E. Acri, M. Bonatto
This is the first part of a series of two articles. In this paper we enumerate and classify the left braces of size $p^2q$ where $p$ and $q$ are distinct prime numbers by the classification of regular subgroups of the holomorph of the abelian groups of the same order. We also provide the formulas that define the constructed braces.
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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…
On two-sided skew braces
本文通过引入弱平凡斜撑并给出其分类，研究双侧斜撑的结构，推广了加法群与乘法群关系的已知结果，并证明两类素与半素斜撑概念在双侧情形下等价。
The aim of this paper is to give a simple construction of the free skew-brace over a set $X$.
We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of Itô's theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang-Baxter equation. Finally, we classify skew braces that contain no non-trivial proper ideals.
We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang-Baxter equation are explored using the structure of skew left braces.
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.
We introduce the notion of Drinfeld twists for both set-theoretical YBE solutions and skew braces. We give examples of such twists and show that all twists between skew braces come from families of isomorphisms between their additive groups. We then describe the relation between these definitions and co-twists on FRT-type Hopf algebras in the category $\mathrm{SupLat}$, and prove that any co-twist on a co-quasitriangular Hopf algebra in $\mathrm{SupLat}$ induces a Drinfeld twist on its remnant skew brace. We go on to classify co-twists on bicrossproduct Hopf algebras coming from groups with un…
Skew braces whose left series is not generated by right-normed star products
本文构造斜括号显式例子，证明其左序列第三项不能由右规范星积生成，揭示斜括号结构性质。
We construct all skew braces of size $pq$ (where $p>q$ are primes) by using Byott's classification of Hopf--Galois extensions of the same degree. For $p\not\equiv 1 \pmod{q}$ there exists only one skew brace which is the trivial one. When $p\equiv 1 \pmod{q}$, we have $2q+2$ skew braces, two of which are of cyclic type (so, contained in Rump's classification) and $2q$ of non-abelian type.
We define combinatorial representations of finite skew braces and use this idea to produce a database of skew braces of small size. This database is then used to explore different concepts of the theory of skew braces such as ideals, series of ideals, prime and semiprime ideals, Baer and Wedderburn radicals and solvability. The paper contains several questions.
Skew braces and Rota-Baxter operators on semi-direct products
本文研究了斜左 brace 与半直积上的 Rota-Baxter 运算符关系，定义了斜左 brace 的平方结构，提供了一种构造 Yang-Baxter 方程新解的方法，并证明了该构造的泛性质。
On a family of simple skew braces
针对非 brace 型单斜 brace 实例稀缺的问题，研究者构造了首个不源自非交换单群的此类无限族，证明满足特定整除条件的素数 p、q 对应两个同构类的 p^p q 阶单斜 brace。
Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
本文研究斜括号中不同幂零性概念的关系，提出零化子幂零斜括号作为幂零群的类比，并在该框架下推广了群论中的若干著名定理，应用于Yang-Baxter方程解的结构分析。
Isoclinism of skew braces
本文定义了skew brace的同倾概念，证明了右幂零性是其同倾不变量，并将该结果应用于Yang-Baxter方程的集合论解理论中。
Skew braces: a brief survey
本文综述斜括号理论，重点探索其与杨-巴克斯特方程组合解的联系，强调该领域的开放问题与猜想，旨在增进对斜括号及其在杨-巴克斯特方程研究中作用的理解并激发后续研究。
The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial braces is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed.
On Grün's lemma for perfect skew braces
本文研究了完美双侧斜格中Grün引理的适用性，发现其在特定条件下成立，但一般情况下不成立。
In this paper we enumerate the skew braces of size $p^2q$ for $p,q$ odd primes by the classification of regular subgroups of the holomorph of the groups of size $p^2q$. In particular, we provide explicit formulas for the skew braces of abelian type.
