Hamiltonian action

Definitions

Weakly Hamiltonian action

A symplectic action of $G$ on $(M,\omega )$ is called weakly Hamiltonian if every fundamental vector field $X_{\xi }$ is a Hamiltonian vector field.

Hamiltonian action

A symplectic action of $G$ on $(M,\omega )$ is called Hamiltonian if it is weakly Hamiltonian, and the diagram below commutes:

i.e. an action is called Hamiltonian if there exists an equivariant moment map.

Motivation

The fundamental vector fields of a symplectic action are symplectic. The goal is to identify an analogous definition for Hamiltonian vector fields where the fundamental vector fields to the action are related to some function $H_{\xi }$ such that

$${\iota }_{{\xi }_M}=d{H}_{\xi }\forall \xi \in \mathfrak{g}.$$

Note that this is already a stronger condition than a symplectic action since each Hamiltonian vector field must be symplectic. However, this definition has issues.

Most importantly, the function $H_{\xi }$ isn’t unique. In fact, there can be infinitely many of them, since $H_{\xi }$ can differ by a constant. Therefore, we aim to pick the function $H_{\xi }$ for each $\xi \in \mathfrak{g}$ such that each of the fundamental vector fields were Hamiltonian?

So, we are looking for some map

$$\begin{array}{rl}\tilde{\mu }:\mathfrak{g}& ⟶C^{\infty }(M)\\ \xi & ⟼H_{\xi }\end{array}$$

such that

$${\iota }_{{\xi }_M}\omega =d{H}_{\xi }.$$

We also want this map to be a Lie algebra homomorphism so that it preserves the Lie bracket $[-,-]$ on $\mathfrak{g}$ and the Poisson bracket $\{-,-\}$ on $C^{\infty }(M)$ seen as a Lie bracket.