From the representation theory of compact Lie groups and their finite subgroups to modular fusion categories associated with affine Lie algebras at some level (or quantum groups at roots of unity) and their module categories.

Abstract: 

Using representation theory of affine Lie algebras, or of quantum groups at roots of unity, one constructs modular fusion categories that have been used for quite a while in various fields of mathematics, as well as in fundamental physics (fusion rules in conformal field theory, WZW models, string theories). In turn, these fusion categories have modules which consitute a kind of quantum analog of the theory of representations for finite subgroups of Lie groups. The purpose of this general talk is to present a few introductory concepts using (classical) representation theory as a guide, without using any results from the theory of affine Lie algebras or quantum groups, and to describe several examples taken from the "quantum Lie subgroups classification", which is known for SU(2), SU(3) and SU(4).