ℤ₂× ℤ₂-symmetric spaces

The notion of a -symmetric space is a generalization of the classical notion of a symmetric space, where a general finite abelian group replaces the group ‫ޚ‬ 2 .We approach the classification of -symmetric space in the case = ‫ޚ‬ 2 × ‫ޚ‬ 2 using recent results on the classification of complex ‫ޚ‬ 2 × ‫ޚ‬ 2graded simple Lie algebras.The notion of a -symmetric space, introduced by R. Lutz [1981], is a generalization of the classical notion of a symmetric space, where a general finite abelian group replaces the group ‫ޚ‬ 2 .The case = ‫ޚ‬ k has also been studied, from the algebraic point of view by V. Kac [1968] and from the point of view of differential geometry by Ledger and Obata [1968], by Kowalski [1980] or by Wolf and Gray [1968] in terms of k-symmetric spaces.In this case, a k-manifold is a homogeneous reductive space and the classification of these varieties is given by the corresponding classification of graded Lie algebras.We approach the classification of -symmetric spaces in the case = ‫ޚ‬ 2 ‫ޚ×‬ 2 in a similar way, using recent results (see [Bahturin et al. 2005]) on the classification of complex ‫ޚ‬ 2 ‫ޚ×‬ 2 -graded simple Lie algebras.

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