Existence

Momentum maps — what are they? G (Lie group) acting on (M,ω) (symplectic manifold) ξ ∈ g ξM — symplectic vector field on M (so LξM ω = 0) Question: Is ξM Hamiltonian? A vector field X on M is Hamiltonian if ω(X,−) = dh for some h ∈ C∞(M) Answer: It depends.... (see later) Assume action is Hamiltonian: ξM is Hamiltonian for every ξ ∈ g. what are they... ctd For each ξ ∈ g we have a function hξ ∈ C∞(M) (unique up to constant). Dependence on ξ is linear (for suitable choices of constants) This gives a map — the momentum map J: M − → g∗ The defining equation is, for m ∈ M and v ∈ TmM, 〈dJm(v), ξ〉 = ω(ξM(m), v). Immediate consequence (the bifurcation lemma): Image(dJm) = g◦m ⊂ g∗... responsible for the famous polytope structure of the imag