Hamiltonian G-space

Definition

A Hamiltonian G-space is the data where is a symplectic manifold, with a [[202405231352|Hamiltonian -action]] with moment map .

Examples

Cotangent lift of action

Let be a Lie group and be a [[202405081540|-manifold]]. Then the cotangent lift of the action

is a Hamiltonian action. We can see this since the map

is a Lie algebra morphism where is the cotangent lift of vector field and is the tautological 1-form on .

Thus, we get the following diagram

which basically boils down to the fact that

where the left-hand side represents the fundamental vector field on the cotangent bundle for a vector and the right-hand side represents the cotangent lift of the fundamental vector field on the base manifold .

So the composition map below is comoment map:

Written another way, the moment map is the map such that

We can see this is the moment map since