Bott type periodicity for the higher octonions

TL;DR

This paper investigates the periodicity properties of higher octonion-like algebras, generalizing classical octonions and Clifford algebras, and establishes a new periodicity pattern similar to that of Clifford algebras.

Contribution

It introduces and analyzes the periodicity of the algebras O_n and O_{p,q}, extending classical algebraic periodicity results to these nonassociative structures.

Findings

Established a periodicity pattern for O_n and O_{p,q} algebras.

Connected the structure of these algebras to cubic forms over Z_2.

Generalized classical algebraic periodicity to higher octonion-like algebras.

Abstract

We study the series of complex nonassociative algebras On and real nonassociative algebras $O_{p,q}$ introduced in [10]. These algebras generalize the classical algebras of octonions and Clifford algebras. The algebras $O_{n}$ and $O_{p,q}$ with $p + q = n$ have a natural $Z_n^2$-grading, and they are characterized by cubic forms over the field $Z_2$. We establish a periodicity for the algebras $O_{n}$ and $O_{p,q}$ similar to that of the Clifford algebras Cln and $Cl_{p,q}$.