On pseudo-Riemannian Lie algebras: a class of new Lie-admissible algebras

TL;DR

This paper redefines pseudo-Riemannian Lie algebras, proves their solvability, and provides classifications for low-dimensional cases, offering new insights and simpler proofs for existing results.

Contribution

It introduces a new definition of pseudo-Riemannian Lie algebras, proves their solvability, and classifies them in dimensions 2 and 3, simplifying previous results.

Findings

All pseudo-Riemannian Lie algebras are solvable.

Provided explicit constructions of Riemann-Lie algebras.

Classified pseudo-Riemannian Lie algebras in dimensions 2 and 3.

Abstract

M. Boucetta introduced the notion of pseudo-Riemannian Lie algebra in [2] when he studied the line Poisson structure on the dual of a Lie algebra. In this paper, we redefine pseudo-Riemannian Lie algebra, which, in essence, is a class of new Lie admissible algebras and prove that all pseudo-Riemannian Lie algebras are solvable. Using our main result and method, we prove some of M. Boucetta's results in [2, 3] in a simple and new way, then we give an explicit construction of Riemann-Lie algebras and a classification of pseudo-Riemannian Lie algebras of dimension 2 and 3.