Representations of twisted q-Yangians

TL;DR

This paper classifies finite-dimensional irreducible representations of twisted q-Yangians related to symplectic Lie algebras, providing explicit descriptions and analogues of classical theorems.

Contribution

It establishes a classification theorem for these representations, parameterized by highest weights or Drinfeld polynomials, and extends PBW theorems to this setting.

Findings

Classified finite-dimensional irreducible representations of twisted q-Yangians.

Provided explicit descriptions for sp(2) case as tensor products of evaluation modules.

Extended PBW theorems and reproduced classification proofs using R-matrix presentation.

Abstract

The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras sp(2n). The representations are parameterized by their highest weights or by their Drinfeld polynomials. In the simplest case of sp(2) we give an explicit description of all the representations as tensor products of evaluation modules. We prove analogues of the Poincare-Birkhoff-Witt theorem for the quantum affine algebra and for the twisted q-Yangians. We also reproduce a proof of the classification theorem for finite-dimensional irreducible representations of the quantum affine algebra by relying on its R-matrix presentation.