Tag - Finite W-algebras

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.

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.

Lewis Topley: Modular representation theory and finite W-algebras

15th May, 2023

Finite W-algebras were introduced by Premet in full generality, and they quickly became quite famous for their many applications in the representation theory of complex semisimple Lie algebras, especially the classification of primitive ideals. However, these algebras first appeared in the representation theory of Lie algebras associated to reductive groups in positive characteristic. In this talk I will survey the history of finite W-algebras in modular representation theory, and explain some of the contributions I have made to the field. The main applications in this talk will be the construction and classification of 'small' modules of Lie algebras.

Alexander Premet: Modular representations of Lie algebras and Humphreys’s Conjecture

25th September, 2020

Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic p and suppose that the derived subgroup of G is simply connected, p is a good prime for the root system of G and the Lie algebra 𝔤=Lie(G) admits a non-degenerate Ad G-invariant symmetric bilinear form. If G is a simple algebraic group of type other than A, the above assumptions mean that p is a good prime for G, i.e. p ≥ 3 if G is of type B, C or D, p ≥ 5 if G is of type G2, F4, E6 or E7, and p ≥ 7 if G is of type E8. If all components of G have type A, B, C, D we set R=ℤ[1/2].  If G has a component of exceptional type but has no components of type E8 we set R=ℤ[1/6]. If G has a component of type E8 we set R=ℤ[1/30]. Given a linear function χ on 𝔤 we denote by Uχ(𝔤) the reduced enveloping algebra of 𝔤 associated with χ. By the Kac-Weisfeiler conjecture (now a theorem), any Uχ(𝔤)-module has dimension divisible by pd(χ) where 2d(χ) is the dimension of the coadjoint G-orbit of χ. In my talk, based on a joint work with Lewis Topley, I'll discuss a natural question raised in the 1990s by Kac, Humphreys and myself and explain that for any χ ∈ 𝔤* the reduced enveloping algebra Uχ(𝔤) has an irreducible module of dimension pd(χ). Forms of finite W-algebras over the ring R and their reductions modulo good primes play a crucial role in our arguments. We also use some recent results on multiplicty-free primitive ideals of U(𝔤𝒞) associated with the rigid nilpotent orbits in complex simple Lie algebras 𝔤𝒞.

Tomoyuki Arakawa: Urod algebra and translation for W-algebras

9th July, 2020

In 2016 Bershtein, Feigin and Litvinov introduced the Urod algebra, which gives a representation-theoretic interpretation of the celebrated Nakajima-Yoshioka blowup equations in the case that the sheaves are of rank two. In this talk we will introduce higher-rank Urod algebras. This is done by constructing translation functors for affine W-algebras.