Integrating categorical and geometric methods in non-semisimple representation theories

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The aim of this project is to solve several open problems in representation theory of Lie algebras and superalgebras via categorical and geometric methods. In particular, we propose to study the category T of tensor representation of sl-infinity. In this proposal this category plays two different roles. In application to infinite-dimensional Lie algebras it is the target category for categorification of boson-fermion correspondence, the Fock space equipped with action of free bosons and fermions is identified with the complexified Grothendieck group of the category T. In application to Lie superalgebras we use the approach originally suggested by Brundan. Namely, we identify the complexified Grothendieck group of the category of representation of a classical supergroup G and certain "tensor" representation M of sl-infinity, and categorify the action of the Chevalley generators of sl-infinity in M by the translation functors. Recent papers of Brundan and Stroppel, Cheng, Lam and Wang, Gruson and the author demonstrate the power of this approach by making essential progress in long standing open problems (such as character formulae, Kazhdan-Lusztig theory for the general linear superalgebras and explicit description of extension groups). We suggest to combine this approach with our results on the structure of individual sl-infinity modules in T to obtain new information about modules over Lie superalgebras: description of tensor products, calculation of superdimension, Kazhdan-Lusztig theory for the orthosymplectic superalgebras. Another goal is to generalize T to the case of fields of positive characteristic, and explore connection with Deligne's tensor categories. We also combine the above approach with geometric methods in representation theory of Lie superalgebras. In particular, we introduce several conjectures relating thick ideals of the tensor category G-mod for a classical supergroup G, equivariant sheaves on the self-commuting cone of G and the socle filtration of the corresponding sl-infinity modules. The last part of proposal addresses the problem of generalizing Borel-Weil-Bott theorem for supergroups. Although partial results in this area were obtained almost 30 years ago, a complete answer is still unknown. Supersymmetry is an important tool in modern theoretical physics. The methods of supersymmetry factor the questions interesting for physicists through the theory of representations of supergroups and superalgebras. There are still a lot of gaps in mathematical foundations in this subject: the methods familiar from representation theory of reductive groups get stuck early since the algebraic structure of representations is significantly more complicated. As new phenomena were discovered, it turned out that they have analogues in other non-semisimple branches of representation theory: modular representations and representations of infinite-dimensional Lie algebras. The latter representations have a wide range of applications: from integrable systems to string theory. We suggest a new approach to a key tool of this theory, vertex operators, by categorification of boson-fermion correspondence. We also plan to study further the duality between representation of infinite-dimensional Lie algebras and classical supergroups.