Quantum groups, q-Boson algebras and quantized Weyl algebras

TL;DR

This paper presents a unified framework for quantum groups, q-Boson algebras, and quantized Weyl algebras, providing new insights into their structure and representations.

Contribution

It introduces a unified construction method and an action of quantum groups on quantized Weyl algebras, offering a conceptual proof of semi-simplicity and classification of simple objects.

Findings

Unified construction of quantum algebra structures

Proof of semi-simplicity of category $\\mathcal{O}(B_q)$

Classification of simple objects in the category

Abstract

We give a unified construction of quantum groups, q-Boson algebras and quantized Weyl algebras and an action of quantum groups on quantized Weyl algebras. This enables us to give a conceptual proof of the semi-simplicity of the category $\mathcal{O}(B_q)$ introduced by T.Nakashima and the classification of all simple objects in it.