arXiv subjects

A. Van Antwerpen

Publications and source records attributed to A. Van Antwerpen.

5 recordsLinked to original sources

arXiv2026

Bijective solutions to the Pentagon Equation

A complete classification of all finite bijective set-theoretic solutions $(S,s)$ to the Pentagon Equation is obtained. First, it is shown that every such solution determines a semigroup structure on the set $S$ that is the direct product $E\times G$ of a semigroup of left zeros $E$ and a group $G$. Next, we prove that this leads to a decomposition of the set $S$ as a Cartesian product $X\times A\times G$, for some sets $X,A$ and to the discovery of a hidden group structure on $A$. Then an unexpected structure of a matched product of groups $A,G$ is found such that the solution $(S,s)$ can be explicitly described as a lift of a solution determined on the set $A\times G$ by this matched product of groups. Conversely, every matched product of groups leads to a family of solutions arising in this way. Moreover, a simple criterion for the isomorphism of two solutions is obtained. These results significantly extend those of Colazzo, Jespers, and Kubat, in their treatment of the involutive case. Furthermore, connections to solutions to the Yang--Baxter Equation and to the theory of skew braces are uncovered. The latter motivate a further investigation of the relationship between solutions to the Yang--Baxter Equation and the Pentagon Equation at the set-theoretical level.

math.GR↗

arXiv2023

Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang-Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces.

math.RA↗

arXiv2022

On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang--Baxter equation

Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions $(X,r)$ and show, for example, that this property is equivalent to the solution associated to the structure monoid $M(X,r)$ (respectively structure group $G(X,r)$) being a multipermuation solution and that $G=G(X,r)$ is solvable of derived length not exceeding the multipermutation level of $(X,r)$ enlarged by one, generalizing results of Gateva-Ivanova and Cameron obtained in the involutive case. Moreover, we also prove that if $X$ is finite and $G=G(X,r)$ is nilpotent, then the torsion part of the group $G$ is finite, it coincides with the commutator subgroup $[G,G]_+$ of the additive structure of the skew left brace $G$ and $G/[G,G]_+$ is a trivial left brace.

math.RA↗

arXiv2020

Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation

We define the radical and weight of a skew left brace and provide some basic properties of these notions. In particular, we obtain a Wedderburn type decomposition for Artinian skew left braces. Furthermore, we prove analogues of a theorem of Wiegold, a theorem of Schur and its converse in the context of skew left braces. Finally, we apply these results to detect torsion in the structure group of a finite bijective non-degenerate set-theoretic solution of the Yang-Baxter equation.

math.RA↗

arXiv2019

Factorizations of skew braces

We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of Itô's theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang-Baxter equation. Finally, we classify skew braces that contain no non-trivial proper ideals.

math.RA↗