Generalized q-Onsager Algebras and Dynamical K-matrices

Abstract

A procedure to construct \(K\)-matrices from the generalized \(q\)-Onsager algebra \(\cO_{q}(\hat{g})\) is proposed. This procedure extends the intertwiner techniques used to obtain scalar (c-number) solutions of the reflection equation to dynamical (non-c-number) solutions. It shows the relation between soliton non-preserving reflection equations or twisted reflection equations and the generalized \(q\)-Onsager algebras. These dynamical \(K\)-matrices are important to quantum integrable models with extra degrees of freedom located at the boundaries: for instance, in the quantum affine Toda field theories on the half-line they yield the boundary amplitudes. As examples, the cases of \(\cO_{q}(a^{(2)}_{2})\) and \(\cO_{q}(a^{(1)}_{2})\) are treated in details.