On Flat Pseudo-Euclidean Nilpotent Lie Algebras

TL;DR

This paper classifies flat pseudo-Euclidean nilpotent Lie algebras, especially of signature (2,n-2), showing they can be constructed via double extensions from simpler algebras and providing explicit classifications and examples.

Contribution

It provides a complete classification of flat pseudo-Euclidean 2-step nilpotent Lie algebras of signature (2,n-2) using double extension methods.

Findings

Center of such algebras is degenerate.

All such algebras can be obtained from abelian Lie algebras via double extension.

Explicit low-dimensional examples are provided.

Abstract

A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature $(2,n-2)$ must be degenerate and all flat pseudo-Euclidean nilpotent Lie algebras of signature $(2,n-2)$ can be obtained by using the double extension process from flat Lorentzian nilpotent Lie algebras. We show also that the center of a flat pseudo-Euclidean 2-step nilpotent Lie algebra is degenerate and all these Lie algebras are obtained by using a sequence of double extension from an abelian Lie algebra. In particular, we determine all flat pseudo-Euclidean 2-step nilpotent Lie algebras of signature $(2,n-2)$. The paper contains also some examples in low dimension.