The octonions as a twisted group algebra

TL;DR

This paper presents a novel perspective on defining octonions as a twisted group algebra over a finite field, simplifying their properties and describing integral orders without complex computations.

Contribution

It introduces a new algebraic formulation of octonions as a twisted group algebra, providing straightforward proofs of their properties and describing integral orders uniformly.

Findings

Octonions can be defined via a basis indexed by  over  with a specific multiplication rule.

Basic properties of octonions follow easily from this new definition.

Existence of sixteen orders of integral octonions containing Gravesian integers is proven without computations.

Abstract

We show that the octonions can be defined as the $\mathbb{R}$-algebra with basis $\lbrace e^x \colon x \in \mathbb{F}_8 \rbrace$ and multiplication given by $e^x e^y = (-1)^{\varphi(x,y)}e^{x + y}$, where $\varphi(x,y) = \operatorname{tr}(y x^6)$. While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. We give a uniform description of the sixteen orders of integral octonions containing the Gravesian integers, and a computation-free proof of their existence.