Indefinite Einstein metrics on nice Lie groups

TL;DR

This paper develops a systematic algebraic approach to construct and classify indefinite signature Einstein metrics on nice nilpotent Lie groups, providing new classifications in dimensions 8 and 9 and establishing their existence in all dimensions ≥8.

Contribution

It introduces a new method for producing and classifying indefinite Einstein metrics on nice nilpotent Lie groups, including explicit classifications in specific dimensions.

Findings

Classified Einstein metrics in dimension 8.

Classified Einstein metrics in dimension 9 under certain conditions.

Proved existence of Einstein nilpotent Lie groups with nonzero scalar curvature in all dimensions ≥8.

Abstract

We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension $\geq 8$.