Deformations of the Exterior Algebra of Differential Forms

TL;DR

This paper investigates how to deform the module structure of the exterior algebra of differential forms so that certain differential operators become module homomorphisms, enabling algebraic analysis of these operators.

Contribution

It provides conditions for deforming the module structure of the exterior algebra to make specific differential operators $S$-linear, and classifies which operators admit such deformations.

Findings

Identified conditions for module structure deformation.

Classified differential operators allowing deformations.

Provided examples illustrating the deformations.

Abstract

Let $D:\Omega\xrightarrow{}\Omega$ be a differential operator defined in the exterior algebra $\Omega$ of differential forms over the polynomial ring $S$ in $n$ variables. In this work we give conditions for deforming the module structure of $\Omega$ over $S$ induced by the differential operator $D$, in order to make $D$ an $S$-linear morphism while leaving the $\mathbb{C}$-vector space structure of $\Omega$ unchanged. One can then apply the usual algebraic tools to study differential operators: finding generators of the kernel and image, computing a Hilbert polynomial of these modules, etc. Taking differential operators arising from a distinguished family of derivations, we are able to classify which of them allow such deformations on $\Omega$. Finally we give examples of differential operators and the deformations that they induce.