Related papers: On the parametrization of solutions of the Yang--B…
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 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…
A connection between the Yang-Baxter relation for maps and the multi-dimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It…
In this article, a system of Yang-Baxter-type matrix equations is studied, $XAX=BXB$, $XBX=AXA$, which "generalizes" the matrix Yang-Baxter equation and exhibits a broken symmetry. We investigate the solutions of this system from various…
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…
In this article, we study unitary rational solutions of the associative Yang-Baxter equation with three spectral parameters. We explain how such solutions arise from the geometry of vector bundles on a cuspidal cubic curve. Moreover, we…
In this paper all eight-vertex type solutions of the colored Yang-Baxter equation dependent on spectral as well as color parameter are given. It is proved that they are composed of three groups of basic solutions, three groups of their…
We present most general one-parametric solutions of the Yang-Baxter equations (YBE) for one spectral parameter dependent $R_{ij}(u)$-matrices of the six- and eight-vertex models, where the only constraint is the particle number conservation…
We obtain two series of spectral parameter dependent solutions to the generalized Yang-Baxter equations (GYBE), for definite types of $N_1^2\times N_2^2$ matrices with general dimensions $N_1$ and $N_2$. Appropriate extensions are presented…
The quantum Yang-Baxter equation admits generalisations to systems of Yang-Baxter type equations called Yang-Baxter systems. Starting from algebra structures, we propose new constructions of some constant as well as the spectral-parameter…
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…
Yang-Baxter system related to quantum doubles is introduced and large class of both continuous and discrete symmetries of the solution manifold are determined. Strategy for solution of the system based on the symmetries is suggested and…
In this paper we investigate the algebraic geometric nature of a solution of the Yang-Baxter equation based on the quantum deformation of the centrally extended $sl(2|2)$ superalgebra proposed by Beisert and Koroteev \cite{BEKO}. We derive…
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.
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…
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…
Several years ago, it was proposed that the usual solutions of the Yang-Baxter equation associated to Lie groups can be deduced in a systematic way from four-dimensional gauge theory. In the present paper, we extend this picture, fill in…
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…
We consider Yang-Baxter equations arising from its associative analog and study corresponding exchange relations. They generate finite-dimensional quantum algebras which have form of coupled ${\rm GL}(N)$ Sklyanin elliptic algebras. Then we…
Let $A$ be a $2\times 2$ matrix over a finite field and consider the Yang-Baxter matrix equation $XAX=AXA$ with respect to $A$. We use a method of computational ideal theory to explore the geometric structure of the affine variety of all…