Symplectic Lie group action

Let $G$ be a Lie group and let $(M,\omega)$ be a symplectic manifold. A smooth action $\Phi:G\times M\to M$ is a symplectic Lie group action if, for every $g\in G$ , the diffeomorphism $\Phi_g(x)=\Phi(g,x)$ is a symplectomorphism:

\Phi_g^*\omega=\omega.

Equivalently, the action homomorphism sends $G$ into the group of symplectomorphisms of $(M,\omega)$ . The definition requires preservation of the specified symplectic form, not merely of its cohomology class or of the associated volume form.

Infinitesimal criterion

For $\xi$ in the Lie algebra of $G$ , let $\xi_M$ be the corresponding infinitesimal generator. A symplectic action satisfies

\mathcal L_{\xi_M}\omega=0.

Since $d\omega=0$ , Cartan's formula makes this equivalent to $d(\iota_{\xi_M}\omega)=0$ . Conversely, this infinitesimal condition implies that the identity component of $G$ acts symplectically. If $G$ is disconnected, its other components must still be checked separately.

Relationship to Hamiltonian actions

A Hamiltonian action is a symplectic action supplied with a moment map whose components have Hamiltonian vector fields equal, up to the adopted sign convention, to the infinitesimal generators. Symplecticity alone only says that the $1$ -forms $\iota_{\xi_M}\omega$ are closed; Hamiltonianity requires them to be exact in a compatible and usually equivariant way. Thus a symplectic action need not be Hamiltonian when first cohomology provides an obstruction.

Examples and non-examples

The standard rotation action of $SO(2)$ on $(\mathbb R^2,dx\wedge dy)$ is symplectic and Hamiltonian. Translation of the symplectic torus by the torus itself is symplectic, but its generating constant $1$ -forms need not be exact, so the action is not Hamiltonian. A diffeomorphism preserving only $\omega^n$ is volume-preserving but need not be symplectic when $\dim M\geq4$ .

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: Chapter 5 on symplectic and Hamiltonian group actions.
Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. DOI record. Relevant: the chapters on Lie group actions and the standard momentum map.