Lie algebra of global derivations

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 derivations

Definition

The Lie algebra of global derivations of a differential manifold is defined as follows:

• As a set, it is the set of all vector fields, or derivations, defined on the whole manifold
• It has the structure of a $\mathbb {R}$-vector space under pointwise addition and scalar multiplication; more generally, it is a module over $C^{\infty }(M)$, the algebra of infinitely differentiable functions, under left multiplication
• It is a Lie algebra: the Lie bracket of two derivations $D_{1}$ and $D_{2}$ is given as:

$D_{1}\circ D_{2}-D_{2}\circ D_{1}$