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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 P8(𝔽) over a field 𝔽, char 𝔽 β‰  3, that contains a primitive cubic root of unity is shown to be isomorphic to the commuting graph of the matrix algebra M3(𝔽). As a consequence, if the field 𝔽 is algebraically closed, then the diameter of the commuting graph for the unique Okubo algebra over 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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Cite

BibTeXRIS
BibTeX
@misc{zhilina2026commuting,
  title = {{Commuting graphs of Okubo algebras}},
  author = {Svetlana Zhilina and Danil Pavlinov},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28006},
  eprint = {2608.28006},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28006},
}
RIS
TY  - UNPB
TI  - Commuting graphs of Okubo algebras
AU  - Svetlana Zhilina
AU  - Danil Pavlinov
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28006
UR  - https://arxiv.org/abs/2608.28006
AB  - 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(𝔽) over a field 𝔽, char 𝔽 β‰  3, that contains a primitive cubic root of unity is shown to be isomorphic to the commuting graph of the matrix algebra M_3(𝔽). As a consequence, if the field 𝔽 is algebraically closed, then the diameter of the commuting graph for the unique Okubo algebra over 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.
ER  -