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 $\{\mathsf{\text{A}},{\mathsf{\text{A}}}^{\ast }\}\in \mathcal{A}$ 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 $\mathcal{A}$. 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 $\mathcal{A}$ 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.