Duncan Laurie (University of Edinburgh) – The structure and representation theory of quantum toroidal algebras

Quantum toroidal algebras Uq(g_tor) are the ‘double affine’ objects within the quantum setting. After reviewing and motivating the more well-studied finite and affine quantum groups, we’ll introduce these quantum toroidal algebras and outline a collection of new results.

First we shall construct an action of the double affine braid group, and derive anti-involutions ψ of Uq(g_tor) that form part of a surprising modular GL(2,Z) symmetry.

Then, using ψ, we will investigate the representation theory of Uq(g_tor), obtaining well-defined tensor products and intertwiners (R-matrices) for an important category of modules.