Bott type periodicity for the higher octonions

Abstract

We study the series of complex nonassociative algebras ${\mathbb{O}}_n$ and real nonassociative algebras ${\mathbb{O}}_{p,q}$ introduced in [10]. These algebras generalize the classical algebras of octonions and Clifford algebras. The algebras ${\mathbb{O}}_n$ and ${\mathbb{O}}_{p,q}$ with $p+q=n$ have a natural ${\mathbb{Z}}_2^n$-grading, and they are characterized by cubic forms over the field ${\mathbb{Z}}_2$. We establish a periodicity for the algebras ${\mathbb{O}}_n$ and ${\mathbb{O}}_{p,q}$ similar to that of the Clifford algebras ${\mathrm{Cl}}_n$ and ${\mathrm{Cl}}_{p,q}$.