Commuting graphs of Okubo algebras

Abstract

Commuting graphs of Okubo algebras are considered, and the problem of their connectivity is studied. The commuting graph of a pseudo-octonion algebra $P_8(\mathbb{F})$ over a field $\mathbb{F}$, $\mathrm{char}\mathbb{F}\ne 3$, that contains a primitive cubic root of unity is shown to be isomorphic to the commuting graph of the matrix algebra $M_3(\mathbb{F})$. As a consequence, if the field $\mathbb{F}$ is algebraically closed, then the diameter of the commuting graph for the unique Okubo algebra over $\mathbb{F}$ equals $4$. It is shown that the commuting graph of the real division Okubo algebra is connected, and its diameter also equals $4$. The proof of this result relies on the fact that, given any two idempotents in an arbitrary Okubo algebra, the intersection of their centralizers is always nonzero.