On the structure of pseudo-Riemannian symmetric spaces
Ines Kath and Martin Olbrich
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Submission date: 23. Aug. 2004
published in: Transformation groups, 14 (2009) 4, p. 847-885
DOI number (of the published article): 10.1007/s00031-009-9071-z
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Following our approach to metric Lie algebras developed in a previous
paper we propose a way of understanding pseudo-Riemannian symmetric spaces
which are not semi-simple. We introduce cohomology sets (called
quadratic cohomology) associated with orthogonal
modules of Lie algebras with involution. Then we construct a functorial assignment
which sends a pseudo-Riemannian symmetric space M to a triple consisting of
(i) a Lie algebra with involution (of dimension much smaller than the dimension
of the transvection group of M),
(ii) a semi-simple orthogonal module of the Lie algebra with involution, and
(iii) a quadratic cohomology class of this module.
That leads to a classification scheme of indecomposable non-simple pseudo-Riemannian symmetric spaces. In addition, we obtain a full classification of symmetric spaces of index 2 (thereby completing and correcting in part earlier classification results due to Cahen/Parker and Neukirchner).