On Einstein Lorentzian nilpotent Lie groups

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.