Representations of D_q(k [x])

TL;DR

This paper classifies irreducible weight modules over the algebra of quantum differential operators on polynomial algebras, extending understanding of module structures in quantum algebra, especially in positive characteristic and roots of unity.

Contribution

It provides a classification of irreducible modules and constructs indecomposable modules in specific cases, advancing the representation theory of quantum differential operators.

Findings

Classified irreducible weight modules over quantum differential operator algebras.

Constructed indecomposable modules in positive characteristic and at roots of unity.

Extended module theory to new algebraic settings.

Abstract

The algebra of quantum differential operators on graded algebras was introduced by V. Lunts and A. Rosenberg. D. Jordan, T. McCune and the second author have identified this algebra of quantum differential operators on the polynomial algebra with coefficients in an algebraically closed field of characteristic zero. It contains the first Weyl algebra and the quantum Weyl algebra as its subalgebras. In this paper we classify irreducible weight modules over the algebra of quantum differential operators on the polynomial algebra. Some classes of indecomposable modules are constructed in the case of positive characteristic and q root of unity.