AI 中文总结
本文研究Okubo代数的交换图连通性，证明特征非3且含本原三次单位根的域上伪八元数代数的交换图与M₃(𝔽)同构，得出代数闭域上Okubo代数及实可除Okubo代数的交换图直径均为4且后者连通，证明基于幂等元中心化子交集非零的性质。
AI 中文摘要
本文研究Okubo代数的交换图及其连通性问题。对于特征char 𝔽≠3且包含本原三次单位根的域𝔽上的伪八元数代数P₈(𝔽)，证明其交换图与矩阵代数M₃(𝔽)的交换图同构。由此可得，若域𝔽是代数闭域，则𝔽上唯一Okubo代数的交换图直径为4；还证明了实可除Okubo代数的交换图是连通的，其直径也为4。该结果的证明依赖于：任意Okubo代数中任意两个幂等元的中心化子交集始终非零。
英文摘要
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.
Journal refS. Zhilina, D. Pavlinov, Commuting graphs of Okubo algebras, J. Math. Sci. 299(6) (2026) 938-950
DOI：10.1007/s10958-026-08484-2
