I'm trying to learn about moment maps in symplectic topology (suppose our Lie group is G with lie algebra g, acting on the symplectic manifold (M,w) by symplectomorphisms). I'm having a hard time, and I've realized this is because I don't have a good conceptual understanding of the lie bracket, either on the lie algebra g, or on the group of symplectomorphisms of (M,w), or on the space of functions C^infty(M,R). Therefore I can't "visualize" the Hamiltonian condition, which requires that the linear map g --> C^infty(M,R), which exists when the action by G is "exact," be a lie algebra homomorphism.

Please tell me how you personally understand/intuit/conceptualize this situation, both the lie bracket stuff and moment maps more generally! Any help is greatly appreciated.