Related papers: Enumeration of set-theoretic solutions to the Yang…

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},…

In {\it Set-theoretical solutions to the quantum Yang-Baxter equation} (Duke Math. J. {\bf 100} (1999), 169--209), Etingof, Schedler and Soloviev introduced, for each non-degenerate involutive set-theoretical solution $(X,\sigma,\tau)$ of…

Racks and quandles are prominent set-theoretical solutions of the Yang-Baxter equation. We enumerate racks and quandles of orders $n\le 13$ up to isomorphism, improving upon the previously known results for $n\le 8$ and $n\le 9$,…

In this paper, we provide techniques to obtain left non-degenerate set-theoretic solutions of the Yang-Baxter equation, drawing on the class of right groups. To this end, we introduce the new algebraic structures of left $RG$-semibraces,…

In this paper, we initiate the study of the interplay between $k$-graphs and the Yang-Baxter equation. For this, we provide two very different perspectives. One one hand, we show that the set of all set-theoretic solutions of the…

We construct $R$-matrices (with a multidimensional spectral parameter) that include additive as well as non-additive parameters. They satisfy the colored Yang-Baxter equation. The solutions depend on a set of commuting operators. They…

We study the solutions of the Yang-Baxter equation associated to nineteen vertex models invariant by the parity-time symmetry from the perspective of algebraic geometry. We determine the form of the algebraic curves constraining the…

We establish a one-to-one correspondence between structure groups of non-degenerate, involutive and braided "set-theoretical" solutions of the quantum Yang-Baxter equation and Garside groups with a certain presentation. Moreover, we show…

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…

Explicit solutions of the quantum Yang-Baxter equation are given corresponding to the non-unitary solutions of the classical Yang-Baxter equation for sl(5).

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 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…

New set-theoretical solutions to the Yang-Baxter Relation are constructed. These solutions arise from the decompositions "in different order" of matrix polynomials and $\theta$-functions. We also construct a "local action of the symmetric…

It is shown how Yang-Baxter maps may be directly obtained from classical counterparts of the star-triangle relations and quantum Yang-Baxter equations. This is based on reinterpreting the latter equation and its solutions which are given in…

A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, ..., x_n \}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$ holds…

We present a systematic technique to construct solutions to the Yang-Baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a…

We establish a correspondence between the invariant subsets of a non-degenerate symmetric set-theoretical solution of the quantum Yang-Baxter equation and the parabolic subgroups of its structure group, equipped with its canonical Garside…

We find all solutions to the constant Yang--Baxter equation $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$ in three dimensions, subject to an additive charge-conservation ansatz. This ansatz is a generalisation of (strict) charge-conservation, for…

We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld…

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 $G(X,r)$.…