Publications and Preprints: Links and Details
Undergraduate coauthors are in italics
On extended Frobenius structures.
( ArXiv )
( Talk Notes )
Joint with
Agustina Czenky,
Jacob Kesten
and Abiel Quinonez
Submitted.
Abstract:
A classical result in quantum topology is that oriented 2-dimensional topological
quantum field theories (2-TQFTs) are fully classified by commutative Frobenius
algebras. In 2006, Turaev and Turner introduced additional structure on Frobenius
algebras, forming what are called extended Frobenius algebras, to classify 2-TQFTs
in the unoriented case. This work provides a systematic study of extended Frobenius
algebras in various settings: over a field, in a monoidal category, and in the framework
of monoidal functors. Numerous examples, classification results, and general
constructions of extended Frobenius algebras are established.
Twists of graded algebras in monoidal categories.
( ArXiv )
Joint with
Fernando Liu Lopez
To appear in Journal of Algebra.
Abstract:
Zhang twists are a common tool for deforming graded algebras over a field
in a way that preserves important ring-theoretic properties. We generalize
Zhang twists to the setting of closed monoidal categories equipped with
their self-enriched structure. Along the way, we prove several key results
about algebraic structures in closed monoidal categories missing from the
literature. We use these to ultimately prove Morita-type results, showcasing
when graded algebras with equivalent categories of graded modules can be
related by Zhang twists.
Reflective centers of module categories and quantum K-matrices.
( ArXiv )
( Video, Chalk Talk )
( Video, Slide Talk )
( Slides )
Joint with
Robert Laugwitz
and Milen Yakimov
Submitted
Abstract:
Our work is motivated by obtaining solutions to the quantum reflection equation (qRE).
To start, given a braided monoidal category \C and \C-module category \M, we introduce a version
of the Drinfeld center \Z(\C) of \C adapted for \M; we refer to this category as the reflective center
\E_\C(\M) of \M. Just like \Z(\C) is a canonical braided monoidal category attached to \C, we show
that \E_\C(\M) is a canonical braided module category attached to \M. We also study when \E_\C(\M)
possesses nice properties such as being abelian, finite, and semisimple.
Our second goal pertains to when \C is the category of modules over a quasitriangular Hopf
algebra H, and \M is the category of modules over an H-comodule algebra A. We show that the
reflective center \E_\C(\M) here is equivalent to a category of modules over (or, is represented by) an
explicit algebra, denoted by R_H(A), which we call the reflective algebra of A. This result is akin
to \Z(\C) being represented by the Drinfeld double Drin(H) of H. We study algebraic properties of
reflective algebras as well.
Our third set of results is also in the Hopf setting above. We show that reflective algebras are
quasitriangular H-comodule algebras, and examine their corresponding quantum K-matrices; this
yields solutions to the qRE. We also establish that the reflective algebra R_H(k) is an initial object
in the category of quasitriangular H-comodule algebras, where k is the ground field. The case when
H is the Drinfeld double of a finite group is illustrated.
Lastly, we study the reflective center \E_\C(\M) as a module category over \Z(\C) in the Hopf setting.
This action induces actions of \C on \E_\C(\M) (including that above), and it gives the reflective algebra
R_H(A) the structure of a Drin(H)-comodule algebra.
On non-counital Frobenius algebras.
( ArXiv )
Joint with
Amanda Hernandez and Harshit Yadav
To appear in Journal of Algebra and its Applications.
Abstract:
A Frobenius algebra is a finite-dimensional algebra A which comes equipped with a coassociative, counital comultiplication map ∆ that is an
A-bimodule map. Here, we examine comultiplication maps for generalizations of Frobenius algebras: finite-dimensional self-injective
(quasi-Frobenius) algebras. We show that large classes of such algebras, including finite-dimensional weak Hopf algebras, come equipped
with a map ∆ as above that is not necessarily counital. We also conjecture that this comultiplicative structure holds for self-injective algebras in general.
Constructing non-semisimple modular categories with local modules.
( ArXiv )
( Talk Notes )
Joint with
Robert Laugwitz
Communications in Mathematical Physics, 403(3) 1363-1409, 2023.
Abstract:
We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again,
modular. This generalizes previous work of A. Kirillov, Jr. and V. Ostrik [Adv. Math. 171 (2002), no. 2] in the semisimple setup. Examples of
non-semisimple modular categories via local modules, as well as connections to the authors' prior work on relative monoidal centers, are provided.
In particular, we classify rigid Frobenius algebras in Drinfeld centers of module categories over group algebras, thus generalizing the classification
by A. Davydov [J. Algebra 323 (2010), no. 5] to arbitrary characteristic.
Filtered Frobenius algebras in monoidal categories.