Some properties of skew braces that are invariant under isoclinism
本文证明了斜支撑的bi-skew、$λ$-homomorphic和inner性质在等倾关系下保持不变。
We study relations between the additive and the multiplicative groups of a two-sided skew brace. In particular, we prove that if the additive group of a two-sided skew brace is finite solvable (respectively, finitely generated nilpotent, finitely generated residually nilpotent, finitely generated residually finite), then the multiplicative group of this skew brace is solvable (respectively, solvable, residually solvable, residually finite). Also, we prove that if the multiplicative group of a two-sided skew brace is nilpotent of nilpotency class $k$, then the additive group of this skew brace …
On $n$-isoclinism of skew braces
本文探讨了斜近环的n-同源性定义，并引入了言语子斜近环和边缘左理想概念。
L. Childs has defined a skew brace $(G, \cdot, \circ)$ to be a bi-skew brace if $(G, \circ, \cdot)$ is also a skew brace, and has given applications of this concept to the equivalent theory of Hopf-Galois structures. The goal of this paper is to deal with bi-skew braces $(G, \cdot, \circ)$ from the yet equivalent point of view of regular subgroups of the holomorph of $(G, \cdot)$. In particular, we find that certain groups studied by T. Kohl, F. Dalla Volta and the author, and C. Tsang all yield examples of bi-skew braces. Building on a construction of Childs, we also give various methods for …
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。
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…
Semidirect products of skew braces
本研究针对双群与斜 brace，探讨了作用、半直积和理想换位子的概念，为相关代数结构研究提供基础。
A generalization of Ito's theorem to skew braces
本文将群论中关于阿贝尔子群乘积的Itô定理推广至skew brace设定，证明了由特定平凡子skew brace乘积构成的skew brace是亚平凡的。
A skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion of brace as introduced by W. Rump. Skew braces can be used to construct solutions of the quantum Yang-Baxter equation. In this article, we introduce a notion of action of a skew brace, and show how it leads to solutions of the closely associated reflection equation.
On dynamical skew braces and skew bracoids
本文研究动力斜括号与斜bracoid的关联，证明技术性假设下动力斜括号生成的箭图Yang–Baxter方程解为辫子群胚即斜bracoid，且连通辫子群胚可平行化同构于动力斜括号，并探讨其组合结构与整数不变量的关系。
In this paper we enumerate the skew braces of non-abelian type of size $p^2q$ for $p,q$ primes with $q>2$ by the classification of regular subgroups of the holomorph of the non-abelian groups of the same order. Since Crespo dealt with the case $q=2$, this paper completes the enumeration of skew braces of size $p^2q$ started in a previous work by the authors. In some cases, we provide also a structural description of the skew braces. As an application, we prove a conjecture posed by V. Bardakov, M. Neshchadim and M. Yadav.
On the Sylow Theorem for Skew Braces
研究有限斜格的Sylow定理，证明当斜格为双侧、双斜、右nilpotent、λ-同态或超可解时成立，并探讨了更多专门情形，证明了广义Hall型定理。
Skew brace extensions, second cohomology and complements
本文研究了斜左括消的扩展、第二上同调及补集，推导了Schur乘子嵌入性质，并引入了最小扩展概念，得出了关于有限斜左括消分裂扩展的简化结果。
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,…
Associativity of the $\star$-product of skew braces
本文证明斜括号中双侧性、$\star$-积结合性与$A\star A$中心性三者中任意两者蕴含第三者，并通过根环给出结合$\star$-积的双侧斜括号的纤维积描述、正合序列及幂零指数界，同时分类相关结构。
Free Skew Braces and Free Solutions of the Yang--Baxter Equation
证明Yang-Baxter方程的自由非退化双射解的结构斜辫子是自由的，并构造任意类的自由右幂零斜辫子，证明其可解词问题及有限解中的律对一切解成立。
Finite skew braces of square-free order and supersolubility
本文研究超可解斜 braces，证明其具有 Sylow 塔并刻画中心幂零理想与多重置换层性质，同时给出判定相关结构斜 brace 是否超可解的算法。
Schur covers of skew braces
本研究建立有限斜 brace 的 Schur 覆盖理论，证明其存在性、不同覆盖的等斜性及提升性质，并给出计算实例，丰富了斜 brace 的代数结构研究。
Relative Rota-Baxter groups and skew left braces
本文探讨了 Relative Rota-Baxter 群与斜左 brace 之间的联系，证明二者在温和限制下存在范畴同构，并提出计算有限群上 Relative Rota-Baxter 算子的高效 GAP 算法及同倾概念。
On the triviality and non-triviality of the automorphism group of a skew brace
本文证明了某些斜近生家族的自同构群平凡性与群阶关系，且构造了阶为2p³的斜近生具有平凡自同构群。
The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces
本文证明了有限斜支架的Schur-Zassenhaus定理和Sylow第三定理，包括Hall理想的子斜支架补的存在性以及Sylow p-子斜支架个数模p同余1的性质。
Soluble skew left braces and soluble solutions of the Yang-Baxter equation
本文引入并研究斜左括号的可解性性质，由此定义Yang-Baxter方程解的可解性，证明非退化集合论可解解可由可解斜左括号刻画，还展示了其丰富理想结构并给出相关实例。
We define the radical and weight of a skew left brace and provide some basic properties of these notions. In particular, we obtain a Wedderburn type decomposition for Artinian skew left braces. Furthermore, we prove analogues of a theorem of Wiegold, a theorem of Schur and its converse in the context of skew left braces. Finally, we apply these results to detect torsion in the structure group of a finite bijective non-degenerate set-theoretic solution of the Yang-Baxter equation.
