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 asssociative commutative algebras.

References in corpus (4)

• Twistor spaces for HKT manifolds
• HKT and OKT Geometries on Soliton Black Hole Moduli Spaces
• Classification of abelian complex structures on 6-dimensional Lie algebras
• Novikov structures on solvable Lie algebras

Cited by in corpus (1)

• Pseudo-Kähler and hypersymplectic structures on semidirect products