Moment map, component Hamiltonians and infinitesimal moment maps

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let a smooth left action of $G$ on a symplectic manifold $(M,\omega )$ be given, with fundamental fields ${\xi }_M$ as in Symplectic and Hamiltonian Lie-group actions. Let $\mu :M\to {\mathfrak{g}}^{\ast }$ be a smooth map to the dual ${\mathfrak{g}}^{\ast }=\mathcal{L}(\mathfrak{g},\mathbb{R})$ (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$). Its components are the smooth functions

${\mu }^{\xi }:M⟶\mathbb{R},{\mu }^{\xi }(p):=⟨\mu (p),\xi ⟩(\xi \in \mathfrak{g}),$

and they depend linearly on $\xi$, because evaluation of a fixed covector is linear. The map $\mu$ is an infinitesimal moment map for the action when the component moment equations

$d{\mu }^{\xi }=-{\iota }_{{\xi }_M}\omega \text{hold for every }\xi \in \mathfrak{g}$

hold; equivalently, when the map $\mathfrak{g}\to C^{\mathrm{\infty }}(M)$, $\xi \mapsto {\mu }^{\xi }$ is linear and each ${\mu }^{\xi }$ is a Hamiltonian function for the vector field $-{\xi }_M$. The infinitesimal moment map is an equivariant moment map when in addition

$\mu (g\cdot p)=g\cdot \mu (p)(g\in G,p\in M)$

for the coadjoint action $g\cdot \alpha =\alpha \circ {\mathrm{Ad}}_{g^{-1}}$ (The coadjoint representation, action and orbits), and a Hamiltonian action is a symplectic action that admits an equivariant moment map.

The distinction between the two notions is deliberate and is used by the nonequivariance lemma later on this page: an infinitesimal moment map is required only to satisfy the differential equations, while equivariance is an additional group-theoretic condition that can genuinely fail. The nonequivariance defect of an infinitesimal moment map is the alternating bilinear map of functions

$c(\xi ,\eta ):=\{{\mu }^{\xi },{\mu }^{\eta }\}-{\mu }^{[\xi ,\eta ]},\xi ,\eta \in \mathfrak{g},$

where $\{,\}$ is the Poisson bracket of the symplectic form. Equivariance of $\mu$ is not built into the definition of an infinitesimal moment map, and no item on this page assumes it unless it is stated. The countable-choice assumption is inherited from Symplectic and Hamiltonian Lie-group actions and is used only there; no choice is made here, and $\mu$ may be replaced by $\mu +c$ for any constant $c\in {\mathfrak{g}}^{\ast }$ without changing any component differential.