Titles and Abstracts

Representation Theory Day

Université Laval, 29 July 2017

10-10:50am, Daniel Nakano (University of Georgia)

Tensor Triangular Geometry and Applications

Tensor triangular geometry as introduced by Paul Balmer is a powerful idea which can be used to extract hidden ambient geometry from a given tensor triangulated category. These geometric structures often arise at the derived/cohomological level and play an important role in understanding the combinatorial property of representations of groups and algebras.

In this talk I will first present a general setting for a compactly generated tensor triangulated category that enables one to classify thick tensor ideals and to determine the Balmer Spectrum. Several examples will be presented to illustrate these beautiful connections which includes the stable module category for finite groups and the derived category of bounded perfect complexes for finitely generated R-modules where R is a commutative Noetherian ring.

For a classical Lie superalgebra g=g_0+g_1, I will show how to construct a Zariski space from a detecting subalgebra f and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional g-modules that are semisimple over g_0. Concrete realizations will be provided for the Lie superalgebra gl(m|n). This answers an old question about finding a geometric object that governs the representation theory for Lie superalgebras. I will also discuss new applications involving quantum groups and affine Lie algebras.

These results represent joint work with B. Boe and J. Kujawa.

11-11:50am, Vera Serganova (University of California, Berkeley)

Finite W-algebras for the Lie Superalgebra Q(n)

A finite W-algebra is a quantization of the Slodowy slice to a nilpotent orbit in the coadjoint representation of a reductive Lie algebra or superalgebra. W-algebras have a lot of applications to representation theory and primitive ideals. We present several general results about finite W-algebras in the supercase and discuss in detail the case of the strange superalgebra Q(n), relating W-algebras with Q-Yangians of Sergeev and Nazarov. This is a joint work with E. Poletaeva.

2-2:50pm, Nicolas Libedinsky (Universidad de Chile)

Recent Developments in Soergel Bimodules

We will start by recalling Hecke algebras and Kazhdan-Lusztig theory. Then we will speak about the Hecke category (Soergel bimodules), the morphisms in this category (light leaves) and how does this fit into the several proofs and disproofs done recently by Geordie Williamson and his collaborators (proof of Kazhdan-Lusztig conjecture, disproof of Lusztig's and James conjectures). We will end by speaking about the new lights appearing in the theory (due mostly to Williamson, but also to Riche, Lusztig, etc).

3-3:20pm, Jie Sun (Michigan Tech University)

Universal Central Extensions of Twisted Current Algebras

Twisted current algebras are fixed point subalgebras of tensor products of Lie algebras and associative algebras under finite group actions. Examples of twisted current algebras include multiloop Lie algebras, twisted forms and equivariant map algebras. In this talk, central extensions of twisted current algebras are constructed and conditions are found under which the construction gives universal central extensions of twisted current algebras.

4-4:50pm, Georgia Benkart (University of Wisconsin, Madison)

Tracing a Path - From Walks on Quivers to Invariant Theory and the Representation Theory of Hopf Algebras

Molien's 1897 formula for the Poincaré series of the polynomial invariants of a finite group has given rise to results in algebraic geometry, coding theory, combinatorics, mathematical physics, and representation theory. This talk will describe an analogue of Molien's formula for tensor invariants. The approach to this and other results discussed in the talk is via walking on McKay quivers, and the associated McKay-Cartan matrix plays a key role. In our work with Klivans and Reiner, we relate McKay-Cartan matrices to chip-firing and sandpile dynamics and show that the matrices are "avalanche-finite." There is beautiful recent generalization of this picture to the representation theory of finite-dimensional Hopf algebras of arbitrary characteristic. The representations of the restricted enveloping algebra of sl(2) over a field of positive characteristic provide an illuminating example.
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