( ArXiv )
( Video, short talk )
( Talk Notes, long talk )
Joint with
Harshit Yadav
International Mathematics Research Notices, 2022, rnac314.
Abstract:
We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result:
A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor,
building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. Next, we produce equivalent conditions for an algebra in a rigid
monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs-Stigner. These two results of
independent interest are then used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric
finite tensor category C is represented by a Frobenius algebra in C. Several directions for further investigation are also proposed.
Algebraic properties of face algebras.
( ArXiv )
( Talk by F. Calderón )
Joint with
Fabio Calderón
Journal of Algebra and its Applications, 2022, 2350076.
Abstract:
Prompted an inquiry of Manin on whether a coacting Hopf-type structure H and an algebra A that is coacted upon share algebraic properties,
we study the particular case of A being a path algebra 𝕜Q of a finite quiver Q and H being Hayashi's face algebra ℌ(Q) attached to Q.
This is motivated by the work of Huang, Wicks, Won, and the second author, where it was established that the weak bialgebra coacting universally
on 𝕜Q (either from the left, right, or both sides compatibly) is ℌ(Q). For our study, we define the Kronecker square Qˆ of Q, and show that ℌ(Q)≅𝕜Qˆ
as unital algebras. Then we obtain ring-theoretic and homological properties of ℌ(Q) in terms of graph-theoretic properties of Q by way of Qˆ.
Constructing non-semisimple modular categories with relative monoidal centers.
( ArXiv )
( Pre-Talk Notes )
( Talk Notes )
Joint with
Robert Laugwitz
International Mathematics Research Notices, 2021, rnab097.
Abstract:
This paper is a contribution to the construction of non-semisimple modular categories. We show that, given such a modular category containing
a modular subcategory the Müger centralizer is also modular. As a consequence, we obtain conditions under which relative monoidal centers give
(non-semisimple) modular categories, and we also show that examples include representation categories of small quantum groups. We further derive
conditions under which representations of more general quantum groups, braided Drinfeld doubles of Nichols algebras of diagonal type, give
(non-semisimple) modular categories.
Universal quantum semigroupoids.
( ArXiv )
( Talk Notes )
Joint with
Hongdi Huang,
Elizabeth Wicks
and Robert Won
Journal of Pure and Applied Algebra., 227(2):107193, 2023.
Abstract:
We introduce the concept of a universal quantum linear semigroupoid (UQSGd), which is a weak bialgebra that coacts on a
(not necessarily connected) graded algebra A universally while preserving grading. We restrict our attention to algebraic
structures with a commutative base so that the UQSGds under investigation are face algebras (due to Hayashi). The UQSGd
construction generalizes the universal quantum linear semigroups introduced by Manin in 1988, which are bialgebras that
coact on a connected graded algebra universally while preserving grading. Our main result is that when A is the path algebra
kQ of a finite quiver Q, each of the various UQSGds intro- duced here is isomorphic to the face algebra attached to Q.
The UQSGds of preprojective algebras and of other algebras attached to quivers are also investigated.
Algebraic structures in group-theoretical fusion categories.
( ArXiv )
( Video )
( Talk Notes )
Joint with
Yiby Morales,
Monique Müller,
Julia Plavnik,
Ana Ros Camacho, and
Angela Tabiri
Algebras and Representation Theory, (2022), pp. 1-33.
Abstract:
It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category
serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize
this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in
group-theoretical fusion categories. This is achieved by providing the `free functor' Φ from a pointed fusion category to
a group-theoretical fusion category with a monoidal structure. Our algebras of interest are then constructed as the image of twisted
group algebras under Φ. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category,
and we establish a Frobenius monoidal structure on Φ as well. As a consequence, our algebras are Frobenius algebras in a group-theoretical
fusion category, and like twisted group algebras in the pointed case, they also enjoy several good algebraic properties.
Algebraic structures in comodule categories over weak bialgebras.
( ArXiv )
( Talk Notes )
Joint with
Elizabeth Wicks
and Robert Won
Communications in Algebra, 50(7):2877–2910, 2022.
Abstract:
For a bialgebra L coacting on a k-algebra A, a classical result states that A is a right L-comodule algebra if and only
if A is an algebra in the monoidal category M^L of right L-comodules; the former notion is formulaic while the latter
is categorical. We generalize this result to the setting of weak bialgebras H. The category M^H admits a monoidal structure
by work of Nill and Böhm-Caenepeel-Janssen, but the algebras in M^H are not canonically k-algebras. Nevertheless, we prove
that there is an isomorphism between the category of right H-comodule algebras and the category of algebras in M^H. We also
recall and introduce the formulaic notion of H coacting on a k-coalgebra and on a Frobenius k-algebra, respectively, and
prove analogous category isomorphism results. Our work is inspired by the physical applications of Frobenius algebras in
tensor categories and by symmetries of algebras with a base algebra larger than the ground field (e.g. path algebras). We
produce examples of the latter by constructing a monoidal functor from a certain corepresentation category of a bialgebra
L to the corepresentation category of a weak bialgebra built from L (a "quantum transformation groupoid"), thereby creating
weak quantum symmetries from ordinary quantum symmetries.
