Marco Castelli, Francesco Catino, Paola Stefanelli
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.
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We develop a method to construct all the indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with a prime-power number of elements and cyclic permutation group. Moreover, we give a complete classification of the indecomposable ones having abelian permutation group and cardinality $pq$ (where $p$ and $q$ are prime numbers not necessarily distinct).
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.
In this paper, we investigate the Nichols algebra $\mathfrak{B}(W_{X,r})$ associated to any non-degenerate involutive solution $(X, r)$ of the Yang-Baxter equation. Infinite examples of finite dimensional Nichols algebras are obtained, including those of dimension $n^m$ with $m$, $n\in\Bbb Z^{\geq 2}$. It turns out that the Nichols algebra $\mathfrak{B}(W_{X, r})$ has interesting relations with multinomial expansion. This is a generalization of the work in arXiv:2103.06489, which built a connection between the Nichols algebras of squared dimension and Pascal's triangle.
On uniconnected solutions of the Yang-Baxter equation and Dehornoy's class
该研究分两部分展开：一是分析置换群为二面体群或极小非循环群的Yang-Baxter方程不可分解对合单连通解，二是研究对合解的Dehornoy类与左braces的关联，给出多类不可分解解的上界与部分精确值。
Involutive (simple) latin solutions of the Yang-Baxter equation and related (left) quasigroups
本文研究了具有正则位移群的杨-巴克方程的蕴含非退化集合论解，描述了不可还原解的块和同余关系，并展示了它们属于拉丁解类别。特别地，我们刻画了具有幂零置换群的简单解，并在位移群为阿贝尔且在总置换群中正规时，用第一韦尔代数进行更精确描述，并对具有最小大小p^p的简单解进行了枚举和分类。
One-generator skew braces and indecomposable set-theoretic solutions to the Yang-Baxter equation
本文研究 Yang–Baxter 方程的单生成元集合论解，建立其与有限单生成元斜括号的精确联系，并推广对合与多重置换解的结果。
This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones among which we especially study two classes of current interest: the simple solutions and those having finite primitive level. In particular, we provide two group-theoretic characterizations of these solutions, involving their permutation groups. Finally, we deal with some open questions.
On the indecomposable involutive solutions of the Yang-Baxter equation of finite primitive level
针对有限本原水平的 Yang-Baxter 方程不可分解对合解，论文用位移群给出群论刻画，枚举小规模解并完全描述本原水平为 2 的情形。
We use Constraint Satisfaction methods to enumerate and construct set-theoretic solutions to the Yang-Baxter equation of small size. We show that there are 321931 involutive solutions of size nine, 4895272 involutive solutions of size ten and 422449480 non-involutive solution of size eight. Our method is then used to enumerate non-involutive biquandles.
Involutive Yang-Baxter groups never act as Frobenius groups
针对 Ramírez 关于二面体置换群对应 Yang-Baxter 集论解基数的猜想，本文证明其对奇数 $n$ 成立、对偶数 $n$ 不成立，并给出有限解置换群不作 Frobenius 群作用的更一般结论。
Motivated by the proof of Rump of a conjecture of Gateva-Ivanova on the decomposability of square-free solutions to the Yang-Baxter equation, we present several other decomposability theorems based on the cycle structure of a certain permutation associated with the solution.
In 1992 Drinfeld posed the question of finding the set theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate solutions on finite sets and groups of $I$-type. A group $\mathcal{G}$ of $I$-type is a group isomorphic to a subgroup of the natural semidirect product of $Fa_n$, the free abelian group of rank $n$, by $Sym_n$, the symmetric group on $n$ letters, so th…
We study involutive set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2. These solutions happen to fall into two classes -- distributive ones and non-distributive ones. The distributive ones can be effectively constructed using a set of abelian groups and a matrix of constants. Using this construction, we enumerate all distributive involutive solutions up to size 14. The non-distributive solutions can be also easily constructed, using a distributive solution and a permutation.
