Related papers: On structure groups of set-theoretic solutions to …
Skew braces play a central role in the theory of set-theoretic non-degenerate solutions of the Yang--Baxter equation, since their algebraic properties significantly affect the behaviour of the corresponding solutions (see for example…
Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $\epsilon>0$, describe the structure of $A$ and…
Quantum doubles of finite group algebras form a class of quasi-triangular Hopf algebras which algebraically solve the Yang--Baxter equation. Each representation of the quantum double then gives a matrix solution of the Yang--Baxter…
The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the $R$ matrix…
Computational methods are an important tool for solving the Yang-Baxter equations(in small dimensions), for classifying (unifying) structures, and for solving related problems. This paper is an account of some of the latest developments on…
We establish a bialgebra structure on Rota-Baxter Lie algebras following the Manin triple approach to Lie bialgebras. Explicitly, Rota-Baxter Lie bialgebras are characterized by generalizing matched pairs of Lie algebras and Manin triples…
A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space…
Let $M(A,I)$ be a free partially commutative monoid with involution and $G(A,I)$ be its quotient group, e.g. a right-angled Artin or Coxeter group. Given a system of word equations over $M(A,I)$ with recognizable constraints with input size…
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…
The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. The first step towards this objective is the introduction of certain generalizations of the familiar shelves and racks…
Connections between set-theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic $R$-matrices are expressed as twists of known solutions. We then focus on reflection and…
We give a complete characterization of all indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level~2. In the first step we present a construction of some family of such solutions and in the second step we…
In this paper all seven-vertex type solutions of the coloured Yang-Baxter equation dependent on spectral as well as coloured parameters are given. It is proved that they are composed of five groups of basic solutions, two groups of their…
In this paper, we investigate the Nichols algebra $\mathfrak{B}(W_{X,r})$ associated to any non-degenerate involutive solution $(X, r)$ of the Yang-Baxter equation. Infinite examples of finite dimensional Nichols algebras are obtained,…
We introduce a new variety of set-theoretic non-associative algebras, P{\l}onka bi-magmas, to describe and classify all solutions of the set-theoretic Yang-Baxter (YB) equation of Baaj-Long-Skandalis (BLS) type. We also study new classes of…
We construct invertible spectral parameter dependent Yang-Baxter solutions ($R$-matrices) by Baxterizing constant non-invertible Yang-Baxter solutions. The solutions are algebraic (representation independent). They are constructed using…
In this paper, the relations between the Yang-Baxter equation and affine actions are explored in detail. In particular, we classify solutions of the Yang-Baxter equations in two ways: (i) by their associated affine actions of their…
R-matrices are the solutions of the Yang-Baxter equation. At the origin of the quantum group theory, they may be interpreted as intertwining operators. Recent advances have been made independently in different directions. Maulik-Okounkov…
In present paper we propose an approach based on examination of the structure of the general solution of equations of the type dy/dx=P(x,y)/Q(x,y), with P and Q polynomials only in y. Under the term structure we mean the dependency…
An augmented rack is a set with a self-distributive binary operation induced by a group action, and has been extensively used in knot theory. Solutions to the Yang-Baxter equation (YBE) have been also used for knots, since the discovery of…