Flat Lorentzian Lie groups: A complete description

TL;DR

This paper provides a complete classification of flat Lorentzian Lie groups, revealing their structure and connection to flat Euclidean Lie algebras, and resolves a longstanding open problem in pseudo-Riemannian geometry.

Contribution

It offers a full structural description of flat Lorentzian Lie groups, identifying their algebraic classes and linking them to flat Euclidean Lie algebras, including low-dimensional classifications.

Findings

Flat Lorentzian Lie groups either have a timelike parallel vector field or are of Kundt type.

All such Lie algebras derive from flat Euclidean Lie algebras via double extension.

Complete classifications are provided for dimensions three and four.

Abstract

In this paper, we establish a complete structural description of flat Lorentzian Lie groups, i.e., Lie groups endowed with a flat left invariant Lorentzian metric, thereby resolving a long-standing open problem in the theory of pseudo-Riemannian Lie groups. Our main result shows that any flat Lorentzian Lie group either admits a timelike parallel left-invariant vector field or is of Kundt type, and that in both cases the underlying Lie algebra falls into one of six explicit classes. A key ingredient of the proof is a refined analysis of the double extension process, which reveals that all flat Lorentzian Lie algebras arise - directly or in a generalized sense - from flat Euclidean ones. As a consequence, we obtain easily a complete classification in dimensions three and four, recovering and unifying several previously known partial results.