Infinitesimal generators of actions Is there a method to obtain an action of an infinite dimensional Lie group starting with its infinitesimal generator  ? 
I'm interested about actions of G on itself . And I was wondering if I can generate actions of G on itself if I know the infinitesimal generator of the action . 
Maybe someone knows some books about this topic. I know the book of P. Olver but I am interested in the general case or the infinite dimensional case.
 A: Given a Lie group acting on a manifold $M$, an infinitesimal generator is a vector field on $M$ that is induced by a one-parameter subgroup of $G$.  Since there is a correspondence between one-parameter subgroups and elements of the Lie algebra $\mathfrak{g}$, we can view the infinitesimal generators of $G$ as being in correspondence with vectors in $\mathfrak{g}$.  In this sense, I think that a different question to ask is if one can create a Lie group from an infinite dimensional Lie algebra.
I'm not sure the answer to this question in the infinite dimensional case, but in the finite dimensional case, this is certainly possible through the exponential map.  It can be shown that there exists a unique simply connected Lie group corresponding to a given (finite dimensional Lie algebra.)  This isn't to say that other Lie groups don't have the same Lie algebra.  On the contrary, the fact that there are generally multiple Lie groups with the same Lie algebra makes the subject rich with interesting mathematics.
