Matrix Representation of Octonions and Generalizations

TL;DR

This paper introduces a novel matrix multiplication framework for special matrices that models octonions and their generalizations, revealing new algebraic structures and insights into Dirac's equation.

Contribution

It defines a new matrix multiplication for specific matrices, leading to representations of octonions and complex numbers, and connects these to the Cayley-Dickson process and Dirac's equation.

Findings

Derived conditions for algebraic associativity and alternativity.

Provided matrix representations of octonions and complex numbers.

Connected matrix representations to Dirac's equation insights.

Abstract

We define a special matrix multiplication among a special subset of $2N\x 2N$ matrices, and study the resulting (non-associative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative non-associative and when they become associative. In particular, these algebras yield special matrix representations of octonions and complex numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac's equation for a free particle. A few other results and remarks arise as byproducts.