Mikhail Khovanov (Columbia University), Brauer Lectures

Lecture 3 Title: Categorification of quantum groups at generic q and at a root of unity

Abstract: Quantum link invariants naturally emerge from actions of quantum groups on tensor powers of their irreducible representations. Categorifications of quantum groups and how they fit into link homology theories will be explained. Categorified quantum groups act on many categories of interest in representation theory and adjacent fields. The quantum parameter q, in the generic situation, becomes a grading shift upon categorification. For the generalization from links to 3-manifolds it is important to understand what happens at the categorified level when q is a root of unity. A possible approach to that problem, via Hopfological algebra, a generalization of homological algebra, will be explained in the talk.