In this chapter we show that the Weyl algebras are members of the family of rings of differential operators. These rings come up in many areas of mathematics: representation theory of Lie algebras, singularity theory and differential equations are some of them.

DEFINITIONS.

Let R be a commutative K-algebra. The ring of differential operators of R is defined, inductively, as a subring of EndK(R). As in the case of the Weyl algebra, we will identify an element a ∈ R with the operator of EndK(R) defined by the rule r ↦ ar, for every r ∈ R.

We now define, inductively, the order of an operator. An operator P ∈ EndK(R) has order zero if [a, P] = 0, for every a ∈ R. Suppose we have defined operators of order < n. An operator P ∈ EndK(R) has order n if it does not have order less than n and [a, P] has order less than n for every a ∈ R. Let Dn(R) denote the set of all operators of EndK(R) of order ≤ n. It is easy to check, from the definitions, that Dn(R) is a K-vector space.

We may characterize the operators of order ≤ 1 in terms of well-known concepts. A derivation of the K-algebra R is a linear operator D of which satisfies Leibniz's rule: D(ab) = aD(b) + bD(a) for every a,b ∈ R. Let DerK(R) denote the K-vector space of all derivations of R. Of course DerK(R) ⊆ EndK(R).