Chapter IV: The classical spaces

Abstract

In this chapter we present a fairly exhaustive collection of examples of “classical” symmetric spaces with twist, and we explain the basic principles of their classification. The classification itself is contained in the tables given in Chapter XII. As noted there (Remark XII.4.8), comparison of the classification of simple real Lie triple systems (carried out by M. Berger, cf. [B57] and the classification of simple real Jordan triple systems (carried out by E. Neher, cf. [Ne80], [Ne8l], and implicitly in another way by B.O. Makarevič, cf. [Ma73] leads to the following two remarkable observations: A. (Existence.) Every non-exceptional irreducible symmetric space has (possibly after a central extension) a Jordan-extension. (A central extension is needed e.g. in the group case of S1(n, ℝ) which does not admit any Jordan extension, but the central extension G1(n, ℝ) does.) B. (Uniqueness.) The number of (local) Jordan extensions of an irreducible non-exceptional symmetric space is either 0, 1, 2 or 3; in most cases it is 1. Summarizing, we can say that the Jordan-Lie functor seems to be fairly close to being injective and surjective. This has already been remarked by E. Neher, cf. [Ne85], where also the exceptional spaces are considered.