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On Doubly Alternative Zero Divisors in Cayley–Dickson Algebras

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Zero divisors of Cayley–Dickson algebras over an arbitrary field 𝔽, char 𝔽 ≠= 2, are studied. It is shown that the zero divisors whose components alternate strongly pairwise and have nonzero norm form hexagonal structures in the zero-divisor graph of a Cayley–Dickson algebra. Properties of the doubly alternative zero divisors at least one of whose components has nonzero norm are established, and explicit forms of their annihilators, orthogonalizers, and centralizers are obtained. Properties of the zero divisors in Cayley–Dickson algebras with anisotropic norm are described, and it is shown that in this case, directed hexagons in the zero-divisor graph can be extended to undirected double hexagons in the orthogonality graph. A criterion of C-equivalence for elements of Cayley–Dickson algebras with anisotropic norm is obtained. Possible values of dimension for the annihilators of elements in Cayley–Dickson algebras are considered.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    S. A. Zhilina

  2. Moscow Institute of Physics and Technology, Dolgoprudny, Russia

    S. A. Zhilina

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  1. S. A. Zhilina
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Correspondence to S. A. Zhilina.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 514, 2022, pp. 18–54.

Translated by S. A. Zhilina.

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Zhilina, S.A. On Doubly Alternative Zero Divisors in Cayley–Dickson Algebras. J Math Sci 272, 496–518 (2023). https://doi.org/10.1007/s10958-023-06444-8

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  • DOI: https://doi.org/10.1007/s10958-023-06444-8

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