Deformations of the exterior algebra of differential forms

Abstract

Let D : Ω → Ω be a differential operator defined in the exterior algebra Ω of differential forms over the polynomial ring S in n variables. In this work we give conditions for deforming the module structure of Ω over S induced by the differential operator D, in order to make D an S-linear morphism while leaving the C-vector space structure of Ω 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 Ω. Finally we give examples of differential operators and the deformations that they induce.