanned by V_{ν}, V_{ν−α}, …, V_{ν−m α}. (39.4.7)

More precisely, W corresponds to the direct sum of these linear subspaces, as a vector space over k, as in Section 33.5. Remember that the linear span of x_{α}, y_{α}, and h_{α} in A is a Lie subalgebra of A that is isomorphic to sl_{2}(k), as a Lie algebra over k. It is easy to see that

(39.4.8) W is a submodule of V, as a module over the linear span of x_{α}, y_{α}, and h_{α} in A,

using the remarks in Section 33.5.

If j is an integer, then

(ν − j α)(h_{α}) = ν(h_{α}) − 2 j, (39.4.9)

and these values are distinct in k, because k has characteristic 0. Of course, the elements of V_{ν−j α} are eigenvectors of ρ_{h_{α}}, with eigenvalue (39.4.9). It follows that any nonzero eigenvector of ρ_{h_{α}} in W has eigenvalue of the form (39.4.9) for some integer j with 0 ≤ j ≤ m, and that the eigenvector is in V_{ν−j α}.

Remember that ν(h_{α}) corresponds to an integer under the natural embedding of Q into k, as in Sections 38.1 and 38.2. We also have that ν ∈ E_{Q} and ν(h_{α}) = λ_{α}(ν), as in Sections 32.2 and 32.3. Thus

ν(h_{α}) = m, (39.4.10)

by (39.4.6), so that (ν − j α)(h_{α}) = m − 2 j (39.4.11)

for every integer j.

Let Z be a nontrivial linear subspace of W that is a submodule of W, as a module over the linear span of x_{α}, y_{α}, and h_{α} in A. Suppose that Z is irreducible as a module over the linear span of x_{α}, y_{α}, and h_{α} in A, as a Lie algebra over k. This implies that Z is isomorphic as a module over the linear span of x_{α}, y_{α}, and h_{α} in A to a module of the form W (m_{Z}) as in the previous section for some nonnegative integer m_{Z}, as in Section 15.9.

Note that m_{Z} is the largest eigenvalue of ρ_{h_{α}} on Z, because of the analogous property of W (m_{Z}). In particular, m_{Z} is an eigenvalue of ρ_{h_{α}} on W, so that

m_{Z} = m − 2 j_{Z} (39.4.12)

for some nonnegative integer j_{Z}, as before. We also have that

2 j_{Z} ≤ m, (39.4.13)

because m_{Z} ≥ 0.

Of course, Z corresponds to the direct sum of one-dimensional eigenspaces of ρ_{h_{α}}, as a vector space over k, with eigenvalues m_{Z} − 2 l, l = 0, 1, …, m_{Z}. Equivalently, these eigenvalues are equal to m−2 j_{Z} −2 l, l = 0, 1, …, m_{Z}. This means that the eigenvalues are as in (39.4.11), with

j_{Z} ≤ j ≤ m − j_{Z}. (39.4.14)

It follows that Z ∩ V_{ν−j α} (39.4.15)

has dimension one when (39.4.14) holds, and that Z corresponds to their direct sum, as a vector space over k.

Weyl’s theorem implies that W corresponds to the direct sum of finitely many irreducible submodules, as a module over the linear span of x_{α}, y_{α}, and h_{α} in A. Each of these irreducible submodules corresponds to the direct sum of its intersections with the V_{ν−j α}’s, as a vector space over k, as in the preceding paragraph.

It follows that V_{ν−j α}, j = 0, 1, …, m, corresponds to the direct sum of its intersections with the irreducible submodules of W whose direct sum is W, as in the preceding paragraph, as a vector space over k. More precisely, V_{ν−j α} corresponds to the direct sum of its intersection with the irreducible submodules of W whose direct sum is W for which the corresponding j_{Z} as in (39.4.12) satisfies (39.4.14).

If j is a nonnegative integer with 2 j ≤ m and Z is as before, then j ≤ m−j_{Z}, by (39.4.13). This means that (39.4.14) reduces to asking that j_{Z} ≤ j. In this case, we get that V_{ν−j α} corresponds to the direct sum of its intersections with the irreducible submodules of W whose direct sum is W for which the corresponding j_{Z} is less than or equal to j, as a vector space over k.

If l is a nonnegative integer with 2 l ≤ m, then let n_{l} be the number of these irreducible submodules of W whose direct sum is W, and for which the corresponding j_{Z} as in is equal to l. It follows that

dim V_{ν−j α} = ∑_{l=0}^{j} n_{l} (39.4.16)

for j = 0, 1, …, [m/2], where [m/2] is the integer part of m/2, as on p120 of [14].

39.5 Some traces of ρ_{x_{α}} ∘ ρ_{z_{α}}

Let us continue with the same notation and hypotheses as in the previous sections. Let l be a nonnegative integer with l ≤ m. We would like to compute the trace of ρ_{x_{α}} ∘ ρ_{z_{α}} on V_{ν−l α}, as on p120f of [14].

Let Z be a nontrivial linear subspace of W that is an irreducible submodule of W, as a module over the linear span of x_{α}, y_{α}, and h_{α}, as in the previous section. Let j_{Z} be as in (39.4.12), and suppose that

j_{Z} ≤ l ≤ m − j_{Z}. (39.5.1)

This implies that Z ∩ V_{ν−l α} has dimension one as a vector space over k, as before. More precisely, Z ∩ V_{ν−l α} is spanned by an eigenvector of ρ_{h_{α}}, with eigenvalue (ν − l α)(h_{α}) = m − 2 l = m_{Z} − 2 (l − j_{Z}). (39.5.2)

39.5. SOME TRACES OF ρ_{X_{α}} ∘ ρ_{Z_{α}} 807

Remember that Z is isomorphic as a module over the linear span of x_{α}, y_{α}, and h_{α} in A to a module of the form W (m_{Z}) in Section 39.3. Using this isomorphism, Z ∩ V_{ν−l α} corresponds to the linear span of v_{j} in W (m_{Z}), with j = l − j_{Z}. This is the same as the span of u_{j} in W (m_{Z}), with j = l − j_{Z}. We have seen that x_{α} · (z_{α} · u_{j}) can be expressed as a multiple of u_{j}, as in (39.3.10).

It follows that the restriction of ρ_{x_{α}} ∘ ρ_{z_{α}} to Z ∩ V_{ν−l α} is equal to multiplication by (m_{Z} − (l − j_{Z})) (l − j_{Z} + 1) (α(t_{b,α})/2). (39.5.3)

This is the same as

(m − l − j_{Z}) (l − j_{Z} + 1) (α(t_{b,α})/2), (39.5.4)

by (39.4.12).

