Complexifications of symmetric spaces and Jordan theory

Bertram, Wolfgang

doi:10.1090/S0002-9947-01-02779-9

Complexifications of symmetric spaces and Jordan theory
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Trans. Amer. Math. Soc. 353 (2001), 2531-2556 Request permission

Abstract:

Generalizing Hermitian and pseudo-Hermitian spaces, we define twisted complex symmetric spaces, and we show that they correspond to an algebraic object called Hermitian Jordan triple products. The main topic of this work is to investigate the class of real forms of twisted complex symmetric spaces, called the category of symmetric spaces with twist. We show that this category is equivalent to the category of all real Jordan triple systems, and we can use a work of B.O. Makarevič in order to classify the irreducible spaces. The classification shows that most irreducible symmetric spaces have exactly one twisted complexification. This leads to open problems concerning the relation of Jordan and Lie triple systems.

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