Representations of Two-parameter Quantum Orthogonal and Symplectic Groups

TL;DR

This paper explores the finite-dimensional representation theory of two-parameter quantum orthogonal and symplectic groups, extending known results and constructing key operators to prove complete reducibility.

Contribution

It extends representation theory results from type A to types B, C, D for two-parameter quantum groups, including constructing R-matrices and Casimir operators.

Findings

Complete reducibility theorem proven for types B, C, D

Construction of R-matrices and quantum Casimir operators

Extension of type A results to other classical types

Abstract

We investigate the finite-dimensional representation theory of two-parameter quantum orthogonal and symplectic groups that we found in [BGH] under the assumption that $rs^{-1}$ is not a root of unity and extend some results [BW1, BW2] obtained for type $A$ to types $B$, $C$ and $D$. We construct the corresponding $R$-matrices and the quantum Casimir operators, by which we prove that the complete reducibility Theorem also holds for the categories of finite-dimensional weight modules for types $B$, $C$, $D$.