Symplectic and Hamiltonian Lie-group actions

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $G$ be a finite-dimensional real Lie group with Lie algebra $\mathfrak{g}=T_{e}G$, let $(M,\omega )$ be a symplectic manifold (Symplectic form and symplectic manifold), and let $G\times M\to M$, $(g,p)\mapsto g\cdot p$, be a smooth left action (Smooth left actions of Lie groups). For $\xi \in \mathfrak{g}$ let ${\xi }_M$ be its fundamental vector field in the library convention

${\xi }_M(p)={\frac{d}{dt}\mid }_0{\exp }_G(-t\xi )\cdot p(p\in M)$

(Fundamental vector fields for a left action). The action is symplectic when

$g^{\ast }\omega =\omega \text{for every }g\in G,$

that is, when every $p\mapsto g\cdot p$ is a symplectomorphism. It is Hamiltonian when it is symplectic and there is a smooth map

$\mu :M⟶{\mathfrak{g}}^{\ast }$

to the algebraic dual ${\mathfrak{g}}^{\ast }=\mathcal{L}(\mathfrak{g},\mathbb{R})$ (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$) such that

$d⟨\mu ,\xi ⟩=-{\iota }_{{\xi }_M}\omega \text{for every }\xi \in \mathfrak{g},$

and $\mu$ is equivariant for the given action on $M$ and the coadjoint action on ${\mathfrak{g}}^{\ast }$ (The coadjoint representation, action and orbits):

$\mu (g\cdot p)=g\cdot \mu (p)(g\in G,p\in M)\mathrm{.}$

Such a $\mu$ is an equivariant moment map for the action, and $(M,\omega ,G,\mu )$ is a Hamiltonian $G$-space.

Two conventions are load-bearing. First, the minus sign in the definition of ${\xi }_M$ enters through the exponential ${\exp }_G(-t\xi )$, not through the moment equation, and $d⟨\mu ,\xi ⟩=-{\iota }_{{\xi }_M}\omega$ is the identity used throughout this page. In the convention that generates $\xi$ by ${\exp }_G(t\xi )$ the same equation reads $d⟨\mu ,\xi ⟩={\iota }_{{\xi }^{\mathrm{\#}}}\omega$, so a source written that way is translated by ${\xi }^{\mathrm{\#}}=-{\xi }_M$ rather than by changing the sign of $\mu$. Second, the coadjoint action is the left action $g\cdot \alpha =\alpha \circ {\mathrm{Ad}}_{g^{-1}}$; with the opposite convention equivariance would be replaced by its inverse.

The map $\mu$ is required to be smooth but not to be a submersion, the action is not required to be free, proper, transitive, or to preserve any additional structure, and $G$ and $M$ may be disconnected; those hypotheses enter only in the theorems that use them. For $G$ with $\mathfrak{g}=0$, in particular for $G$ discrete, a Hamiltonian action is exactly a symplectic action and $\mu$ is the constant map to the zero-dimensional dual. Here ${\mathrm{A}\mathrm{C}}_{\omega }$ is countable choice; it is used exactly through the supplied fundamental-vector-field construction, which itself invokes countable choice, and no further choice is made in this definition.