Moment Maps

The moment map is a fundamental tool linking Hamiltonian group actions on symplectic manifolds to the dual of the Lie algebra. It generalizes classical conserved quantities (linear and angular momentum) to arbitrary symmetry groups.

Definition and Properties

Definition5.1Moment Map

Let $(M, \omega)$ be a symplectic manifold with a Hamiltonian action of a Lie group $G$ . A moment map is a smooth map $\mu: M \to \mathfrak{g}^*$ satisfying:

For each $\xi \in \mathfrak{g}$ , the function $\mu_\xi = \langle \mu, \xi \rangle: M \to \mathbb{R}$ is a Hamiltonian for the infinitesimal generator $X_\xi$ : $d\mu_\xi = \iota_{X_\xi}\omega$ .
Equivariance: $\mu(g \cdot x) = \mathrm{Ad}_g^* \cdot \mu(x)$ for all $g \in G, x \in M$ .

Definition5.2Comoment Map

The comoment map $\tilde{\mu}: \mathfrak{g} \to C^\infty(M)$ is defined by $\tilde{\mu}(\xi) = \mu_\xi$ . The equivariance of $\mu$ is equivalent to $\tilde{\mu}$ being a Lie algebra homomorphism: $\tilde{\mu}([\xi, \eta]) = \{\tilde{\mu}(\xi), \tilde{\mu}(\eta)\}$ .

ExampleMoment Map on $T^*G$

The left $G$ -action on $T^*G$ by cotangent lifts is Hamiltonian with moment map $\mu_L: T^*G \to \mathfrak{g}^*$ given by $\mu_L(\alpha_g) = (dR_g)^* \alpha_g$ (right-trivialization). This identifies $T^*G$ with $G \times \mathfrak{g}^*$ .

Existence and Uniqueness

RemarkObstructions to Existence

A moment map exists if and only if the action is Hamiltonian, which requires the infinitesimal generators to be Hamiltonian vector fields. The obstruction lies in $H^1(M; \mathbb{R})$ (whether $\iota_{X_\xi}\omega$ is exact) and in $H^2(\mathfrak{g}; \mathbb{R})$ (whether the comoment map can be made a Lie algebra homomorphism). When both vanish, $\mu$ exists and is unique up to a constant in $(\mathfrak{g}^*)^G$ .

Definition5.3Hamiltonian $G$-Space

A Hamiltonian $G$ -space is a triple $(M, \omega, \mu)$ where $(M, \omega)$ is a symplectic manifold, $G$ acts on $M$ preserving $\omega$ , and $\mu: M \to \mathfrak{g}^*$ is an equivariant moment map. This is the standard setup for symplectic reduction and geometric quantization.