It is sometimes demanded that a Hamiltonian group action $G \times M \to M$ allow for a Lie algebra homomorphism from $\mathfrak{g}$ to $C^\infty(M)$ with the Poisson bracket.

Is there a natural theory of infinity-dimensional Lie groups for which (a) a subgroup $H$ of the symplectomorphism group of $M$ has Lie algebra given by $C^\infty(M)$, or $C^\infty(M)$ modulo $H^0(M)$, and (b) any map of Lie algebras exponentiates to one of Lie groups (perhaps with the condition that the domain Lie group be simply connected)?

For (a), one might imagine the group of diffeomorphisms generated by time-1 flows of Hamiltonian vector fields, closed under composition and perhaps also under a natural norm. I do not know if there is a sense in which this generates the usual group of Hamiltonian symplecomorphisms.

For (b), I am woefully underequipped.

The motivation for this question is to confirm a philosophical reason for the definition of Hamiltonian action, or to simplify the definition. For if the answer to both (a) and (b) is affirmative, a "Lie group map" $G \to H$ would result in a linear map $\mathfrak{g} \to C^\infty(M) = C^\infty(M, R)$, and hence by smooth adjointness a smooth map $M \to hom(\mathfrak{g},R) = \mathfrak{g}^\vee$ satisfying the requirements of being a(n equivariant) momentum map.