2 00 8 Z 2 × Z 2 - symmetric spaces

The notion of a Γ-symmetric space is a generalization of the classical notion of a symmetric space, where a general finite abelian group Γ replaces the group Z2. The case Γ = Z k has also been studied, from the algebraic point of view by V.Kac [14] and from the point of view of the differential geometry by Ledger, Obata [17], Kowalski [16] or Wolf-Gray [20] 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 Γ-symmetric space was introduced by R.Lutz in [18]. We approach the classification of such spaces in the case Γ = Z2 × Z2 using recent results (see [2]) on the classification of complex Z2 × Z2-graded simple Lie algebras.