When first defining (or, rather, recalling the definition of) Lie algebras I mentioned that the bracket makes each element of a Lie algebra act by derivations on itself. We can actually say a bit more about this.
First off, we need an algebra over a field . This doesn’t have to be associative, as our algebras commonly are; all we need is a bilinear map . In particular, Lie algebras count.
Now, a derivation of is firstly a linear map from back to itself. That is, , where this is the algebra of endomorphisms of as a vector space over , not the endomorphisms as an algebra. Instead of preserving the multiplication, we impose the condition that behave like the product rule:
It’s easy to see that the collection is a vector subspace, but I say that it’s actually a Lie subalgebra, when we equip the space of endomorphisms with the usual commutator bracket. That is, if and are two derivations, I say that their commutator is again a derivation.
This, we can check:
We’ve actually seen this before. We identified the vectors at a point on a manifold with the derivations of the (real) algebra of functions defined in a neighborhood of , so we need to take the commutator of two derivations to be sure of getting a new derivation back.
So now we can say that the mapping that sends to the endomorphism lands in because of the Jacobi identity. We call this mapping the “adjoint representation” of , and indeed it’s actually a homomorphism of Lie algebras. That is, . The endomorphism on the left-hand side sends to , while on the right-hand side we get . That these two are equal is yet another application of the Jacobi identity.
One last piece of nomenclature: derivations in the image of are called “inner”; all others are called “outer” derivations.