Uniqueness and ambiguity of moment maps - Imported Knowlpedia Content

Statement

Let a Lie group $G$ act symplectically on $M$ , and let $\mu,\nu:M\to\mathfrak g^*$ be moment maps for the same action and sign convention. Their difference $\delta=\nu-\mu$ is locally constant: on each connected component $C\subseteq M$ , there is a covector $c_C\in\mathfrak g^*$ with $\delta|_C=c_C$ . If both maps are equivariant, then

c_{gC}=\operatorname{Ad}_g^*c_C.

In particular, when $M$ is connected, the ambiguity is addition of a single coadjoint-invariant covector $c\in(\mathfrak g^*)^G$ . These shifts describe every possible choice: once one moment-map value is fixed on each component, no further freedom remains.

Proof

For every $\xi\in\mathfrak g$ , the two moment-map identities give

d\langle\nu-\mu,\xi\rangle =\iota_{\xi_M}\omega-\iota_{\xi_M}\omega=0.

Each scalar component $\langle\delta,\xi\rangle$ is therefore locally constant, so $\delta$ is locally constant as a map to the finite-dimensional vector space $\mathfrak g^*$ . Equivariance of both maps gives

\delta(gx)=\operatorname{Ad}_g^*\delta(x),

which is exactly the stated relation among componentwise constants. This standard uniqueness argument is given in Ortega and Ratiu, §4.2.

Structure and consequences

For connected $G$ , a coadjoint-invariant covector annihilates the commutator algebra $[\mathfrak g,\mathfrak g]$ . Hence a connected semisimple group admits no nonzero ambiguity of this kind. For an abelian group the coadjoint action is trivial, so every constant covector gives another equivariant moment map.

Normalization conditions remove some or all of the ambiguity. For example, prescribing $\mu(x_0)$ at one point of each connected component determines the moment map uniquely when those prescribed values are compatible with stabilizers and equivariance.

Conventions and scope

References

Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. DOI record. Relevant: §4.2, momentum-map equivariance and ambiguity.
Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: Chapter 5, moment maps and Hamiltonian actions.