Real Semisimple Lie Groups

Abstract

Our study of real semisimple Lie groups and algebras is based on the theory of complex semisimple Lie groups developed in Ch. 4. This is possible because the complexification of a real semisimple Lie algebra is also semisimple (see 1.4.7). However, the correspondence between real and complex semisimple Lie algebras established with the help of the complexification is not one-to-one; any complex semisimple Lie group has at least two non-isomorphic real forms. As it turns out, to describe the real forms of a given complex semisimple Lie algebra g is the same as to classify the involutive automorphisms of g up to conjugacy in Aut g. This classification is easily obtained from the results of 4.4. The global classification of real semisimple Lie groups makes use of the so-called Cartan decomposition of these groups which also plays an important role in various applications of the Lie group theory.