Representations and Lie Theory Seminar

Autumn 2018

Time: Wednesday, 16:15 - 17:15
Location: MW 154

Schedule of talks:


August 29 Staff (après Etingof-Varchenko) Dynamical constructions in representation theory I. Fusion operator
September 19 Staff (après Etingof-Varchenko) Dynamical constructions in representation theory II. Weyl group
October 2 (Tuesday) 3-4PM

Unusual time! Same place!

Andrea Appel The KZ-Casimir system and braided Coxeter categories
October 10 No seminar Autumn break
October 24 Xiaomeng Xu Stokes phenomenon, Gelfand-Zeitlin systems and Frobenius manifolds
October 31
November 7 Petr Pushkar Quantum K theory of the Grassmannian and the Baxter operator
November 14 Alex Weekes Yangians and KLR algebras
November 21 No seminar Thanksgiving break
December 5 You Qi On a tensor product categorification at prime roots of unity


August 29: In this series of talks, we will go over the construction of a few dynamical operators which emerge naturally in the representation theory of Lie algebras. I will explain what the fusion operator is and why it satisfies the Knizhnik--Zamolodchikov (partial differential) equation. We will focus on the (dynamical) Weyl group symmetries of the fusion operator in a follow up talk.

September 19: In this part, I will present the definition and basic properties of a dynamical Weyl group action on a given representation of a simple Lie algebra (and time permitting, affine Lie algebra). I will not assume anything from the first talk.

October 2: It is well-known that the monodromy data of the KZ equations of a semisimple Lie algebra are encoded by a non-trivial braided tensor structure on the category of its deformation modules. In this talk, I will introduce the equivariant Casimir connection of a symmetrizable Kac-Moody algebra and I will then show that the monodromy data of the associated KZ-Casimir joint system are described by the axiomatic of braided Coxeter categories, which are, informally, braided tensor categories carrying an action of a given generalized braid group on their objects. This is based on joint works with V. Toledano Laredo.

October 24: This talk will include a general introduction to a linear differential system with irregular singularities, and its relation with Gelfand-Zeitlin systems, Yang-Baxter equations, and deformations of Frobenius manifolds.

November 7: In this talk I will define the quantum K-theory of the Grassmannian and show its connection to representation theory of quantum groups and quantum integrable systems. In particular, the Baxter operator will be identified with operators of quantum multiplication by quantum tautological classes via Bethe equations. Based on joint works with P.Koroteev, A.Smirnov and A.Zeitlin.

November 14: In joint work with J. Kamnitzer, P. Tingley, B. Webster and O. Yacobi, we've studied the representation theory of algebras called truncated shifted Yangians. Recently, we proved that this representation theory is governed by some algebras closely related to KLR algebras. In this talk, I'll describe this connection through a few simple examples. I'll also describe some applications to some (possibly) more familiar problems in representation theory, like the study of Gelfand-Tsetlin modules over U(gl(n)), and the calculation of characters of simple modules for Yangians.

December 5: Motivated by finding a categorical analogue of conformal blocks, we explain a formalism of extending a given categorical quantum group representation on a Weyl module to a certain tensor product representations. In particular, equipped with p-differential graded structures, the machinery gives rise to a categorification of certain tensor product representations of Weyl modules at prime roots of unity. This is based on joint work and work in progress with M. Khovanov and J. Sussan.

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