Structure groupoids of quiver-theoretic Yang-Baxter maps
本文研究了有向图理论Yang-Baxter映射的结构群oids，证明其为Garside群oids，并构造了非退化解，探讨了主同型类型解的特性。
In this paper we show that all indecomposable nondegenerate set-theoretical solutions to the Quantum Yang-Baxter equation on a set of prime order are affine, which allows us to give a complete and very simple classification of such solutions. This result is a natural application of the general theory of set-theoretical solutions to the quantum Yang-Baxter equation. It is also a generalization of the corresponding statement for involutive set-theoretical solutions proved in an earlier paper of P.E. and A.S. with T.Schedler. In order to prove our main result, we use the classification theory dev…
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.
Indecomposable involutive set-theoretical solutions to the Yang-Baxter equation of size $p^2$
本文通过分析相关括号结构，完整分类了当 $p$ 为素数时大小为 $p^2$ 的不可分解对合非退化集论 Yang-Baxter 方程解。
Involutive Yang-Baxter: cabling, decomposability, Dehornoy class
本文基于 Rump 定理与 cabling 运算开发新机制，用于生成 Yang-Baxter 方程对合解的可分解性测试，不仅给出了近期定理的初等证明与原创结果，还利用 braces 语言为 Dehornoy 类提供了概念性解释。
Simple solutions of the Yang-Baxter equation of cardinality $p^n$
该研究构造了两类Yang-Baxter方程的集合论解，同时肯定回答了Castelli提出的相关存在性问题。
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.…
Novel non-involutive solutions of the Yang-Baxter equation from (skew) braces
本文利用（斜）brace构造Yang-Baxter方程的新型非对合解，给出逐元素解、逆解双射及Drinfeld扭，并指出相关量子代数为拟三角拟双代数。
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 $p^2q$.
On the cabling of non-involutive set-theoretic solutions of the Yang--Baxter equation
本文通过在 Yang–Baxter 幺半群中实施缆线化，将该方法推广到非对合双射非退化集论解，并建立函子性、可分解性判据及无平方解的应用结论。
It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly-${\mathbb Z}$ groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted …
We investigate a class of non-involutive solutions of the Yang-Baxter equation which generalize self-distributive (derived) solutions. In particular, we study generalized multipermutation solutions in this class. We show that the Yang-Baxter (permutation) groups of such solutions are nilpotent. We formulate the results in the language of biracks.
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…
On indecomposable involutive solutions to the Yang-Baxter equation whose squaring map is a $p$-cycle
探讨Ramírez和Vendramin提出的关于平方映射为对换或\(3 -\)循环的不可分解对合解的问题，研究\(T\)为\(p -\)循环时的情况，在特定假设下给出否定答案，还表明拉丁解情况更严格。
Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group
该研究建立 Yang-Baxter 方程集合论解与置换斜 brace 的性质关联，提出匹配的多重置换水平变体，关联轨道数与生成元、研究自同构群，还给出带阿贝尔置换群的有限不可分解对合解的描述及3级水平下的同构类数目。
Indecomposable set-theoretical solutions to the Yang--Baxter equation on a set of size $pq$
本文针对基数为 $pq$（$p,q$ 为不同素数）的情形，借助 Cedó 与 Okniński 的结果刻画 Yang–Baxter 方程不可分解集合论解的同构分类。
Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. We generalize Rump's braces to the non-commutative setting and use this new structure to study not necessarily involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation. Based on results of Bachiller and Catino and Rizzo, we develop an algorithm to enumerate and construct classical and non-classical braces of small size up to isomorphism. This algorithm is used to produce a database of braces of small size. The paper contains several open problems, questi…
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].
A first aim of this paper is to give sufficient conditions on left non-degenerate bijective set-theoretic solutions of the Yang-Baxter equation so that they are non-degenerate. In particular, we extend the results on involutive solutions obtained by Rump in [36] and answer in a positive way to a question posed by Cedó, Jespers, and Verwimp [19, Question 4.2]. Moreover, we develop a theory of extensions for left non-degenerate set-theoretic solutions of the Yang-Baxter equation that allows one to construct new families of set-theoretic solutions.
