Related papers: Skew left braces and the Yang-Baxter equation
The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers, and that in an arbitrary…
Employing the algebraic structure of the left brace and the dynamical extensions of cycle sets, we investigate a class of indecomposable involutive set-theoretic solutions of the Yang-Baxter equation having specific imprimitivity blocks.…
We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra.
In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite…
We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use set-theoretic solutions to construct solutions to the parameter-dependent reflection…
We study indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with cyclic permutation groups (cocyclic solutions). In particular, we show that there is no one-to-one correspondence between indecomposable cocyclic…
We improve Algorithm 5.1 of [Math. Comp. {\bf 86} (2017), 2519-2534] for computing all non-isomorphic skew left braces, and enumerate left braces and skew left braces of orders up to 868 with some exceptions. Using the enumerated data, we…
Combining the notions of braces and relative Rota-Baxter operators on groups in connection with the Yang-Baxter equation and a factorization theorem of Lie groups from integrable systems, relative Rota-Baxter operators on braces and…
We generalize Nichita, Popovici and Tanasa solutions of the Braid equation to quasi-Yang-Baxter equation. We define quasi-braided Lie algebras in an additive monoidal category as a natural generalization of Majid's braided Lie algebra…
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…
In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many…
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…
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…
In this paper we introduce a procedure that, given a solution to the Yang-Baxter equation as input, produces a stochastic (or Markovian) solution to (a possibly dynamical version of) the Yang-Baxter equation. We then apply this…
We have found some new solutions of both rational and trigonometric types by rewriting Yang-Baxter equation as a triple product equation in a vector space of matrices.
Nipotency of skew braces is related to certain types of solutions of the Yang-Baxter equation. This paper delves into the study of centrally nilpotent skew braces. In particular, we study their torsion theory (Section 4.1) and we introduce…
Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This…
We investigate two sub-classes of skew bracoids, the first consists of those we term almost a brace, meaning the multiplicative group decomposes as a certain semi-direct product, and then those that are almost classical, which additionally…
We present solutions for the (constant and spectral-parameter) Yang-Baxter equations and Yang-Baxter systems arising from algebra structures and discuss about their symmetries. In the last section, we present some applications.
Self-distributive (SD) structures form an important class of solutions to the Yang--Baxter equation, which underlie spectacular knot-theoretic applications of self-distributivity. It is less known that one go the other way round, and…