Octonionic presentation for the Lie group $SL(2,{\mathbb O})$

TL;DR

This paper provides an octonionic framework to describe the Lie group $SL(2,\mathbb{O})$, revealing its structure through generators, isomorphisms, and connections to Lorentz algebras, enriching the understanding of octonionic Lie groups.

Contribution

It introduces an octonionic presentation of $SL(2,\mathbb{O})$, characterizes its generators, and constructs explicit isomorphisms with Lorentz algebras for various division algebras.

Findings

$SL(2,\mathbb{O})$ is generated by invertible, determinant-preserving transformations.

Explicit isomorphisms between $\mathfrak{sl}(2,\mathbb{K})$ and Lorentz algebras are constructed.

Characterization of generators of $G_2$ is provided.

Abstract

The purpose of this paper is to provide an octonionic description of the Lie group $SL(2,{\mathbb O})$. The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from $\mathfrak{h}_2({\mathbb O})$ onto itself. An interesting characterization is given for the generators of $G_2$. Also, explicit isomorphisms are constructed between the Lie algebras $\mathfrak{sl}(2,{\mathbb K})$, for ${\mathbb K}={\mathbb R}, {\mathbb C}, {\mathbb H}, {\mathbb O}$, and their corresponding Lorentz algebras.