Tag - Kac-Moody algebras

Jon Brundan: Isomeric Heisenberg and Kac–Moody categorification

29th May, 2024

The isomeric Heisenberg category acts naturally on a number of abelian categories appearing in the representation theory of the isomeric supergroup Q(n), and also on representations of Sergeev’s algebra which is related to the double covers of symmetric groups. I will explain an efficient way to convert an action of the isomeric Heisenberg category on these and other abelian categories into an action of a corresponding super Kac–Moody 2-category. To properly understand the odd simple root indexed by the element zero of the ground field requires the theory of odd symmetric functions developed by Ellis, Khovanov and Lauda, the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka, and the covering quantum groups defined and studied by Clark and Wang.

Peng Shan: Modularity for W-algebras and affine Springer fibres

31st August, 2023

We will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibres. We will also explain how to match the modular group action on the characters with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. This is based on joint work with Dan Xie and Wenbin Yan.

Kaiming Zhao: Simple smooth modules

29th May, 2023

Let L be a graded Lie algebra by integers with k-th homogenous space Lk where k are integers. An L-module V is called a smooth module if any vector in V can be annihilated by Lk for all sufficiently large k. Smooth modules for affine Kac-Moody algebras were introduced and studied by Kazhdan and Lusztig in 1993. I will show why this class of modules should be studied and what results are known now. An easy characterization for simple smooth modules for some Lie algebras will be provided.

Slava Futorny: Free Field Constructions for Affine Kac-Moody Algebras

15th October, 2021

Classical free field realizations of affine Kac-Moody algebras (introduced by M.Wakimoto, B.Feigin and E.Frenkel) play an important role in quantum field theory. B.Cox initiated the study of free field realizations for the non-standard Borel subalgebras which led to an important class of intermediate (or parabolic) Wakimoto modules. A uniform construction of such realizations will be discussed based on a joint work with L.Krizka and P.Somberg.

Mikhail Kotchetov: Fine gradings on classical simple Lie algebras

8th July, 2021

Gradings by abelian groups have played an important role in the theory of Lie algebras since its beginning: the best known example is the root space decomposition of a semisimple complex Lie algebra, which is a grading by a free abelian group (the root lattice). Involutive automorphisms or, equivalently, gradings by the cyclic group of order 2, appear in the classification of real forms of these Lie algebras. Gradings by all cyclic groups were classified by V. Kac in the late 1960s and applied to the study of symmetric spaces and affine Kac-Moody Lie algebras.

In the past two decades there has been considerable interest in classifying gradings by arbitrary groups on algebras of different varieties including associative, Lie and Jordan. Of particular importance are the so-called fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism, although not in a unique way. If the ground field is algebraically closed and of characteristic 0, then the classification of fine abelian group gradings on an algebra (up to equivalence) is the same as the classification of maximal quasitori in the algebraic group of automorphisms (up to conjugation). Such a classification is now known for all finite-dimensional simple complex Lie algebras.

In this talk I will review the above mentioned classification and present a recent joint work with A. Elduque and A. Rodrigo-Escudero in which we classify fine gradings on classical simple real Lie algebras.

Maria Gorelik: Depths and cores in the light of DS-functors

25th March, 2021

The Duflo-Serganova functors DS are tensor functors relating representations of different Lie superalgebras. In this talk I will consider the behaviour of various invariants, such as the defect, the dual Coxeter number, the atypicality and the cores, under the DS-functor. I will introduce a notion of depth playing the role of defect for algebras and atypicality for modules. I will mainly concentrate on examples of symmetrizable Kac-Moody and Q-type superalgebras.

Milen Yakimov: Quantum cluster algebras at roots of unity and discriminants

3rd October, 2020

Cluster algebras were invented by Fomin and Zelevinsky twenty years ago. Since then they have played an important role in a number of settings in combinatorics, geometry, representation theory and topology. We will introduce a notion of root of unity quantum cluster algebras which are PI algebras, and will show that they have large canonical central subalgebras isomorphic to the original cluster algebras. These are far reaching generalizations of the De Concini-Kac-Procesi central subalgebras that appear in the study of the irreducible representations of big quantum groups. We will describe a general theorem computing the discriminants of these algebras. In a special situation it yields a formula for the discriminants of the quantum unipotent cells at roots of unity associated to all symmetrizable Kac-Moody algebras.