momentum map

Let $(M,\omega )$ be a symplectic manifold, $G$ a Lie group acting on that manifold, $𝔤$its Lie algebra, and ${𝔤}^{*}$ the dual of the Lie algebra. This action induces a map$\alpha :𝔤\to 𝔛(M)$ where $𝔛(M)$ is the Lie algebra of vector fields on $M$, such that$\exp (tX)(m)={\rho }_t(m)$ where $\rho$ is the flow of $\alpha (X)$. Then a moment map$\mu :M\to {𝔤}^{*}$ for the action of $G$ is a map such that

$$H_{\mu (X)}=\alpha (X).$$

Here $\mu (X)(m)=\mu (m)(X)$, that is, $\mu (m)$ is a covector, so we applyit to the vector $X$ and get a scalar function $\mu (X)$, and $H_{\mu (X)}$ is its Hamiltonian vector field.

Generally, the moment maps we are interested in are equivariant with respect to the coadjointaction, that is, they satisfy

$${\mathrm{Ad}}_g^{*}\circ \mu =\mu \circ g.$$