Algebraic structure of the Lorentz and of the Poincar\'e Lie algebras

TL;DR

This paper investigates the algebraic structure and simplicity conditions of Lorentz and Poincaré Lie algebras over various fields and rings, revealing their ideal structure, automorphisms, and derivations.

Contribution

It provides a detailed analysis of the ideal structure and automorphism groups of Lorentz and Poincaré algebras over different fields and rings, including new simplicity criteria.

Findings

Lorentz algebras are simple over fields without square root of -1.

Lorentz and Poincaré algebras are not simple in characteristic 2.

Automorphism groups are characterized for these algebras over various fields.

Abstract

We start with the Lorentz algebra $ L=o_{R}(1,3)$ over the reals and find a suitable basis $B$ relative to which the structure constants are integers. Thus we consider the $Z$-algebra $L_{Z}$ which is free as a $Z$-module and its $Z$-basis is $B$. This allows us to define the Lorentz type algebra $L_K:=L_{Z}\otimes_{Z} K$ over any field $K$. In a similar way, we consider Poincar\'e type algebras over any field $K$. In this paper we study the ideal structure of Lorentz and of Poincar\'e type algebras over different fields. It turns out that Lorentz type algebras are simple if and only if the ground field has no square root of $-1$. Thus, they are simple over the reals but not over the complex. Also, if the ground field is of characteristic $2$ then Lorentz and Poincar\'e type algebras are neither simple nor semisimple. We extend the study of simplicity of the Lorentz algebra to the case of a ring of scalars where we have to use the notion of $m$-simplicity (relative to a maximal ideal $m$ of the ground ring of scalars). The Lorentz type algebras over a finite field $F_q$ where $q=p^n$ and $p$ is odd are simple if and only if $n$ is odd and $p$ of the form $p=4k+3$. In case $p=2$ then the Lorentz type algebra are not simple. Once we know the ideal structure of the algebras, we get some information of their automorphism groups. For the Lorentz type algebras (except in the case of characteristic $2$) we describe the affine group scheme of automorphisms and the derivation algebras. For the Poincar\'e algebras we restrict this program to the case of an algebraically closed field of characteristic other than $2$.