Moment maps as Poisson maps - Imported Knowlpedia Content

Statement

Let $G$ act on the symplectic manifold $(M,\omega)$ , using $\iota_{X_f}\omega=df$ , and let $\mu:M\to\mathfrak g^*$ be an equivariant moment map. Equip $\mathfrak g^*$ with the plus Lie–Poisson bracket determined by

\{\ell_\xi,\ell_\eta\}_{\mathfrak g^*}(\alpha) =\langle\alpha,[\xi,\eta]\rangle, \qquad \ell_\xi(\alpha)=\langle\alpha,\xi\rangle .

Then $\mu$ is a Poisson map. Conversely, a moment map that is Poisson is infinitesimally equivariant; if $G$ is connected, this infinitesimal condition implies coadjoint equivariance. Equivalently, pullback by $\mu$ preserves Poisson brackets of smooth functions. The forward implication requires no connectedness hypothesis.

Proof idea

Write $\mu^\xi=\ell_\xi\circ\mu$ . The moment-map identity says $X_{\mu^\xi}=\xi_M$ . Equivariance differentiates to the bracket identity

\{\mu^\xi,\mu^\eta\}_M=\mu^{[\xi,\eta]}.

This verifies preservation of the Lie–Poisson bracket on linear functions. The Leibniz rule and a local-coordinate argument then extend the equality to arbitrary smooth functions on $\mathfrak g^*$ . Conversely, the Poisson property applied to linear functions gives the displayed infinitesimal identity. See Marsden and Ratiu, §12.4 and Ortega and Ratiu, §4.2.

Coadjoint orbits

The symplectic leaves of the plus Lie–Poisson structure are coadjoint orbits equipped with the compatible Kirillov–Kostant–Souriau form. Consequently, an equivariant moment map sends Hamiltonian vector fields generated by its components to the corresponding Lie–Poisson Hamiltonian fields and maps each $G$ -orbit into a coadjoint orbit.

Conventions and scope

References

Jerrold E. Marsden and Tudor S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Texts in Applied Mathematics 17, Springer, 1999. DOI record. Relevant: §12.4, equivariant momentum maps and Poisson maps.
Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. DOI record. Relevant: §4.2, infinitesimal equivariance and momentum-map brackets.