Related papers: Classification of eight-vertex solutions of the co…

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…

A unified approach is applied in the consideration of the multi-parametric (colored) Yang-Baxter equations (YBE) and the usual YBE with two-parametric R-matrices, relying on the existence of the arbitrary functions in the general solutions.…

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…

Most of the set-theoretical solutions of the Yang-Baxter equation studied in the past years were non-degenerate multipermutation solutions. For degenerate solutions, a correct definition of multipermutation solutions has not been…

In this note we study `a la Baxter [1] the possible integrable manifolds of the asymmetric eight-vertex model. As expected they occur when the Boltzmann weights are either symmetric or satisfy the free-fermion condition but our analysis…

We study involutive set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2. These solutions happen to fall into two classes -- distributive ones and non-distributive ones. The distributive ones can be effectively…

We introduce new methods to describe admissible states of the six-vertex and the eight-vertex lattice models of statistical mechanics. For the six-vertex model, we view the admissible states as differential forms on a grid graph. This…

We study all five-, six-, and one eight-vertex type two-state solutions of the Yang-Baxter equations in the form $A_{12} B_{13} C_{23} = C_{23} B_{13} A_{12}$, and analyze the interplay of the `gauge' and `inversion' symmetries of these…

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

In this paper we derive from arguments of string scattering a set of eight tetrahedron equations, with different index orderings. It is argued that this system of equations is the proper system that represents integrable structures in three…

Two types of Yang-Baxter systems play roles in the theoretical physics -- constant and colour dependent. The constant systems are used mainly for construction of special Hopf algebra while the colour or spectral dependent for construction…

We classify all regular solutions of the Yang-Baxter equation of eight-vertex type. Regular solutions correspond to spin chains with nearest-neighbour interactions. We find a total of four independent solutions. Two are related to the usual…

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…

The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang-Baxter equation. Use of the 2-dimensional representations recovers the six-vertex model solution. Solutions in arbitrary dimensions, which are…

In the classification of solutions of the Yang--Baxter equation, there are solutions that are not deformations of the trivial solution (essentially the identity). We consider the algebras defined by these solutions, and the corresponding…

For any algebra two families of coloured Yang-Baxter operators are constructed, thus producing solutions to the two-parameter quantum Yang-Baxter equation. An open problem about a system of functional equations is stated. The matrix forms…

Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set $X$ and a function r:X x X --> X x X which satisfies the braid relation. We…

Given an arbitrary choice of two sets of nonzero Boltzmann weights for $n$-color lattice models, we provide explicit algebraic conditions on these Boltzmann weights which guarantee a solution (i.e., a third set of weights) to the…

This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones among which we…

We study the Yang-Baxter equation for the $R$-matrices of the six-vertex model. We analyze the solutions and give new parametrizations of the Yang-Baxter equation. In particular, we find the maximal commutative families of parametrized…