On the structure of pseudo-Riemannian symmetric spaces

TL;DR

This paper introduces a new cohomology-based framework for classifying non-semisimple pseudo-Riemannian symmetric spaces, including a complete classification of those with index 2, refining earlier results.

Contribution

It develops quadratic cohomology for Lie algebras with involution and provides a functorial classification scheme for indecomposable non-simple pseudo-Riemannian symmetric spaces.

Findings

Classification scheme for indecomposable non-simple pseudo-Riemannian symmetric spaces

Complete classification of symmetric spaces of index 2

Refinement and correction of earlier classification results

Abstract

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).