§4. The moment map 17

An important example of a Poisson G-manifold is the space

(g∗,

c)

considered in Section 2.2 with the map g →

C∞(g∗)

defined by fX

g∗

(F ) =

F, X .

Theorem 4. The Poisson G-manifold

(g∗,

c) is a universal (final) object

in the category P(G).

This means that for any object (M, f(·)

M

) there exists a unique morphism

μ from

(

M, f(·)

M

)

to

(

g∗,

f(·)

g∗

)

, namely, the so-called moment map defined

by

(21) μ(m), X = fX

M

(m).

Proof. A direct corollary of the property (20) of a morphism in the category

P(G).

Lemma 6. Let M be a homogeneous Poisson G-manifold. Then the mo-

ment map is a covering of a coadjoint orbit.

Proof. First, the image μ(M) is a homogeneous submanifold in

g∗,

i.e. a

coadjoint orbit Ω ⊂ g∗.

Second, the transitivity of the G-action on M implies that the rank of the

Jacobi matrix for μ is equal to dim M. Hence, μ is locally a diffeomorphism

and globally a covering of Ω.

It is worthwhile to discuss here the relation between homogeneous Pois-

son G-manifolds and homogeneous symplectic G-manifolds.

From Lemma 6 we see that all homogeneous Poisson G-manifolds are ho-

mogeneous symplectic G-manifolds. We show here that the converse is also

true when coadjoint orbits are simply connected. In general, the converse

statement becomes true after minor corrections.

This I call the universal property of coadjoint orbits.

Any symplectic manifold (M, σ) has a canonical Poisson structure c

(Appendix II.3.2). But if a Lie group G acts on M and preserves σ, it

does not imply that (M, c) is a Poisson G-manifold. Indeed, there are two

obstacles for this:

1. Topological obstacle. The Lie field LX , X ∈ g, is locally a skew

gradient of some function fX , but this function may not be defined globally.

To overcome this obstacle, we can consider an appropriate covering M

of M where all fX , X ∈ g, are single-valued. It may happen that the initial

group G is not acting on M and must be replaced by some covering group

G.