Left-invariant Lorentzian flat metrics on Lie groups

TL;DR

This paper classifies Lorentzian flat Lie algebras by analyzing their structure, providing new proofs, and constructing all such algebras with degenerate center up to dimension 6, advancing understanding of pseudo-Riemannian geometry on Lie groups.

Contribution

It offers a new proof of Milnor's classical result, reduces the classification problem, and constructs all Lorentzian flat Lie algebras with degenerate center up to dimension 6.

Findings

Reduction of Lorentzian flat Lie algebra classification to those with trivial or degenerate center

Use of double extension process to construct Lorentzian flat Lie algebras with degenerate center

Complete list of Lorentzian flat Lie algebras with degenerate center up to dimension 6

Abstract

We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of Lorentzian flat Lie algebras to those with trivial center or those with degenerate center. We show that the double extension process can be used to construct all Lorentzian flat Lie algebras with degenerate center generalizing a result of Aubert-Medina on Lorentzian flat nilpotent Lie algebras. Finally, we give the list of Lorentzian flat Lie algebras with degenerate center up to dimension 6.