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arXiv · 2608.28006

Commuting graphs of Okubo algebras

Svetlana Zhilina·Danil Pavlinov
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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 $\mathbb{F}$, $\mathrm{char} \, \mathbb{F} \neq 3$, 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.

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Svetlana Zhilina, Danil Pavlinov. 2026-08-28. Commuting graphs of Okubo algebras. https://doi.org/10.1007/s10958-026-08484-2

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