Nonunimodular Lorentzian flat Lie algebras

TL;DR

This paper characterizes nonunimodular Lorentzian flat Lie algebras as double extensions of Riemannian flat Lie algebras and classifies all such algebras up to dimension 4.

Contribution

It provides a complete characterization of nonunimodular Lorentzian flat Lie algebras using double extension methods and offers a classification for low-dimensional cases.

Findings

Nonunimodular Lorentzian flat Lie algebras are double extensions of Riemannian flat Lie algebras.

Complete classification of these algebras up to dimension 4.

Unimodularity is equivalent to geodesic completeness of the metric.

Abstract

A Lorentzian flat Lie group is a Lie group $G$ with a flat left invariant metric $\mu$ with signature $(1,n-1)=(-,+,\ldots,+)$. The Lie algebra $\mathfrak{g}=T_eG$ of $G$ endowed with $\langle\;,\;\rangle=\mu(e)$ is called flat Lorentzian Lie algebra. It is known that the metric of a flat Lorentzian Lie group is geodesically complete if and only if its Lie algebra is unimodular. In this paper, we characterise nonunimodular Lorentzian flat Lie algebras as double extensions (in the sense of Aubert-Medina \cite{Aub-Med}) of Riemannian flat Lie algebras. As application of this result, we give all nonunimodular Lorentzian flat Lie algebras up to dimension 4.