本文同时运用群的匹配对理论和大括号理论研究了杨-巴克斯特方程（YBE）的集合论解。本文证明了对称群（辫状对合群）与左支撑概念之间的密切关系，并得到了有限多置换水平的对称群与相应的左支撑的新结果.引入了对称群$(G,r)$的一个新的不变量\emph{the derived chain of ideals of} $G$，它给出了关于$G$的收缩递归过程的精确信息.证明了有限多置换水平$m$的对称群$(G,r)$是可解群，其可解长度至多为$m$ .每个集合论解$(X,r)$的YBE，我们关联两个不变序列的对称群：（i）序列的派生对称群;（ii）序列的派生置换群，并探讨这些明确的描述递归过程的收缩。我们发现新的标准是必要的和充分的要求，$(X, r)$是一个多置换的解决方案。
英文摘要：
We involve simultaneously the theory of matched pairs of groups and the theory of braces to study set-theoretic solutions of the Yang-Baxter equation (YBE). We show the intimate relation between the notions of a symmetric group (a braided involutive group) and a left brace, and find new results on symmetric groups of finite multipermutation level and the corresponding braces. We introduce a new invariant of a symmetric group $(G,r)$, \emph{the derived chain of ideals of} $G$, which gives a precise information about the recursive process of retraction of $G$. We prove that every symmetric group $(G,r)$ of finite multipermutation level $m$ is a solvable group of solvable length at most $m$. To each set-theoretic solution $(X,r)$ of YBE we associate two invariant sequences of symmetric groups: (i) the sequence of its derived symmetric groups; (ii) the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction. We find new criteria necessary and sufficient to claim that $(X, r)$ is a multipermutation solution.
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,…
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…
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].
Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group
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。
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）
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.
We examine links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify new quantum groups associated to set-theoretic solutions coming from braces. We also construct a novel class of quantum discrete integrable systems and we derive symmetries for the corresponding periodic transfer matrices.
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 study the Yang-Baxter equation for the $R$-matrices of the six-vertex model. We analyze the solutions and give new parametrizations of the Yang-Baxter equation. In particular, we find the maximal commutative families of parametrized solutions which generalize the $R$-matrices from the affine quantum (super)-groups. Then we give a new parametrization of the Yang-Baxter equation by a groupoid of non-free-fermionic matrices. In the appendix, we study the general algebraic structure of the solutions of the Yang-Baxter and formulate a conjecture that extends the conjecture by Brubaker, Bump, and…
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.
L. N. Childs defined a bi-skew brace to be a skew brace such that if we swap the role of the two operations, then we find again a skew brace. In this paper, we give a systematic analysis of bi-skew braces. We study nilpotency and solubility, and connections between bi-skew braces and set-theoretic solutions of the Yang--Baxter equation. Further, we deal with Byott's conjecture in the case of bi-skew braces, and we use bi-skew braces as a tool to solve a classification problem proposed by L. Vendramin. In the final part, we investigate brace blocks, defined by A. Koch to be families of group op…
In this paper we study the problem of classification of indecomposable solutions of the Yang-Baxter equation. Using a scheme proposed by Bachiller, Cedó, and Jespers, and recent advances in the classification of braces we classify all indecomposable solutions with some particular permutation groups. We do this for all groups of size $pq$, all abelian groups of size $p^2q$ and all dihedral groups of size $p^2q$.
The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie bialgebra leads to the introduction the mock-Lie Yang-Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang-Baxter equation on a Lie algebra. Note that a skew-symmetric solution of mock-Lie Yang-Baxter equation gives a mock-Lie bialgebra. Finally, the notation of $\mathcal O$-operators are studied to construct skew-symmetric solution of mock-Lie Yang-Baxter equation.
