Temperley-Lieb K-matrices

TL;DR

This paper investigates boundary integrability of vertex models linked to the Temperley-Lieb algebra and quantum groups, providing systematic solutions to boundary Yang-Baxter equations with explicit free parameter counts.

Contribution

It introduces a systematic method to construct solutions to boundary Yang-Baxter equations for various affine Lie algebra-based vertex models, revealing the number of free parameters in these solutions.

Findings

Found a 2n^2+1 free parameter solution for A_1^{(1)} and C_n^{(1)} models.

Discovered a 2n^2+2n+1 free parameter solution for A_1^{(1)} spin-n, B_n^{(1)}, and D_n^{(1)} models.

Provided explicit solutions for boundary integrability in models associated with specific affine Lie algebras.

Abstract

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 ${\cal U}_{q}[X_{n}]$ for the affine Lie algebras $X_{n}$ = $A_{1}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$ and $D_{n}^{(1)}$. A systematic computation method is used to constructed solutions of the boundary Yang-Baxter equations. We find a $2n^{2}+1$ free parameter solution for $A_{1}^{(1)} $ spin-$(n-1/2)$ and $ C_{n}^{(1)}$ vertex models. It turns that for $A_{1}^{(1)} $ spin-$n$, $ B_{n}^{(1)}$ and $D_{n}^{(1)}$ vertex models, the solution has $2n^{2}+2n+1$ free parameters.