Algebraic Structure of the Lorentz and of the Poincaré Lie Algebras

Abstract

We start with the Lorentz algebra $\text{\large$\mathfrak{L}$}=\text{\large$\mathfrak{o}$}_{\mathbb{R}}(1,3)$ over the reals and find a suitable basis $B$ such that the structure constants relative to it are integers. Thus we consider the $\mathbb{Z}$-algebra $\text{\large$\mathfrak{L}$}_{\mathbb{Z}}$ which is free as a $\mathbb{Z}$-module of which $B$ is $\mathbb{Z}$-basis. This allows us to define the Lorentz type algebra $\text{\large$\mathfrak{L}$}_K:=\text{\large$\mathfrak{L}$}_{\mathbb{Z}}\otimes_{\mathbb{Z}} K$ over any field $K$. In a similar way, we consider Poincaré type algebras over any field $K$. In this paper we study the ideal structure of Lorentz and of Poincaré 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 Poincaré 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 $\text{\large$\mathfrak{m}$}$-simplicity (relative to a maximal ideal $\text{\large$\mathfrak{m}$}$ of the ground ring of scalars). The Lorentz type algebras over a finite field $\mathbb{F}_q$ 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 algebras 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 Poincaré algebras we restrict this program to the case of an algebraically closed field of characteristic other than 2.