arXiv subjects

Charlotte Verwimp

Publications and source records attributed to Charlotte Verwimp.

5 recordsLinked to original sources

arXiv2022

Finite idempotent set-theoretic solutions of the Yang--Baxter equation

It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies of a group) and some maps $q$ and $φ_x$ on $X$, for $x\in X$. This structure turns out to be a group precisely when the associated structure monoid is cancellative and all the maps $φ_x$ are equal to an automorphism of this group. Equivalently, the structure algebra $K[M(X,r)]$ is right Noetherian, or in characteristic zero it has to be semiprime. The structure algebra always is a left Noetherian representable algebra of Gelfand--Kirillov dimension one. To prove these results it is shown that the structure semigroup $S(X,r)$ has a decomposition in finitely many cancellative semigroups $S_u$ indexed by the diagonal, each $S_u$ has a group of quotients $G_u$ that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that $X$ equals the diagonal is fully described by a single permutation on $X$.

math.RA↗

arXiv2022

Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses

To determine and analyze arbitrary left non-degenerate set-theoretic solutions of the Yang-Baxter equation (not necessarily bijective), we introduce an associative algebraic structure, called a YB-semitruss, that forms a subclass of the category of semitrusses as introduced by Brzeziński. Fundamental examples of YB-semitrusses are structure monoids of left non-degenerate set-theoretic solutions and (skew) left braces. Gateva-Ivanova and Van den Bergh introduced structure monoids and showed their importance (as well as that of the structure algebra) for studying involutive non-degenerate solutions. Skew left braces were introduced by Guarnieri, Vendramin and Rump to deal with bijective non-degenerate solutions. Hence, YB-semitrusses also yield a unified treatment of these different algebraic structures. The algebraic structure of YB-semitrusses is investigated, and as a consequence, it is proven, for example, that any finite left non-degenerate set-theoretic solution of the Yang-Baxter equation is right non-degenerate if and only if it is bijective. Furthermore, it is shown that some finite left non-degenerate solutions can be reduced to non-degenerate solutions of smaller size. The structure algebra of a finitely generated YB-semitruss is an algebra defined by homogeneous quadratic relations. We prove that it often is a left Noetherian algebra of finite Gelfand-Kirillov dimension that satisfies a polynomial identity, but in general, it is not right Noetherian.

math.RA↗

arXiv2022

Corrigendum and Addendum to "Structure monoids of set-theoretic solutions of the Yang--Baxter equation"

One of the results in our article, which appeared in Publ. Mat. 65 (2021), 499--528, is that the structure monoid $M(X,r)$ of a left non-degenerate solution $(X,r)$ of the Yang-Baxter Equation is a left semi-truss, in the sense of Brzeziński, with an additive structure monoid that is close to being a normal semigroup. Let $η$ denote the least left cancellative congruence on the additive monoid $M(X,r)$. It is then shown that $η$ also is a congruence on the multiplicative monoid $M(X,r)$ and that the left cancellative epimorphic image $\bar{M}=M(X,r)/η$ inherits a semi-truss structure and thus one obtains a natural left non-degenerate solution of the Yang-Baxter equation on $\bar{M}$. Moreover, it restricts to the original solution $r$ for some interesting classes, in particular if $(X, r)$ is irretractable. The proof contains a gap. In the first part of the paper we correct this mistake by introducing a new left cancellative congruence $μ$ on the additive monoid $M(X,r)$ and show that it also yields a left cancellative congruence on the multiplicative monoid $M(X,r)$ and we obtain a semi-truss structure on $M(X,r)/μ$ that also yields a natural left non-degenerate solution. In the second part of the paper we start from the least left cancellative congruence $ν$ on the multiplicative monoid $M(X,r)$ and show that it also is a congruence on the additive monoid $M(X,r)$ in case $r$ is bijective. If, furthermore, $r$ is left and right non-degenerate and bijective then $ν=η$, the least left cancellative congruence on the additive monoid $M(X,r)$, extending an earlier result of Jespers, Kubat and Van Antwerpen to the infinite case.

math.RA↗

arXiv2021

Structure monoids of set-theoretic solutions of the Yang-Baxter equation

Given a set-theoretic solution $(X,r)$ of the Yang--Baxter equation, we denote by $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A'=A'(X,r)$, the left, respectively right, derived structure monoid of $(X,r)$. It is shown that there exist a left action of $M$ on $A$ and a right action of $M$ on $A'$ and 1-cocycles $π$ and $π'$ of $M$ with coefficients in $A$ and in $A'$ with respect to these actions respectively. We investigate when the 1-cocycles are injective, surjective or bijective. In case $X$ is finite, it turns out that $π$ is bijective if and only if $(X,r)$ is left non-degenerate, and $π'$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r) $ is left non-degenerate, in particular $π$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(\bar M, \bar r)$ on the least cancellative image $\bar M= M(X,r)/η$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/η$, for example when $(X,r)$ is irretractable, then $\bar r$ is an extension of $r$. It also is shown that non-degenerate irretractable solutions necessarily are bijective.

math.RA↗

arXiv2020

Skew lattices and set-theoretic solutions of the Yang-Baxter equation

In this paper we discuss and characterize several set-theoretic solutions of the Yang-Baxter equation obtained using skew lattices, an algebraic structure that has not yet been related to the Yang-Baxter equation. Such solutions are degenerate in general, and thus different from solutions obtained from braces and other algebraic structures. Our main result concerns a description of a set-theoretic solution of the Yang-Baxter equation, obtained from an arbitrary skew lattice. We also provide a construction of a cancellative and distributive skew lattice on a given family of pairwise disjoint sets.

math.QA↗