Generalised Einstein metrics on Lie groups

TL;DR

This paper systematically studies generalized Einstein metrics on Lie groups, providing classifications in Riemannian and Lorentzian cases, especially for solvable and almost Abelian Lie groups, using a new algebraic reformulation.

Contribution

It introduces a new algebraic reformulation for classifying generalized Einstein metrics on Lie groups and completes classifications for solvable and almost Abelian cases in various dimensions.

Findings

Full classification of solvable generalized Einstein Lie groups in the Riemannian case.

Classification of almost Abelian generalized Einstein Lie groups in the Lorentzian case with δ=0.

Complete classification in four dimensions for both Riemannian and Lorentzian cases.

Abstract

We continue the systematic study of left-invariant generalised Einstein metrics on Lie groups initiated in arXiv:2206.01157. Our approach is based on a new reformulation of the corresponding algebraic system. For a fixed Lie algebra $\mathfrak{g}$, the unknowns of the system consist of a scalar product $g$ and a $3$-form $H$ on $\mathfrak{g}$ as well as a linear form $\delta$ on $\mathfrak{g}\oplus\mathfrak{g}^*$. As in arXiv:2206.01157, the Lie bracket of $\mathfrak{g}$ is considered part of the unknowns. In the Riemannian case, we show that the generalised Einstein condition always reduces to the commutator ideal and we provide a full classification of solvable generalised Einstein Lie groups. In the Lorentzian case, under the additional assumption $\delta=0$, we classify -- up to one case -- all almost Abelian generalised Einstein Lie groups. We then particularize to four dimensions and provide a full classification of generalised Einstein Riemannian Lie groups as well as generalised Einstein Lorentzian Lie groups with $\delta =0$ and non-degenerate commutator ideal.