Endocabling of involutive solutions to the Yang-Baxter equation, with an application to solutions whose diagonal is a cyclic permutation
本文引入内扭结技术，通过排列括号的$λ$-自同态变形Yang-Baxter方程的对合非退化集合论解，并证明对角映射为循环且大小为素数幂的解具有有限多重排列层级或可迭代收缩至大小为4的唯一解。
In this paper, we introduce a new sequence $\bar{N}_m$ to find a new estimation of the cardinality $N_m$ of the minimal involutive square-free solution of level $m$. As an application, using the first values of $\bar{N}_m$, we improve the estimations of $N_m$ obtained by Gateva-Ivanova and Cameron and by Lebed and Vendramin. Following the approach of the first part, in the last section we construct several new counterexamples to the Gateva-Ivanova's Conjecture.
Coprime extensions of indecomposable solutions to the Yang-Baxter equation
该研究引入Yang-Baxter方程集合论解的扭扩张与互素扩张概念，证明不可分解解的互素扩张均可表为扭扩张，并借此结合已有结构定理完整刻画了阶为三个不同素数乘积的不可分解解，还讨论了相关上同调理论。
We study involutive non-degenerate set-theoretic solutions (X,r) of the Yang-Baxter equation on a finite set X. The emphasis is on the case where (X,r) is indecomposable, so the associated permutation group acts transitively on X. One of the major problems is to determine how such solutions are built from the imprimitivity blocks; and also how to characterize these blocks. We focus on the case of so called simple solutions, which are of key importance. Several infinite families of such solutions are constructed for the first time. In particular, a broad class of simple solutions of order p^2, …
Given a left brace $G$, a method to construct all the involutive, non-degenerate set-theoretic solutions $(Y,s)$ of the YBE, such that $\mathcal{G}(Y,s)\cong G$ is given. This method depends entirely on the brace structure of $G$.
We prove that a finite non-degenerate involutive set-theoretic solution (X,r) of the Yang-Baxter equation is a multipermutation solution if and only if its structure group G(X,r) admits a left ordering or equivalently it is poly-(infinite cyclic).
The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation recently has been reduced to the problem of describing all the left braces. In particular, the classification of all finite left braces is fundamental in order to describe all finite such solutions of the Yang-Baxter equation. In this paper we continue the study of finite simple left braces with the emphasis on the application of the asymmetric product of left braces in order to construct new classes of simple left braces. We do not only construct new classes but also we interpret …
We develop a theory of extensions for involutive and nondegenerate solutions of the set-theoretic Yang-Baxter equation and use it to produce new families of solutions. As an application we construct an infinite family of counterexamples to a conjecture of Gateva-Ivanova related to the retractability of square-free solutions.
In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. In particular, we prove that the matched product of two solutions $r_S$ and $r_T$ is of finite order if and only if $r_S$ and $r_T$ are. Furthermore, we show that with sufficient information on $r_S$ and $r_T$ we can precisely establish the order of the matched product. Finally, we prove that if $B…
This paper explores the structure groups $G_{(X,r)}$ of finite non-degenerate set-theoretic solutions $(X,r)$ to the Yang-Baxter equation. Namely, we construct a finite quotient $\overline{G}_{(X,r)}$ of $G_{(X,r)}$, generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if $X$ injects into $G_{(X,r)}$, then it also injects into $\overline{G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of $G_{(X,r)}$. We show that multipermu…
We describe several methods of constructing R-matrices that are dependent upon many parameters, for example unitary R-matrices and R-matrices whose entries are functions. As an application, we construct examples of R-matrices with prescribed singular values. We characterise some classes of indecomposable set-theoretic solutions of the quantum Yang-Baxter equation (QYBE) and construct R-matrices related to such solutions. In particular, we establish a correspondence between one-generator braces and indecomposable, non-degenerate involutive set-theoretic solutions of the QYBE, showing that such …
Yang-Baxter representations of the infinite symmetric group
本文研究了量子 Yang-Baxter 方程酉对合解定义的无限对称群表示，通过 Young 图对和算子代数技术给出了 Yang-Baxter 特征标与等价类的完整分类及正规形式。
In this paper, we initiate the study of the interplay between $k$-graphs and the Yang-Baxter equation. For this, we provide two very different perspectives. One one hand, we show that the set of all set-theoretic solutions of the Yang-Baxter equation is a special class of single-vertex $k$-graphs. As a consequence, we construct an infinite family of large solutions of the Yang-Baxter equation from an arbitrarily given one. On the other hand, we prove that all single-vertex $k$-graphs are YB-semigroups of square-free, involutive solutions of the Yang-Baxter equation. Other various connections a…
Algebraic structures in set-theoretic Yang-Baxter & reflection equations
该研究围绕集合论Yang-Baxter方程与反射方程的解展开，梳理brace代数性质，分析对合解的参数化解与量子代数、Drinfeld扭与Yangian的关联，推导反射相关的代数关系并证明特殊反射映射下存在反射有限子代数。
Yang-Baxter extremal characters of wreath products of finite groups with the infinite symmetric group
本文研究了有限群与无限对称群的缠绕积的极值杨-巴克斯特特征，通过关联表示与参数集，揭示了极值特征与参数之间的对应关系。
Braces were introduced by Rump as a generalization of Jacobson radical rings. It turns out that braces allow us to use ring-theoretic and group-theoretic methods for studying involutive solutions to the Yang-Baxter equation. If braces are replaced by skew braces, then one can use similar methods for studying not necessarily involutive solutions. Here we collect problems related to (skew) braces and set-theoretic solutions to the Yang-Baxter equation.
