Title: Quantum groups from quantum character varieties

Date: 10/19/2016

Time: 2:30 PM - 3:50 PM

Place: C304 Wells Hall

Speaker: Alexander Shapiro, University of Toronto

It was shown by Fock and Goncharov that moduli spaces of local systems on decorated surfaces provide examples of cluster varieties and thus admit canonical quantizations. I will describe a joint work with Gus Schrader where we embed the quantum group U_q(sl_n) into the quantized moduli space of $SL(n)$-local systems on a punctured disk with two marked points. This embedding endows the quantum group with a system of quantum cluster coordinates. It also allows us to realize the adjoint action of the $R$-matrix as a half Dehn twist of a twice punctured disk and factor it into a sequence of cluster mutations. If time permits, I will discuss applications of our construction to the representation theory.
The talk will be preceded by a 20-25 minutes refresher on quantum groups. I will recall the definition and discuss how one might think of a quantum group as a quantization of a Poisson-Lie group on one hand, and a deformation of a universal enveloping algebra on the other. The main part of the talk will be independent of the first one, all necessary definitions will be reintroduced.