## Research

#### Comments/ Questions/ Concerns are very much welcomed!

__ Publications and Preprints __ *Coauthors, Abstract, and some Talks/Notes are provided below*

On quantum groups associated to non-Noetherian regular algebras of dimension 2

Actions of some pointed Hopf algebras on path algebras of quivers

Semisimple Hopf actions on Weyl algebras

Pointed Hopf actions on fields, I

Poincare-Birkoff-Witt deformations of smash product algebras from Hopf actions on Koszul algebras

The universal enveloping algebra of the Witt algebra is not noetherian

Quantum binary polyhedral groups and their actions on quantum planes

Semisimple Hopf actions on commutative domains

Hopf actions on filtered regular algebras

Hopf actions and Nakayama automorphisms

Representation theory of three-dimensional Sklyanin algebras

Degenerate Sklyanin algebras and generalized twisted homogeneous coordinate rings

__ Other Slides/ Talks/ Handouts/ Works __

Quantum Symmetry (Various talks Spring 2015)

Hopf algebra actions on noncommutative algebras (SACNAS 2014)

Actions of finite dimensional Hopf algebras on commutative domains (Poisson 2014)

Examples of Hopf algebras and noncommutative regular algebras

### Publications and Preprints

On quantum groups associated to non-Noetherian regular algebras of dimension 2

Joint with
__Xingting Wang__

__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 (Slides)

Joint with
__Ryan Kinser__

*Submitted*

__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 (Notes from Talk)

Joint with
__Juan Cuadra__ and
__Pavel Etingof__

*Submitted*

__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 (Notes from Talk)

Joint with
__Pavel Etingof__

*To appear in Transformation Groups*

__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 (Notes from Talk)

Joint with
__Sarah Witherspoon__

*To appear in Algebra & Number Theory *

__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 (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 (Slides)

Joint with
__Kenneth Chan__,
__Ellen Kirkman__, and
__James Zhang__

*To appear in Journal für die Reine und Angewandte Mathematik (Crelle's Journal) *

__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 (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

Joint with
__Kenneth Chan__,
__Yanhua Wang__, and
__James Zhang__

*Journal of Algebra* (1) 397 (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 (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 (Talk)

*Nuclear Physics B* (1) 860 (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

*Journal of Algebra* (7) 322 (2009) pp. 2508-2527 [pages 1-24 in the above link].

Corrigendum: (1) 356 (2012), 275-282 [pages 25-31].

__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.

### Slides/ Talks

Quantum Symmetry (Prezi presentation) (slides)

Various talks, Spring 2015

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