# Women in Noncommutative Algebra and Representation Theory (WINART) (16w5023)

## Organizers

Georgia Benkart (University of Wisconsin-Madison)

Ellen Kirkman (Wake Forest University)

Susan Montgomery (University of Southern California)

Chelsea Walton (Temple University)

## Objectives

The goal of the workshop, Women in Noncommutative Algebra and Representation Theory (WINART), is to facilitate collaborations among women mathematicians in these areas of research. Out of the 42 participants, approximately half will be graduate students and postdoctoral researchers who received their Ph.D. degree after 2009.

The compelling need for the workshop is the underrepresentation of women at major research conferences and workshops in the areas of the WINART workshop. For example, out of the total number of speakers at the events mentioned in the next paragraph (excluding the Connections for Women workshops at MSRI), under 20% were women. The aim of the proposed event is to create an avenue for the participants, especially the early-career ones, to maintain a sustainable research career and to acquaint participants with exciting new directions of research.

The timeliness of this proposal is prompted by the success of the MSRI semester-long program Noncommutative Algebraic Geometry and Representation Theory in Spring 2013, and also several recent/ upcoming workshops and conferences including: Infinite Dimensional Hopf Algebras mini-workshop (Oberwolfach, April 2014); Interactions between Algebraic Geometry and Noncommutative Algebra conference (Oberwolfach, May 2014); Geometric Representation Theory conference (Cargsé, June 2014); Noncommutative Algebraic Geometry workshop (Shanghai, August 2014); Algebraic Lie Theory and Representation Theory conference (Edinburgh, September 2014); and Geometric Representation Theory semester-long program (MSRI, Fall 2014).

The format for WINART will be similar to past BIRS conferences for women in mathematics (e.g., Women in Numbers 2008, 2011, 2014; Women in Topology; Algebraic Combinatorixx). Namely, the conference will be research intensive with about 8-10 lectures.

Approximately six months prior to the conference, the participants will be divided up into 7-9 research groups, led by 1-2 research leaders. The research leaders, in consultation with the group members, will choose a stimulating project/ topic for the group and will send out relevant details and a list of reading materials before the conference. The group will then meet before the conference remotely to discuss directions to pursue during the conference. Several follow-up meetings after the workshop are expected and likely will include special sessions at AMS or CMS meetings.

At the conference, each group leader will be invited to give an introductory talk on her area of research. These talks will occur during the first two days of the conference. Mid-week, there will be a panel discussion on Collaboration. Lastly, there will be 15-minute reports given by a representative from each research group on the final day of the workshop. The summary will recap the group's progress for the week and will list directions for further investigation. The rest of the time will be dedicated to research. BIRS is an ideal setting for this workshop, as members from different groups can discuss their progress and questions over shared meals.

The list of research group leaders are: Martina Balagovic, Karin Baur, Georgia Benkart, Maria Gorelik, Susan Montgomery, Sonia Natale, Rosa Orellana, Maria Redondo, Vera Serganova, Anne Shepler, Susan Sierra, Andrea Solotar, Catharina Stroppel, Gordana Todorov, Michaela Vancliff, and Sarah Witherspoon. Building on the expertise of the volunteer leaders, the following topics in noncommutative algebra and representation theory to be featured at the conference.

**(Balagovic & Stroppel) Categorification and coideal subalgebras.** Categorification is a process of replacing some set-theoretic object (e.g., a module over an algebra) by a category-theoretic object (e.g., a category), so that the original object and information about it gets refined. Given a categorification, the original object can be easily recovered (e.g., by taking the Grothendieck group of the category), but finding and constructing a categorification is a difficult task. One of the first examples of categorification was the work of Khovanov, who used it to obtain refined knot invariants by categorifying the Jones polynomial. This approach was generalized by Stroppel and coauthors using categories arising in Lie theory (category O). Catharina Stoppel's work on categorification has applications in knot homology, gives new results about multiplicity formulas and graded branching rules and establishes unexpected equivalences (for instance between representations of Lie superalgebras and perverse sheaves on Grassmannians). The general structure of quantum symmetric pairs is one of the research focuses of Martina Balagovic (joint with Stefan Kolb). Stroppel and Balagovic are both interested in categorifying quantum symmetric pairs and their representations.

