Research

Summary of Research Work

Theme
Paper #
(Rep. Theory of) Noncom. Graded Algebras
1, 2, 7, 14, 17, 25, 26, 28
Genuine Quantum Symmetry
  3, 4, 6, 9, 11, 16
No Quantum Symmetry
 3, 4, 5, 10, 13, 15, 19, 20
Quantum Invariant Theory
 4, 6, 9, 22, 23, 31
Universal Quantum Groups
 12, 21
Deformations
 8, 18, 24, 27, 29
Algebras in Monoidal Categories
 32, 33, 34, 35

.

Summary of Expository Work

On Noncommutative Algebra
30

.

List of Publications and Preprints

Coauthors, Abstracts, and some Talks/Notes are provided below

  1. (35) Algebraic structures in group-theoretical fusion categories
  2. (34) Algebraic structures in comodule categories over weak bialgebras
  3. (33) Tensor algebras in finite tensor categories
  4. (32) Braided commutative algebras over quantized enveloping algebras
  5. (31) Noncommutative Knörrer periodicity and noncommutative Kleinian singularities
  6. (30) An Invitation to Noncommutative Algebra
  7. (29) Gelfand-Kirillov dimension of cosemisimple Hopf algebras
  8. (28) On the quadratic dual of the Fomin-Kirillov algebras
  9. (27) PBW deformations of quadratic monomial algebras
  10. (26) Poisson geometry and representations of PI 4-dimensional Sklyanin algebras
  11. (25) The Poisson geometry of the 3-dimensional Sklyanin algebras
  12. (24) Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p^3 and pq^2
  13. (23) McKay correspondence for semisimple Hopf actions on regular graded algebras, II
  14. (22) McKay correspondence for semisimple Hopf actions on regular graded algebras, I
  15. (21) On quantum groups associated to a pair of preregular forms
  16. (20) Finite dimensional Hopf actions on algebraic quantizations
  17. (19) Finite dimensional Hopf actions on deformation quantizations
  18. (18) PBW deformations of braided products
  19. (17) Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2
  20. (16) Pointed Hopf actions on fields, II
  21. (15) Finite dimensional Hopf actions on Weyl algebras
  22. (14) Maps from the enveloping algebra of the positive Witt algebra to regular algebras
  23. (13) Hopf coactions on commutative algebras generated by a quadratically independent comodule
  24. (12) On quantum groups associated to non-Noetherian regular algebras of dimension 2
  25. (11) Actions of some pointed Hopf algebras on path algebras of quivers
  26. (10) Semisimple Hopf actions on Weyl algebras
  27. (9) Pointed Hopf actions on fields, I
  28. (8) Poincare-Birkhoff-Witt deformations of smash product algebras from Hopf actions on Koszul algebras
  29. (7) The universal enveloping algebra of the Witt algebra is not noetherian
  30. (6) Quantum binary polyhedral groups and their actions on quantum planes
  31. (5) Semisimple Hopf actions on commutative domains
  32. (4) Hopf actions on filtered regular algebras
  33. (3) Hopf actions and Nakayama automorphisms
  34. (2) Representation theory of three-dimensional Sklyanin algebras
  35. (1) Degenerate Sklyanin algebras and generalized twisted homogeneous coordinate rings

Survey Talks and Other Works

  • (I) Symmetries in Algebra (GROW conference 2019)
  • (H) Artin-Schelter regular algebras and their quantum symmetries (course notes)
  • (G) Monoidal Categories "Cheat Sheet"
  • (F) Quantum Symmetry...and more (AWM Symposium address)
  • (E) Quantum Symmetry (Various survey talks)
  • (D) No Quantum Symmetry (US-Mexico conf. June 2016)
  • (C) Hopf algebra actions on noncommutative algebras (SACNAS 2014)
  • (B) Actions of finite dimensional Hopf algebras on commutative domains (Poisson 2014)
  • (A) Examples of Hopf algebras and noncommutative regular algebras
  • (0) Thesis


Publications and Preprints: Links and Details

Undergraduate coauthors are in italics

Algebraic structures in group-theoretical fusion categories.
( ArXiv link ) ( Talk 1 Notes ) ( Talk 2 Notes ) ( Talk 2 Video )
Joint with Yiby Morales, Monique Müller, Julia Plavnik, Ana Ros Camacho, and Angela Tabiri
submitted

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 link ) ( Notes from Talk + Pre-Talk )
Joint with Elizabeth Wicks and Robert Won
submitted

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 link ) ( Notes from Talk )
Joint with Pavel Etingof and Ryan Kinser
to appear in International Mathematics Research Notices

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 link ) ( Talk ) ( Notes from Talk )
Joint with Robert Laugwitz
Submitted

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 link )
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 link )
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 link )
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 link ) ( Notes from Talk )
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 link )
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 link ) ( Notes from Talk - 20 mins ) ( Notes from Talk - 50 mins )
Joint with Xingting Wang and Milen Yakimov
Submitted

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 link ) ( Notes from Talk )
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 link ) ( Slides )
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 link ) ( Notes from Pre-Talk ) ( Notes from Talk )
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 link ) ( Notes from Pre-Talk ) ( Notes from Talk )
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 link ) ( Talk ) ( Notes from Talk )
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 link ) ( Notes from Talk )
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 link )
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 link ) ( Slides )
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 link ) ( 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 link ) ( Talk ) ( Notes from Talk )
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 link ) ( Notes from Talk )
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 link )
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 link ) ( Notes from Talk )
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 link ) ( Slides )
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 link ) ( Notes from Talk )
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 link ) ( Talk ) ( Notes from Talk )
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 link ) ( Notes from Talk )
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 link ) ( Notes from Talk )
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 link ) (Slides)
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 link ) ( Notes from Talk )
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 link )
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 link ) ( Talk )
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.
( Article ) ( Talk )
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

Symmetries in Algebra (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: [EGNO] Etingof, Gelaki, Nikshych, Ostrik's text on Tensor Categories

Quantum Symmetry...and more
AWM Symposium address (slides of math portion of the talk). Rice University, April 2019

Quantum Symmetry ( Talk ) (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 (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 (notes)
US-Mexico Noncommutative Algebra, Representation Theory, and Categorification conference at USC, June 2016  

Hopf algebra actions on noncommutative algebras
Survey for SACNAS, October 2014

Actions of finite dimensional Hopf algebras on commutative domains
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