Suppose that 2 l ≤ m. (39.5.5)

This implies that l ≤ m − j_{Z}, by (39.4.13), so that (39.5.1) reduces to the condition that j_{Z} ≤ l, as before. Remember that W corresponds to the direct sum of finitely many irreducible submodules, as a module over the linear span of x_{α}, y_{α}, and h_{α} in A, as in the previous section. We have also seen that V_{ν−l α} corresponds to the direct sum of its intersections with these irreducible submodules of W, as a vector space over k. More precisely, in this case V_{ν−l α} corresponds to the direct sum of its intersections with the irreducible submodules of W whose direct sum is W and for which the corresponding j_{Z} is less than or equal to l.

If j is a nonnegative integer with 2 j ≤ m, then n_{j} is the number of the irreducible submodules of W whose direct sum is W and for which the corresponding j_{Z} is equal to j, as before. Remember that ρ_{x_{α}} ∘ ρ_{z_{α}} maps V_{ν−l α} into itself, as in the previous section. The trace of the restriction of ρ_{x_{α}} ∘ ρ_{z_{α}} to V_{ν−l α} is equal to

tr_{V_{ν−l α}}(ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{l} n_{j} (m − l − j) (l − j + 1) (α(t_{b,α})/2). (39.5.6)

More precisely, for each j = 0, 1, …, l, there are n_{j} terms in the trace of the form (39.5.4), with j_{Z} = j.

Observe that n_{j} = dim V_{ν−j α} − dim V_{ν−(j−1) α} (39.5.7)

for each j = 0, 1, …, [m/2], by (39.4.16). This uses (39.4.4) when j = 0, which says that dim V_{ν+α} = 0. (39.5.8)

It follows that

(39.5.9) tr_{V_{ν−l α}}(ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{l}(dim V_{ν−j α} − dim V_{ν−(j−1) α})
·(m − l − j) (l − j + 1) (α(t_{b,α})/2).

The right side of (39.5.9) is equal to

∑_{j=0}^{l}(dim V_{ν−j α}) (m − l − j) (l − j + 1) (α(t_{b,α})/2) (39.5.10)

minus

∑_{j=0}^{l}(dim V_{ν−(j−1) α}) (m − l − j) (l − j + 1) (α(t_{b,α})/2). (39.5.11)

Note that the j = 0 term in (39.5.11) is equal to 0, by (39.5.8). It is easy to see that (39.5.11) is equal to

∑_{j=0}^{l}(dim V_{ν−j α}) (m − l − j − 1) (l − j) (α(t_{b,α})/2), (39.5.12)

because the j = l term in this sum is equal to 0.

Observe that

(m − l − j) (l − j + 1) − (m − l − j − 1) (l − j) (39.5.13)

= (m − l − j) (l − j) + (m − l − j)
−(m − l − j) (l − j) + (l − j)
= m − 2 j

for each j = 0, 1, …, l. Using this, we get that

tr_{V_{ν−l α}}(ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{l}(dim V_{ν−j α}) (m − 2 j) (α(t_{b,α})/2). (39.5.14)

More precisely, the right side is obtained by taking (39.5.10) minus (39.5.12), and using (39.5.13) for the factors in the middle of the terms in the sums. Remember that this holds when 2 l ≤ m, as in (39.5.5).

39.6 The case where 2 l > m

We continue with the same notation and hypotheses as in the previous sections. In particular, l is a nonnegative integer with l ≤ m, as in the preceding section, and we suppose that 2 l > m (39.6.1)

in this section. We would like to compute the trace of ρ_{x_{α}} ∘ ρ_{z_{α}} on V_{ν−l α}, as on p121 of [14].

Let us check that

dim V_{ν−j α} = dim V_{ν−(m−j) α} (39.6.2)

39.6. THE CASE WHERE 2 L > M 809

for j = 0, 1, …, m. Remember that the α-string of weights through ν is as in (39.4.5). The α-string through ν is mapped to itself by the symmetry σ_{α} on E_{R} that maps Φ onto itself, as in Section 38.12. More precisely, σ_{α} maps ν − j α to ν −(m−j) α for each j = 0, 1, …, m, by (39.4.6). This implies (39.6.2), because the dimensions of the weight spaces are invariant under the action of the Weyl group of Φ on the weights of V, as in Section 38.7.

Let Z be a nontrivial linear subspace of W that is an irreducible submodule of W, as a module over the linear span of x_{α}, y_{α}, and h_{α} in A, as in Section 39.4. If j_{Z} is as in (39.4.12), then 2 j_{Z} ≤ m, as in (39.4.13), so that j_{Z} < l, by (39.6.1). This means that (39.5.1) holds exactly when l ≤ m − j_{Z}, which is the same as saying that j_{Z} ≤ m − l. (39.6.3)

In this case, Z ∩ V_{ν−l α} has dimension one as a vector space over k, as in the previous two sections. The restriction of ρ_{x_{α}} ∘ ρ_{z_{α}} to Z ∩ V_{ν−l α} is equal to multiplication by (39.5.4), as before.

Remember that W corresponds to the direct sum of finitely many irreducible submodules, as a module over the linear span of x_{α}, y_{α}, and h_{α} in A, as in Section 39.4. We also have that V_{ν−l α} corresponds to the direct sum of its intersections with these irreducible submodules of W, as a vector space over k, as before. In this case we get more precisely that V corresponds to the direct sum of its intersections with the irrducible submodules of W whose direct sum is W and for which the corresponding j_{Z} satisfies (39.6.3). Note that

m − l < m/2, (39.6.4)

by (39.6.1).

If j is a nonnegative integer with j ≤ m − l, then j < m/2, by (39.6.4). Remember that n_{j} is the number of the irreducible submodules of W whose direct sum is W and for which the corresponding j_{Z} is equal to j, as in the previous two sections. Of course, ρ_{x_{α}} ∘ ρ_{z_{α}} maps V_{ν−l α} into itself, as in Section 39.4. The trace of the restriction of ρ_{x_{α}} ∘ ρ_{z_{α}} to V_{ν−l α} is equal to

tr_{V_{ν−l α}}(ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{m−l} n_{j} (m − l − j) (l − j + 1) (α(t_{b,α})/2). (39.6.5)

Indeed, for each j = 0, 1, …, m − l, there are n_{j} terms in the trace of the form (39.5.4), with j_{Z} = j, as before.

Observe that the j = m − l term in the sum on the right side of (39.6.5) is equal to 0. Thus

tr_{V_{ν−l α}}(ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{m−l−1} n_{j} (m − l − j) (l − j + 1) (α(t_{b,α})/2). (39.6.6)

More precisely, if l = m, then the sum on the right should be interpreted as being equal to 0.

