The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple $(S,+, \cdot)$ with $(S,+)$ a semigroup and $(S, \cdot)$ an inverse semigroup satisfying the relation $a \left(b + c\right) = a b + a\left(a^{-1} + c\right)$, for all $a,b,c \in S$, where $a^{-1}$ is the inverse of $a$ in $(S, \cdot)$. In particular, we give several constructions of inverse semi-braces which allow for obtaining solutions that are different from those until known.
Involutive non-degenerate set theoretic solutions of the Yang-Baxter equation are considered, with a focus on finite solutions. A rich class of indecomposable and irretractable solutions is determined and necessary and sufficient conditions are found in order that these solutions are simple. Then a link between simple solutions and simple left braces is established, that allows us to construct more examples of simple solutions. In particular, the results answer some problems stated in the recent paper [13].
Braces were introduced by W. Rump in 2006 as an algebraic system related to the quantum Yang-Baxter equation. In 2017, L. Guarnieri and L. Vendramin defined for the same purposes a more general notion of a skew left brace. Recently, L. Guo, H. Lang, Y. Sheng [arXiv:2009.03492] gave a definition of what is a Rota-Baxter operator on a group. We connect these two notions as follows. It is shown that every Rota-Baxter group gives rise to a skew left brace. Moreover, every skew left brace can be injectively embedded into a Rota-Baxter group. When the additive group of a skew left brace is complete,…
Employing the algebraic structure of the left brace and the dynamical extensions of cycle sets, we investigate a class of indecomposable involutive set-theoretic solutions of the Yang-Baxter equation having specific imprimitivity blocks. Moreover, we study one-generator left braces of multipermutation level 2.
In the first part of this paper, we investigate the retraction of finite uniconnected involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation by means of left braces, giving a precise description in some cases. In the core of the paper, we also use left braces to classify all the uniconnected involutive non-degenerate set-theoretic solutions having odd size and a Z-group permutation group. As an application, we classify all the uniconnected involutive non-degenerate solutions having odd square-free size.
We consider involutive, non-degenerate, finite set theoretic solutions of the Yang-Baxter equation. Such solutions can be always obtained using certain algebraic structures that generalize nil potent rings called braces. Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent preliminary results. We first identify the generic form of the twists associated to set theoretic solutions and we show that these twists are admissible, i.e. they satisfy a certain co-cycle condition. These findings are also valid for Baxterized solutions of th…
Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group
In this paper we present a characterization of finite simple involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation by means of left braces and we provide some significant examples.
Left semi-braces and solutions to the Yang-Baxter equation
设$r:X^{2}\rightarrow X^{2}$是杨-巴克斯特方程在有限集$X$上的集合论解. Gateva-Ivanova和Van den Bergh证明，如果$r$是非退化和对合的，那么代数$K\langle x \in X \mid xy =uv \mbox{ if } r(x,y)=(u,v)\rangle$与交换多项式代数在多个变量中共享许多性质;特别是这个代数是Noether代数，满足多项式恒等式，并且具有正整数的Gelfand-Kirillov维数。Lebed和Vendramin最近将这一结果推广到任意非退化双射解。这样的解决方案自然与有限斜左括号。在本文中，我们将证明一个类似的结果为任意的解决方案$r_B$，是与左半括号$B$，这样的解决方案可以退化，甚至可以是幂等的。为了做到这一点，我们首先描述了这样的半括号，我们证明了一些分解结果扩展结果的Catino，Colazzo和Stefanelli。
Indecomposable involutive non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation of cardinality $p_1\cdots p_n$, for different prime numbers $p_1,\ldots, p_n$, are studied. It is proved that they are multipermutation solutions of level $\leq n$. In particular, there is no simple solution of a non-prime square-free cardinality. This solves a problem stated in [F. Cedó, J. Okniński, Constructing finite simple solutions of the Yang-Baxter equation, Adv. Math. 391 (2021), 107968] and provides a far reaching extension of several earlier results on indecomposability of solutions.…
Free Skew Braces and Free Solutions of the Yang--Baxter Equation
Vrije Universiteit Brussel（布鲁塞尔自由大学）；Université libre de Bruxelles（布鲁塞尔自由大学）；Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；Maynooth University（梅努斯大学）
