Title: Quantum groups from quantum character varieties

Date: 10/19/2016

Time: 2:30 PM - 3:50 PM

Place: C304 Wells Hall

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.