Momentum Mappings

The momentum map is essentially due to Lie, [5], pp. 300–343. The modern notion is due to Kostant [3], Souriau [9], and Kirillov [2]. The setting for the moment mapping is a smooth symplectic manifold (M,ω) or even a Poisson manifold (M,P ) with the Poisson bracket on functions {f, g} = P (df, dg) (where P = ω : T M → TM is the Poisson tensor). To each function f there is the associated Hamiltonian vector field Hf = P (df) ∈ X(M,P ), where X(M,P ) is the Lie algebra of all locally Hamiltonian vector fields Y ∈ X(M) satisfying LY P = 0 for the Lie derivative. Let (M,ω) be a symplectic manifold for some time. Then this can be subsumed into the following exact sequence of Lie algebra homomorphisms