Tensor algebras in finite tensor categories.
( ArXiv )
( Talk Notes )
Joint with
Pavel Etingof
and Ryan Kinser
International Mathematics Research Notices, 2021, rnz332.
Abstract:
This paper introduces methods for classifying actions of finite-dimensional Hopfalgebras on
path algebras of quivers, and more generally on tensor algebras T_B(V) where B is a semisimple
k-algebra and V is a B-bimodule. We do this by working within the broader framework of finite
(multi-)tensor categories \C, parameterizing tensor algebras in \C in terms of \C-module
categories. We utilize this parametrization to obtain two classification results for actions
of semisimple Hopf algebras: the first for actions which preserve the ascending filtration on
tensor algebras, and the second for actions which preserve the descending filtration on
completed tensor algebras. Extending to more general fusion categories, we illustrate our
parameterization result for tensor algebras in the pointed fusion categories Vec_G^ω and in
group-theoretical fusion categories, especially for the representation category
\C = Rep(H_8) of the Kac-Paljutkin Hopf algebra. Finally returning to path algebras of quivers,
we give criteria for an indecomposable semisimple algebra in a group-theoretical fusion
category to be commutative upon applying a fiber functor.
Braided commutative algebras over quantized enveloping algebras.
( ArXiv )
( Video )
( Talk Notes )
Joint with
Robert Laugwitz
Transformation Groups 26 (2021), pp. 957-993.
Abstract:
We produce braided commutative algebras in braided monoidal categories by generalizing Davydov's full center construction
of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal
centers \Z_\B(\C) from algebras in B-augmented monoidal categories \C, where such \C and \Z_\B(\C) are defined by the first author
in previous work. Here, \B is an arbitrary braided monoidal category; Davydov's (and previous works of others) take place
in the special case when \B is the category of vector spaces Vect_k over a field k. Since key examples of relative monoidal
centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras
over such quantum groups.
One application of our work is that we produce Morita invariants for algebras in \B-augmented monoidal
categories. Moreover, for a large class of \B-augmented monoidal categories, our braided commutative algebras arise as a braided
version of centralizer algebras. This generalizes the fact that centers of algebras in Vect_k serve as Morita invariants.
Many examples are provided throughout.
Noncommutative Knörrer periodicity and noncommutative Kleinian singularities.
( ArXiv )
Joint with
Andrew Conner,
Ellen Kirkman, and
W. Frank Moore
J. Algebra 540 (2019), pp. 234–273.
Abstract:
We establish a version of Knörrer's Periodicity Theorem in the context of noncommutative invariant theory.
Namely, let A be a left noetherian AS-regular algebra, let f be a normal and regular element of A of positive degree,
and take B=A/(f). Then there exists a bijection between the set of isomorphism classes of indecomposable non-free
maximal Cohen-Macaulay modules over B and those over (a noncommutative analog of) its second double branched cover
(B^#)^#. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first
three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by
the second and fourth authors with Chan and Zhang.
An Invitation to Noncommutative Algebra.
( ArXiv )
EDGE Program’s Impact on the Mathematics Community and Beyond. Springer, Cham, 2019. pp. 339–366.
Abstract:
This is a brief introduction to the world of Noncommutative Algebra aimed for
advanced undergraduate and beginning graduate students.
Gelfand-Kirillov dimension of cosemisimple Hopf algebras.
( ArXiv )
Joint with
Alexandru Chirvasitu
and Xingting Wang
Proceedings of the American Mathematical Society 147 (2019), pp. 4665–4672.
Abstract:
In this note, we compute the Gelfand-Kirillov dimension of cosemisimple Hopf algebras that arise as deformations of a
linearly reductive algebraic group. Our work lies in a purely algebraic setting and generalizes results of Goodearl-Zhang (2007),
of Banica-Vergnioux (2009), and of D'Andrea-Pinzari-Rossi (2017).
On the quadratic dual of the Fomin-Kirillov algebras.
( ArXiv )
( Talk Notes )
Joint with
James Zhang
Transactions of the American Mathematical Society 372 (2019), pp. 3921–3945.
Abstract:
We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual)
E_n^! of the Fomin-Kirillov algebras E_n; these algebras are connected ℕ-graded and are defined for n≥2.
