Octonionic representations of Clifford algebras and triality

TL;DR

This paper extends Clifford algebra representations using octonions, introducing octonionic spinors and revealing a $ ext{S}_3 imes SO(8)$ structure in triality automorphisms, thus advancing the understanding of algebraic symmetries in mathematical physics.

Contribution

It develops a novel framework for representing Clifford algebras with octonions, addressing non-associativity and non-commutativity, and explores the implications for spinors and automorphisms.

Findings

Octonionic representations lead to new notions of octonionic spinors.

The triality automorphisms exhibit a $ ext{S}_3 imes SO(8)$ structure.

The framework addresses algebraic challenges posed by octonions.

Abstract

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest $\perm_3 \times SO(8)$ structure in this framework.