Poincar\'e and sl(2) algebras of order 3

TL;DR

This paper initiates a classification of Lie algebras of order 3, focusing on those based on sl(2,C) and the Poincaré algebra, and explores their deformations and contractions.

Contribution

It provides the first comprehensive classification of Lie algebras of order 3 related to sl(2,C) and Poincaré algebras, including deformation and contraction frameworks.

Findings

Classification of Lie algebras of order 3 based on sl(2,C) and Poincaré algebra

Development of deformation theory for Lie algebras of order 3

Analysis of contractions for these algebras

Abstract

In this paper we initiate a general classification for Lie algebras of order 3 and we give all Lie algebras of order 3 based on $\mathfrak{sl}(2,\mathbb C)$ and $\mathfrak{iso}(1,3)$ the Poincar\'e algebra in four-dimensions. We then set the basis of the theory of the deformations (in the Gerstenhaber sense) and contractions for Lie algebras of order 3.