Matched pairs and Yang-Baxter operators
本文研究Hopf代数上匹配对诱导的Yang-Baxter算子，给出其对合性的等价条件，肯定回答Ferri-Sciandra开放问题，并分类8维非半单Hopf代数上的匹配对，证明其算子均对合。
Given a skew left brace $B$, a method is given to construct all the non-degenerate set-theoretic solutions $(X,r)$ of the Yang Baxter equation such that the associated permutation group $\mathcal{G}(X,r)$ is isomorphic, as a skew left brace, to $B$. This method depends entirely on the brace structure of $B$. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.
Reflections to set-theoretic solutions of the Yang-Baxter equation
本文通过探索双射非退化解与导出解的联系，确定杨-巴克斯特方程的反射，简化了特定情况下的反射研究，扩展了对合非退化解的已有结果至更一般的解，并提供了定义反射的方法及小阶斜括相关的数值数据。
On Nichols algebras associated to Near-rack solutions of the Yang-Baxter equation
本文研究Yang-Baxter方程近架解的Nichols代数，给出t-等价存在的充分条件，构造有限维Nichols代数，并分类对合近架解对应的所有有限维Nichols代数。
Braces were introduced by Rump as a promising tool in the study of the set-theoretic solutions of the Yang-Baxter equation. It has been recently proved that, given a left brace $B$, one can construct explicitly all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation such that the associated permutation group is isomorphic, as a left brace, to $B$. It is hence of fundamental importance to describe all simple objects in the class of finite left braces. In this paper we focus on the matched product decompositions of an arbitrary finite left brace. This is used to con…
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…
Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. In this work we study the braided group $G=G(X,r)$ of an involutive square-free solution $(X,r)$ of finite order $n$ and cyclic index $p=p(X,r)$ and the group algebra $\textbf{k} [G]$ over a field $\textbf{k}$. We show that $G$ contains a $G$-invariant normal subgroup $\mathcal{F}_p$ of finite index $p^n$, $\mathcal{F}_p$ is isomorphic to th…
A new family of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin, who first gave a counterexample to Gateva-Ivanova's Strong Conjecture. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova's Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova. It is proved that…
Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This family includes the original solution and the classical derived solution. All these solutions induce isomorphic actions of the braid group/monoid on $X^n$. The structure monoids of $r$ and $r^{(k)}$ are related by an explicit bijective $1$-cocycle-like map. We thus turn reflections into a tool for studying YBE solutions, rather than a side object of study. …
We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions we also construct admissible Drinfeld twists similar to the set-theoretic ones, …
Segre products and Segre morphisms in a class of Yang-Baxter algebras
本文研究了 Yang-Baxter 方程的有限非退化对合集合论解所对应的二次 Yang-Baxter 代数，显式给出了 Segre 积的单生成子与二次关系表示，引入了 Segre 映射的类似物并求出其像与核，且结果与经典交换情形一致。
Self-distributive structures, braces & the Yang-Baxter equation
本文研究了自分配结构、括号与Yang-Baxter方程的关系，探讨了rack和quandle的代数性质，以及与Baxter化反演集合论解相关的量子代数和可积性。
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