Tag - Representation theory

Tomoyuki Arakawa: Hilbert Schemes of the points in the plane and quasi-lisse vertex superalgebras

29th August, 2023

For each complex reflection group Γ one can attach a canonical symplectic singularity ℳΓ. Motivated by the 4D/2D duality discovered by Beem et al., Bonetti, Menegheli and Rastelli conjectured the existence of a supersymmetric vertex operator algebra WΓ whose associated variety is isomorphic to ℳΓ. We prove this conjecture when the complex reflection group Γ is the symmetric group SN, by constructing a sheaf of ℏ-adic vertex algebras on the Hilbert schemes of N points in the plane. In physical terms, the vertex operator algebra WSN corresponds, by the 4D/2D duality, to the 4-dimensional N=4 super Yang-Mills theory with gauge group SLN.

Nate Harman: Stability Phenomena in Representation Theory

28th August, 2023

For various natural sequences of groups, such as the general linear groups GLn or symmetric groups Sn, certain aspects of their representation theory act the same for all sufficiently large n. A classic example of this is Schur-Weyl duality, which gives a uniform description of degree d representations of GLn, provided n is at least d. I will discuss this and other examples of stability phenomena in representation theory, and how this sort of stabilization manifests itself in other areas of mathematics.

Thomas Creutzig: Tensor categories of modules of W-algebras

28th August, 2023

Let V be an affine vertex algebra of some simple Lie algebra 𝔤 and some level. Let KL be the category of V-modules whose conformal weight spaces are integrable 𝔤-modules. A famous result of Kazhdan and Lusztig tells us that for almost all levels KL is a braided tensor category and as such equivalent to a category of weight modules of the quantum group Uq(𝔤) of 𝔤 for suitable q.

It is desired to have similar results for suitable categories of W-algebras and superalgebras. In particular one wants to understand tensor structure and equivalences to quantum supergroups.

I will outline how to prove such statements and illustrate this in some examples.

Gurbir Dhillon: Geometric representations of affine W-algebras

28th August, 2023

We will discuss how to realize categories of highest weight representations of affine W-algebras as categories of perverse sheaves on affine flag manifolds, and the modules for affine Hecke algebras which they categorify. Applications, e.g. to character formulas and the Kac-Roan-Wakimoto conjecture, will be discussed. For principal nilpotent elements, this was worked out jointly with Raskin, and the general case is work in progress with Arakawa.

Eric Vasserot: Non-symmetric quantum groups and critical cohomology of quiver varieties

28th August, 2023

We realize the quantum loop groups and shifted quantum loop groups of arbitrary types, possibly non-symmetric, using critical K-theory. This gives a generalization of Nakajima’s construction of symmetric quantum loop groups via quiver varieties to non-symmetric types. This also yields a geometric realization of some simple modules, in particular the Kirillov-Reshethikin modules, and the tensor product of prefundamental modules.

Csaba Schneider: Computing invariants of some nilpotent Lie algebras

21st August, 2023

I will present some interesting computations concerning polynomial and rational invariants of nilpotent Lie algebras. I will say more about standard filiform Lie algebras which appear to have the highest level of complication among the small-dimensional algebras. I will outline an implementable algorithm for the computation of generators of the field of rational invariants.

Lleonard Rubio y Degrassi: Hochschild cohomology groups under gluing idempotents

7th August, 2023

Stable equivalences occur frequently in the representation theory of finite-dimensional algebras; however, these equivalences are poorly understood. An interesting class of stable equivalences is obtained by ‘gluing’ two idempotents. More precisely, let A be a finite-dimensional algebra with a simple projective module and a simple injective module. Assume that B is a subalgebra of A having the same Jacobson radical. Then B is constructed by identifying the two idempotents belonging to the simple projective module and to the simple injective module, respectively. In this talk we will compare the first Hochschild cohomology groups of finite-dimensional monomial algebras under gluing two arbitrary idempotents (hence not necessarily inducing a stable equivalence). As a corollary, we will show that stable equivalences obtained by gluing two idempotents provide 'some functoriality' to the first Hochschild cohomology, that is, HH1(A) is isomorphic to a quotient of HH1(B).

Ziqing Xiang: Quantum wreath product

28th July, 2023

The classical wreath product G ≀ Σd is a semidirect product Gd ⋊ Σd with Σd acting on Gd by permutations. We deform this classical wreath product by deforming G into an associative algebra B, deforming Σd into a Hecke algebra, and deforming the action. The result is called a quantum wreath product B ≀ H(d). Many variants of Hecke algebras can be viewed as quantum wreath products, hence could be treated in a unified manner.

In this talk, we will discuss necessary and sufficient conditions for quantum wreath products to have a basis of suitable size. We will also discuss some other structural results, the Schur algebras of these quantum wreath products, and their representations.