Alicja Smoktunowicz, Ryszard R. Andruszkiewicz
We find all explicit involutive solutions $X \in \mathbb C^{n \times n}$ of the Yang-Baxter-like matrix equation $AXA=XAX$, where $A \in \mathbb C^{n \times n}$ is a given involutory matrix. The construction is algorithmic.
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In this paper, we consider Rota-Baxter operators on involutive associative algebras. We define cohomology for Rota-Baxter operators on involutive algebras that governs the formal deformation of the operator. This cohomology can be seen as the Hochschild cohomology of a certain involutive associative algebra with coefficients in a suitable involutive bimodule. We also relate this cohomology with the cohomology of involutive dendriform algebras. Finally, we show that the standard Fard-Guo construction of the functor from the category of dendriform algebras to the category of Rota-Baxter algebras…
Non-involutive solutions of the Yang-Baxter equation of multipermutation level 2
本文研究多重排列层级2且非2-约化的Yang-Baxter方程集合论解，提出基于无平方2-约化解与两个双射的构造方法，并给出枚举算法，完成大小不超过6的全部解的分类。
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 multipermutation 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 ab…
Set-theoretic solutions to the Yang-Baxter equation have been studied extensively by means of related algebraic systems such as cycle sets and braces, dynamical versions of which have also been developed. No work focuses on set-theoretic solutions to the Hom-Yang-Baxter equation (HYBE for short). This paper investigates set-theoretic solutions to HYBE and associated algebraic system, called Hom-cycle sets. We characterize left non-degenerate involutive set-theoretic solutions to HYBE and Hom-cycle sets, and establish their relations. We discuss connections among Hom-cycle sets, cycle sets, lef…
On a class of involutive Yang-Baxter groups
本文研究了Sylow子群幂零类至多为2的有限可解群是否为IYB-group的问题，并证明了在幂零剩余为$\Q_8$-free或Sylow $2$-子群同构于$Q_8$的条件下，此类群均为IYB-group。
Diagonals of solutions of the Yang-Baxter equation
该文研究Yang-Baxter方程非对合集论解的对角映射，证明非退化解的对角映射是交换双射，解决了相关公开问题，还推广了多置换解的刻画与可分解性结果。
In {\it Set-theoretical solutions to the quantum Yang-Baxter equation} (Duke Math. J. {\bf 100} (1999), 169--209), Etingof, Schedler and Soloviev introduced, for each non-degenerate involutive set-theoretical solution $(X,σ,τ)$ of the Yang-Baxter equation, the equivalence relation $\sim$ defined on the set $X$ and they considered a new non-degenerate involutive induced \emph{retraction} solution defined on the quotient set $X^{\sim}$. It is well known that translating set-theoretical non-degenerate solutions of the Yang-Baxter equation into the universal algebra language we obtain an algebra c…
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。
In this paper, we mainly present some new solutions of the Hom-Yang-Baxter equation from Hom-algebras, Hom-coalgebras and Hom-Lie algebras, respectively. Also, we prove that these solutions are all self-inverse and give some examples. Finally, we introduce the notion of Hom-Yang-Baxter systems and obtain two kinds of Hom-Yang-Baxter systems.
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。
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。
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。
Let $L$ be a simple anti-commutative algebra. In this paper we prove that a non skew-symmetric solution of the classical Yang-Baxter equation on $L$ with $L$-invariant symmetric part induces on $L$ a Rota-Baxter operator of a non-zero weight.
We study possible connections between Rota-Baxter operators of non-zero weight and non-skew-symmetric solutions of the classical Yang-Baxter equation on finite-dimensional quadratic Lie algebras. The particular attention is made to the case when for a solution $r$ the element $r+τ(r)$ is $L$-invariant.
We give a self-contained proof that a skew left brace yields a solution of the Yang-Baxter equation.
Isotopes of biracks and Zhang twists of algebras
本文介绍了$\mathbb{N}^p$-分级birrack的同位素概念，并研究了其与$\mathbb{N}^p$-分级Yang-Baxter代数张扭曲的关系，证明了由分配解确定的Yang-Baxter代数是多项式代数的张扭曲。
We introduce the notion of associative (BiHom-)Yang-Baxter pair of weight $(λ,γ)$ which can provide the solution to the double curved Rota-Baxter (BiHom-)system. Equivalent characterizations of (quasitriangular) covariant BiHom-bialgebra are given. We also prove that associative BiHom-Yang-Baxter equation of weight $-1$ can be obtained by the unitary quasitriangular covariant BiHom-bialgebra. At last, we present two approaches to construct (BiHom-)pre-Lie modules from Rota-Baxter (BiHom-)paired modules.
