Classification of the algebras ${\mathbb{O}}_{p,q}$

Abstract

We study a series of real nonassociative algebras ${\mathbb{O}}_{p,q}$ introduced in $[5]$. These algebras have a natural ${\mathbb{Z}}_2^n$-grading, where $n=p+q$, and they are characterized by a cubic form over the field ${\mathbb{Z}}_2$. We establish all the possible isomorphisms between the algebras ${\mathbb{O}}_{p,q}$ preserving the structure of ${\mathbb{Z}}_2^n$-graded algebra. The classification table of ${\mathbb{O}}_{p,q}$ is quite similar to that of the real Clifford algebras ${\mathrm{Cl}}_{p,q}$, the main difference is that the algebras ${\mathbb{O}}_{n,0}$ and ${\mathbb{O}}_{0,n}$ are exceptional.

15 pages, 2 figures