|
Zapiski Nauchnykh Seminarov POMI, 2020, Volume 496, Pages 61–86
(Mi znsl7014)
|
|
|
|
This article is cited in 2 scientific papers (total in 2 papers)
Relation graphs of the sedenion algebra
A. E. Gutermanabc, S. A. Zhilinabc a Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region
b Lomonosov Moscow State University
c Moscow Center for Fundamental and Applied Mathematics
Abstract:
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 $4$.
Key words and phrases:
Cayley–Dickson algebras, sedenions, relation graphs, connected components.
Received: 01.10.2020
Citation:
A. E. Guterman, S. A. Zhilina, “Relation graphs of the sedenion algebra”, Computational methods and algorithms. Part XXXIII, Zap. Nauchn. Sem. POMI, 496, POMI, St. Petersburg, 2020, 61–86
Linking options:
https://www.mathnet.ru/eng/znsl7014 https://www.mathnet.ru/eng/znsl/v496/p61
|
Statistics & downloads: |
Abstract page: | 105 | Full-text PDF : | 45 | References: | 10 |
|