Noetherian algebras of quantum differential operators

TL;DR

This paper proves that certain algebras of quantum differential operators are Noetherian and describes their structure, including their simplicity and domain properties, for polynomial and quantum n-space cases.

Contribution

It establishes the Noetherian property for quantum differential operator algebras and characterizes their algebraic structure in the quantum n-space context.

Findings

The algebra of quantum differential operators on a polynomial algebra is Noetherian.

The algebra of quantum differential operators on quantum n-space is a simple Noetherian domain.

The structure is described as a skew group algebra over a quantized Weyl algebra.

Abstract

We consider algebras of quantum differential operators, for appropriate bicharacters on a polynomial algebra in one indeterminate and for the coordinate algebra of quantum $n$-space for $n\geq 3$. In the former case a set of generators for the quantum differential operators was identified in work by the first author and T. C. McCune but it was not known whether the algebra is Noetherian. We answer this question affirmatively, setting it in a more general context involving the behaviour Noetherian condition under localization at the powers of a single element. In the latter case we determine the algebra of quantum differential operators as a skew group algebra of the group $\mathbb{Z}^n$ over a quantized Weyl algebra. It follows from this description that this algebra is a simple right and left Noetherian domain.