Representations of Two-Parameter Quantum Groups and Schur-Weyl Duality

TL;DR

This paper classifies simple modules for two-parameter quantum groups related to gl_n and sl_n, and establishes a Schur-Weyl duality analogue linking these quantum groups to Hecke algebras.

Contribution

It provides a complete classification of finite-dimensional simple modules and proves a Schur-Weyl duality analogue for two-parameter quantum groups.

Findings

Finite-dimensional simple modules classified

Schur-Weyl duality analogue established

Centralizer algebra is a quotient of a Hecke algebra

Abstract

We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n, and give a complete reducibility result. These quantum groups have a natural n-dimensional module V. We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the quantum group action on the k-fold tensor power of V is a quotient of a Hecke algebra for all n and is isomorphic to the Hecke algebra in case n\geq k.