We establish that the algebra E_n^! is module-finite over its center (so, satisfies a polynomial identity),
is Noetherian, and has Gelfand-Kirillov dimension ⌊n/2⌋ for each n≥2. We also observe that E_n^! is not prime for n≥3.
By a result of Roos, E_n is not Koszul for n≥3, so neither is E_n^! for n≥3. Nevertheless, we prove that E_n^! is Artin-Schelter
(AS-)regular if and only if n=2, and that E_n^! is both AS-Gorenstein and AS-Cohen-Macaulay if and only if n=2,3. We also show
that the depth of E_n^! is ≤1 for each n≥2, conjecture we have equality, and show this claim holds for n=2,3. Several other
directions for further examination of E_n^! are suggested at the end of this article.
PBW deformations of quadratic monomial algebras.
( ArXiv )
Joint with
Zachary Cline,
Andrew Estornell and
Matthew Wynne
Communications in Algebra 47, no. 7 (2019), pp. 2670-2688.
Abstract:
A result of Braverman and Gaitsgory from 1996 gives necessary and sufficient conditions for a filtered algebra to be a
Poincaré-Birkhoff-Witt (PBW) deformation of a Koszul algebra. The main theorem in this paper establishes conditions equivalent to
the Braverman-Gaitsgory Theorem to efficiently determine PBW deformations of quadratic monomial algebras. In particular, a graphical
interpretation is presented for this result, and we discuss circumstances under which some of the conditions of this theorem need not be checked.
Several examples are also provided. Finally, with these tools, it is then shown that each quadratic monomial algebra admits a nontrivial PBW
deformation.
Poisson geometry and representations of PI 4-dimensional Sklyanin algebras.
( ArXiv )
( Talk Notes, technical ) ( Talk Notes, broad)
( Video, broad)
Joint with
Xingting Wang and
Milen Yakimov
Selecta Mathematica, 27:99 (2021).
Abstract:
Take S to be a 4-dimensional Sklyanin (elliptic) algebra that is module-finite over its center Z; thus, S is PI. Our first result is the construction of
a Poisson Z-order structure on S such that the induced Poisson bracket on Z is non-vanishing. We also provide the explicit Jacobian structure of this bracket,
leading to a description of the symplectic core decomposition of the maximal spectrum Y of Z. We then classify the irreducible representations of S by combining
(1) the geometry of the Poisson order structures, with (2) algebro-geometric methods for the elliptic curve attached to S, along with (3) representation-theoretic
methods using line and fat point modules of S. Along the way, we improve results of Smith and Tate obtaining a description the singular locus of Y for such S. The
classification results for irreducible representations are in turn used to determine the zero sets of the discriminants ideals of these algebras S.
Poisson geometry of PI three-dimensional Sklyanin algebras.
( ArXiv ) ( Talk Notes, technical ) ( Talk Notes, broad)
( Video, broad)
Joint with
Xingting Wang and
Milen Yakimov
Proceedings of the London Mathematical Society 118, no. 6, (2019), pp. 1471-1500.
Abstract:
We give the 3-dimensional Sklyanin algebras S that are module-finite over their center Z the structure of a Poisson Z-order
(in the sense of Brown-Gordon). We show that the induced Poisson bracket on Z is non-vanishing and is induced by an explicit
potential. The \mathbb{Z}_3 \times \Bbbk^\times-orbits of symplectic cores of the Poisson structure are determined (where the group acts on S by
algebra automorphisms). In turn, this is used to analyze the finite-dimensional quotients of S by central annihilators: there are
3 distinct isomorphism classes of such quotients in the case (n,3)≠1 and 2 in the case (n,3)=1, where n is order of the elliptic curve
automorphism associated to S. The Azumaya locus of S is determined, extending results of Walton for the case (n,3)=1.
Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p^3 and pq^2.
( ArXiv ) ( Talk Notes )
Joint with
Adriana Mejía Castaño,
Susan Montgomery,
Sonia Natale, and
Maria D. Vega
Journal of Pure and Applied Algebra 222, no. 7, (2018) pp. 1643-1669.
Abstract:
Let p and q be distinct prime numbers. We study the Galois objects and
cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension p^3
and of dimension pq^2. We obtain that the p+1 non-isomorphic self-dual semisimple Hopf algebras of dimension
p^3 classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8
dimensional Kac-Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita
equivalence classes among semisimple Hopf algebras of odd dimension p^3, established by
the third-named author in an appendix.
McKay correspondence for semisimple Hopf actions on regular graded algebras, II.
