# Lie algebra of first-order differential operators

From Diffgeom

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 be a differential manifold. Let be the algebra of infinitely differentiable functions on . The Lie algebra of first-order differential operators is defined as follows:

- As a set, it is the set of all maps from to , 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 -vector space structure is by pointwise addition and scalar multiplication.
- There is a natural -bimodule structure, by composition. In other words, acts on a first-order differential operator by:

where denotes multiplication by . Similarly, the right action is given by: