Lie Groups

Abstract

In this chapter we introduce Lie groups, which are smooth manifolds that are also groups in which multiplication and inversion are smooth maps. Besides providing many examples of interesting manifolds themselves, they are essential tools in the study of more general manifolds, primarily because of the role they play as groups of symmetries of other manifolds. We begin with the definition of Lie groups and some of the basic structures associated with them, and then present a number of examples. Next we study Lie group homomorphisms, which are group homomorphisms that are also smooth maps. Then we introduce Lie subgroups (subgroups that are also smooth submanifolds), which lead to a number of new examples of Lie groups. After explaining these basic ideas, we introduce actions of Lie groups on manifolds, which are the primary raison d’être of Lie groups. At the end of the chapter, we briefly touch on group representations.