# Lie algebra of global derivations

From Diffgeom

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 -vector space under pointwise addition and scalar multiplication; more generally, it is a module over , the algebra of infinitely differentiable functions, under left multiplication
- It is a Lie algebra: the Lie bracket of two derivations and is given as: