Generic Gelfand-Tsetlin Representations of $U_q^{\text{tw}}(\mathfrak{so}_3)$ and $U_q^{\text{tw}}(\mathfrak{so}_4)$

TL;DR

This paper constructs and analyzes infinite-dimensional Gelfand-Tsetlin representations for certain twisted quantum groups, extending finite-dimensional models and providing conditions for their irreducibility and structure.

Contribution

It introduces generic Gelfand-Tsetlin representations for $U_q^{tw}(so_3)$ and $U_q^{tw}(so_4)$, generalizing prior finite-dimensional and classical constructions.

Findings

Constructed infinite-dimensional representations for the quantum groups.

Provided sufficient conditions for irreducibility of these representations.

Established an upper bound for the length of the representations using Casimir elements.

Abstract

We construct generic Gelfand-Tsetlin representations of the $\imath$quantum groups $U_q^{\text{tw}}(\mathfrak{so}_3)$ and $U_q^{\text{tw}}(\mathfrak{so}_4)$. These representations are infinite-dimensional analogs to the finite-dimensional irreducible representations provided by Gavrilik and Klimyk. They are quantum analogs of generic Gelfand-Tsetlin representations constructed by Mazorchuk. We give sufficient conditions for irreducibility and provide an upper bound for the length with the help of Casimir elements found by Molev, Ragoucy, and Sorba.