Related papers: Almost classical skew bracoids
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 investigate structural and rigidity properties of \emph{Lie skew braces} (LSBs), objects essentially known in the literature as \emph{post--Lie groups}, obtained by endowing a manifold with two compatible group laws that share the same…
In this paper, we study the Hopf-Galois structures on a finite Galois extension whose Galois group $G$ is an almost simple group in which the socle $A$ has prime index $p$. Each Hopf-Galois structure is associated to a group $N$ of the same…
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…
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…
This work initiates a systematic study of the class of quasi bijective and quasi non-degenerate solutions to the set-theoretic Yang-Baxter equation. The motivation stems from the observation that solutions that arise from dual weak braces…
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…
Relative Rota-Baxter groups are generalisations of Rota-Baxter groups and introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota-Baxter groups with skew left braces, which are well-known to…
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…
Rota-Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum…
We give a self-contained proof that a skew left brace yields a solution of the Yang-Baxter equation.
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,…
Let k be a field. Let also (F, G) be a matched pair of groups. We give necessary and sufficient conditions on a pair (\sigma, \tau) of 2-cocycles in order that the crossed product algebra and the crossed coproduct coalgebra…
We produce novel non-involutive solutions of the Yang-Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from braces and skew braces, and surprisingly in the case of braces they are not…
Let $(H,R)$ be a quasitriangular weak Hopf algebra over a field $k$. We show that there is a braided monoidal equivalence between the Yetter-Drinfeld module category $^H_H\mathscr{YD}$ over $H$ and the category of comodules over some…
The adjunction between coalgebras and Hopf algebras, first described by Takeuchi, allows one to prove that the semi-abelian category of cocommutative Hopf algebras has enough $\mathcal E$-projective objects with respect to the class…
Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been…
The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse…
We formulate the generation of finite dimensional pointed Hopf algebras by group-like elements and skew-primitives in geometric terms. This is done through a more general study of connected and coconnected Hopf algebras inside a braided…
We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square…