Geodesic
On doubly alternative zero divisors in Cayley–Dickson algebras
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXV, Tome 514 (2022), pp. 18-54 Cet article a éte moissonné depuis la source Math-Net.Ru

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Zero divisors of Cayley–Dickson algebras over an arbitrary field $\mathbb{F}$, $\mathrm{char}\, \mathbb{F} \neq 2$, 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 doubly alternative zero divisors at least one of whose components has nonzero norm are established, and explicit forms of their annihilators, othogonalizers, and centralizers are obtained. 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. 
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     title = {On doubly alternative zero divisors in {Cayley{\textendash}Dickson} algebras},
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S. A. Zhilina. On doubly alternative zero divisors in Cayley–Dickson algebras. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXV, Tome 514 (2022), pp. 18-54. http://geodesic.mathdoc.fr/item/ZNSL_2022_514_a1/

[1] A. E. Guterman, S. A. Zhilina, “Grafy otnoshenii veschestvennykh algebr Keli–Diksona”, Zap. nauchn. semin. POMI, 472, 2018, 44–75

[2] A. E. Guterman, S. A. Zhilina, “Grafy otnoshenii algebry sedenionov”, Zap. nauchn. semin. POMI, 496, 2020, 61–86

[3] A. E. Guterman, S. A. Zhilina, “Kontr-algebry Keli–Diksona: dvazhdy alternativnye deliteli nulya i grafy otnoshenii”, Fund. prikl. mat., 23:3 (2020), 95–129 | MR

[4] K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, A. I. Shirshov, Koltsa, blizkie k assotsiativnym, Nauka, M., 1978 | MR

[5] S. A. Zhilina, “Grafy otnoshenii algebry kontrsedenionov”, Zap. nauchn. semin. POMI, 482 (2019), 87–113

[6] J. C. Baez, “The octonions”, Bull. Amer. Math. Soc. (N.S.), 39 (2001), 145–205 | DOI | MR

[7] L. Bentz, D. Tray, “Subalgebras of the split octonions”, Adv. Appl. Clifford Alg., 28:2 (2018), 40 | DOI | MR

[8] D. K. Biss, D. Dugger, D. C. Isaksen, “Large annihilators in Cayley–Dickson algebras”, Comm. Algebra, 36:2 (2008), 632–664 | DOI | MR

[9] D. K. Biss, D. Dugger, D. C. Isaksen, “Large annihilators in Cayley–Dickson algebras II”, Bol. Soc. Mat. Mex., 13:2 (2007), 269–292 | MR

[10] G. Dolinar, B. Kuzma, N. Stopar, “Characterization of orthomaps on the Cayley plane”, Aeq. Math., 92:2 (2018), 243–265 | DOI | MR

[11] G. Dolinar, B. Kuzma, N. Stopar, “The orthogonality relation classifies formally real simple Jordan algebras”, Comm. Algebra, 48:6 (2020), 2274–2292 | DOI | MR

[12] P. Eakin, A. Sathaye, “On automorphisms and derivations of Cayley–Dickson algebras”, J. Algebra, 129:2 (1990), 263–278 | DOI | MR

[13] N. Jacobson, “Composition algebras and their automorphisms”, Rend. Circ. Mat. Palermo, 7 (1958), 55–80 | DOI | MR

[14] B. Kuzma, “Dimensions of complex Hilbert spaces are determined by the commutativity relation”, J. Operator Theory, 79:1 (2018), 201–211 | MR

[15] K. McCrimmon, A taste of Jordan algebras, Springer-Verlag, New York, 2004 | MR

[16] G. Moreno, “The zero divisors of the Cayley–Dickson algebras over the real numbers”, Bol. Soc. Mat. Mex. (tercera ser.), 4:1 (1998), 13–28 | MR

[17] G. Moreno, “Alternative elements in the Cayley-Dickson algebras”, Topics in Mathematical Physics, General Relativity and Cosmology in Honor of Jerzy Plebañski, World Sci. Publ, Hackensack, NJ, 2006, 333–346 | DOI | MR

[18] G. Moreno, Constructing zero divisors in the higher dimensional Cayley–Dickson algebras, 2005, arXiv: math/0512517

[19] R. Moufang, “Zur Structur von Alternativekörpern”, Math. Ann., 110 (1935), 416–430 | DOI | MR

[20] A. Pixton, “Alternators in the Cayley–Dickson algebras”, Forum Math., 21:5 (2009), 853–869 | DOI | MR

[21] R. D. Schafer, “On the algebras formed by the Cayley–Dickson process”, Amer. J. Math., 76:2 (1954), 435–446 | DOI | MR

[22] S. Zhilina, “Orthogonality graphs of real Cayley–Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons”, Int. J. Algebra Comput., 31:4 (2021), 663–689 | DOI | MR

[23] S. Zhilina, “Orthogonality graphs of real Cayley–Dickson algebras. Part II: The subgraph on pairs of basis elements”, Int. J. Algebra Comput., 31:4 (2021), 691–725 | DOI | MR