Differential and holomorphic differential operators on noncommutative algebras

TL;DR

This paper develops sheaves of differential operators on noncommutative algebras, exploring their algebraic structures, relations, and the extension to holomorphic operators, advancing the understanding of noncommutative geometry.

Contribution

It introduces a framework for sheaves of differential operators on noncommutative algebras, including zero curvature quotients and relations for Hopf algebras with non-bicovariant structures.

Findings

Sheaves of differential operators form the center of the bimodule category.

Symbols of differential operators are defined but not extensively studied.

Holomorphic differential operators are considered without zero curvature quotients.

Abstract

This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of as category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considered, though without the quotient to ensure zero curvature.