Lie algebra of global derivations

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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 \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