On the structure of pseudo-Riemannian symmetric spaces

By Ines Kath, M. Olbrich. 2004

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Abstract.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 semisimple. 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 semisimple 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 nonsimple 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 and Parker and to Neukirchner).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4d584464bcf4a8b5e8ae09e1f45d68afc875249f