As in the previous section, we can use (39.5.7) to express the sum as

∑_{j=0}^{m−l−1}(dim V_{ν−j α}) (m − l − j) (l − j + 1) (α(t_{b,α})/2) (39.6.7)

minus

∑_{j=0}^{m−l−1}(dim V_{ν−(j−1) α}) (m − l − j) (l − j + 1) (α(t_{b,α})/2). (39.6.8)

The j = 0 term in (39.6.8) is equal to 0, by (39.5.8), as before. One can check that (39.6.8) is equal to

∑_{j=0}^{m−l−1}(dim V_{ν−j α}) (m − l − j − 1) (l − j) (α(t_{b,α})/2), (39.6.9)

because the j = m − l − 1 term in this sum is equal to 0. Of course, all of these sums should be interpreted as being equal to 0 when l = m, as in the preceding paragraph.

As before, we can use (39.5.13) to get that

tr_{V_{ν−l α}}(ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{m−l−1}(dim V_{ν−j α}) (m − 2 j) (α(t_{b,α})/2). (39.6.10)

The right side is obtained by taking (39.6.7) minus (39.6.9), and using (39.5.13) for the factors in the middle of the terms in the sums again. The sum on the right should be interpreted as being equal to 0 when l = m, as usual.

39.7 Some related expressions for traces

Let us continue with the same notations and hypotheses as in the previous sections. Let us check that

tr_{V_{ν−l α}}(ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{l}(dim V_{ν−j α}) (m − 2 j) (α(t_{b,α})/2) (39.7.1)

for every l = 0, 1, …, m, as on p121 of [14]. This is the same as (39.5.14), which we have seen holds when 2 l ≤ m.

If 2 l > m, then l > m − l, and we would like to verify that

∑_{j=m−l}^{l}(dim V_{ν−j α}) (m − 2 j) = 0. (39.7.2)

It is easy to see that (39.7.1) follows from (39.6.10) and (39.7.2). Observe that

m − 2 (m − j) = −(m − 2 j) (39.7.3) for every j. This implies that

(dim V_{ν−(m−j) α}) (m − 2 (m − j)) = −(dim V_{ν−j α}) (m − 2 j) (39.7.4)

for j = 0, 1, …, m, by (39.6.2). One can get (39.7.2) from (39.7.4), using also the fact that the j = m/2 term in the sum on the left side of (39.7.2) is equal to 0 when m is even.

Let α ∈ Φ be given again, and suppose now that ν ∈ A′_{0} is any weight of V with respect to A_{0}. We do not ask that ν +α not be a weight of V, as in (39.4.4). As in Section 38.12, we can take r, q to be the largest integers such that ν − r α and ν + q α are weights of V, so that q, r ≥ 0, by hypothesis. Remember that r − q = λ_{α}(ν), as before. Thus the previous remarks hold for ν + q α in place of ν, and with m = r + q = λ_{α}(ν + q α). (39.7.5)

It follows that

tr_{V_{ν+(q−l) α}}(ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{l}(dim V_{ν+(q−j) α}) (m − 2 j) (α(t_{b,α})/2) (39.7.6)

for every l = 0, 1, …, m, as in (39.7.1). In particular, we can take l = q, to get that

tr_{V_{ν}} (ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{q}(dim V_{ν+(q−j) α}) (m − 2 j) (α(t_{b,α})/2). (39.7.7)

Equivalently,

tr_{V_{ν}} (ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{q}(dim V_{ν+j α}) (m − 2 (q − j)) (α(t_{b,α})/2). (39.7.8)

Observe that

m − 2 (q − j) = r − q + 2 j = λ_{α}(ν) + 2 j = ν(h_{α}) + 2 j, (39.7.9)

where the last step is as in Sections 32.2 and 32.3. This implies that

tr_{V_{ν}} (ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{q}(dim V_{ν+j α}) (ν(h_{α}) + 2 j) (α(t_{b,α})/2). (39.7.10)

If j > q, then ν + j α is not a weight of V, so that dim V_{ν+j α} = 0. Thus

tr_{V_{ν}} (ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{∞}(dim V_{ν+j α}) (ν(h_{α}) + 2 j) (α(t_{b,α})/2), (39.7.11)

where the sum on the right reduces to a finite sum. This corresponds to some remarks on p121 of [14].

812 CHAPTER 39. DIMENSIONS OF WEIGHT SPACES

39.8 Related identities for suitable ν

Let us continue with the same notation and hypotheses as in the previous sections, and let Υ be as in Section 32.3. Let α ∈ Φ be given again, and suppose that ν ∈ A′_{0} has the property that (39.8.1) ν + j_{0} α is a weight of V with respect to A_{0} for some integer j_{0}. This implies that ν + j_{0} α ∈ Υ, as in Section 38.12. It follows that ν ∈ Υ, because Υ is a group with respect to addition that contains Φ.

Let q_{0}, r_{0} be the maximal and minimal integers such that (39.8.2) ν + q_{0} α, ν + r_{0} α are weights of V with respect to A_{0}, respectively. Thus r_{0} ≤ j_{0} ≤ q_{0}. If j ∈ Z and r_{0} ≤ j ≤ q_{0}, then (39.8.3) ν + j α is a weight of V with respect to A_{0}, as in Section 38.12. Equivalently, ν + r_{0} α, …, ν + q_{0} α, (39.8.4) is the α-string through any of its terms, as before.

Of course, σ_{α}(ν + j α) = ν + j α − (λ_{α}(ν) + j λ_{α}(α)) α = ν − (λ_{α}(ν) + j) α (39.8.5) for every j ∈ Z. Remember that the set of weights of V is invariant under the Weyl group of Φ, as in Section 38.7. This means that σ_{α} maps (39.8.4) onto itself, in the opposite order, as before. In particular, σ_{α}(ν + r_{0} α) = ν + q_{0} α (39.8.6) It follows that −r_{0} − q_{0} = λ_{α}(ν), (39.8.7) which could also be obtained from the analogous statement in Section 38.12.

Note that dim V_{ν+j α} = dim V_{σ_{α}(ν+j α)} (39.8.8) for every j ∈ Z, because the dimensions of the weight spaces are invariant under elements of the Weyl group of Φ, as in Section 38.7 again. Thus dim V_{ν+j α} = dim V_{ν−(λ_{α}(ν)+j) α} (39.8.9) for every j ∈ Z, which is the same as (39.6.2) when q_{0} = 0. We also have that λ_{α}(ν) − 2 (λ_{α}(ν) + j) = −(λ_{α}(ν) + 2 j) (39.8.10) for every j ∈ Z. It follows that

(dim V_{ν−(λ_{α}(ν)+j) α}) (λ_{α}(ν) − 2 (λ_{α}(ν) + j)) (39.8.11)

= −(dim V_{ν+j α}) (λ_{α}(ν) + 2 j)

for every j ∈ Z, which is the same as (39.7.4) when q_{0} = 0.

