Normal Subgroups, Nilpotent and Solvable Lie Groups

Abstract

In this chapter, we address structural aspects of Lie groups. Here an important issue is to see that for any closed normal subgroup N of a Lie group G, the quotient G/N carries a canonical Lie group structure, so that we may consider N and G/N as two pieces into which G decomposes. With this information, we then address the canonical factorization of a morphism of Lie groups into a surjective, a bijective, and an injective one. In particular, we describe some tools to calculate fundamental groups of Lie groups and homogeneous spaces.