2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143503In this review, I describe a recent approach based on the representation theory of the $q-$Onsager algebra which is used to derive exact results for the XXZ open spin chain. The complete spectrum and eigenstates are obtained as rational functions of a single variable which discrete values correspond to the roots of a certain characteristic polynomial. Comments and open problems are also presented.High Energy Physics - TheoryMathematical PhysicsQuantum Algebratext