A Counterexample to Byott's Conjecture for Finite Skew Braces
构建加法群可解、乘法群不可解的有限斜括号，其乘法群商同构于\(\PSL_2(7)\)，反驳拜奥特猜想。
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.
Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their…
The structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented
本文证明了与Yang-Baxter方程的有限非退化解相关联的结构斜撑具有有限表示性质。
On the multiplicative group of a two-sided skew brace of solvable type
研究双侧斜近生的乘法群在加法群可解时的有限商群性质，证明其可解性，并扩展Nasybullov定理至任意可解类型双侧斜近生。
Finiteness conditions on skew braces and solutions of the Yang-Baxter equation
研究了斜近形的有限性条件，揭示了其与Yang-Baxter方程解之间的关系，以及有限分解因子和强左理想在其中的作用。
Categories of skew left braces and trifactorised groups
本文旨在深化斜左括号与三因子化群之间的联系，该联系编码了斜左括号的结构、商及同态信息。
Deformed solutions of the Yang-Baxter equation associated to dual weak left $\star$-braces
本文引入双弱左星brace和平方斜左brace，并证明其分布器变形映射为Yang-Baxter方程的解，扩展了相关研究成果。
Skew braces with no proper left ideals
本文研究有限左单斜辫子的部分分类，其中左单斜辫子定义为没有非平凡真左理想的斜辫子，并与有限域扩张上的极小Hopf-Galois结构一一对应。
On simple compact Lie skew braces
本文从全局和微分视角研究李斜 brace 的简单性，证明紧致连通简单李斜 brace 的结构特性，并构造非紧致例子展示其非刚性。
On Dedekind Skew Braces
本文研究Dedekind斜辫子，证明有限情形中心幂零，推广至局部循环群，并给出相关解的刻画。
Rota--Baxter operators and skew left brace structures over Heisenberg Group
本文通过求解定义方程分类了3维Heisenberg李代数上weight 1的Rota--Baxter算子，并利用指数双射将其诱导至Heisenberg群，进而枚举了该群上所有的skew left brace结构。
We classify all skew braces of Heisenberg type for a prime number $ p>3 $. Furthermore, we determine the automorphism group of each one of these skew braces (as well as their socle and annihilator). Hence, by utilising a link between skew braces and Hopf-Galois theory, we can determine all Hopf-Galois structures of Heisenberg type on Galois field extensions of fields of degree $ p^{3} $.
Chain conditions on skew braces and solutions of the Yang-Baxter Equation
研究满足理想极大或极小条件的斜辫环，类比Hall和McLain的群论结果，分析可解性和局部幂零性对有限生成性的作用，并引入Yang-Baxter方程非退化集论解的有限性和链条件。
Carter-like theorems for skew braces
本文引入 Carter 子斜辫子，分析其结构限制，并证明在特定条件下有限可解斜辫子及有限多重置换斜辫子中存在此类子结构。
Constructing skew left braces whose additive group has trivial centre
本文完整刻画了加法群具有平凡中心的有限斜左括号的所有可能乘法群，改进了Tsang的早期结果，并回答了其在2024年Ischia群论会议上提出的公开问题。
On a Cauchy theorem for finite skew braces
本文证明了有限双侧斜格和双斜格中存在Cauchy定理，并探讨了相关结构结果。
Commuting probability of skew left braces
研究斜左 brace 的交换概率，证明其上限为 3/4，且 3/4 是唯一在 (5/8,1) 区间内出现的有理数，同时探讨了其性质与应用。
Free commutative skew braces
本文构造了任意生成集上的自由交换斜括号，将其嵌入有理函数并用线性方程刻画像，从而简化元素比较。
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