Searcharxiv⌕ Search

arXiv subjects

Svetlana Zhilina

Publications and source records attributed to Svetlana Zhilina.

12 recordsLinked to original sources
arXiv2026

Commuting graphs of Okubo algebras

Commuting graphs of Okubo algebras are considered, and the problem of their connectivity is studied. The commuting graph of a pseudo-octonion algebra $P_8(\mathbb{F})$ over a field $\mathbb{F}$, $\mathrm{char} \, \mathbb{F} \neq 3$, that contains a primitive cubic root of unity is shown to be isomorphic to the commuting graph of the matrix algebra $M_3(\mathbb{F})$. As a consequence, if the field $\mathbb{F}$ is algebraically closed, then the diameter of the commuting graph for the unique Okubo algebra over $\mathbb{F}$ equals $4$. It is shown that the commuting graph of the real division Okubo algebra is connected, and its diameter also equals $4$. The proof of this result relies on the fact that, given any two idempotents in an arbitrary Okubo algebra, the intersection of their centralizers is always nonzero.

math.RA↗
arXiv2026

Orthogonality graphs of real Cayley-Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons

We study zero divisors whose components alternate strongly pairwise and construct oriented hexagons in the zero divisor graph of an arbitrary real Cayley-Dickson algebra. In case of the algebras of the main sequence, the zero divisor graph coincides with the orthogonality graph, and any hexagon can be extended to a double hexagon. We determine the multiplication table of the vertices of a double hexagon. Then we find a sufficient condition for three elements to generate an alternative subalgebra of an arbitrary Cayley-Dickson algebra. Finally, we consider those zero divisors whose components are both standard basis elements up to sign. We classify them and determine necessary and sufficient conditions under which two such elements are orthogonal.

math.RA↗
arXiv2026

Orthogonality graphs of real Cayley-Dickson algebras. Part II: The subgraph on pairs of basis elements

We consider zero divisors of an arbitrary real Cayley-Dickson algebra such that their components are both standard basis elements. We construct inductively the orthogonality graph on these elements. Then we show that, if we restrict our attention to at least $16$-dimensional algebras, two algebras are isomorphic if and only if their graphs are isomorphic. We also provide an algorithm to retrieve the Cayley-Dickson parameters of an algebra from its graph.

math.RA↗
arXiv2026

Relation graphs of the sedenion algebra

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

math.RA↗
arXiv2026

On doubly alternative zero divisors in Cayley-Dickson algebras

Zero divisors of Cayley-Dickson algebras over an arbitrary field $\mathbb{F}$, $\mathrm{char} \, \mathbb{F} \neq 2$, 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.

math.RA↗
arXiv2026

Diameter of the commutativity graph of the real sedenions

The commutativity graph of the real sedenion algebra is considered. It is shown that those elements whose imaginary part is not a zero divisor correspond to isolated vertices of this graph. All other elements form a connected component whose diameter equals $3$.

math.RA↗
arXiv2026

On orthogonality graphs of Okubo algebras

The orthogonality graph of an Okubo algebra with isotropic norm over an arbitrary field $\mathbb{F}$ is considered. Its connected components are described, and their diameters are computed. It is shown that there exist at most two shortest paths between any pair of vertices, and the conditions under which the shortest path is unique are determined.

math.RA↗
arXiv2024

Birkhoff-James classification of norm's properties

For an arbitrary normed space $\mathcal X$ over a field $\mathbb F \in \{ \mathbb R, \mathbb C \}$, we define the directed graph $Γ(\mathcal X)$ induced by Birkhoff-James orthogonality on the projective space $\mathbb P(\mathcal X)$, and also its nonprojective counterpart $Γ_0(\mathcal X)$. We show that, in finite-dimensional normed spaces, $Γ(\mathcal X)$ carries all the information about the dimension, smooth points, and norm's maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian $C^\ast$-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph $Γ_0(\mathcal{R})$ of a (real or complex) Radon plane $\mathcal{R}$ is isomorphic to the graph $Γ_0(\mathbb F^2, \|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.

math.FA↗
arXiv2023

On the lengths of descendingly flexible and descendingly alternative algebras

We introduce the classes of descendingly flexible and descendingly alternative algebras over an arbitrary field $\mathbb{F}$. We suggest a new method based on the sequence of differences between the dimensions of the linear spans of words, which allows us to obtain upper bounds on the lengths of these algebras. We also present an example of an algebra of arbitrarily large dimension such that these bounds are achieved on it asymptotically.

math.RA↗
arXiv2023

On the lengths of Okubo algebras

We compute the lengths of two particular cases of (possibly non-unital) composition algebras, namely, standard composition algebras and Okubo algebras over an arbitrary field $\mathbb{F}$. These results finish the complete description of lengths of symmetric composition algebras and finite-dimensional flexible composition algebras.

math.RA↗
arXiv2022

Roots and Critical Points of Polynomials over Cayley--Dickson Algebras

We study the roots of polynomials over Cayley--Dickson algebras over an arbitrary field and of arbitrary dimension. For this purpose we generalize the concept of spherical roots from quaternion and octonion polynomials to this setting, and demonstrate their basic properties. We show that the spherical roots (but not all roots) of a polynomial $f(x)$ are also roots of its companion polynomial $C_f(x)$ (defined to be the norm of $f(x)$). For locally-complex Cayley--Dickson algebras, we show that the spherical roots of $f'(x)$ (defined formally) belong to the convex hull of the roots of $C_f(x)$, and we also prove that all roots of $f'(x)$ are contained in the snail of $f(x)$, as defined by Ghiloni and Perotti for quaternions. The latter two results generalize the classical Gauss--Lucas theorem to the locally-complex Cayley--Dickson algebras, and we also generalize Jensen's classical theorem on real polynomials to this setting.

math.RA↗