2004/08/18 by Ines Kath, Kath, Ines, Martin Olbrich +1 · 2 citations
Mathematics · #17B56 #53C35 #53C50 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Representation Theory (math.RT) #math.DG #math.RT #msc:17B56 #msc:53C35 #msc:53C50
paper · pdf · doi:10.48550/arxiv.math/0408249
37 pages
arxiv created 2004/08/18 · openalex publication_date 2004/08/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Following our approach to metric Lie algebras developed in math.DG/0312243 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 math.DG/0301326).