On the ${\mathcal{U}}_q[sl(2)]$ Temperley-Lieb reflection matrices

This work concerns the boundary integrability of the spin-s ${\mathcal{U}}_q[sl(2)]$ 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 $2s(s+1)+3\mathrm{/}2$ 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.