Pseudo-K\"ahler Lie algebras with abelian complex structures

TL;DR

This paper classifies pseudo-Kähler Lie algebras with abelian complex structures using algebraic data and analyzes their curvature properties, including conditions for Einstein metrics, through a method of double extensions.

Contribution

It provides a complete algebraic classification of these Lie algebras and characterizes their geometric curvature properties, especially Einstein conditions, via associative algebra data.

Findings

Lie algebras are classified by pairs (U,H) with U a complex commutative associative algebra and H a compatible hermitian form.

The curvature properties of associated pseudo-Kähler metrics are explicitly studied.

A method of double extensions describes all such Lie algebras inductively.

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.