Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds

Abstract

The aim of this paper is to construct left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups. We consider the class of \(\mathfrak {z}\)-standard Sasaki solvable Lie algebras of dimension \(2n+3\), which are in one-to-one correspondence with pseudo-Kähler nilpotent Lie algebras of dimension 2n endowed with a compatible derivation, in a suitable sense. We characterize the pseudo-Kähler structures and derivations giving rise to Sasaki–Einstein metrics. We classify \(\mathfrak {z}\)-standard Sasaki solvable Lie algebras of dimension \(\le 7\) and those whose pseudo-Kähler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudo-Iwasawa type.