Lie Groups and Algebraic Groups

Abstract

Hermann Weyl, in his famous book The Classical Groups, Their Invariants and Representations [164], coined the name classical groups for certain families of matrix groups. In this chapter we introduce these groups and develop the basic ideas of Lie groups, Lie algebras, and linear algebraic groups. We show how to put a Lie group structure on a closed subgroup of the general linear group and determine the Lie algebras of the classical groups. We develop the theory of complex linear algebraic groups far enough to obtain the basic results on their Lie algebras, rational representations, and Jordan–Chevalley decompositions (we defer the deeper results about algebraic groups to Chapter 11). We show that linear algebraic groups are Lie groups, introduce the notion of a real form of an algebraic group (considered as a Lie group), and show how the classical groups introduced at the beginning of the chapter appear as real forms of linear algebraic groups.