Let us check that

∑_{j=−∞}^{∞}(dim V_{ν+j α}) (λ_{α}(ν) + 2 j) = 0. (39.8.12)

More precisely, the sum on the left reduces to a finite sum, because dim V_{ν+j α} = 0 unless ν + j α is a weight of V. If λ_{α}(ν) is an even integer, then the term in the sum corresponding to j = −λ_{α}(ν)/2 is automatically equal to 0. If j ∈ Z and j ≠ −λ_{α}(ν)/2, then the terms in the sum corresponding to j and to −(λ_{α}(ν) + j) cancel with each other, because of (39.8.11). This means that the terms corresponding to j > −λ_{α}(ν)/2 and j < −λ_{α}(ν)/2 cancel in the sum.

Remember that λ_{α}(ν) = ν(h_{α}), as in Sections 32.2 and 32.3. Thus (39.8.12) is the same as saying that

∑_{j=−∞}^{∞}(dim V_{ν+j α}) (ν(h_{α}) + 2 j) = 0. (39.8.13)

In fact, this holds for every ν ∈ A′_{0}. Indeed, if ν ∈ A′_{0} and ν +j α is not a weight of V for any j ∈ Z, then dim V_{ν+j α} = 0 for every j ∈ Z, and (39.8.13) holds trivially. This basically corresponds to (10) on p122 of [14].

If ν ∈ A′_{0} is not a weight of V, then

∑_{j=1}^{∞}(dim V_{ν+j α}) (ν(h_{α}) + 2 j) = 0, (39.8.14)

where the sum on the left reduces to a finite sum as before. This holds trivially when ν + j α is not a weight of V for any positive integer j, as in the preceding paragraph. Otherwise, suppose that (39.8.1) holds for some j_{0} ≥ 1, and let r_{0} be as in (39.8.2). In this case, r_{0} ≥ 1, because of (39.8.3). This means that (39.8.14) follows from (39.8.13), as on p121f of [14].

39.9 Some remarks about dual bases

We continue with the same notation and hypotheses as in the previous sections. Remember that Δ is a base for Φ, and that h_{α}, α ∈ Δ, is a basis for A_{0}, as a vector space over k.

If β ∈ Δ, then there is a unique μ_{β} ∈ A′_{0} such that

(39.9.1)  μ_{β}(h_{α}) = 1 when α = β
= 0 for every α ∈ Δ with α ≠ β.

814 CHAPTER 39. DIMENSIONS OF WEIGHT SPACES The μ_{β}’s, β ∈ Δ, form a basis for A′_{0}, as a vector space, which is the dual basis associated to h_{α}, α ∈ Δ.

If ν ∈ A′_{0}, then there is a unique t_{b,ν} ∈ A_{0} such that ν(w) = b(w, t_{b,ν}) (39.9.2) for every w ∈ A_{0}, because the restriction of b(·, ·) to A_{0} is nondegenerate, as in Section 39.2. More precisely, ν ↦ t_{b,ν} is a one-to-one linear mapping from A′_{0} onto A_{0}.

In Section 39.2, we took w_{β}, β ∈ Δ, to be the basis for A_{0} that is dual to h_{α}, α ∈ Δ, with respect to the bilinear form b(·, ·) on A_{0}. If α, β ∈ Δ, then μ_{β}(h_{α}) = b(h_{α}, w_{β}) (39.9.3) by construction. This means that b(h_{α}, t_{b,μ_{β}}) = b(h_{α}, w_{β}) (39.9.4) for every α, β ∈ Δ, by (39.9.2). It follows that t_{b,μ_{β}} = w_{β} (39.9.5) for every β ∈ Δ.

If ν ∈ A′_{0}, then

ν(h_{α}) = ∑_{β∈Δ} ν(h_{β}) μ_{β}(h_{α}) (39.9.6)

for every α ∈ Δ. This implies that

ν = ∑_{β∈Δ} ν(h_{β}) μ_{β}, (39.9.7)

as elements of A′_{0}. Thus

t_{b,ν} = ∑_{β∈Δ} ν(h_{β}) t_{b,μ_{β}} = ∑_{β∈Δ} ν(h_{β}) w_{β}. (39.9.8)

If ν_{1}, ν_{2} ∈ A′_{0}, then put b′(ν_{1}, ν_{2}) = b(t_{b,ν_{1}}, t_{b,ν_{2}}). (39.9.9) This defines a nondegenerate bilinear form on A′_{0}, as in Section 17.10. Note that b′(ν_{1}, ν_{2}) = ν_{2}(t_{b,ν_{1}}) (39.9.10) for every ν_{1}, ν_{2} ∈ A′_{0}, by (39.9.2). It follows that

b′(ν_{1}, ν_{2}) = ∑_{β∈Δ} ν_{1}(h_{β}) ν_{2}(w_{β}) (39.9.11)

39.10. TRACES AND B′ 815

for every ν_{1}, ν_{2} ∈ A′_{0}, by (39.9.8). This corresponds to some remarks on p121 of [14].

If α ∈ Φ, then b′(α, α) = α(t_{b,α}) ≠ 0, (39.9.12)

as in Section 39.2. We also have that

h_{α} = (2/α(t_{b,α})) t_{b,α} = (2/b′(α, α)) t_{b,α}, (39.9.13)

because h_{α} is a multiple of t_{b,α}, and α(h_{α}) = 2. If ν ∈ A′_{0}, then we get that

ν(h_{α}) = (2/α(t_{b,α})) ν(t_{b,α}) = (2/α(t_{b,α})) b′(α, ν). (39.9.14)

39.10 Traces and b′

Let us continue with the same notation and hypotheses as in the previous sections, and let ν ∈ A′_{0} be a weight of V with respect to A_{0} again. If w ∈ A_{0}, then the action of w on V_{ν} is the same as multiplication by ν(w), by definition of V_{ν}. If α ∈ Δ, then it follows that ρ_{h_{α}} ∘ ρ_{w_{α}} corresponds to multiplication by ν(h_{α}) ν(w_{α}) on V_{ν}. In particular, ρ_{h_{α}} ∘ ρ_{w_{α}} maps V_{ν} into itself, with trace

tr_{V_{ν}} (ρ_{h_{α}} ∘ ρ_{w_{α}}) = (dim V_{ν}) ν(h_{α}) ν(w_{α}). (39.10.1)

This implies that

∑_{α∈Δ} tr_{V_{ν}} (ρ_{h_{α}} ∘ ρ_{w_{α}}) = (dim V_{ν}) ∑_{α∈Δ} ν(h_{α}) ν(w_{α}). (39.10.2)

Thus

∑_{α∈Δ} tr_{V_{ν}} (ρ_{h_{α}} ∘ ρ_{w_{α}}) = (dim V_{ν}) b′(ν, ν), (39.10.3)

where b′(·, ·) is as in (39.9.9), by (39.9.11). This corresponds to (8) on p121 of [14].

