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arXiv:1210.8449 · Journal of Statistical Mechanics Theory and Experiment

On the multiparametric ${\mathcal{U}}_{q}[{D}_{n+1}^{(2)}]$ vertex model

Ricardo S Vieira, A Lima-Santos

Published Feb 5, 2013• 5 citations
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Abstract

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 a non-trivial generalization of the K -matrix solutions of the vertex model when distinct reflections and extra free parameters are admissible.

DOI
10.1088/1742-5468/2013/02/p02011

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On the multiparametric Uq[Dn+1(2)]{\mathcal{U}}_{q}[{D}_{n+1}^{(2)}]Uq​[Dn+1(2)​] vertex model
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 a non-trivial generalization of the K -matrix solutions of the vertex model when distinct reflections and extra free parameters are admissible.
In this work, we employ the algebraic-differential method recently developed by the author to solve the Yang-Baxter equation for arbitrary fifteen-vertex models satisfying the ice-rule. We show that there are four different families of such regular RRR matrices containing several free-parameters. The corresponding reflection KKK matrices, solutions of the boundary Yang-Baxter equation, were also found and classified. We found that there are three different families of regular KKK matrices, regardless of what RRR matrix we choose.
This work concerns studies of boundary integrability of the vertex models from representations of the Temperley–Lieb algebra associated with the quantum group 𝒰 q [ X n ] for the affine Lie algebras X n = , , and . A systematic computation method is used to construct solutions of the boundary Yang–Baxter equations. We find a 2 n 2 + 1 free parameter solution for spin-( n − 1/2) and vertex models. It turns out that for spin- n , and vertex models, the solution has 2 n 2 + 2 n + 1 free parameters.
Abstract 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 elements, however, can be regarded as independent variables and eliminated from the systems, after which, two systems of polynomial equations are obtained in their place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nevertheless, the remaining unknowns can be found by solving a few simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work, we consider the Yang–Baxter equation for ( n + 1) (2 n + 1)-vertex models with a generalization based on the A n −1 symmetry. This differential approach allows us to solve the Yang–Baxter equation in a systematic way.