On Einstein Lorentzian nilpotent Lie groups

TL;DR

This paper classifies Lorentzian Einstein metrics on nilpotent Lie groups, showing that low-dimensional cases have degenerate centers and providing the first examples of Ricci-flat Lorentzian nilpotent Lie algebras with nondegenerate centers.

Contribution

It offers a complete classification of Lorentzian Einstein metrics on nilpotent Lie groups up to dimension 5 and introduces new Ricci-flat examples with nondegenerate centers.

Findings

All nilpotent Lie groups up to dimension 5 with Lorentzian Einstein metrics have degenerate centers.

Nilpotent Lie algebras with non-zero scalar curvature have nondegenerate Euclidean centers.

First examples of Ricci-flat Lorentzian nilpotent Lie algebras with nondegenerate centers.

Abstract

In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie groups up to dimension $5$ endowed with a Lorentzian Einstein left invariant metric have degenerate center and we use this fact to give a complete classification of these metrics. We show that if $\mathfrak{g}$ is the Lie algebra of a nilpotent Lie group endowed with a Lorentzian left invariant Einstein metric with non zero scalar curvature then the center $Z(\mathfrak{g})$ of $\mathfrak{g}$ is nondegenerate Euclidean, the derived ideal $[\mathfrak{g},\mathfrak{g}]$ is nondegenerate Lorentzian and $Z(\mathfrak{g})\subset[\mathfrak{g},\mathfrak{g}]$. We give the first examples of Ricci-flat Lorentzian nilpotent Lie algebra with nondegenerate center.