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arXiv:1011.2891 · Journal of Statistical Mechanics Theory and Experiment

On the \mathcal {U}_{q}[sl(2)] Temperley–Lieb reflection matrices

A. Lima-Santos

Published Jan 6, 2011• 7 citations
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Abstract

This work concerns the boundary integrability of the spin- Temperley–Lieb model. A systematic computation method is used to construct the solutions of the boundary Yang–Baxter equations. For s half-integer, a general 2 s ( s + 1) + 3/2 free parameter solution is presented. It turns that for s integer, the general solution has 2 s ( s + 1) + 1 free parameters. Moreover, some particular solutions are discussed.

DOI
10.1088/1742-5468/2011/01/p01009

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This work concerns the boundary integrability of the spin- Temperley–Lieb model. A systematic computation method is used to construct the solutions of the boundary Yang–Baxter equations. For s half-integer, a general 2 s ( s + 1) + 3/2 free parameter solution is presented. It turns that for s integer, the general solution has 2 s ( s + 1) + 1 free parameters. Moreover, some particular solutions are discussed.
This work concerns to the studies of boundary integrability of the vertex models from representations of the Temperley-Lieb algebra associated with the quantum group Uq[Xn]{\cal U}_{q}[X_{n}]Uq​[Xn​] for the affine Lie algebras XnX_{n}Xn​ = A1(1)A_{1}^{(1)}A1(1)​, Bn(1)B_{n}^{(1)}Bn(1)​, Cn(1)C_{n}^{(1)}Cn(1)​ and Dn(1)D_{n}^{(1)}Dn(1)​. A systematic computation method is used to constructed solutions of the boundary Yang-Baxter equations. We find a 2n2+12n^{2}+12n2+1 free parameter solution for $A_{1}^{(1)} $ spin-(n−1/2)(n-1/2)(n−1/2) and $ C_{n}^{(1)}$ vertex models. It turns that for $A_{1}^{(1)} $ spin-nnn, $ B_{n}^{(1)}$ and Dn(1)D_{n}^{(1)}Dn(1)​ vertex models, the solution has 2n2+2n+12n^{2}+2n+12n2+2n+1 free parameters.
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.
The Yang-Baxter equation appear in various situations in physics and mathematics. For example it arises as a consistency condition in integrable models. The reflection equation (boundary Yang-Baxter equation) is a generalization of the Yang-Baxter equation to systems with a boundary. A further generalization to systems with defects which admits both reflection and transmission can be made, which results in reflection-transmission Yang-Baxter equations.In this thesis the Yang-Baxter equation and the reflection equation are presented. Representations of the Temperley-Lieb algebra and the blob algebra are used to construct matrices which solve the respective equations. For the reflection-transmission Yang-Baxter equations, steps toward a solution are taken by using a similar approach as for the first two cases, namely by finding an algebra whose representations can be used to construct matrices which solve the equations.