Let α ∈ Φ be given, and let us rewrite (39.7.11) using b′(·, ·). Namely,

tr_{V_{ν}} (ρ_{x_{α}} ∘ ρ_{z_{α}}) = ∑_{j=0}^{∞}(dim V_{ν+j α}) b′(α, ν + j α), (39.10.4)

by (39.9.12) and (39.9.14). This corresponds to (7) on p121 of [14].

Let c_{A}(b) be the Casimir element of UA associated to b, as in Section 39.1. The action of c_{A}(b) on V leads to a linear mapping c_{A,V} (b) from V into itself. Using the expression (39.2.13) for c_{A}(b), we get that

c_{A,V} (b) = ∑_{α∈Δ} ρ_{h_{α}} ∘ ρ_{w_{α}} + ∑_{α∈Φ} ρ_{x_{α}} ∘ ρ_{z_{α}}. (39.10.5)

Thus

(39.10.6) tr_{V_{ν}} c_{A,V} (b) = (dim V_{ν}) b′(ν, ν)

+ ∑_{α∈Φ} ∑_{j=0}^{∞}(dim V_{ν+j α}) b′(α, ν + j α),

by (39.10.3) and (39.10.4). This corresponds to some remarks on p121 of [14].

Observe that

∑_{α∈Φ} b′(α, ν) = 0, (39.10.7)

because the term corresponding to any α ∈ Φ cancels with the term corresponding to −α. This permits us to drop the j = 0 term from the sum on the right side of (39.10.6), to get that

(39.10.8) tr_{V_{ν}} c_{A,V} (b) = (dim V_{ν}) b′(ν, ν)

+ ∑_{α∈Φ} ∑_{j=1}^{∞}(dim V_{ν+j α}) b′(α, ν + j α).

This corresponds to some more remarks on p121 of [14].

If ν ∈ A′_{0} is not a weight of V, then

∑_{j=1}^{∞}(dim V_{ν+j α}) b′(α, ν + j α) = 0 (39.10.9)

for every α ∈ Φ. This follows from (39.8.14), using (39.9.12) and (39.9.14). Of course, V_{ν} = {0} in this case, so that the left side of (39.10.8) and the first term on the right side are equal to 0. In fact, (39.10.8) also holds under these conditions, because of (39.10.9). This corresponds to some remarks on p121f of [14].

Similarly, if ν ∈ A′_{0} and α ∈ Φ, then

∑_{j=−∞}^{∞}(dim V_{ν+j α}) b′(α, ν + j α) = 0. (39.10.10)

This follows from (39.8.13), using (39.9.12) and (39.9.14) again. Equivalently,

∑_{j=1}^{∞}(dim V_{ν−j α}) b′(−α, ν − j α) (39.10.11)

= (dim V_{ν}) b′(α, ν) + ∑_{j=1}^{∞}(dim V_{ν+j α}) b′(α, ν + j α).

Remember that Δ is a base for Φ, and that Φ^{+} is the set of positive roots in Φ with respect to Δ. The sum over α ∈ Φ in the right side of (39.10.8) can be expressed as

∑_{α∈Φ^{+}} ∑_{j=1}^{∞}(dim V_{ν+j α}) b′(α, ν + j α) (39.10.12)

+ ∑_{α∈Φ^{+}} ∑_{j=1}^{∞}(dim V_{ν−j α}) b′(−α, ν − j α).

Thus we can use (39.10.11) to get that

(39.10.13) tr_{V_{ν}} c_{A,V} (b) = (dim V_{ν}) b′(ν, ν) + ∑_{α∈Φ^{+}}(dim V_{ν}) b′(α, ν)

+2 ∑_{α∈Φ^{+}} ∑_{j=1}^{∞}(dim V_{ν+j α}) b′(α, ν + j α).

Put

δ = \frac{1}{2} ∑_{α∈Φ^{+}} α, (39.10.14)

which was denoted ρ in Section 30.6. Using this, we get that

(39.10.15) tr_{V_{ν}} c_{A,V} (b) = (dim V_{ν}) b′(ν + 2 δ, ν)

+2 ∑_{α∈Φ^{+}} ∑_{j=1}^{∞}(dim V_{ν+j α}) b′(α, ν + j α).

This corresponds to some remarks on p122 of [14].

39.11 Using an algebraically closed field

We continue with the same notations and hypotheses as in the previous sections, and suppose also for the moment that k is algebraically closed.

Remember that the Casimir element c_{A}(b) of UA associated to b is in the center of UA, as in Section 39.1. This implies that the associated linear mapping c_{A,V} (b) on V commutes with the action of any element of A on V.

Remember that V is supposed to be irreducible as a module over A, as in Section 39.4. Thus we can use Schur’s lemma to get that

c_{A,V} (b) = c(A, b, V) I_{V} (39.11.1)

for some c(A, b, V) ∈ k, where I_{V} is the identity mapping on V, as in Section 6.14.

Of course, V is supposed to have positive finite dimension as a vector space over k too. This implies that V has a primitive or maximal vector of some weight μ ∈ A′_{0}, as in Section 38.1. It follows that V is standard cyclic of weight μ, because V is irreducible as a module over A, as in Section 33.10.

Remember that the weight space V_{μ} corresponding to μ has dimension one as a vector space over k, as in Section 33.8. This implies that

tr_{V_{μ}}c_{A,V} (b) = c(A, b, V). (39.11.2)

If α ∈ Φ^{+} and j ∈ Z_{+}, then μ + j α is not a weight of V, as in Section 33.8 again. Thus we can use (39.10.15) to get that

c(A, b, V) = b′(μ + 2 δ, μ). (39.11.3)

This corresponds to some remarks on p122 of [14].

Otherwise, if we do not ask that k be algebraically closed, then let k_{1} be an algebraically closed field that contains k as a subfield, as usual. We can get a vector space Ṽ over k_{1} from V as in Section 33.2, as well as a Lie algebra (Ã, [·, ·]_{A}) over k_{1}. Remember that the action of A on V has a unique extension to an action of Ã on Ṽ so that Ṽ becomes a module over Ã, as a Lie algebra over k_{1}, as before. We have also seen that the conditions on A discussed in Section 33.1 lead to analogous properties of Ã, as in Section 33.4.

Let ˜(A_{0}) be the linear subspace of Ã that corresponds to A_{0}, and let ˜(A_{0})′ be the dual of ˜(A_{0}), as a vector space over k_{1}, as before. If ν ∈ A′_{0}, then let ν̃

be the unique extension of ν to an element of ˜(A_{0})′.

Remember that V is standard cyclic of some weight μ ∈ A_{0}, as a module over A, as in Section 38.2. This uses the hypotheses that V have positive finite dimension as a vector space over k, and that V is irreducible as a module over A, as in Section 39.4.

It is easy to see that a primitive or maximal vector in V of weight μ is also primitive or maximal of weight μ̃ in Ṽ, as a module over Ã. Using this, one can check that Ṽ is standard cyclic of weight μ̃, as a module over Ã. This implies that Ṽ is irreducible as a module over Ã, because Ṽ has finite dimension as a vector space over k_{1}, as in Section 33.9.

