Relation graphs of the sedenion algebra

A. E. Guterman   ; S. A. Zhilina

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIII, Tome 496 (2020), pp. 61-86

Résumé

Let $\mathbb{S}$ denote the algebra of sedenions and let $\Gamma_O(\mathbb{S})$ denote its orthogonality graph. We observe that any pair of zero divisors in $\mathbb{S}$ produces a double hexagon in $\Gamma_O(\mathbb{S})$. The set of vertices of a double hexagon can be extended to a basis of $\mathbb{S}$ that has a convenient multiplication table. We explicitly describe the set of vertices of an arbitrary connected component of $\Gamma_O(\mathbb{S})$ and find its diameter. Then we establish a bijection between the connected components of $\Gamma_O(\mathbb{S})$ and the lines in the imaginary part of the octonions. Finally, we consider the commutativity graph of the sedenions and discover that all elements whose imaginary parts are zero divisors belong to the same connected component, and its diameter lies in between $3$ and $4$.

Bibliographie

[1] J. C. Baez, “The Octonions”, Bull. Amer. Math. Soc. (N.S.), 39 (2002), 145–205 | DOI | MR | Zbl

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

[3] 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 | Zbl

[4] H. C. Brown, Structure of zero divisors, and other algebraic structures, in higher dimensional real Cayley–Dickson algebras, Doct. Diss., 2088, 1972 | MR

[5] R. E. Cawagas, “On the structure and zero divisors of the Cayley–Dickson sedenion algebra”, Disc. Math. General Algebra Appl., 24 (2004), 251–265 | DOI | MR | Zbl

[6] K.-C. Chan, D. Ž. Ðoković, “Conjugacy classes of subalgebras of the real sedenions”, Canad. Math. Bull., 49 (2006), 492–507 | DOI | MR | Zbl

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

[8] A. E. Guterman, S. A. Zhilina, “Kontr-algebry Keli–Diksona: dvazhdy alternativnye deliteli nulya i grafy otnoshenii”, Fund. prikl. matem., 2020 (to appear)

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

[10] S. H. Khalil, P. Yiu, “The Cayley-Dickson algebras, a theorem of A. Hurwitz, and quaternions”, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform., 24 (1997), 117–169 | MR | Zbl

[11] R. P. C. de Marrais, The 42 assessors and the box-kites they fly: diagonal axis-pair systems of zero-divisors in the sedenions'16 dimensions, 2000, arXiv: math/0011260

[12] R. P. C. de Marrais, Flying higher than a box-kite: kite-chain middens, sand mandalas, and zero-divisor patterns in the $2^n$-ions beyond the sedenions, 2002, arXiv: math/0207003

[13] R. P. C. de Marrais, Box-kites III: Quizzical quaternions, mock octonions, and other zero-divisor-suppressing “sleeper cell” structures in the sedenions and $2^n$-ions, 2004, arXiv: math/0403113

[14] K. McCrimmon, A Taste of Jordan Algebras, Springer-Verlag, New York, 2004 | MR | Zbl

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

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

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

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

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

[20] S. A. Zhilina, “Orthogonality graphs of real Cayley–Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons”, Int. J. Algebra Comput. (to appear)