A multiparameter family of irreducible representations of the quantum plane and of the quantum Weyl algebra

TL;DR

This paper constructs a broad family of irreducible representations for the quantum plane and Weyl algebra over any field, clarifying conditions for isomorphism and weight properties when the deformation parameter is not a root of unity.

Contribution

It introduces a new multiparameter family of irreducible representations for quantum algebra structures, expanding understanding of their classification and properties.

Findings

Constructed irreducible representations over arbitrary fields.

Determined conditions for isomorphism between representations.

Identified when representations are weight representations.

Abstract

We construct a family of irreducible representations of the quantum plane and of the quantum Weyl algebra over an arbitrary field, assuming the deformation parameter is not a root of unity. We determine when two representations in this family are isomorphic, and when they are weight representations, in the sense of Bavula.