The bilinear form b(·, ·) on A considered in Section 39.2 has a unique extension to a bilinear form b̃(·, ·) on Ã. One can verify that this extension satisfies the analogues of the conditions on b(·, ·) mentioned earlier. The basis for A and corresponding dual basis for A with respect to b(·, ·) mentioned in Section 39.2 can be used in Ã too.

One can use b̃(·, ·) to get an associated Casimir element c̃_{A}(b̃) as in Section 39.1. This can be expressed in terms of the bases for Ã mentioned in the preceding paragraph as in Section 39.2. This leads to a linear mapping c̃_{_{A},_{V}}(b̃) from V into itself, as in the previous section. This linear mapping can be expressed in terms of the bases for Ã mentioned in the preceding paragraph in the same way as before. In particular, c̃_{_{A},_{V}}(b̃) is the same as the natural

extension of c_{A,V} (b) to a mapping from Ṽ into itself that is linear over k_{1}. Because k_{1} is algebraically closed, we can use Schur’s lemma to get that

c̃_{_{A},_{V}}(b̃) = c̃(Ã, b̃, Ṽ) I_{V} (39.11.4)

for some c̃(Ã, b̃, Ṽ) ∈ k_{1}, where I_{V} is the identity mapping on Ṽ, as before. We also get that

tr_{V_{μ}}c̃_{_{A},_{V}}(b̃) = c̃(Ã, b̃, Ṽ), (39.11.5)

where Ṽ_{μ} is the weight space in Ṽ associated to μ̃, as before.

If γ ∈ ˜(A_{0})′, then there is a unique element t̃_{_{b},γ} of ˜(A_{0}) such that

γ(w) = b̃(w, t̃_{_{b},γ}) (39.11.6)

for every w ∈ ˜(A_{0}), because the restriction of b̃(·, ·) to ˜(A_{0}) is nondegenerate,

as before. Note that γ ↦ t̃_{_{b},γ} is a one-to-one linear mapping from ˜(A_{0})′ onto

˜(A_{0}), as before. If ν ∈ A′_{0}, and ν̃ is its unique extension to an element of ˜(A_{0})′, then it is easy to see that

t̃_{_{b},_{ν}} = t_{b,ν}. (39.11.7)

If γ_{1}, γ_{2} ∈ ˜(A_{0})′, then put

b̃′(γ_{1}, γ_{2}) = b̃(t̃_{_{b},γ_{1}}, t̃_{_{b},γ_{2}}), (39.11.8)

which defines a bilinear form on ˜(A_{0})′, as before. If ν_{1}, ν_{2} ∈ A′_{0} and ν̃_{1}, ν̃_{2} are

their unique extensions to elements of ˜(A_{0})′, then

b̃′(ν̃_{1}, ν̃_{2}) = b′(ν_{1}, ν_{2}), (39.11.9)

by (39.11.7).

Remember that the analogue Φ̃ of Φ for Ã and ˜(A_{0}) consists of α̃, α ∈ Φ, as in Section 33.4. Similarly, the collection Δ̃ of α̃, α ∈ Δ, is a base for Φ̃, as before. We have also seen that the collection Φ̃^{+} of positive roots in Φ̃ with respect to Δ̃ consists exactly of β̃, β ∈ Φ^{+}. Thus the analogue of (39.10.14) for Φ̃ and Δ̃ is

δ̃ = \frac{1}{2} ∑_{α∈Φ^{+}} α̃. (39.11.10)

This is the same as the unique extension of (39.10.14) to an element of ˜(A_{0})′.

Remember that V is standard cyclic of weight μ ∈ A′_{0}, and that Ṽ is standard cyclic of weight μ̃. It follows that

c̃(Ã, b̃, Ṽ) = b̃′(μ̃ + 2 δ̃, μ̃), (39.11.11)

as in (39.11.3). This implies that

c̃(Ã, b̃, Ṽ) = b′(μ + 2 δ, μ), (39.11.12)

by (39.11.9). In particular, this means that c̃(Ã, b̃, Ṽ) is an element of k. Remember that c̃_{_{A},_{V}}(b̃) is the same as the natural extension of c_{A,V} (b) to a

mapping from Ṽ to itself that is linear over k. Thus we get that (39.11.1) holds, where c(A, b, V) is as in (39.11.3), by (39.11.4) and (39.11.12).

39.12 Freudenthal’s formula

Let us continue with the same notation and hypotheses as in the previous sections, where k is not asked to be algebraically closed. Thus

c_{A,V} (b) = b′(μ + 2 δ, μ) I_{V} , (39.12.1)

by (39.11.1), (39.11.3), and their extensions to the case where k is not necessarily algebraically closed.

If ν ∈ A′_{0}, then we get that

tr_{V_{ν}} c_{A,V} (b) = (dim V_{ν}) b′(μ + 2 δ, μ). (39.12.2)

Combining this with (39.10.15), we obtain that

(dim V_{ν}) b′(μ + 2 δ, μ) = (dim V_{ν}) b′(ν + 2 δ, ν)

+2 ∑_{α∈Φ^{+}} ∑_{j=1}^{∞}(dim V_{ν+j α}) b′(α, ν + j α). (39.12.3)

Equivalently, this means that

(dim V_{ν}) (b′(μ + 2 δ, μ) − b′(ν + 2 δ, ν)) (39.12.4)

= 2 ∑_{α∈Φ^{+}} ∑_{j=1}^{∞}(dim V_{ν+j α}) b′(α, ν + j α).

This is Freudenthal’s formula, as on p122 of [14].

Of course, (39.12.4) is trivial unless ν + j α is a weight of V for some nonnegative integer j. In this case, ν ∈ Υ, as in Section 39.8. In particular, this means that ν ∈ E_{Q}, as in Section 32.3.

Suppose from now on in this section that

(39.12.5) b(·, ·) is the Killing form b_{A}(·, ·) on A.

Note that this satisfies the conditions mentioned in Section 39.2, including nondegeneracy, because A is semisimple.

Let b′_{A}(·, ·) be the corresponding bilinear form on A′_{0}, as in Section 39.9. If ν_{1}, ν_{2} ∈ Φ, then b′_{A}(ν_{1}, ν_{2}) corresponds to an element of Q, under the natural embedding of Q into k, as in Section 17.11. This implies that b′_{A}(ν_{1}, ν_{2}) corresponds to an element of Q when ν_{1}, ν_{2} ∈ E_{Q}.

Let (·, ·)_{E_{Q}} be the restriction of b′_{A}(·, ·) to E_{Q}, considered as taking values in Q, as in Section 17.12. More precisely, this is a bilinear form on E_{Q}, as a vector space over Q. Note that (·, ·)_{E_{Q}} is symmetric on E_{Q}, because b_{A}(·, ·) is symmetric on A, and thus b′_{A}(·, ·) is symmetric on A′_{0}.