( ArXiv )
( Pre-Talk Notes )
( Talk Notes )
Joint with
Kenneth Chan,
Ellen Kirkman, and
James Zhang
Journal of Noncommutative Geometry 13, no. 1, (2019), pp. 87–114.
Abstract:
We continue our study of the McKay Correspondence for grading preserving actions of semisimple Hopf algebras
H on (noncommutative) Artin-Schelter regular algebras A. Here, we establish correspondences between module categories
over A^H, over A#H, and over End_{A^H} A. We also study homological properties of (endomorphism rings of) maximal
Cohen-Macaulay modules over A^H
McKay correspondence for semisimple Hopf actions on regular graded algebras, I.
( ArXiv )
( Pre-Talk Notes )
( Talk Notes )
Joint with
Kenneth Chan,
Ellen Kirkman, and
James Zhang
Journal of Algebra 508 (2018), pp. 512-538.
Abstract:
In establishing a more general version of the McKay correspondence,
we prove Auslander’s theorem for actions of semisimple Hopf algebras
H on noncommutative Artin-Schelter regular algebras A of global dimension
two, where A is a graded H-module algebra, and the Hopf action on A is inner
faithful with trivial homological determinant. We also show that each fixed
ring A^H under such an action arises an analogue of a coordinate ring of a
Kleinian singularity.
On quantum groups associated to a pair of preregular forms.
( ArXiv )
( Video )
( Talk Notes )
Joint with
Alexandru Chirvasitu
and Xingting Wang
Journal of Noncommutative Geometry 13, no. 1, (2019), pp. 115–159.
Abstract:
We define the universal quantum group H that preserves a pair of Hopf comodule maps, whose underlying vector
space maps are preregular forms defined on dual vector spaces. This generalizes the construction of Bichon and
Dubois-Violette (2013), where the target of these comodule maps are the ground field. We also recover the quantum
groups introduced by Dubois-Violette and Launer (1990), by Takeuchi (1990), by Artin, Schelter, and Tate (1991),
and by Mrozinski (2014), via our construction. As a consequence, we obtain an explicit presentation of a universal
quantum group that coacts simultaneously on a pair of N-Koszul Artin-Schelter regular algebras with arbitrary
quantum determinant.
Finite dimensional Hopf actions on algebraic quantizations.
( ArXiv )
( Talk Notes )
Joint with
Pavel Etingof
Algebra and Number Theory 10, no. 10, (2016), pp. 2287-2310.
Abstract:
Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra
[CEW1, CEW2], we showed that a semisimple Hopf action on a Weyl algebra
over a polynomial algebra k[z_1,...,z_s] factors through a group action, and this in fact holds for any finite
dimensional Hopf action if s=0. We also generalized these results to finite dimensional Hopf actions on algebras
of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations
of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical
symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties),
finite W-algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of
abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic
number theory, due to A. Perucca.
Finite dimensional Hopf actions on deformation quantizations.
( ArXiv )
Joint with
Pavel Etingof
Proceedings of the American Mathematical Society 145 (2017), pp. 1917-1925.
Abstract:
We study when a finite dimensional Hopf action on a quantum formal deformation A of a commutative domain A_0
(i.e., a deformation quantization) must factor through a group algebra. In particular, we show that this
occurs when the Poisson center of the fraction field of A_0 is trivial.
PBW deformations of braided products.
( ArXiv )
( Talk Notes )
Joint with
Sarah Witherspoon
Journal of Algebra 504 (2018), pp. 536-567.
Abstract:
We present new examples of deformations of smash product algebras that arise from Hopf algebra actions on
pairs of module algebras. These examples involve module algebras that are Koszul, in which case a PBW theorem
we established previously applies. Our construction generalizes several ‘double’ constructions appearing in the
literature, including Weyl algebras and some types of Cherednik algebras, and it complements the braided double
construction of Bazlov and Berenstein. Many suggestions of further directions are provided at the end of the work.
Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2.
( ArXiv )
( Code )
Joint with
Daniel J. Reich
Involve, a Journal of Mathematics 11-4 (2018), 585-608.
Abstract:
The representation theory of a three-dimensional Sklyanin algebra S depends on its (noncommutative projective algebro-)
geometric data: an elliptic curve E in \mathbb{P}^2, and an automorphism \sigma of E given by translation by a point.
Indeed, by a result of Artin-Tate-van den Bergh, we have that S is module-finite over its center if and only if \sigma
has finite order. In this case, all irreducible representations of S are finite-dimensional, of at most dimension
|\sigma| by a result of the second author.
In this work, we determine explicitly all irreducible representations of S associated to \sigma of order 2, up to equivalence.
