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تسجيل مستخدم جديدPrimitive set-theoretic solutions of the Yang-Baxter equation
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To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${mathcal G}(X,r)$ acting on $X$. We prove that every primitive permutation group of this type is of prime order $p$. Moreover, $(X,r)$ is then a so called permutation solution determined by a cycle of length $p$. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.
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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 ex
ist a left action of $M$ on $A$ and a right action of $M$ on $A$ and 1-cocycles $pi$ and $pi$ 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 $pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $pi$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r) $ is left non-degenerate, in particular $pi$ 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 $bar M= M(X,r)/eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/eta$, 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.
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 s
emitrusses as introduced by Brzezinski. 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.
Given a finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation and a field $K$, the structure $K$-algebra of $(X,r)$ is $A=A(K,X,r)=Klangle Xmid xy=uv mbox{ whenever }r(x,y)=(u,v)rangle$. Note that $A=oplus_{ngeq 0} A_n$ is
a graded algebra, where $A_n$ is the linear span of all the elements $x_1cdots x_n$, for $x_1,dots ,x_nin X$. One of the known results asserts that the maximal possible value of $dim (A_2)$ corresponds to involutive solutions and implies several deep and important properties of $A(K,X,r)$. Following recent ideas of Gateva-Ivanova cite{GI2018}, we focus on the minimal possible values of the dimension of $A_2$. We determine lower bounds and completely classify solutions $(X,r)$ for which these bounds are attained in the general case and also in the square-free case. This is done in terms of the so called derived solution, introduced by Soloviev and closely related with racks and quandles. Several problems posed in cite{GI2018} are solved.
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 s
ome 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.
Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation are discussed and many consequences are derived. In particular, for each positive integer $n$ a finite square-free multiperm
utation solution of the Yang-Baxter equation with multipermutation level $n$ and an abelian involutive Yang-Baxter group is constructed. This answers a problem of Gateva-Ivanova and Cameron. It is also proved that finite non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation whose associated involutive Yang-Baxter group is abelian are retractable in the sense of Etingof, Schedler and Soloviev. Earlier the authors proved this with the additional square-free hypothesis on the solutions. Retractability of solutions is also proved for finite square-free non-degenerate involutive set-theoretic solutions associated to a left brace.
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