Hamiltonian Group Actions
The definition of the moment map requires to be exact. In practice it is useful to make an even stronger assumption. The G-action is said to be Hamiltonian if and only if the following conditions hold. First, for every ξ in the one-form is exact, meaning that it equals for some smooth function
If this holds, then one may choose the to make the map linear. The second requirement for the G-action to be Hamiltonian is that the map be a Lie algebra homomorphism from to the algebra of smooth functions on M under the Poisson bracket.
If the action of G on (M, ω) is Hamiltonian in this sense, then a moment map is a map such that writing defines a Lie algebra homomorphism satisfying . Here is the vector field of the Hamiltonian, defined by
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