§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.
Previous Page Next Page