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.
In this article, we study properties as simplicity, solvability and nilpotency for Lie bracket algebras arising from Leavitt path algebras, based on the talented monoid of the underlying graph. We show that graded simplicity and simplicity of the Leavitt path algebra can be connected via the Lie bracket algebra. Moreover, we use the Gelfand-Kirillov dimension for the Leavitt path algebra for a classification of nilpotency and solvability.
Using Bieberbach groups we study multipermutation involutive solutions to the Yang-Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right $p$-nilpotent skew brace…
This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them.
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 …
It is proved that any Rota---Baxter operator of zero weight on Jordan algebra of a nondegenerate bilinear symmetric form is nilpotent of index less or equal three. We state exact value of nilpotency index on simple Jordan algebra of Clifford type over fields $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{Z}_p$. For $\mathbb{Z}_p$, we essentially use the results from number theory concerned quadratic residues and Chevalley---Warning theorem.
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 no…
We introduce non-degenerate solutions of the Yang-Baxter equation in the setting of symmetric monoidal categories. Our theory includes non-degenerate set-theoretical solutions as basic examples. However, infinite families of non-degenerate solutions (that are not of set-theoretical type) appear. As in the classical theory of Etingof, Schedler and Soloviev, non-degenerate solutions are classified in terms of invertible 1-cocycles. Braces and matched pairs of cocommutative Hopf algebras (or braiding operators) are also generalized to the context of symmetric monoidal categories and turn out to b…
We study set-theoretic solutions $(X,r)$ of the Yang-Baxter equations on a set $X$ in terms of the induced left and right actions of $X$ on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid $S(X,r)$ and we show that $r$ extends as a solution $(S(X,r),r_S)$. Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids…
We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}(2)$ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf algebra $w\mathfrak{sl}_{q}(2)$ and a $J$-weak Hopf algebra $v\mathfrak{sl}_{q}(2)$ which are studied in detail. It is shown that the monoids of group-like elements of $w\mathfrak{sl}_{q}(2)$ and $v\mathfrak{sl}_{q}(2)$ are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from $w\mat…
For a finite involutive non-degenerate solution $(X,r)$ of the Yang--Baxter equation it is known that the structure monoid $M(X,r)$ is a monoid of I-type, and the structure algebra $K[M(X,r)]$ over a field $K$ share many properties with commutative polynomial algebras, in particular, it is a Noetherian PI-domain that has finite Gelfand--Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid $M(X,r)$ and the algebra $K[M(X,r)]$ is much more complicated than in the involutive case, we provide some deep insights. …
We define the concept of weak pseudotwistor for an algebra $(A, μ)$ in a monoidal category $\mathcal{C}$, as a morphism $T:A\otimes A\rightarrow A\otimes A$ in $\mathcal{C}$, satisfying some axioms ensuring that $(A, μ\circ T)$ is also an algebra in $\mathcal{C}$. This concept generalizes the previous proposal called pseudotwistor and covers a number of exemples of twisted algebras that cannot be covered by pseudotwistors, mainly examples provided by Rota-Baxter operators and some of their relatives (such as Leroux's TD-operators and Reynolds operators). By using weak pseudotwistors, we introd…
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 soluti…
We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota-Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota-Baxter hierarchy. We conjecture that the operator relations are a noncommutative Groebner basis for the ideal they generate.
Young tableaux carry an associative product, described by the Schensted algorithm. They thus form a monoid $\mathbf{Pl}$, called \emph{plactic}. It is central in numerous combinatorial and algebraic applications. In this paper, the tableaux product is shown to be completely determined by a braiding $σ$ on the (much simpler!) set of columns $\mathbf{Col}$. Here a \emph{braiding} is a set-theoretic solution to the Yang--Baxter equation. As an application, we identify the Hochschild cohomology of $\mathbf{Pl}$, which resists classical approaches, with the more accessible braided cohomology of $(\…
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 struct…