Let (·, ·)_{E_{R}} be the natural extension of (·, ·)_{E_{Q}} to a bilinear form on E_{R}, as a vector space over R. Of course, (·, ·)_{E_{R}} is symmetric on E_{R}, and in fact it is an inner product on E_{R}, as a vector space over R, as in Section 17.13.

If α ∈ Φ, then the symmetry on E_{R} with vector α that maps Φ onto itself is the same as the reflection on E_{R} associated to α with respect to (·, ·)_{E_{R}}, as in Section 22.1. This means that (·, ·)_{E_{R}} is invariant under the Weyl group of Φ.

Remember that V is standard cyclic of weight μ ∈ A′_{0}, as in the previous section. In particular, μ is a weight of V, so that μ ∈ Υ, and thus μ ∈ E_{Q}. Of course, δ ∈ E_{Q} too, and it is easy to see that

b′_{A}(μ + 2 δ, μ) − b′_{A}(ν + 2 δ, ν) = (μ + 2 δ, μ)_{E_{R}} − (ν + 2 δ, ν)_{E_{R}}
(39.12.6)  = (μ + δ, μ + δ)_{E_{R}} − (ν + δ, ν + δ)_{E_{R}}

for every ν ∈ E_{Q}. Combining this with (39.12.4), we get that

(dim V_{ν}) ((μ + δ, μ + δ)_{E_{R}} − (ν + δ, ν + δ)_{E_{R}}) (39.12.7)

= 2 ∑_{α∈Φ^{+}} ∑_{j=1}^{∞}(dim V_{ν+j α}) (α, ν + j α)_{E_{R}}

for every ν ∈ E_{Q}.

Let Π = Π(V) be the set of ν ∈ A′_{0} such that ν is a weight of V, so that Π ⊆ Υ. Remember that Π is saturated as a subset of Υ, with highest weight μ, as in Section 38.12.

If ν ∈ Π, then

(ν + δ, ν + δ)_{E_{R}} ≤ (μ + δ, μ + δ)_{E_{R}}, (39.12.8)

with equality only when ν = μ, as in Section 30.11. This means that the left side of (39.12.7) is a positive number times the dimension of V_{ν} when ν ∈ Π and ν ≠ μ. This corresponds to some remarks on p122 of [14].

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Index

A((T)), 167 A[[T]], 81, 167 A[[T_{1}, …, T_{n}]], 111 A[T], 81 A[T_{1}, …, T_{n}], 111 A^{op}, 140 A_{d}[T_{1}, …, T_{n}], 120 A_{X}(k) = A_{X}, 561 A_{X}^{n}(k) = A_{X}^{n}, 561 absolute convergence, 19 absolute value functions, 3

ultrametric, 4 absolute values, 185 absolutely simple Lie algebras, 336 absorbing sets, 22 abstract Jordan decompositions, 295 abstract weights, 636 ad-diagonalizable elements, 294 ad-nilpotent elements, 197 ad-semisimple elements, 294 adjoint mappings, 18, 46, 48, 65, 67,

238, 255 adjoints of differential operators, 742 algebra automorphisms, 28, 34, 35 algebra homomorphisms, 28, 34, 35 algebra isomorphisms, 28, 34, 35 algebras, 27

associated to magmas, 561 α-string through β, 416 α-string through ν, 794 anti-self-adjointness, 34, 73, 327

linear mappings, 47, 254 antisymmetric mappings,

bilinear, 27, 43 linear, 43 multilinear, 149 archimedean absolute value functions,

4 archimedean property, 186 Assoc_{X}(k) = Assoc_{X}, 565 Assoc_{X}^{n}(k) = Assoc_{X}^{n}, 567 associative algebras, 28 associative bilinear mappings, 138

balanced sets, 22 balanced subgroups, 24 Banach algebras, 58 Banach spaces, 12 bases for free modules, 545 bases of root systems, 398 bilinear actions, 125 bilinear forms, 43 bilinear mappings, 20, 27 BL(V), 52 BL(V, W), 13 BL(V_{1}, …, V_{n}; W), 150 Borel subalgebras, 471, 479 bounded bilinear mappings, 20 bounded linear mappings, 13 bounded multilinear mappings, 150

C, 3 c(X, W), 163 c_{00}(X, W), 163 C_{A}(E), 153 Cartan matrix, 412 Cartan subalgebras, 374 Casimir elements, 280, 282, 283, 800 Cauchy products, 79 Cauchy–Schwarz inequality, 17 center, 153, 154 centralizer, 153 Chevalley algebras, 630 Chevalley bases, 621 closed balls, 3 commutative algebras, 28 commutative Lie algebras, 30 complete reducibility, 276 complex-linear mappings, 16 complex-linear subspaces, 323 complexifications, 323 conjugate-linear mappings, 16, 253 conjugation in k[\sqrt{−a}], 252 convergent series, 18 Coxeter graphs, 414

of root systems, 414

D_{n}(A), 213 d_{n}(A), 213 degree, 723 derivations, 32 derivatives, 107, 115, 728 derived algebra, 191 derived series, 191 descending central series, 194 diagonal automorphisms, 530, 603, 619 diagonal map, 555 diagonal matrices, 213 diagonalizable linear mappings, 217 direct products, 77 direct sums, 77 discrete absolute value functions, 7 discrete metric, 2 dominant integral weights, 675 dominant weights, 639 dual systems, 394 Dynkin diagrams, 433

E(A), 507 eigenvalues, 695 eigenvectors, 693, 695 Engel subalgebras, 369 equivalent absolute value functions, 6 Euler operator, 734 extended formal power series, 186

flags, 201 formal characters, 795 formal differential operators, 117, 682,

731 homogeneous, 122, 737 formal Laurent polynomials, 724 homogeneous, 735 formal Laurent series, 723 homogeneous, 735 formal polynomials, 81, 111, 532 homogeneous, 120 formal power series, 81, 111 free algebras, 561 free associative algebras, 565 free Lie algebras, 562 free magmas, 560 Freudenthal’s formula, 820 fundamental representations, 792 fundamental group (root systems), 639 fundamental modules, 792 fundamental root systems, 398 fundamental weights, 637

g_{n}(k), 123

general linear algebra, 39, 40 general linear group, 51 generators and relations, 573 GL(V), 51 gl(V), 40 GL_{n}(A), 51 gl_{n}(A), 39 graded algebra associated to UA, 544

