Finite dimensional representations of $U_q(C(n+1))$ at arbitrary $q$

TL;DR

This paper develops a systematic method to construct finite dimensional irreducible representations of the quantum supergroup $U_q(C(n+1))$, showing their relation to classical Lie superalgebra representations and exploring special cases at roots of unity.

Contribution

It introduces a systematic construction method for irreps of $U_q(C(n+1))$ and characterizes their deformation from classical Lie superalgebra irreps, including at roots of unity.

Findings

Finite dimensional irreps are deformations of classical Lie superalgebra irreps.

At roots of unity, all irreps are finite dimensional with additional types like atypical and cyclic.

Explicit study of all irreps of $U_q(C(2))$ is provided.

Abstract

A method is developed to construct irreducible representations(irreps) of the quantum supergroup $U_q(C(n+1))$ in a systematic fashion. It is shown that every finite dimensional irrep of this quantum supergroup at generic $q$ is a deformation of a finite dimensional irrep of its underlying Lie superalgebra $C(n+1)$, and is essentially uniquely characterized by a highest weight. The character of the irrep is given. When $q$ is a root of unity, all irreps of $U_q(C(n+1))$ are finite dimensional; multiply atypical highest weight irreps and (semi)cyclic irreps also exist. As examples, all the highest weight and (semi)cyclic irreps of $U_q(C(2))$ are thoroughly studied.