Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras

Let g be a complex simple finite-dimensional Lie algebra and Uq({g}) the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional Uq({g}) -module gives rise to a family of trigonometric solutions of Cherednik’s generalized reflection equation. These depend on the choice of a quantum affine symmetric pair Uq(k)⊂ Uq({g}) . Our result relies on the construction of universal K-matrices for arbitrary quantum symmetric pairs we obtained in [Represent. Theory 26 , 764–824 (2022)] and the fact that every irreducible Uq({g}) -module is generically irreducible under restriction to Uq(k) . In the case of small modules and Kirillov–Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations.