Fundamental Groups of Split Real Kac-Moody Groups

Abstract

The structure of maximal compact subgroups in semisimple Lie groups was investigated by Cartan and, later, Mostow: In 1949, Mostow gave a new proof of a Cartan´s theorem stating that a connected semisimple Lie group G is a topological product of a maximal compact subgroup K and a Euclidean space, implying in particular that G and K have isomorphic fundamental groups. Subsequent case-by-case analysis provided the isomorphism types of these maximal compact subgroups and their fundamental groups.Starting in the 1940´s, Dynkin diagrams have been used to describe the structure of simple Lie groups. Dynkin diagrams correspond to Cartan matrices, and a generalization of this concept led to the theory of Kac-Moody algebras and, in particular, their associated Kac-Moody groups, developed by Kac and Tits. Kac-Moody groups endowed with the Kac-Peterson topology have been extensively investigated by Köhl and Hartnick.The aim of this thesis is to determine the fundamental group of any algebraically simply connected semisimple split real topological Kac-Moody group associated to a symmetrizable generalized Cartan matrix. We present a uniform result which makes it possible to determine the fundamental group of such a group - and, in particular, of any algebraically simply connected split real simple Lie group - directly from its Dynkin diagram.