Lie Groups and Algebras: Differential Geometric Approach

Abstract

In Chapter 3 we constructed a local Lie group from a linear Lie algebra by exponentiating the matrices in the algebra. By Ado’s theorem, this method succeeds in obtaining all local Lie groups; but Lie’s original methods involved the integration of overdetermined systems of partial differential equations. The classical solution of the problem is quite involved, though Pontryagin gives a fairly concise treatment of it. The calculations are simplified considerably by Cartan’s use of the calculus of differential forms (exterior calculus). We shall develop the subject in this chapter, comparing both the classical and modern approaches, and giving efficient proofs of the basic results using the exterior calculus. This chapter is included purely for its historical interest, and is independent of the rest of the book.