This is achieved via an algorithm in Maple. Moreover, we discuss a geometric parametrization of equivalence classes of these
irreducible representations by depicting the Azumaya locus of S over its center. We also tailor our algorithm to recover
well-known results about irreducible representations of the skew polynomial ring \mathbb{C}_{-1}[x,y].
Pointed Hopf actions on fields, II.
( ArXiv link ) ( Talk )
Joint with
Pavel Etingof
Journal of Algebra 460 (2016), pp. 253-283.
Abstract:
This is a continuation of the authors' study of finite-dimensional pointed Hopf algebras H which act inner faithfully
on commutative domains. As mentioned in Part I of this work, the study boils down to the case where H acts inner faithfully
on a field. These Hopf algebras are referred to as Galois-theoretical.
In this work, we provide classification results for finite-dimensional pointed Galois-theoretical Hopf algebras H
of finite Cartan type. Namely, we determine when such H of type A_1^{\times r} and some H of rank two possess the
Galois-theoretical property. Moreover, we provide necessary and sufficient conditions for Reshetikhin twists of small
quantum groups to be Galois-theoretical.
Finite dimensional Hopf actions on Weyl algebras.
( ArXiv )
( Video )
( Talk Notes )
Joint with
Juan Cuadra and
Pavel Etingof
Advances in Mathematics 302 (2016), pp. 25-39.
Abstract:
We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of
characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum
symmetries. This improves a previous result by the authors, where the statement was established for semisimple H.
The proof relies on a refinement of the method previously used: namely, considering reductions of the action of H on
A modulo prime powers rather than primes. We also show that the result holds, more generally, for algebras of differential
operators. This gives an affirmative answer to a question posed by the last two authors.
Maps from the enveloping algebra of the positive Witt algebra to regular algebras.
( ArXiv )
( Talk Notes )
Joint with
Susan Sierra
Pacific J. Math. 284, no. 2 (2016), pp. 475–509.
Abstract:
We construct homomorphisms from the universal enveloping algebra of the positive (part of the) Witt algebra to several
different Artin-Schelter regular algebras, and determine their kernels and images. As a result, we produce elementary proofs
that the universal enveloping algebras of the Virasoro algebra, the Witt algebra, and the positive Witt algebra are neither
left nor right noetherian.
Hopf coactions on commutative algebras generated by a quadratically independent comodule.
( ArXiv )
Joint with
Pavel Etingof,
Debashish Goswami, and
Arnab Mandal
Communications in Algebra 45, no. 8, (2017) pp. 3410–3412.
Abstract:
Let A be a commutative unital algebra over an algebraically closed field k of characteristic not equal to 2,
whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations.
Let Q be a Hopf algebra that coacts on A inner-faithfully, while leaving V invariant. We prove that
Q must be commutative when either:
(i) the coaction preserves a non-degenerate bilinear form on V; or (ii) Q is co-semisimple, finite-dimensional,
and char(k)=0.
On quantum groups associated to non-Noetherian regular algebras of dimension 2.
( ArXiv )
( Talk Notes )
Joint with
Xingting Wang
Mathematische Zeitschrift 284, no. 1, (2016) pp. 543–574.
(Typo in Example 2.18(c):
q^{−2m} b D^m = D^m b and q^{2m} c D^m = D^m c
should be
q^{2m} b D^m = D^m b and q^{-2m} c D^m = D^m c)
Abstract: We investigate homological and ring-theoretic properties of universal quantum linear groups
that coact on Artin-Schelter regular algebras A(n) of global dimension 2, especially with central homological
codeterminant (or central quantum determinant). As classified by Zhang, the algebras A(n) are connected \N-graded algebras
that are finitely generated by n indeterminants of degree 1, subject to one quadratic relation. In the case when
the homological codeterminant of the coaction is trivial, we show that the quantum group of interest, defined independently
by Manin and by Dubois-Violette and Launer, is Artin-Schelter regular of global dimension 3 and also skew Calabi-Yau
(homologically smooth of dimension 3). For central homological codeterminant, we verify that the quantum groups are
Noetherian and have finite Gelfand-Kirillov dimension precisely when the corresponding comodule algebra A(n) satisfies
these properties, that is, if and only if n=2. We have similar results for arbitrary homological codeterminant if we
require that the quantum groups are involutory. We also establish conditions when Hopf quotients of these quantum groups,
that also coact on A(n), are cocommutative.
Actions of some pointed Hopf algebras on path algebras of quivers.
( ArXiv )
( Talk Notes )
Joint with
Ryan Kinser
Algebra and Number Theory 10, no. 1 (2016) pp. 117-154.
(Only => of Lemma 2.5 holds, which is needed for our main results.
<= direction requires faithful G(T(n))-action, along with x not acting by scalar multiple of 1-g.
