New Special Einstein Pseudo-Riemannian Metrics on Solvable Lie Algebras

TL;DR

This paper presents a method to construct Einstein pseudo-Kähler and para-Kähler metrics on solvable Lie algebras, classifies certain low-dimensional cases, and discusses potential for generating more special Einstein pseudo-Riemannian metrics.

Contribution

It introduces a concrete procedure for constructing Einstein pseudo-Kähler and para-Kähler metrics on solvable Lie algebras and classifies specific low-dimensional cases.

Findings

Existence of Einstein metrics on certain solvable Lie algebras.

Classification of rank-one pseudo-Iwasawa extensions of type-(Nil4).

Identification of metrics on generalized Heisenberg Lie algebra.

Abstract

We exhibit a concrete procedure to construct Einstein pseudo-K\"ahler and para-K\"ahler metrics on solvable Lie algebras. We apply this method to classify all the rank-one pseudo-Iwasawa extensions of type-(Nil4) nilsoliton in low dimension. We prove that such metrics exists on the rank-one pseudo-Iwasawa extension of the generalized Heisenberg Lie algebra. Further ideas and suggestions to produce more special Einstein pseudo-Riemannian metrics are exposed.