Algebra/Topology Seminar

Speaker: Ralf Köhl

Title: Comparing Lie groups and Kac-Moody groups

Abstract: Kac-Moody groups can be thought of as infinite-dimensional Chevalley groups. Over finite fields they provide a wealth of examples and counterexamples for geometric group theory, as they are lattices inside their locally compact completions, just like k[t,1/t] is a lattice diagonally embedded inside k((t)) \times k((1/t)). Over the real numbers they are of interest to theoretical physics, notably supergravity.

In my talk I focus on Kac-Moody groups over the reals. Lacking concrete feasible models, the first step is to gain some intuition for Kac-Moody groups. To this end I will study Lie groups from a point of view that can be generalized to Kac-Moody groups, as colimits of suitable amalgams of groups. Then I will highlight various recent developments in the theory of Kac-Moody groups, e.g. computing their fundamental groups. In passing we will arrive at a nice combinatorial approach to the fundamental group that also explains why the fundamental groups of split real semisimple Lie groups are what they are.