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 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
Canonical name	MomentumMap
Date of creation	2013-03-22 13:14:36
Last modified on	2013-03-22 13:14:36
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	4
Author	bwebste (988)
Entry type	Definition
Classification	msc 53D20
Synonym	moment map