So Ex 3.13 should be shorter, and Ex. 7.7 should be omitted. Rest of results remain unchanged.)
Abstract: We classify Hopf actions of Taft algebras T(n) on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful Z_n-action (by directed graph automorphisms) also admits inner faithful actions of a Taft algebra. Several examples for actions of the Sweedler algebra T(2) and for actions of T(3) are presented in detail. We then extend the results on Taft algebra actions on path algebras to actions of the Frobenius-Lusztig kernel u_q(sl2), and to actions of the Drinfeld double of T(n).
Semisimple Hopf actions on Weyl algebras.
( ArXiv )
( Talk Notes )
Joint with
Juan Cuadra and
Pavel Etingof
Advances in Mathematics 282 (2015), pp. 47-55.
Abstract: We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.
Pointed Hopf actions on fields, I.
( ArXiv ) ( Video )
( Talk Notes )
Joint with
Pavel Etingof
Transformation Groups 20, no. 4 (2015), pp. 985-1013.
Abstract: Actions of semisimple Hopf algebras H over an algebraically closed field of characteristic zero on commutative domains were classified recently by the authors. The answer turns out to be very simple- if the action is inner faithful, then H has to be a group algebra. The present article contributes to the non-semisimple case, which is much more complicated. Namely, we study actions of finite dimensional (not necessarily semisimple) Hopf algebras on commutative domains, particularly when H is pointed of finite Cartan type.
The work begins by reducing to the case where H acts inner faithfully on a field; such a Hopf algebra is referred to as Galois-theoretical. We present examples of such Hopf algebras, which include the Taft algebras, u_q(sl_2), and some Drinfeld twists of other small quantum groups. We also give many examples of finite dimensional Hopf algebras which are not Galois-theoretical. Classification results on finite dimensional pointed Galois-theoretical Hopf algebras of finite Cartan type will be provided in the sequel, Part II, of this study.
Poincare-Birkhoff-Witt deformations
of smash product algebras from Hopf actions on Koszul algebras
( ArXiv)
( Talk Notes )
Joint with
Sarah Witherspoon
Algebra and Number Theory 8, no. 7 (2014) pp. 1701-1731.
Abstract: Let H be a Hopf algebra and let B be a Koszul H-module algebra. We provide necessary
and sufficient conditions for a filtered algebra to be a Poincare-Birkhoff-Witt (PBW) deformation of the smash product algebra
B#H. Many examples of these deformations are given.
The universal enveloping algebra of the Witt algebra is not noetherian.
( ArXiv )
( Talk Notes )
Joint with
Susan Sierra
Advances in Mathematics 262 (2014), pp. 239-260.
Abstract: This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal enveloping algebra of the Witt (or centerless Virasoro) algebra is not noetherian. To show this, we prove our main result: the universal enveloping algebra of the positive part of the Witt algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (noncommutative) birationally commutative projective surfaces.
As a consequence of our main result, we also show that the enveloping algebras of many other infinite dimensional Lie algebras are
not noetherian. These Lie algebras include the Virasoro algebra and all infinite dimensional Z-graded simple Lie algebras of polynomial growth.
Quantum binary polyhedral groups and their actions on quantum planes.
( ArXiv )
( Talk Notes )
Joint with
Kenneth Chan,
Ellen Kirkman, and
James Zhang
Journal für die Reine und Angewandte Mathematik (Crelle's Journal) 2016, no. 719 (2016) pp. 211–252.
Abstract: We classify quantum analogues of actions of finite subgroups G of SL_2(k) on commutative polynomial rings k[u,v]. More precisely, we produce a classification of pairs (H,R), where H is a finite dimensional Hopf algebra that acts inner faithfully and preserves the grading of an Artin-Schelter regular algebra R of global dimension two.
Remarkably, the corresponding invariant rings R^H share similar regularity and Gorenstein properties as the invariant rings
k[u,v]^G in the classic setting. We also present several questions and directions for expanding this work in noncommutative invariant theory.
Semisimple Hopf actions on commutative domains.
( ArXiv )
( Talk Notes )
Joint with
Pavel Etingof
Advances in Mathematics 251C (2014), pp. 47-61.
Abstract: Let H be a semisimple Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if A arises as an H-module algebra via an inner faithful H-action, then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen.
The main results of this article extend to working over k of positive characteristic. On the other hand,
we obtain results on Hopf actions on Weyl algebras as a consequence of the main theorem.
Hopf actions on filtered regular algebras.
( ArXiv )
Joint with
Kenneth Chan,
Yanhua Wang, and
James Zhang
Journal of Algebra 397, no. 1 (2014), pp. 68-90.
