Sheaf of differential operators

This article defines a basic construct that makes sense on any differential manifold
View a complete list of basic constructs on differential manifolds

Definition

The sheaf of differential operators of a differential manifold can be defined in many ways:

• It is the sheaf-theoretic algebra of differential operators for the sheaf of differentiable functions (in other words, if one views the sheaf of differentiable functions abstractly as a sheaf of $\R$-algebras, then the sheaf of differential operators is the sheaf-theoretic analogues of its algebra of differential operators)
• It associates to every open set, the algebra of differential operators of the $\R$-algebra of differentiable functions on that open set. Restriction maps are defined in the usual fashion.
• It is the sheaf generated, on every open set, by the algebra of differentiable functions, and the derivations

Construction and structure

Construction starting from derivations and multiplications

For simplicity, we restrict to the case where the open subset is the whole manifold $M$. A similar argument works for every open subset of the manifold.

We start with the algebra of infinitely differentiable functions of the manifold $M$. This is a commutative $\R$-algebra, and is denoted as $C^\infty(M)$.

We now look at the set of all derivations from $C^\infty(M)$ to itself. This is not an algebra, since the composite of two derivations need not be a derivation. It, however, turns out to be a Lie algebra, and is precisely the Lie algebra of vector fields on $M$. This is a left $C^\infty(M)$-module: post-composing a derivation with a multiplication gives a derivation. In other words:

If $f$ is a function and $D$ is a derivation, then $fD$, defined as:

$g \mapsto (f)(Dg)$

is also a derivation.

However, composing on the right with a function multiplication does not yield a derivation:

$g \mapsto D(fg)$

is not a derivation.

Instead of looking at the derivations alone, it makes sense to look at the $C^\infty(M)$-module comprising those operators that can be obtained as the sum of a derivation and a multiplication. In other words, maps of the form:

$g \mapsto Dg + fg$

It turns out that this is a $C^\infty(M)$-bimodule, and is termed the module of first-order differential operators (the corresponding sheaf is termed the sheaf of first-order differential operators).

The first-order differential operators are not closed under composition. In other words, a composite of first-order differential operators is not necessarily a first-order differential operator. The algebra generated by first-order differential operators under composition, is precisely the algebra of differential operators.

Natural filtration

The algebra of differential operators on $M$ has a natural filtration, in terms of the order of a differential operator. A differential operators is said to have order $\le r$ if it can be expressed as a sum of differential operators, each of which is a composite of at most $r$ first-order differential operators.

There is no natural gradation to the algebra of differential operators, because there is no way to separate a pure order $r$ component of a differential operator.