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

Abstract

Let $\mathfrak{g}$ be a complex simple finite-dimensional Lie algebra and $U_q(\hat{\mathfrak{g}})$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q(\hat{\mathfrak{g}})$-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 $U_q(\mathfrak{k})\subset U_q(\hat{\mathfrak{g}})$. 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 $U_q(\hat{\mathfrak{g}})$-module is generically irreducible under restriction to $U_q(\mathfrak{k})$. In the case of small modules and Kirillov–Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations.