Noncommutative differential operators, Sobolev spaces and the centre of a category

TL;DR

This paper develops a framework for noncommutative differential operators and Sobolev spaces on modules with connection, revealing their algebraic structure and centrality within a categorical context.

Contribution

It introduces noncommutative differential operators, defines Sobolev spaces for modules with connection, and shows the tensor algebra of vector fields forms a central algebra in the bimodule connection category.

Findings

Differential operators form a unital associative algebra over a noncommutative algebra.

Noncommutative Sobolev spaces are defined for modules with connection and Hermitian structure.

Tensor algebra of vector fields lies in the centre of the bimodule connection category.

Abstract

We consider differential operators over a noncommutative algebra $A$ generated by vector fields. These are shown to form a unital associative algebra of differential operators, and act on $A$-modules $E$ with covariant derivative. We use the repeated differentials given in the paper to give a definition of noncommutative Sobolev space for modules with connection and Hermitian inner product. The tensor algebra of vector fields, with a modified bimodule structure and a bimodule connection, is shown to lie in the centre of the bimodule connection category ${}_A\mathcal{E}_A$, and in fact to be an algebra in the centre. The crossing natural transformation in the definition of the centre of the category is related to the action of the differential operators on bimodules with connection.