Pseudo-K\"ahler and pseudo-Sasaki structures on Einstein solvmanifolds

TL;DR

This paper constructs and classifies specific Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups, linking pseudo-K"ahler structures with Sasaki-Einstein metrics and providing classifications up to dimension 7.

Contribution

It introduces a new class of Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups and classifies them in low dimensions, expanding understanding of geometric structures on Lie groups.

Findings

Classified $rak z$-standard Sasaki solvable Lie algebras up to dimension 7.

Identified pseudo-K"ahler structures leading to Sasaki-Einstein metrics.

Constructed Einstein metrics that are standard but not of pseudo-Iwasawa type.

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-K\"ahler nilpotent Lie algebras of dimension $2n$ endowed with a compatible derivation, in a suitable sense. We characterize the pseudo-K\"ahler structures and derivations giving rise to Sasaki-Einstein metrics. We classify $\mathfrak z$-standard Sasaki solvable Lie algebras of dimension $\leq 7$ and those whose pseudo-K\"ahler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudo-Iwasawa type.