Irreducible representations of the quantum Weyl algebra at roots of unity given by matrices

TL;DR

This paper classifies all irreducible matrix solutions of the quantum Weyl algebra at roots of unity, including singular cases, completing previous work that focused only on nonsingular solutions.

Contribution

It extends prior classification results by including and characterizing singular irreducible matrix solutions of the quantum Weyl algebra at roots of unity.

Findings

Complete classification of irreducible matrix solutions, including singular cases.

Identification of all solutions up to equivalence.

Extension of previous results to a broader set of solutions.

Abstract

To describe the representation theory of the quantum Weyl algebra at an $l$th primitive root $\gamma$ of unity, Boyette, Leyk, Plunkett, Sipe, and Talley found all nonsingular irreducible matrix solutions to the equation $yx-\gamma xy=1$, assuming $yx\neq xy$. In this note, we complete their result by finding and classifying, up to equivalence, all irreducible matrix solutions $(X, Y)$, where $X$ is singular.