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We study the exact solution of an $N$-state vertex model based on therepresentation of the $U_q[SU(2)]$ algebra at roots of unity with diagonal openboundaries. We find that the respective reflection equation provides us onegeneral class of diagonal $K$-matrices having one free-parameter. We determinethe eigenvalues of the double-row transfer matrix and the respective Betheansatz equation within the algebraic Bethe ansatz framework. The structure ofthe Bethe ansatz equation combine a pseudomomenta function depending on afree-parameter with scattering phase-shifts that are fixed by the roots ofunity and boundary variables.

The spectrum of an open vertex model based on the U_q[SU(2)] at roots of unity