TITLE: Is there an infinite dimensional Lie group associated to the Lie algebra of all vector fields on a manifold?
QUESTION [12 upvotes]: Since the space $\Gamma(TM)$ of all vector fields on a smooth manifold $M$ is a real Lie algebra with respect to the usual commutator bracket, I was curious if in fact it is the Lie algebra of some infinite dimensional Lie group? Would it be the Lie algebra of the automorphism group $\text{Aut}(M)$, if the latter group can be realized as some appropriate infinite dimensional manifold?
More generally, can any infinite dimensional Lie algebra be realized as the Lie algebra of an infinite dimensional Lie group? If not, can this be "corrected" by considering a larger category of generalized smooth spaces, such as diffeological spaces, or Frolicher spaces?
I personally know almost nothing about infinite dimensional Lie groups and manifolds; I have only studied the finite-dimensional theory. This is just something I was curious about.

REPLY [3 votes]: To add to Qiaochu Yuan's answer, I found out that in synthetic differential geometry (which admits a well-adapted model containing the category of smooth manifolds as a fully subcategory) it is easy to prove that the Lie algebra of vector fields is associated to the diffeomorphism group. Working in such a smooth topos, let $M$ be a space, $R$ the line object, and $D=\{d \in R : d^2 = 0 \}$ the "walking tangent vector". 
The tangent bundle is $M^D \to M$ where the projection is the evaluation at zero map. By currying (product-exponential adjunction), a vector field on $M$ is an infinitesimal transformation of the space; more precisely a map $X: D \to M^M$ such that $X(0)=1_M$. In other words, the vector fields $\mathfrak{X}(M)$ on $M$ coincide with the tangent space at the identity to the monoid $M^M$ of endomorphisms of $M$. Moreover, if $M$ is infinitesimally linear, then any vector field $X$ on $M$ satisfies $X(-d) \circ X(d) = 1_M$ and is thus invertible. So we have particular $T_1 (M^M) = T_1 \mathrm{Aut}(M)$.
So $\mathfrak{X}(M)$ is the Lie algebra of both the endomorphism monoid of $M$, and its diffeomorphism group (provided $M$ is infinitesimally linear).
If you're curious about SDG, Kock's excellent introductory text is available online for free.