A construction of the free skew brace

左序列不由右规范星积生成的斜括号

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。

Skew braces of size $pq$

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.

On skew braces and their ideals (with an Appendix by Agata Smoktunowicz)

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.

斜左 brace 与半直积上的 Rota-Baxter 运算符

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。

关于一类单斜 brace（斜 brace）

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。

斜括号的幂零性与Yang-Baxter方程的多置换解

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。

skew brace的同倾

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。

斜括号：简要综述

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。

Cohomology, Extensions and Automorphisms of Skew Braces

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.

关于完美斜左格的Grün引理

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。

Skew braces of size $p^2q$

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.

斜支撑在等倾下不变的一些性质

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。

Connections between properties of the additive and the multiplicative groups of a two-sided skew brace

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 …

关于斜近环的n-同源性

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。

Bi-Skew Braces and Regular Subgroups of the Holomorph

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 …

通过斜 brace 研究 Yang-Baxter 方程的解，及其在具有阿贝尔置换群的不可分解对合解中的应用

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。

Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces

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…

斜 brace 的半直积

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。

Ito定理向skew braces的推广

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。

Actions of skew braces and set-theoretic solutions of the reflection equation

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.

论动力斜括号与斜bracoid

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。

Skew braces of size $p^2q$ II: non-abelian type

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.

关于斜格的Sylow定理

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。

斜左括消扩展、第二上同调与补集

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。

Rota-Baxter groups, skew left braces, and the Yang-Baxter equation

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,…

斜括号的$\star$-积的结合性

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。

自由斜辫子与Yang-Baxter方程的自由解

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。

Skew braces of size $p^2q$ I: abelian type

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.

无平方因子阶的有限斜 braces 与超可解性

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。

斜 brace 的 Schur 覆盖

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 群与斜左 brace

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。

关于斜近生的自同构群平凡性与非平凡性的研究

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。

有限斜支架的Schur-Zassenhaus定理与Sylow第三定理

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。

可解斜左括号与Yang-Baxter方程的可解解

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。

Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation

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.

关于有限斜括号的拜奥特猜想的一个反例

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。

Solutions of the Yang-Baxter equation associated to skew left braces, with applications to racks

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.

On skew braces (with an appendix by N. Byott and L. Vendramin)

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…

与Yang-Baxter方程的有限非退化解相关联的结构斜撑是有限表示的

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。

关于可解类型的双侧斜近生的乘法群

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。

关于斜近形的有限性条件与Yang-Baxter方程的解

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。

斜左括号与三因子化群的范畴

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。

变形的Yang-Baxter方程解与双弱左星 brace 的关联

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。

无真左理想的斜辫子

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。

关于简单紧致李斜 brace 的研究

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。

关于Dedekind斜辫子

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。

Heisenberg 群上的 Rota--Baxter 算子与斜左 brace 结构

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。

Skew Braces and Hopf-Galois Structures of Heisenberg Type

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} $.

斜辫环上的链条件与Yang-Baxter方程的解

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 型定理用于斜辫子（skew braces）

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。

构造加法群具有平凡中心的斜左括号

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。

关于有限斜格的Cauchy定理

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。

斜左 brace 的交换概率

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。

自由交换斜括号

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。