**(Baur & Todorov) Cluster categories.** Cluster categories were introduced by Gordana Todorov in joint work with Aslak Buan, Robert Marsh, Marcus Reineke, and Idun Reiten in the seminal paper "Tilting Theory and Cluster Combinatorics" to study the combinatorics of cluster algebras. This work provided new and remarkable insights on the representation theory of hereditary Artin algebras. The study of cluster categories also has a connection to the study of Calabi Yau categories. For instance, it was shown in the work of Claire Amiot, Idun Reiten, and Gordana Todorov that large classes of Calabi-Yau categories are equivalent to generalized cluster categories as triangulated categories. Karin Baur, in joint work with Robert Marsh, established a beautiful geometric description of another generalization of a cluster category: m-cluster categories introduced by Bernhard Keller. Recent work of Baur and Todorov provides several connections between cluster categories and other important objects arising in algebra and physics. Todorov, in joint work with Kiyoshi Igusa and Jerzy Weyman, has investigated the relationship between cluster categories and the study of periodic trees and semi-invariants and maximal green sequences. Baur, in joint work with Alastair King and Robert Marsh, established a correspondence between cluster categories of Grassmannians and dimer models.

**(Benkart & Orellana) Diagram algebras, tensor invariants, representation theory.** The centralizer algebra of transformations, that commute with the action of a group or algebra on tensor powers of a representation, can often be realized as an algebra of diagrams. Important examples include the Temperley-Lieb algebras of statistical mechanics, the partition algebras, and the Iwahori-Hecke algebras. These centralizer (diagram) algebras have a rich combinatorics, as well as many connections with representation theory, knot and link invariants, categorification, and quantum computing. Georgia Benkart is an expert on centralizer algebras and their connections with crystal bases, web bases, symmetric functions, and representation theory. Her recent work determines, in a very general setting, an expression for the Poincare series for the tensor invariants as a quotient of two determinants, and uses that to develop new connections between the tensor invariants for the finite subgroups of SU2 and the polynomial invariants of Weyl groups via McKay's celebrated correspondence. Orellana has tackled the longstanding problem of determining the Kronecker coefficients, described by Richard Stanley as "one of the main problems in the combinatorial representation theory of the symmetric group." Her recent work with Bowman and DeVisscher, offers wonderful new insights to the problem and a brand new approach using partition algebras; this work began at the 2011 BIRS workshop, "Algebraic Combinatorial".

**(Gorelik & Serganova) Representations of Lie superalgebras.** Supersymmetry, an important tool in modern theoretical physics, has long motivated the theory of representations of supergroups and superalgebras. Maria Gorelik and Vera Serganova are experts in the representation theory of Lie superalgebras; each have authored over 20 articles on this subject alone. Gorelik's contributions include establishing the structure of the ghost centre of a Lie superalgebra, and most recently, computing of characters of relatively integrable irreducible highest weight modules over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras (joint with Victor Kac). Moreover, Vera Serganova has been a leader in study of Lie superalgebras for 30 years. Some of her many contributions consists of establishing the character formula for atypical representations of the Lie superalgebras gl(m|n), computing the characters of finite-dimensional simple modules over a basic classical Lie superalgebra (joint with Caroline Gruson), and providing formulae for Shapovalov's determinants for the Poisson Lie superalgebras (joint with Maria Gorelik). Her recent interests includes study of the category T of tensor representations of sl∞.

