On the ${\cal{U}}_{q}[sl(2)]$ Temperley-Lieb reflection matrices

TL;DR

This paper systematically constructs boundary solutions for the spin-s ${ m U}_q[sl(2)]$ Temperley-Lieb model, revealing parameter-rich solutions for both half-integer and integer spins, advancing understanding of boundary integrability.

Contribution

It introduces a systematic method to derive boundary Yang-Baxter solutions for the ${ m U}_q[sl(2)]$ Temperley-Lieb model, providing explicit solutions with many free parameters for different spin cases.

Findings

General solutions with many free parameters for half-integer spins.

Explicit solutions with fewer parameters for integer spins.

Discussion of particular boundary solutions.

Abstract

This work concerns the boundary integrability of the spin-s ${\cal{U}}_{q}[sl(2)]$ Temperley-Lieb model. A systematic computation method is used to constructed the solutions of the boundary Yang-Baxter equations. For $s$ half-integer, a general $2s(s+1)+3/2$ free parameter solution is presented. It turns that for $s$ integer, the general solution has $2s(s+1)+1$ free parameters. Moreover, some particular solutions are discussed.