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Complexifications of symmetric spaces and Jordan theory


Author: Wolfgang Bertram
Journal: Trans. Amer. Math. Soc. 353 (2001), 2531-2556
MSC (2000): Primary 17C36, 53C15; Secondary 22E15, 53B35
DOI: https://doi.org/10.1090/S0002-9947-01-02779-9
Published electronically: February 15, 2001
MathSciNet review: 1814081
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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. Makarevic 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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Additional Information

Wolfgang Bertram
Affiliation: Institut Elie Cartan, Département de Mathématiques, Université Henri Poincaré (Nancy I), B.P. 239, 54506 Vandœuvre-les-Nancy Cedex, France
Email: bertram@iecn.u-nancy.fr

DOI: https://doi.org/10.1090/S0002-9947-01-02779-9
Keywords: Symmetric space, complexification, Jordan and Lie triple system
Received by editor(s): February 19, 1998
Published electronically: February 15, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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