Deformed Dolan-Grady relations in quantum integrable models

A new hidden symmetry is exhibited in the reflection equation and related quantum integrable models. It is generated by a dual pair of operators {A,A∗}∈A subject to q-deformed Dolan-Grady relations. Using the inverse scattering method, a new family of quantum integrable models is proposed. In the simplest case, the Hamiltonian is linear in the fundamental generators of A. For general values of q, the corresponding spectral problem is quasi-exactly solvable. Several examples of two-dimensional massive/massless (boundary) integrable models are reconsidered in light of this approach, for which the fundamental generators of A are constructed explicitly and exact results are obtained. In particular, we exhibit a dynamical Askey-Wilson symmetry algebra in the (boundary) sine-Gordon model and show that asymptotic (boundary) states can be expressed in terms of q-orthogonal polynomials.