# Simple weight modules over the quantum Schr\"{o}dinger algebra

## TL;DR

This paper classifies all simple weight modules with finite-dimensional weight spaces over the quantum Schrödinger algebra when q is not a root of unity, revealing four distinct classes of modules.

## Contribution

It determines the center of the quantum Schrödinger algebra and classifies simple modules with finite-dimensional weight spaces, introducing four new classes of modules.

## Key findings

- Four classes of simple modules identified: dense, highest weight, lowest weight, and twisted modules.
- The center of the quantum Schrödinger algebra is explicitly determined.
- Classification applies when q is not a root of unity.

## Abstract

In the present paper, using the technique of localization, we determine the center of the quantum Schr\"{o}dinger algebra $\S_q$ and classify simple modules with finite-dimensional weight spaces over $\S_q$, when $q$ is not a root of unity. It turns out that there are four classes of such modules: dense $U_q(\mathfrak{sl}_2)$-modules, highest weight modules, lowest weight modules, and twisted modules of highest weight modules.