On doubly alternative zero divisors in Cayley-Dickson algebras

Abstract

Zero divisors of Cayley-Dickson algebras over an arbitrary field $\mathbb{F}$, $\mathrm{char}\mathbb{F}\ne 2$, are studied. It is shown that zero divisors, whose components alternate strongly pairwise and have nonzero norm, form hexagonal structures in the zero divisor graph of a Cayley-Dickson algebra. The properties of doubly alternative zero divisors, at least one of whose components has nonzero norm, are established, and an explicit form of their annihilators, orthogonalizers, and centralizers is obtained. The properties of 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 annihilators of elements of Cayley-Dickson algebras are considered.

References in corpus (5)

• The zero divisors of the Cayley-Dickson algebras over the real numbers
• Constructing zero divisors in the higher dimensional Cayley-Dickson algebras
• Relation graphs of the sedenion algebra
• Orthogonality graphs of real Cayley-Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons
• Orthogonality graphs of real Cayley-Dickson algebras. Part II: The subgraph on pairs of basis elements

Cited by in corpus (1)

• Diameter of the commutativity graph of the real sedenions