Pseudo-K\"ahler 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-K\"ahler 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-K\"ahler Lie algebras can be inductively described by a certain method of double extensions applied to the associated complex asssociative commutative algebras.
Cite
@article{arxiv.1206.3464,
title = {Pseudo-K\"ahler Lie algebras with abelian complex structures},
author = {Ignacio Bajo and Esperanza Sanmartín},
journal= {arXiv preprint arXiv:1206.3464},
year = {2015}
}