Related papers: Finite skew braces of square-free order and supers…

We define isoclinism of skew braces and present several applications. We study some properties of skew braces that are invariant under isoclinism. For example, we prove that right nilpotency is an isoclinism invariant. This result has…

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…

This paper examines the connections between (relative) Rota--Baxter groups, skew left braces, and enlargements of these structures on naturally associated semi-direct products. Given a skew left brace, we define a new skew left brace,…

We introduce left and right series of left semi-braces. This allows to define left and right nilpotent left semi-braces. We study the structure of such semi-braces and generalize some results, known for skew left braces, to left…

We study the class of one-generator solutions to the Yang-Baxter equation, extending some recent results concerning the classes of involutive and multipermutation solutions. Moreover we show the precise relationship between indecomposable…

We study simple set-theoretic solutions of the Yang-Baxter equation that are finite and non-degenerate. Such retractable solutions are fully described and to investigate the irretracble solutions we give a new algebraic method. Our approach…

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…

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…

The aim of this short note is to completely answer Questions 2.34 and 2.35 of arXiv:1806.01127. In particular, we show that a finite strong-nil skew brace $B$ of abelian type need not be right-nilpotent, but that this is the case if~$B$ is…

Assume that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that $C_{G}(F)=1$. In this paper, we investigate this situation and prove that if $C_G(H)$ is supersoluble and $C_{G'}(H)$…

The main objective of this article is to initiate a detailed structure theory of left nilpotent skew braces $B$ of class $2$, i.e. skew braces with $B^3 = 0$. We prove that if $B$ is of nilpotent type, then $B$ is centrally nilpotent. In…

Let $r:X^{2}\rightarrow X^{2}$ be a set-theoretic solution of the Yang-Baxter equation on a finite set $X$. It was proven by Gateva-Ivanova and Van den Bergh that if $r$ is non-degenerate and involutive then the algebra $K\langle x \in X…

A dual weak brace is an algebraic structure $\left(S,\,+,\,\circ\right)$ including skew braces and giving rise to a set-theoretic solution of the Yang-Baxter equation. We show that such a map belongs to a family of set-theoretic solutions,…

A skew brace $A = (A,\cdot,\circ)$ is said to be \textit{left-simple} if $A\neq1$ and it has no left ideal other than $1$ and $A$. The purpose of this paper is to give a partial classification of the finite left-simple skew braces. A result…

We develop the theory of Schur covers of finite skew braces. We prove the existence of at least one Schur cover. We also compute several examples. We prove that different Schur covers are isoclinic. Finally, we prove that Schur covers have…

A skew brace is a ring-like and group-like algebraic structure that was introduced in the study of set-theoretic solutions to the Yang-Baxter equation. In this survey paper, we shall consider the left series, right series, socle series, and…

Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is Malcev nilpotent. Applying this characterization to solutions coming from racks, we…

The main result of this paper is an explicit construction of the free commutative skew brace -- that is, a skew brace whose circle group is commutative -- on an arbitrary generating set $X$. We embed this object into a set of rational…

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…

In this paper, we study the irreducible representations of skew braces of order \( pq \), which is equivalent to studying the representation theory of groups of order \( p^2q^2 \) arising from skew left braces, where \( p > q \) are primes.…