Tag - Representations of Lie algebras

Bojko Bakalov: An operadic approach to vertex algebras and Poisson vertex algebras

8th December, 2022

I will start by reviewing the notions of vertex algebra, Poisson vertex algebra, and Lie conformal algebra, and their relations to each other. Then I will present a unified approach to all these algebras as Lie algebras in certain pseudo-tensor categories, or equivalently, as morphisms from the Lie operad to certain operads. As an application, I will introduce a cohomology theory of vertex algebras similarly to Lie algebra cohomology, and will show how it relates to the cohomology of Poisson vertex algebras and of Lie conformal algebras.

Jonathan Kujawa: Support varieties for Lie superalgebras  

26th July, 2022

Support varieties are a method which uses cohomology to bring commutative algebra and algebraic geometry to places where it doesn't obviously belong. Lie superalgebras are a graded analogue of Lie algebras and they seem well adapted this technology. In this talk, I will give an overview of recent efforts to use support varieties to study the representation theory of Lie superalgebras.

Ivan Penkov: New analogues of category 𝒪 for the Lie algebra 𝔰𝔩(∞)

5th May, 2022

I will recall several highest weight categories for sl(∞) studied in the past decade, and will then report on the newest highest weight categories introduced by P. Zadunaisky. A main point is the use a non-obvious Borel subalgebra plus a semi-large annihilator condition. As a side effect, the new categories produce interesting and challenging combinatorics.

Scott Larson: A Categorification of the Lusztig-Vogan Module

4th April, 2022

Continuous actions of real reductive groups are often studied by first linearizing the action to spaces related to functions, then using algebra via Lie algebras and compact groups (cf. Gelfand, Harish-Chandra, Vogan). This paradigm essentially simplifies to the easier problem of studying a complex algebraic group K acting on flag varieties. K-orbit closures are important for representation theory, are generalizations of Schubert varieties, and certain properties are explicitly determined via equivariant resolutions of singularities. In joint work with Anna Romanov, we provide a geometric and algebraic categorification of the Lusztig-Vogan module using the equivariant derived category. Our methods allow us to compute cohomology of all fibres of resolutions constructed quite generally and generalize Soergel bimodule techniques from complex to real reductive algebraic groups.

David Galban: Cohomology and Representation Theory for Lie Superalgebras

14th March, 2022

This talk will consist of two parts. In the first, I will describe the cohomology groups for the subalgebra 𝔫+ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu, essentially with these calculations essentially providing the first steps towards an analogue of Kostant’s theorem for Lie superalgebras. In the second part, based on joint work with Nakano, I will analyze the sheaf cohomology groups RI indBG L𝔣(λ), where L𝔣(λ) is an irreducible representation for the detecting subalgebra 𝔣, providing analogues for the BBW theorem and Kempf’s vanishing theorem for sufficiently large λ.

Joanna Meinel: Decompositions of tensor products: Highest weight vectors from branching

14th December, 2021

We consider tensor powers of the natural 𝔰𝔩n-representation, and we look for descriptions of highest weight vectors therein: We discuss explicit formulas for n=2, a recursion for n=3, and for bigger n we demonstrate how Jucys-Murphy elements allow us to compute highest weight vectors (both in theory and in practice using Sage).

Bojko Bakalov: On the Cohomology of Vertex Algebras and Poisson Vertex Algebras

17th October, 2021

Following Beilinson and Drinfeld, we describe vertex algebras as Lie algebras for a certain operad of n-ary chiral operations. This allows us to introduce the cohomology of a vertex algebra V as a
Lie algebra cohomology. When V is equipped with a good filtration, its associated graded is a Poisson vertex algebra. We relate the cohomology of V to the variational Poisson cohomology studied previously by De Sole and Kac.

Tamanna Chatterjee: Parity Sheaves Arising from Graded Lie Algebras I

30th August, 2021

Let G be a complex, connected, reductive, algebraic group, and χ : ℂ× → G be a fixed cocharacter that defines a grading on 𝔤, the Lie algebra of G. Let G0 be the centralizer of χ(ℂ×). Here I will talk about G0-equivariant parity sheaves on the n-graded piece, 𝔤n. For the first half we will spend on building the background of derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. We want to study DbG0(𝔤n, k) for characteristic of k is positive. For that we will dive into the results of Lusztig in characteristic 0 in the graded setting. The main result from Lusztig is that every perverse sheaf occurs as a direct summand of the parabolic induction of the simple perverse sheaf associated to some cuspidal pair. The goal of the second talk will be to extend this result into positive characteristic.

Daniele Rosso: Fixed rings of twisted generalized Weyl algebras

12th April, 2021

Twisted generalized Weyl algebras (TGWAs) are a large family of algebras that includes several algebras of interest for ring theory and representation theory, like Weyl algebras and quotients of the enveloping algebra of 𝔰𝔩2. In this work, we study invariants of TGWAs under diagonal graded automorphisms. Under certain conditions, we are able to show that the fixed ring of a TGWA by such an automorphism is again a TGWA. We apply this theorem to study properties of the fixed ring, such as the Noetherian property and simplicity. We also look at the behavior of simple weight modules for TGWAs when restricted to the action of the fixed ring.