Clas Löfwall: Rings of differential operators

Abstract.

We begin with assuming that R is a commutative k-algebra and give the definition of the ring of differential operators on R. We study the Weyl algebra, which is the ring of differential operators on the polynomial algebra \(k[x_1,\ldots,x_n]\) and mention some results of how the Weyl algebra can be used to compute the ring of differential operators on related algebras. Following an article by Lunts and Rosenberg, we modify the definition to cover also non-commutative algebras. In fact, they define what they call "the differential part" of any R-bimodule, which then is applied to the bimodule of $k$-endomorphisms of R to obtain the ring of differential operators on R. They also consider the graded case, which I rather will call the "super" case, since it is based on the "super" bracket \([a,b]=ab-(-1)^{|a||b|}ba\). They prove that the ring of differential operators on the exterior algebra (considered as a super algebra) is analogous to the Weyl algebra. We will study the simplest case, the exterior algebra on two generators and compare the ring of differential operators in the two cases, the super and the non-super case.