Born Lie algebras

TL;DR

This paper classifies low-dimensional Lie algebras with Born structures, showing they can be constructed via bicross products from pseudo-Riemannian Lie algebras, and analyzes their curvature properties.

Contribution

It introduces a classification of low-dimensional Born Lie algebras and explores their geometric and curvature characteristics.

Findings

All 4-dimensional Lie algebras with Born structures are classified.

Six-dimensional nilpotent Lie algebras with integrable Born structures are characterized.

Curvature properties of the associated pseudo-Riemannian metrics are analyzed.

Abstract

We show that every Born Lie algebra can be obtained by the bicross product construction starting from two pseudo-Riemannian Lie algebras. We then obtain a classification of all Lie algebras up to dimension four and all six-dimensional nilpotent Lie algebras admitting an integrable Born structure. Finally, we study the curvature properties of the pseudo-Riemannian metrics of the integrable Born structures obtained in our classification results.