symplectic action

Definition

Let be a Lie group that acts on a symplectic manifold smoothly. We call this action symplectic if for all action map is a symplectomorphism, i.e.

Relation to fundamental vector fields

For acting symplecticly on , every fundamental vector field for is a symplectic vector field.

Proof

The flow of the vector field is by definition. However, we know that for every , and that the action is symplectic, so

so the flow is a symplectomorphim for every .

Therefore, we know that all the generating vector fields of a symplectic action are closed, i.e.