Generalized q-Onsager Algebras and Dynamical K-matrices

A procedure to construct $K$-matrices from the generalized $q$-Onsager algebra ${\text{cO}}_q(\widehat{g})$ 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 ${\text{cO}}_q(a_2^{(2)})$ and ${\text{cO}}_q(a_2^{(1)})$ are treated in details.