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.
Best Answer
Morally speaking, the Lie algebra of vector fields is the Lie algebra of $\text{Diff}(M)$, the diffeomorphism group of $M$. The relationship between these is less tight than in the finite-dimensional case: for example,
It is also not true that infinite-dimensional Lie algebras are Lie algebras of infinite-dimensional Lie groups in general (see, for example, this MO question), and I have the impression that this is a genuine phenomenon and not just an artifact of working with too restrictive of a notion of infinite-dimensional Lie group, although I don't know much about this.