Given a symplectic manifold , there is the group of Hamiltonian symplectomorphisms acting on . If is prequantizable this lifts to the group of quantomorphisms, both of them covering the diffeomorphisms of :
quantomorphisms Hamiltonian symplectomorphisms diffeomorphisms .
A Hamiltonian action of a Lie group on is an action by quantomorphisms, hence a Lie group homomorphism
See (Brylinski, prop. 2.4.10).
In the literature this is usually discussed at the infinitesimal level, hence for the corresponding Lie algebras:
smooth functions+Poisson bracket Hamiltonian vector fields vector fields
Now an (infinitesimal) Hamiltonian action is a Lie algebra homomorphism
Dualizing, the homomorphism is equivalently a linear map
which is a homomorphism of Poisson manifolds. This is called the moment map of the (infinitesimal) Hamiltonian -action.
Warning The lift from to above, hence from the existence of Hamiltonians to an actual choice of Hamiltonians is in general indeed a choice. There may be different choices. In the literature the difference between and (or of their Lie theoretic analogs) is not always clearly made.
By (Atiyah-Bott), the action of a Lie algebra on a symplectic manifold is Hamiltonian if and only if the symplectic form has a (basic, closed) extension to equivariant de Rham cohomology.
A comprehensive account is in (see around section 2.1)
The perspective on Hamiltonian actions in terms of maps to extensions, infinitesimally and integrally, is made explicit in prop. 2.4.10 of
- Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser (1993)
The characterization in equivariant cohomology is due to
Generalization to Hamiltonian actions by a Lie algebroid (instead of just a Lie algebra) is discussed in
- Rogier Bos, Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids (arXiv:math.SG/0604027)