Pseudo-Kähler Lie algebras with Abelian complex structures

Abstract

We study Lie algebras endowed with an Abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U, H) where U is a complex commutative associative algebra and H is a sesquilinear Hermitian form on U which verifies certain compatibility conditions with respect to the associative product on U. The Riemannian and Ricci curvatures of the associated pseudo-Kähler metric are studied and a characterization of those Lie algebras which are Einstein but not Ricci flat is given. It is seen that all pseudo-Kähler Lie algebras can be inductively described by a certain method of double extensions applied to the associated complex associative commutative algebras.