Moment map

Remarks

Example: let the circle $S^1 = \R / 2\pi\Z$ act on $\R^2$ by rotations, with $\omega = dx \wedge dy$. Identify $\mathfrak{g}$ and $\mathfrak{g}^*$ with $\R$; the generator is $\xi_M = -y\,\partial_x + x\,\partial_y$. Let $\mu(x, y) = -\frac{1}{2}(x^2 + y^2)$. Then $\iota_{\xi_M} \omega = -y\,dy - x\,dx = d\mu$, and clause 2 holds because $G$ is abelian and $\mu$ is rotation invariant.

Non-example: the translations of the torus $T^2 = \R^2 / \Z^2$ in the $x$ direction preserve $\omega = dx \wedge dy$, but there is no moment map. Clause 1 would need $\iota_{\partial_x} \omega = dy$ to be the differential of a function on $T^2$, and $dy$ is closed but not exact there.

For a connected group, clause 2 can be replaced by an infinitesimal condition. It says that $\xi \mapsto \mu^\xi$ is a Lie algebra homomorphism into functions with the Poisson bracket, up to the sign convention. When $G$ is abelian, a moment map can be changed by adding any constant in $\mathfrak{g}^*$. Many authors use the opposite sign convention in clause 1.