Lie algebra of first-order 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

This article gives a global construction for a differential manifold. There exists a sheaf analog of it, that associates a similar construct to every open subset. This sheaf analog is termed: sheaf of first-order differential operators

Definition

Let $M$ be a differential manifold. Let $C^{\infty }(M)$ be the algebra of infinitely differentiable functions on $M$. The Lie algebra of first-order differential operators is defined as follows:

• As a set, it is the set of all maps from $C^{\infty }(M)$ to $C^{\infty }(M)$, that can be expressed as the sum of a derivation, and pointwise multiplication by a function. The derivation can be thought of as the pure first-order part, and the scalar multiplication can be thought of as the zeroth
• The $\mathbb {R}$-vector space structure is by pointwise addition and scalar multiplication.
• There is a natural $C^{\infty }(M)$-bimodule structure, by composition. In other words, $f\in C^{\infty }(M)$ acts on a first-order differential operator $d$ by:

$d\mapsto m(f)\circ d$

where $m(f)$ denotes multiplication by $f$. Similarly, the right action is given by:

$d\mapsto d\circ m(f)$