Chapter XIII: Further topics

Abstract

The aim of these notes was to introduce and to define the geometric objects corresponding to (real) Jordan structures, to present a fairly exhaustive description of the main examples and to give in this way a self-contained “geometric” introduction to Jordan theory. However, we have not even started to develop a “structure theory” in the usual sense which should contain the following elements: - A root theory for symmetric spaces with twist: relate the Peirce decomposition w.r.t. a complete system of orthogonal tripotents (cf. [Lo75], [Ne82]) with the root theory for symmetric spaces - see [Lo85] for the case of compact spaces. We have developed the theory as far as possible without root theory in order to convince the reader that in Jordan theory much can be done before root theory really becomes necessary - in contrast to Lie theory where root theory is an almost indispensable tool. - A representation theory: relate representations of Lie groups and symmetric spaces to representations of Jordan objects. With this aim in mind, we have in our presentation paid some attention on functorial questions - they show the way one has to go if one wants to understand what a geometric representation theory of Jordan objects should look like. Although there exists not yet even the outline of a theory, we can already see that (at least in the semisimple case) homomorphisms quite often have “good” properties in the sense that they carry over to Jordan-extensions, to conformal Lie algebras or even -groups, etc. The representation theory of Hermitian symmetric spaces is treated by Lie theoretic methods in [Sa80]; for some interesting results on representations of Jordan algebras cf. [Cl92].