Nichols Algebras and Their Interactions with Lie Theory, Hopf Algebras and Tensor Categories (15w5053)

Arriving in Banff, Alberta Sunday, September 6 and departing Friday September 11, 2015

Organizers

(Universidad Nacional de Cordoba)

Pavel Etingof (Massachusetts Institute of Technology)

(University of Marburg)

(University of Washington)

(Texas A&M University)

(University of Washington)

Objectives

The goal of the proposed workshop is to bring together experts in several different subjects all intrinsically related to Nichols algebras via the underlying techniques and ideas coming from Hopf algebras, Lie theory, conformal field theory and combinatorics. On one hand, there will be participants working in the rapidly developing structural theory of Nichols algebras and related Hopf algebras who produced multiple outstanding results in recent years, including classification results of Andruskiewitsch, Schneider, and Heckenberger. On the other hand, we will bring specialists working in representation theory, finite tensor categories, cohomology, and invariant theory who would benefit from interaction with those having expertise in Nichols algebras as well as provide new directions and context for the theory of Nichols algebras. Each area that we intend to represent at the workshop is currently under active development as indicated, in particular, by a selected list of a few of the recent relevant meetings: {it Colloquium on Hopf Algebras, Quantum Groups and Tensor Categories}, La Falda (Argentina), 2009; {it Conference in Hopf Algebras and Noncommutative Algebra}, Sde Boker (Israel), 2010; {it Conformal Field Theories and Tensor Categories}, Beijing (China), 2011; Hopf Algebras and Tensor Categories, Almer'ia (Spain), 2011; Noncommutative Invariant Theory, Seattle (USA), 2012; {it Cohomology and Support in Representation Theory}, Seattle (USA), 2012; Nichols Algebras and Weyl Groupoids, Oberwolfach (Germany), 2012; Interactions between Noncommutative Algebra, Representation Theory, and Algebraic Geometry, Berkeley (USA), 2013. Despite the growing number of people working in these subjects and a healthy number of respective conferences, our proposed workshop will be unique in bringing together experts interested in these deeply interrelated areas with newly emerging connections which have not been explored to their full potential. We propose to discuss the recent progress, open problems and future directions in the following areas with the specific objectives of developing the already forged connections between them and establishing new ones. (1) Structure and finite-dimensionality of Nichols algebras. Heckenberger classified the finite- dimensional Nichols algebras of diagonal type (that is, coming from finite abelian groups). The classification, as understood presently, splits into several important classes: Nichols algebras of Cartan type, essentially related to the finite quantum groups of Lusztig; Nichols algebras related to finite quantum supergroups, and a third class related to modular contragredient Lie superalgebras. Furthermore, Angiono described the defining relations of the Nichols algebras in Heckenberger's list. These results open the way to several developments, some already achieved and some in progress, concerning representation theory, classification of pointed Hopf algebras, finite generation of cohomology rings, and applications to tensor categories. In parallel, there were very interesting developments in Nichols algebras coming from finite nonabelian groups, mainly from two ideas: to use systematically the language and structure of racks in the formulation of the problems, and to attach a Weyl groupoid to the Nichols algebra of a decomposable Yetter-Drinfeld module. Albeit the general structure of this class of Nichols algebras is still to be unveiled, there are many significant results in this direction: on one hand, new examples were found, on the other, it was shown that several classes of groups simply do not admit finite-dimensional Nichols algebras. (2) Cohomology of Nichols algebras and Hopf algebras. The open problem of finite generation of the cohomology ring of a finite dimensional Hopf algebra is central for various applications in representation theory such as construction of a theory of support varieties. For group algebras of finite groups in positive characteristic, finite generation results go back to the classical work of Golod, Venkov and Evens over 50 years ago. Friedlander and Suslin proved the fundamental theorem on the finite generation of cohomology for any finite dimensional cocommutative Hopf algebra in 1997. In 1993, Ginzburg and Kumar proved finite generation of cohomology of small quantum groups, using techniques developed by Friedlander and Parshall for restricted enveloping algebras in 1983. In 2004 Etingof and Ostrik formulated a conjecture on finite generation of cohomology for any finite tensor category which encompasses categories of representations of finite dimensional Hopf algebras. The conjecture is still open in general, with recent progress on some pointed Hopf algebras by Mastnak, Pevtsova, Schauenburg, and Witherspoon. Their work illuminated the close relationship between the problem of