**(Montgomery & Natale) Fusion categories and finite-dimensional Hopf algebras.** The program to classify finite-dimensional complex Hopf algebras has made great advances over the last two decades, especially for pointed Hopf algebras and semisimple Hopf algebras. A question receiving much attention of late is whether all semisimple Hopf algebras can be obtained from group algebras by a handful of constructions: bosonizations, extensions, Hopf duals, and twists. Sonia Natale answered this question affirmatively for semisimple Hopf algebras of low dimension. The categorical version of this program, the classification of fusion categories, is under active investigation by Sonia Natale and others. Fusion categories are of independent interest in several areas of mathematics, including conformal field theory and operator algebras. Moreover, Susan Montgomery has been a leading figure in the study of finite-dimensional Hopf algebras and their representations for 30 years. She has authored over 40 papers on the subject and wrote "Hopf algebras and their actions on rings" which is widely considered to be an influential text of the field. Her current research interests include the study of Frobenius-Schur indicators for semisimple Hopf algebras and fusion categories.

**(Redondo & Solotar) Hochschild (co)homology and linear categories.** Hochschild cohomology of associative algebras is important in many areas of mathematics: such as ring theory, group theory, representation theory, mathematical physics, homotopy theory and topology. It is well known that homological methods are useful in order to study associative algebras and to understand their properties. Even though Hochschild cohomology is defined easily in terms of very basic linear algebra, it has important invariance properties, for example it is invariant under derived, stable, and Morita equivalences. The Hochschild cohomology space has a cup product and a graded Lie bracket. One wants to understand its structure as a graded vector space, as a ring and as a graded Lie algebra. One might aim for a presentation by generators and relations, though this appears to be a very difficult problem in general. In fact there are just a few examples of computations of the ring structure of Hochschild cohomology completed. María Julia Redondo and Andrea Solotar are both experts in Hochschild (co)homology, particularly in its connection to (Galois and universal) coverings of linear categories. Further, their joint work with several authors investigates the Hochschild (co)homology of many important classes of algebras, which includes cluster-tilted algebras, generalized Weyl algebras, incidence algebras, string algebras, and Yang-Mills algebras.

**(Shepler & Witherspoon) Algebraic deformations and homological algebra.** Many algebras of interest are deformations of simpler, better-understood algebras, with Hochschild cohomology recording the possible deformations and providing a tool for understanding them. Anne Shepler and Sarah Witherspoon employ homological and combinatorial techniques to study deformations of algebras formed from group actions on other algebras, such as polynomial algebras, and generalizations. The resulting semidirect product algebras and their deformations arise in combinatorics, geometry, and representation theory, and go by various names such as symplectic reflection algebras, graded Hecke algebras, and rational Cherednik algebras. Deformations of universal enveloping algebras, Ore algebras, and Sridharan algebras extended by groups give other examples. Many deformations retain some of the structure of the semidirect product algebras from which they come, such as (PBW) bases of monomials that harken back to the original Poincare-Birkhoff-Witt Theorem on universal enveloping algebras. This fact helps in finding answers to questions about the algebraic structure, representations, and cohomology of the deformed algebras.

**(Sierra & Vancliff) Noncommutative projective algebraic geometry.** This field was launched in the 1980s to help classify Artin-Schelter regular (AS-regular) algebras. Such algebras are graded rings that share many homological properties with commutative polynomial rings. Since then, the global techniques in projective algebraic geometry have proved to be useful in understanding noncommutative geometric objects and in classifying algebras of low dimension. One long-standing project is to classify AS-regular algebras of global dimension four. Michaela Vancliff has a successful program in classifying subclasses of such algebras and studying their representations. In so doing, Vancliff, in joint work with Tom Cassidy, introduced a quantized version of graded Clifford algebras, so-called graded skew Clifford algebras, which yield examples of AS-regular algebras of arbitrary finite global dimension and which provide candidates for generic AS-regular algebras of global dimension four. Another long-standing goal is to classify noncommutative graded domains of low Gelfand-Kirillov (GK) dimension. Susan Sierra has made several significant contributions to the GK dimension 3 case, in particular classifying birationally commutative GK-3 graded domains. Sierra, in joint work with Chelsea Walton, employed this machinery to prove that the universal enveloping algebra of the Witt algebra is not Noetherian, answering a 20+-year old question.