(Z/2Z x Z/2Z)-symmetric spaces

Abstract

The notion of a $\Gamma $-symmetric space is a generalization of the classical notion of a symmetric space, where a general finite abelian group $\Gamma $ replaces the group $Z_2$. The case $\Gamma =\Z_k$ has also been studied, from the algebraic point of view by V.Kac \cite{VK} and from the point of view of the differential geometry by Ledger, Obata, Kowalski or Wolf - Gray in terms of $k$-symmetric spaces. In this case, a $k$-manifold is an homogeneous reductive space and the classification of these varieties is given by the corresponding classification of graded Lie algebras. The general notion of a $\Gamma $-symmetric space was introduced by R.Lutz. We approach the classification of such spaces in the case $\Gamma=Z_2^2$ using recent results on the classification of complex $Z_2^2$-graded simple Lie algebras.