finite generation of cohomology of some types of Hopf algebras and the deep structural properties of Nichols algebras. Bringing together experts in Nichols algebras and in homological algebra will lead to advances in the finite generation problem for many families of Hopf algebras. Such progress will enhance our understanding of various geometric invariants and the global structure of the categories of representations for finite dimensional Hopf algebras. (3) Hopf actions on Artin-Schelter algebras. One fundamental concept in noncommutative projective geometry is that of an Artin-Schelter (AS) regular algebra, introduced in the late 1980s by Artin and Schelter in their seminal paper that began the classification of the noncommutative regular algebras of dimension three (the quantum projective planes). This classification was completed by Artin, Tate and Van den Bergh in the early 1990s. Since then, more and more AS regular algebras and algebras with similar properties have been discovered in different subjects of mathematics such as noncommutative algebraic geometry and the theories of Hopf algebras, quantum groups, and Nichols algebras. A program investigating Hopf actions on AS regular algebras was initiated by Kirkman, Kuzmanovich and Zhang in 2010, that extended earlier work of Jorgensen and Zhang and of Jing and Zhang on group actions on AS regular algebras. The research in this area is closely related to the representation theory of Hopf algebras, homological aspects of noncommutative algebra and noncommutative algebraic geometry. Recently the classification of finite dimensional Hopf actions on AS regular algebras of dimension 2 was completed by Chan, Kirkman, Walton, Wang and Zhang. There are many open questions: one is to find an appropriate version of McKay correspondence for Hopf actions and another is the classification of finite dimensional Hopf actions on AS regular algebras of dimension 3 (or higher). Answers will require deep structural knowledge of the Hopf algebras involved, such as that obtained through the theory of Nichols algebras. (4) Noncommutative invariant theory: Invariant theory for Hopf actions. Classical invariant theory of commutative polynomial rings contains many beautiful results. In particular, from a homological algebra point of view, the following results are fundamental: Noether's theorem on integrality over invariant subrings, the Shephard-Todd-Chevalley Theorem (a criterion for a fixed subring to be a polynomial ring), the Watanabe Theorem (a criterion for a fixed subring to be Gorenstein) and the Kac- Watanabe-Gordeev Theorem (a criterion for a fixed subring to be a complete intersection ring). In the noncommutative world, Hopf algebra actions are more natural than group actions. Invariant theory of finite dimensional Hopf algebra actions on noncommutative algebras includes Etingof's very recent result generalizing the theorem of Noether and building on Skryabin's earlier work on Hopf actions on commutative algebras. On AS regular algebras, invariant theory has been partially developed by Kirkman, Kuzmanovich and Zhang, and includes noncommutative versions of classical results such as a Watanabe theorem for Hopf actions. One immediate goal is to establish Shephard-Todd-Chevalley and Kac-Watanabe- Gordeev theorems in the Hopf setting, where only partial results exist thus far. One also wants to understand well the actions of particular types of Hopf algebras, such as pointed Hopf algebras, in which case the underlying Nichols algebras are important. Due to the lack of comprehensive understanding of noncommutative actions on noncommutative spaces, much of the territory remains to be explored. Interaction among those working in noncommutative invariant theory and those working in the structure theory of some types of Hopf algebras, such as those arising from Nichols algebras, will be very fruitful. Any progress is also potentially useful in approaching questions regarding the structure of cohomology of Hopf algebras, where actions and invariants appear.(5) Tensor categories. One of the most important applications of finite quantum groups is an alternative construction of the fusion categories arising from the WZWN-model. This construction and the corresponding identification involve quite a few delicate and difficult tools from representation theory, vertex algebras and tensor category theory. It is expected that finite-dimensional Nichols algebras will produce new fusion categories by following the pattern of the construction just mentioned. In classifying various types of tensor categories, one is led to those all of whose simple objects are invertible, that is those that are representations of basic quasi-Hopf algebras. In case these quasi-Hopf algebras are Hopf algebras, dually one obtains pointed Hopf algebras, and again Nichols algebras arise. One is interested as well in the Drinfeld centers and module categories for these tensor categories, and invariant theory in this context. Nichols algebras of diagonal type were also used recently as the input of the definition of logarithmic conformal field theories by Semikhatov and collaborators. Discussions on these lines would be of great value for further developments.