Equivariant moment map - Imported Knowlpedia Content

Definition

Let a Lie group $G$ act symplectically on $(M,\omega)$ , and let $\mu:M\to\mathfrak g^*$ be a moment map. It is an equivariant moment map if

\mu(g\cdot x)=\operatorname{Ad}_g^*\mu(x)

for every $g\in G$ and $x\in M$ , where $\operatorname{Ad}^*$ is the coadjoint action. Thus $\mu$ satisfies both the differential moment-map identity for every $\xi\in\mathfrak g$ and the global compatibility condition intertwining the two $G$ -actions.

Infinitesimal criterion

Differentiating equivariance at the identity gives

d\mu_x\bigl(\xi_M(x)\bigr)=\operatorname{ad}_\xi^*\mu(x).

With the coadjoint convention $\operatorname{Ad}_g^*\lambda=\lambda\circ\operatorname{Ad}_{g^{-1}}$ , this formula uses the corresponding infinitesimal coadjoint action. If $G$ is connected, the infinitesimal identity for every $\xi$ implies the global equivariance condition Ortega and Ratiu, §4.2.

Equivariance defect

For a nonequivariant moment map, the difference

\sigma(g)=\mu(g\cdot x)-\operatorname{Ad}_g^*\mu(x)

is independent of $x$ when $M$ is connected and defines a group cocycle with values in $\mathfrak g^*$ . Changing $\mu$ by a constant changes this cocycle by a coboundary. Consequently, the obstruction to choosing an equivariant moment map is a cohomology class rather than merely a poor normalization.

Consequences and conventions

Equivariance makes inverse images of coadjoint-invariant subsets $G$ -invariant and is the hypothesis normally used in Hamiltonian reduction. Sign conventions for the fundamental vector field, the moment-map equation, and the infinitesimal coadjoint action vary together in the literature; the global equation in the core is convention-independent once the cited coadjoint action has been fixed.

References

J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. Springer DOI record. Relevant: §4.2.
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1990. Cambridge DOI record. Relevant: Chapter 3.