Double extension of flat pseudo-Riemannian $F$-Lie algebras

TL;DR

This paper introduces a new algebraic construction called double extension for flat pseudo-Riemannian $F$-Lie algebras, providing a method to generate and classify certain Lorentzian and scalar product nilpotent Lie algebras.

Contribution

It defines the concept of flat pseudo-Riemannian $F$-Lie algebras and constructs their double extensions, offering a framework for classifying specific Lorentzian and scalar product nilpotent Lie algebras.

Findings

Double extension generates all weakly flat Lorentzian bi-nilpotent $F$-Lie algebras with light-cone subspaces.

Applicable to nilpotent Lie algebras with flat scalar products of signature $(2,n-2)$.

Constructs Poisson algebras compatible with flat scalar products.

Abstract

We define the concept of a flat pseudo-Riemannian $F$-Lie algebra and construct its corresponding double extension. This algebraic structure can be interpreted as the infinitesimal analogue of a Frobenius Lie group devoid of Euler vector fields. We show that the double extension provides a framework for generating all weakly flat Lorentzian non-abelian bi-nilpotent $F$-Lie algebras possessing one dimensional light-cone subspaces. A similar result can be established for nilpotent Lie algebras equipped with flat scalar products of signature $(2,n-2)$ where $n\geq 4$. Furthermore, we use this technique to construct Poisson algebras exhibiting compatibility with flat scalar products.