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

We consider a complex simple finite dimensional Lie algebra and its associated quantum affine algebra. We show that every irreducible finite dimensional module over the quantum affine algebra naturally produces a family of trigonometric solutions of Cherednik’s generalized reflection equation. The resulting families depend on the choice of a quantum affine symmetric pair subalgebra inside the quantum affine algebra. The proof combines two main inputs. First, we use the existence of universal K matrices for arbitrary quantum symmetric pairs established in earlier work. Second, we prove that an irreducible finite dimensional module over the quantum affine algebra is, for generic parameters, still irreducible after restriction to the chosen symmetric pair subalgebra. Together, these ingredients allow us to pass from universal boundary data to concrete operator valued solutions of reflection type equations in every irreducible representation. As further applications, we specialize to small modules and to Kirillov Reshetikhin modules, where our construction yields new explicit solutions of both the standard and the transposed reflection equations.