Simple Modules over Second Quantum Weyl Algebra

TL;DR

This paper investigates the structure of the second quantum Weyl algebra at roots of unity, explicitly determining its PI degree and classifying all simple modules, thus solving a key problem in noncommutative algebra.

Contribution

It provides the first complete classification of simple modules over the multiparameter second quantum Weyl algebra at roots of unity.

Findings

PI degree of the algebra is explicitly determined

Complete classification of simple modules is achieved

The results solve a known open problem in noncommutative algebra

Abstract

In this article, we study the multiparameter second quantum Weyl algebra at roots of unity. In this setting, the algebra is a polynomial identity (PI) algebra, and the dimension of its simple modules is bounded above by its PI degree. We explicitly determine the PI degree and provide a complete classification of simple modules. This classification offers a comprehensive solution to $\href{https://doi.org/10.1007/978-3-030-19486-4_23}{\text{Problem 2: C, Walton (2019)- An Invitation to Noncommutative Algebra}}$ for the second quantum Weyl algebra.