Noncompact Semi-Simple Groups

Abstract

This chapter addresses noncompact semi-simple Lie groups. The starting point is the construction of the Cartan and Iwasawa decompositions of the Lie groups. Either decomposition shows that the differentiable manifold underlying a noncompact semi-simple and connected Lie group G is the product of some Euclidean space with a connected Lie group K whose Lie algebra is compact. Due to these splittings, the question of describing the universal covering $\widetilde {G}$ of G reduces to determining the universal covering of K, which has been done earlier.