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
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