Octonionic Representations of GL(8,R) and GL(4,C)

TL;DR

This paper introduces a novel octonionic operator framework that facilitates the representation of complex matrix groups and Clifford algebra, overcoming nonassociativity challenges.

Contribution

It develops left-right octonionic barred operators to represent GL(8,R) and GL(4,C), establishing new links between octonions and matrix groups.

Findings

Reproduces GL(8,R) using octonionic operators

Establishes connection between octonionic operators and 4x4 complex matrices

Provides an octonionic representation of 4D Clifford algebra

Abstract

Octonionic algebra being nonassociative is difficult to manipulate. We introduce left-right octonionic barred operators which enable us to reproduce the associative GL(8,R) group. Extracting the basis of GL(4,C), we establish an interesting connection between the structure of left-right octonionic barred operators and generic 4x4 complex matrices. As an application we give an octonionic representation of the 4-dimensional Clifford algebra.