Real Forms

Abstract

In the last chapter we classified all the semi-simple algebras over the complex numbers; but each complex algebra in general has several real forms. For example, we saw in Chapter 1 that sl(2, R) and su(2) are distinct real forms of the complex algebra sl(2, C). That is, there is no real isomorphism between them, yet they have the same complexification. The task of classifying the real forms of the semi-simple algebras is more involved than that of classifying the complex forms themselves. It was first done by Cartan in 1914. Later, in 1926–28 he used this classification to determine all the symmetric spaces. A complete discussion of the real forms and their relationship to the symmetric spaces is given in the well known book of Helgason, and we shall give only a brief introduction here.