Lie Groups and Homogeneous Spaces

Abstract

In this chapter we will give a brief introduction to Lie groups and homogeneous spaces. We first define the notions of Lie groups and Lie algebras and show the relationship between Lie groups and Lie algebras. In particular, we will give some methods to compute the Lie algebra of a Lie group. Many examples of Lie groups and Lie algebras, including the general linear Lie groups, orthogonal Lie groups, symplectic groups, and spin groups, as well as their Lie algebras, are given in Sect. 2.1. In Sect. 2.2, we recall the method of Lie transformation groups, which is one of the most important tools in differential geometry. Section 2.3 is devoted to introducing the structure and classification of complex semisimple Lie algebras. In Sect. 2.4, we collect some important results on homogeneous Riemannian manifolds. Finally, in Sect. 2.5, we present the theory of Riemannian symmetric spaces.