# momentum map

Let $(M,\omega)$ be a symplectic manifold  , $G$ a Lie group acting on that manifold  , $\mathfrak{g}$ its Lie algebra  , and $\mathfrak{g}^{*}$ the dual of the Lie algebra. This action induces a map $\alpha:\mathfrak{g}\to\mathfrak{X}(M)$ where $\mathfrak{X}(M)$ 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 $\mu:M\to\mathfrak{g}^{*}$ 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 apply it 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 coadjoint action, that is, they satisfy

 $\mathrm{Ad}^{*}_{g}\circ\mu=\mu\circ g.$
Title momentum map MomentumMap 2013-03-22 13:14:36 2013-03-22 13:14:36 bwebste (988) bwebste (988) 4 bwebste (988) Definition msc 53D20 moment map