Moment-map component - Imported Knowlpedia Content

Definition

Let $G$ act on a symplectic manifold $(M,\omega)$ , let $\mu:M\to\mathfrak g^*$ be a moment map, and fix $\xi\in\mathfrak g$ . The moment-map component in the direction $\xi$ is the smooth function

\mu^\xi:M\to\mathbb R,\qquad \mu^\xi(p)=\langle\mu(p),\xi\rangle.

With the conventions

\xi_M(p)=\left.\frac{d}{dt}\right|_0\exp(t\xi)\mathbin{\cdot}p, \qquad d\mu^\xi=\iota_{\xi_M}\omega,

the function $\mu^\xi$ is a Hamiltonian function for the infinitesimal generator $\xi_M$ . It records the Hamiltonian associated with one infinitesimal symmetry.

Linearity and reconstruction

The assignment $\xi\mapsto\mu^\xi$ is linear from $\mathfrak g$ to $C^\infty(M)$ . Conversely, a linear family of functions $H_\xi$ determines a map $\mu:M\to\mathfrak g^*$ by $\langle\mu(p),\xi\rangle=H_\xi(p)$ . Thus the component equations for all $\xi$ are equivalent to the single $\mathfrak g^*$ -valued moment-map equation.

Equivariance and constants

If $\mu$ is equivariant for the coadjoint action, then

\mu^\xi(g\mathbin{\cdot}p)=\mu^{\operatorname{Ad}_{g^{-1}}\xi}(p).

Adding a constant $c\in\mathfrak g^*$ shifts each component by $\langle c,\xi\rangle$ without changing its differential. Equivariance is preserved precisely when $c$ is fixed by the coadjoint action.

Examples and conventions

For the cotangent-lifted action on $T^*Q$ , the standard component is $\mu^\xi(q,\alpha)=\alpha(\xi_Q(q))$ . On a coadjoint orbit with its KKS form, the inclusion moment map has $\mu^\xi(\lambda)=\langle\lambda,\xi\rangle$ . Authors using $d\mu^\xi=-\iota_{\xi_M}\omega$ obtain the negative components relative to the present convention.

References

Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics 222, Birkhäuser, 2004. DOI record. Relevant: Chapter 4, the standard momentum map and its component functions.
Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: Chapter 5, Hamiltonian actions and moment maps.