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
- 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: