New Lie algebras over the group $\mathbb Z_2^3$

TL;DR

This paper introduces a broad class of Lie algebras over the group ^3, generalizing known structures like the exceptional ^3 Lie algebras, and explores their properties and classifications.

Contribution

It demonstrates that many Lie algebras can be viewed as generalized group algebras over ^3, extending previous classifications and including solvable and nilpotent cases.

Findings

Identification of new Lie algebra structures over ^3

Application of graded contraction classifications to these algebras

Examples of solvable and nilpotent Lie algebras of various dimensions

Abstract

A new structure, based on joining copies of a group by means of a \emph{twist}, has recently been considered to describe the brackets of the two exceptional real Lie algebras of type $G_2$ in a highly symmetric way. In this work we show that these are not isolated examples, providing a wide range of Lie algebras which are generalized group algebras over the group $\mathbb{Z}_2^3$. On the one hand, some orthogonal Lie algebras are quite naturally generalized group algebras over such group. On the other hand, previous classifications on graded contractions can be applied to this context getting many more examples, involving solvable and nilpotent Lie algebras of dimensions 32, 28, 24, 21, 16 and 14.