Hermitian forms, 47, 254 highest weights, 646, 702 Hilbert spaces, 17 Hom(A, B), 27 Hom_{k}(A, B), 27 homomorphisms

between algebras, 28, 34, 35 between Lie algebras, 30 between modules, 26, 133, 134 between representations, 133, 134

ideals, 41, 42

identity matrix, 38 indecomposability, 699 inner automorphism, 306 inner derivations, 291 inner products, 16, 241, 255 Int A, 305 integral domains, 653 integral weights, 675 intertwining mappings, 127, 133, 134 invariant bilinear mappings, 64 invariant elements, 154 invariant sesquilinear forms, 67 inverse roots, 393 inverse systems, 394 invertible elements, 50 invertible linear mappings, 60 involutions, 34, 35, 328 irreducible representations, 143 irreducible root systems, 415 irreducible sets, 655, 657 isometric linear mappings, 60 isomorphic ordered fields, 187 isomorphic representations, 133, 134 isomorphic root systems, 388

Jacobi identity, 29

kernels of module homomorphisms, 41 killing form, 158

L(V), 40 L(V, W), 40 L(V_{1}, …, V_{n}; W), 148 L_{k}(V_{1}, …, V_{n}; W), 148 L_{X}(k) = L_{X}, 562 L_{X}^{n}(k) = L_{X}^{n}, 567 lengths in the Weyl group, 452 lengths of m-tuples, 546 Lie algebra homomorphisms, 30 Lie algebras, 30 Lie bracket, 30 Lie subalgebras, 30 Lie’s theorem, 206 linear mappings, 27 locally nilpotent linear mappings, 590 long roots, 420 lower central series, 194 LP_{A}(T_{1}, …, T_{n}), 724 LP_{A,d}(T_{1}, …, T_{n}), 735 LP_{A,d}^{l,+}(T_{1}, T_{2}), 745 LP_{A,d}^{l,−}(T_{1}, T_{2}), 745 LS_{A}(T_{1}, …, T_{n}), 723 LS_{A,d}(T_{1}, …, T_{n}), 735 LS_{A,d}^{l,+}(T_{1}, T_{2}), 745 LS_{A,d}^{l,−}(T_{1}, T_{2}), 745

M_{n}(A), 37 M_{X}, 560 magma homomorphisms, 560 magmas, 560 maximal roots, 421 maximal vector of weight μ, 695 maximal vector of weight λ, 312 metrics, 2 module automorphisms, 64 module homomorphisms, 26, 133, 134 module isomorphisms, 27, 133, 134 modules,

over associative algebras, 130 over k, 26 over Lie algebras, 131

multi-indices, 110, 532 multilinear mappings, 148 multiplicative identity elements, 28 multiplicative seminorms, 52 multiplicities, 693

N (A), 507

nilpotent elements of associative alge-

bras, 197 nilpotent Lie algebras, 194 non-archimedean absolute value func-

tions, 4 nondegenerate bilinear forms, 45 nondegenerate sesquilinear forms, 48,

254 nonnegative self-adjoint operators, 270 normalizer, 198 norms, 11

open balls, 2 open sets, 3 opposite algebra, 140 ordered fields, 184 ordered rings, 184 orthogonal transformations, 62 orthogonal vectors, 241, 255 Ostrowski’s theorems, 6, 7

P UA, 557 P. Hall families, 571 p-adic absolute value, 4 p-adic integers, 9 p-adic metric, 4 p-adic numbers, 6 polarization, 680 poles of bounded order, 168, 170, 173,

174, 176 positive formal series, 186 positive roots, 400 positive self-adjoint operators, 270 primitive elements of UA, 555 primitive of weight λ, 312 primitive of weight μ, 695 products of submodules, 190

Q, 4 Q_{p}, 6

R, 3 R_{+}, 7 radicals of bilinear mappings, 160 radicals of Lie algebras, 193 ranks of Lie algebras, 526, 659 ranks of root systems, 386 real parts, 262 real-bilinear mappings, 324 real-linear subspaces, 323 real-linear mappings, 16, 323 reduced expressions, 452 reduced root systems, 386 reductive Lie algebras, 234 reflections, 384 regular elements of Lie algebras, 659 regular linear functionals, 407 regular semisimple elements, 662 replicas, 224 representations,

of associative algebras, 129 of Lie algebras, 131

residue fields, 10 root lattices, 636 root strings, 416 root systems, 385

type A_{2}, 432, 435

type A_{n}, 424, 432
type B_{n}, 426
type BC_{n}, 427
type C_{n}, 426
type D_{n}, 429
type E_{6}, 449
type E_{7}, 447
type E_{8}, 445
type F_{4}, 440
type G_{2}, 436, 437

roots, 386

S_{n}(A), 213 s_{n}(A), 213 s_{n}(k), 124 saturated sets of weights, 646 scalar matrices, 213 Schur’s lemma, 143 self-adjoint linear mappings, 47, 254 self-adjoint sets, 244, 258 self-adjointness, 34, 73, 327 self-normalizing Lie subalgebras, 199 semi-ultrametrics, 2 semi-ultranorms, 11 semimetrics, 2 seminorms, 11 semisimple Lie algebras, 193 semisimple linear mappings, 294 semisimple modules, 276 Serre’s theorem, 586 sesquilinear forms, 47, 254 short roots, 420 simple Lie algebras, 229 simple modules, 143 simple root systems, 398 simple roots, 398 SL(V), 51 sl(V), 41 SL_{n}(A), 51 sl_{n}(A), 39 solvable Lie algebras, 191 special linear algebra, 39, 41 special linear group, 51 standard Borel subalgebras, 479 standard cyclic modules, 697 strong ad-nilpotence, 506 strongly dominant weights, 639 subalgebras, 28, 30, 480 submodules, 26, 132, 133 submultiplicative seminorms, 52 subrepresentations, 132 sum of root systems, 415 supports of functions, 163 symmetric algebras, 542 symmetric mappings,

bilinear, 27, 43 linear, 43 multilinear, 149

symmetric tensors, 166 symmetries, 383

T V, 535 T^{n}V, 166 T_{n,r}(A), 203 t_{n,r}(A), 203 tensor algebra, 535 tensor powers, 166 tensor products, 162, 704 termwise eventual agreement, 82, 168 termwise eventually constant, 82, 168 toral subalgebras, 341 totally real linear subspaces, 323 trace, 39, 41, 157 transpose matrices, 38 trivial absolute value function, 3 trivial ultranorm, 12

UA, 536 U_{l}(F), 201 ultrametric absolute values, 4 ultrametrics, 2 ultranorms, 11 unitary transformations, 62 universal enveloping algebras, 536

Vandermonde determinant, 680 vanishing to some order, 82, 120 Verma modules, 709 weight α, 342 weight λ, 312 weight μ, 693 weight spaces, 312, 693 weights, 312, 693 Weyl chamber, 408 Weyl group, 387 Weyl relations, 491 Weyl’s theorem, 278, 285

Z, 9

Z(A), 154 Z_{+}, 4 Z_{p}, 9 Zariski topology, 655