Abstract: We study finite dimensional Hopf algebra actions on so-called filtered Artin-Schelter regular algebras of dimension n,
particularly on those of dimension 2. The first Weyl algebra is an example of such on algebra with n=2, for instance.
Results on the Gorenstein condition and on the global dimension of the corresponding fixed subrings are also provided.
Hopf actions and Nakayama automorphisms.
( ArXiv )
( Video )
Joint with
Kenneth Chan and
James Zhang
Journal of Algebra 409 (2014), pp. 26-53.
Abstract: Let H be a Hopf algebra with antipode S, and let A be an N-Koszul Artin-Schelter regular algebra.
We study connections between the Nakayama automorphism of A and S^2 of H when H coacts on A inner-faithfully.
Several applications pertaining to Hopf actions on Artin-Schelter regular algebras are given.
Representation theory of three-dimensional Sklyanin algebras.
( ArXiv )
( Video )
Nuclear Physics B 860, no. 1 (2012), pp. 167-185.
(Minor computational correction in Section 5)
Abstract: We determine the dimensions of the irreducible representations of the Sklyanin algebras with global dimension 3. This contributes to the study of marginal deformations of the N=4 super Yang-Mills theory in four dimensions in supersymmetric string theory. Namely, the classification of such representations is equivalent to determining the vacua of the aforementioned deformed theories.
We also provide the polynomial identity degree for the Sklyanin algebras that are module finite over their center. The Calabi-Yau geometry of these algebras is also discussed.
Degenerate Sklyanin algebras and generalized twisted homogeneous coordinate rings
( Article )
Journal of Algebra 322, no. 7 (2009) pp. 2508-2527 [pages 1-24 in link].
Corrigendum: 356, no. 1
(2012), 275-282 [pages 25-31 in link].
Abstracts: [Article] In this work, we introduce the point parameter ring B, a generalized twisted homogeneous coordinate ring associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of Artin-Tate-van den Bergh, namely that B is generated in degree one and thus is a factor of the corresponding degenerate Sklyanin algebra.
[Corrigendum] There is an error in the computation of the truncated point schemes of the degenerate Sklyanin algebra S(1,1,1); they are larger than was claimed in Proposition 3.13 of the above paper. We provide a description of the correct truncated point schemes. Results about the corresponding point parameter ring associated to these schemes are given afterward.
Survey Talks and Other Works: Links and Details
Frobenius Algebras Galore
- LAGOON: Leicester Algebra and Geometry Open ONline, July 2021
(Video)
- Blackwell-Tapia Conference, Nov 2021
(Video)
Quantum Symmetry, from an algebraic point of view
(Video)
(Talk Notes)
U. Chicago, Colloquium, May 2021
Deformation Theory "Cheat Sheet"
Resources:
- Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke's text on Poisson Structures
- Pavel Etingof's text on Calogero-Moser Systems and Representation Theory (Chapter 3)
and lecture notes on Exploring noncommutative algebra via deformation theory
- Sarah Witherspoon's book on Hochschild Cohomology for Algebras
Symmetries in Algebra (Talk Notes)
Graduate Research Opportunities for Women (GROW) conference, October 2019
Artin-Schelter regular algebras and their quantum symmetries (course notes)
- Introduction
- Lecture I
- Lecture II
- Lecture III
- Lecture IV
- Lecture V
- Terminology Handout
- Examples Handout
Event: Hopf Algebras and Tensor Categories CIMPA Research School, Cordoba, Argentina, 2019.
Monoidal Categories "Cheat Sheet"
Resource: Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik's text on Tensor Categories
Quantum Symmetry...and more (Talk Notes)
AWM Symposium address. Rice University, April 2019
Quantum Symmetry (Video)
(Prezi presentation)
(slides)
AMS Denver special session plenary talk (Fall 2016)
Annual Math Conference at Miami U. Ohio (Fall 2017)
(The presentation and slides above contain more than what was covered in the talks; I skipped some of the prepared material.)
Quantum Symmetry (Video)
(Talk Notes)
Modular Categories- Their Representations, Classification, and Applications at BIRS-Oaxaca, August 2016
Quantum Symmetry
(Prezi presentation) (Slides)
Various survey talks, Spring 2015
No Quantum Symmetry (Talk Notes)
US-Mexico Noncommutative Algebra, Representation Theory, and Categorification conference at USC, June 2016
Hopf algebra actions on noncommutative algebras (Talk Notes)
Survey for SACNAS, October 2014
Actions of finite dimensional Hopf algebras on commutative domains
(Talk Notes)
Poisson Geometry conference, U. Illinois Urbana-Champaign, August 2014
Examples of Hopf algebras and noncommutative regular algebras (Handout)
For talks on Hopf actions on noncommutative regular algebras
Thesis
