Lie Groups and Lie Algebras

Abstract

The objective of this chapter is to introduce the concepts of Lie groups and their Lie algebras. The Lie algebra $\mathfrak {g}$ of a Lie group G is defined as the space of invariant vector fields (left or right, depending on choice), with bracket given by the Lie bracket of vector fields. The flows of invariant vector fields establish the exponential map $\exp :\mathfrak {g}\rightarrow G$, which is the main link between $\mathfrak {g}$ and G. These constructions make exhaustive use of results about vector fields on manifolds and their Lie brackets. A short collection of such results can be found in Appendix A. Adjoint representations are another tool for linking Lie groups and their Lie algebras. The formulas involving such representations are developed in this chapter. They are used along the whole theory.