Related papers: Abelian fixed point free endomorphisms and the Yan…
We show that the epimorphism problem is solvable for targets that are virtually cyclic or a product of an Abelian group and a finite group.
It is known that the local Yang--Baxter equation is a generator of potential solutions to Zamolodchikov's tetrahedron equation. In this paper, we show under which additional conditions the solutions to the local Yang--Baxter equation are…
Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such…
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.
We propose that the Yang-Baxter deformation of the symmetric space sigma-model parameterized by an r-matrix solving the homogeneous (classical) Yang-Baxter equation is equivalent to the non-abelian dual of the undeformed model with respect…
We use Constraint Satisfaction methods to enumerate and construct set-theoretic solutions to the Yang-Baxter equation of small size. We show that there are 321931 involutive solutions of size nine, 4895272 involutive solutions of size ten…
Let $G$ be a finite nonabelian group, and let $\psi:G\to G$ be a homomorphism with abelian image. We show how $\psi$ gives rise to two Hopf-Galois structures on a Galois extension $L/K$ with Galois group (isomorphic to) $G$; one of these…
Given a finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation and a field $K$, the structure $K$-algebra of $(X,r)$ is $A=A(K,X,r)=K\langle X\mid xy=uv \mbox{ whenever }r(x,y)=(u,v)\rangle$. Note that…
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…
To determine and analyze arbitrary left non-degenerate set-theoretic solutions of the Yang-Baxter equation (not necessarily bijective), we introduce an associative algebraic structure, called a YB-semitruss, that forms a subclass of the…
In this paper we consider families of multiparametric $R$-matrices to make a systematic study of the boundary Yang-Baxter equations in order to discuss the corresponding families of multiparametric $K$-matrices. Our results are indeed…
In this letter we present constant solutions to the tetrahedron equations proposed by Zamolodchikov. In general, from a given solution of the Yang-Baxter equation there are two ways to construct solutions to the tetrahedron equation. There…
We study noninvolutive set-theoretic solutions $(X,r)$ of the Yang-Baxter equations in terms of the properties of the canonically associated algebraic objects-the braided monoid $S(X,r)$, the quadratic Yang-Baxter algebra $A= A(\textbf{k},…
Starting from a quantum dilogarithm over a Pontryagin self-dual LCA group $A$, we construct an operator solution of the Yang-Baxter equation generalizing the solution of the Faddeev-Volkov model. Based on a specific choice of a subgroup…
We extend the cabling method by Lebed, Ram\'irez and Vendramin from involutive to bijective non-degenerate set-theoretic solutions of the Yang--Baxter equation by working in the Yang--Baxter monoid $M(X,r)$ rather than the group $G(X,r)$.…
Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set-theoretic solutions of the well-known Yang-Baxter equation. The…
We study direct products of free-abelian and free groups with special emphasis on algorithmic problems. After giving natural extensions of standard notions into that family, we find an explicit expression for an arbitrary endomorphism of…
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 completely determine the free parts of the set-theoretic Yang-Baxter (co)homology groups of finite cyclic biquandles, along with fully computing the torsion subgroups of their 1st and 2nd homology groups. Furthermore, we provide upper…
The notion of a geometric crystal was introduced by A.Berenstein and D.Kazhdan, motivated by the needs of representation theory of p-adic groups. It was shown by A.Braverman, A.Berenstein, and D.Kazhdan that some particular geometric…