Representations of twisted q-Yangians

Abstract

The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with $${\mathfrak{gl}_N}$$ . We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras $${\mathfrak{sp}_{2n}}$$ . The representations are parameterized by their highest weights satisfying certain dominance-type conditions. In the simplest case of $${\mathfrak{sp}_2}$$ , we give an explicit description of all the representations as tensor products of evaluation modules. We give new proofs of the (well-known) Poincaré–Birkhoff–Witt theorem for the quantum affine algebra and for the twisted q-Yangians in their RTT-presentations. We also reproduce Tarasov’s proof of the classification theorem for finite-dimensional irreducible representations of the quantum affine algebra by relying on its R-matrix presentation.