Lie algebra of first-order differential operators

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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


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 \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)