2 - Calculus on Lie groups

Summary

In this chapter we examine briefly the details of the technical definition of a Lie group. This chapter can be skipped on a first reading of this book. Eventually, however, taking a small amount of time to be familiar with the the concepts involved will pay major dividends when it comes to understanding the proofs of the key theorems.

By definition, Lie groups are locally Euclidean, so we can use tools we know and love from calculus to study functions, vector fields and so on that can be defined on them. Thus, we study differentiation on a Lie group. There are at least three important cases to consider. The first involves understanding the intrinsic definition of tangent vectors. These ideas inform every other understanding of a tangent vector, so we do that first. A second and simpler line of argument is strictly for matrix presentations, while a third treats tangent vectors as linear, first order differential operators. We will need all three.

The major theorem we prove is that the set of tangent vectors at any given point g ∈ G is in one-to-one correspondence with the set of one parameter subgroups of G. After a discussion of the exponential map in its various guises, we end the chapter with a discussion of concepts analogous to tangent vectors, one parameter subgroups and the exponential map for transformation groups.