Planar Prop of Differential Operators

TL;DR

This paper introduces a new framework for differential operators on associative algebras using Hochschild cohomology, revealing their structure as a planar prop with multiple inputs and outputs.

Contribution

It defines the differential operators as a Hochschild cohomology-based planar prop and establishes a connection with automorphisms of trivial associative deformations.

Findings

D(A) has a planar prop structure.

Surjective symbol map for formally smooth algebras.

Constructs a map from automorphisms to differential operators.

Abstract

We propose a definition of differential operators of an associative algebra $A$ in the spirit of Hochschild cohomology. Specifically we define $D(A)$ as the zero cohomology of a certain bicomplex formed by Hom-spaces $\mathrm{Hom}(A^{\otimes q}, A^{\otimes p})$. We show that it has a structure of a planar prop, i.e. each differential operator has multiple inputs and outputs and they can be composed along planar graphs. Furthermore, for a formally smooth algebra we have the surjective symbol map from $D(A)$ to the space of poly-derivations. We also consider another planar prop $E(A)$ generated by automorphisms of the trivial associative deformation of $A$ over the completion of a free associative algebra. We construct a natural map from $E(A)$ to $D(A)$ and identify its image.