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…
Associative Yang-Baxter equation arises in different areas of algebra, e.g., when studying double quadratic Poisson brackets, non-abelian quadratic Poisson brackets, or associative algebras with cyclic 2-cocycle (anti-Frobenius algebras). Precisely, faithful representations of anti-Frobenius algebras (up to isomorphism) are in one-to-one correspondence with skew-symmetric solutions of associative Yang-Baxter equation (up to equivalence). Following the work of Odesskii, Rubtsov and Sokolov and using computer algebra system Sage, we found some constant skew-symmetric solutions of associative Yan…
$ω$-Lie bialgebras and $ω$-Yang-Baxter equation
本文引入了等价于Manin三元组与匹配对的乘法$ω$-李双代数，研究了$ω$-Yang-Baxter方程，并基于由左对称代数构造的$ω$-$\mathcal{O}$-算子给出了该方程的斜对称解。
Yang-Baxter equations and relative Rota-Baxter operators for left-Alia algebras associated to invariant theory
本文研究left-Alia双代数的构造理论，引入left-Alia Yang-Baxter方程，证明其反对称解可生成三角双代数，并通过相对Rota-Baxter算子与pre-left-Alia代数提供该方程的反对称解。
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。
Yang-Baxter permutation group actions on distributive Yang-Baxter algebras
本文研究分配Yang-Baxter代数的置换群作用，证明其同构于斜多项式代数，刻画反射群条件并描述不变子代数，建立Auslander定理并给出pertinency下界。
In this article, we give a few classes of solutions for the Yang-Baxter type matrix equation, $AXA=XAX$. We provide all solutions for the cases when $A$ is equivalent to a Jordan block or has precisely two Jordan blocks. We also have given a few general properties of the solutions of the YB-equation.
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 soluti…
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。
Spectral-parameter dependent Yang-Baxter operators and Yang-Baxter systems from algebra structures
对任意代数，构造两族有色Yang-Baxter算子，从而得到双参数量子杨-巴克斯特方程的解.提出了一个关于函数方程组的公开问题。计算了这些算子在二维和三维代数上的矩阵形式。给出了其中一个族的FRT双代数。导出了单参数量子杨-巴克斯特方程的解，构造了一个Yang-Baxter系统.
New constructions of Yang-Baxter systems
量子杨-巴克斯特方程允许推广到称为杨-巴克斯特系统的杨-巴克斯特型方程系统。从代数结构出发，我们提出了一些常数以及谱参数依赖的Yang-Baxter系统的新结构。此外，我们还明确地给出了二维常数型的交换代数结构。
The Strongly Symmetric Elements and Solutions of Yang-Baxter Equation
证明了所有强对称元都是李代数中的常数经典杨-巴克斯特方程或代数中的量子杨-巴克斯特方程的解.另外，还得到了特征为2的域$k$上具有dim $L \le 3$的李代数$L$中常数经典杨-巴克斯特方程（CYBE）的所有解
A new method to construct involutive non-degenerate set-theoretic solutions $(X^n,r^{(n)})$ of the Yang-Baxter equation from an initial solution $(X,r)$ is given. Furthermore, the permutation group $\mathcal{G}(X^n,r^{(n)})$ associated to the solution $(X^n,r^{(n)})$ is isomorphic to a subgroup of $\mathcal{G}(X,r)$, and in many cases $\mathcal{G}(X^n,r^{(n)})\cong \mathcal{G}(X,r)$.
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 …
Garside groups and Yang-Baxter equation
该研究建立了特定加赛德群与量子杨-巴克斯特方程特定解的结构群的一一对应，并通过Δ-纯加赛德群刻画了不可分解解，连接了群论与量子代数领域。
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。
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。
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-B…
We introduce the concept of an extended O-operator that generalizes the well-known concept of a Rota-Baxter operator. We study the associative products coming from these operators and establish the relationship between extended O-operators and the associative Yang-Baxter equation, extended associative Yang-Baxter equation and generalized Yang-Baxter equation.
We explicitly determine all Rota-Baxter operators (of weight zero) on $sl(2,C)$ under the Cartan-Weyl basis. For the skew-symmetric operators, we give the corresponding skew-symmetric solutions of the classical Yang-Baxter equation in $sl(2,C)$, confirming the related study by Semenov-Tian-Shansky. In general, these Rota-Baxter operators give a family of solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebra $sl(2,C) \ltimes_{{\rm ad}^{\ast}} sl(2,C)^{\ast}$. They also give rise to 3-dimensional pre-Lie algebras which in turn yield solutions of the classical Yang-Baxt…
In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative) Rota-Baxter operator on a Leibniz algebra and construct the graded Lie algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements. By these structures and the twisting theory of twilled Leibniz algebras, we further define the classical Leibniz Yang-Baxter equation, classical Leibniz r-matrices and triangular Leibniz bialgebras. Finally, w…
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.
Motivated by recent work on Hom-Lie algebras and the Hom-Yang-Baxter equation, we introduce a twisted generalization of the classical Yang-Baxter equation (CYBE), called the classical Hom-Yang-Baxter equation (CHYBE). We show how an arbitrary solution of the CYBE induces multiple infinite families of solutions of the CHYBE. We also introduce the closely related structure of Hom-Lie bialgebras, which generalize Drinfel'd's Lie bialgebras. In particular, we study the questions of duality and cobracket perturbation and the sub-classes of coboundary and quasi-triangular Hom-Lie bialgebras.
Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures
本文研究群和李群上权为0的相对Rota-Baxter算子，并建立了其与预群、括号、Yang-Baxter方程集合论解及$T$-结构之间的显式关系。
We obtain a simple family of solutions to the set-theoretic Yang-Baxter equation, one which depends only on considering special endomorphisms of a finite group. We show how such an endomorphism gives rise to two non-degenerate solutions to the Yang-Baxter equation, solutions which are inverse to each other. We give concrete examples using dihedral, alternating, symmetric, and metacyclic groups.
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