§4. The moment map 21
The classification of coadjoint orbits reduces in this case to the problem of
classification of symmetric matrices up to transformations
S
gtSg,
g Sp(2n, R).

Remark 4. The problems arising in Examples 7 and 8 are particular
cases of the following general problem: classification of a pair (S, A) where
S is a symmetric matrix and A is an antisymmetric matrix with respect to
simultaneous linear transformations:
(S, A)
(gtSg, gtAg),
g GL(n, R).

Example
8.∗
Let M be a compact smooth simply connected 3-dimen-
sional
manifold5
with a given volume form vol. Let G = Diff(M, vol) be the
group of volume preserving diffeomorphisms of M.
The role of the Lie algebra g = Lie(G) is played by the space Vect(M, vol)
of all divergence-free vector fields on M. We recall that the divergence of a
vector field ξ with respect to a volume form vol is a function div ξ on M
such that Lξ(vol) = div ξ · vol. Here is the Lie derivative along the field
ξ. Using the identity (see formula (16) in Appendix II.2.3)
= d + d
we obtain that iξvol = dθξ where θξ is some 1-form on M defined modulo
exact forms (differentials of functions). Now any smooth map K :
S1
M
defines a linear functional FK on g:
(24) FK (ξ) =
S1
K∗(θξ).
(It is clear that adding a differential of a function to θξ does not change
the value of the integral.) Moreover, the functional FK does not change if
we reparametrize
S1
so that the orientation is preserved. In other words, it
depends only on the oriented curve
K(S1).
We see that the classification of coadjoint orbits in this particular case
contains as a subproblem the classification of oriented knots in M up to a
volume preserving isotopy.
Example
9.∗
Let G =
Diff+(S1)
denote the group of orientation pre-
serving diffeomorphisms of the circle, and let G be its simply connected
5The
famous Poincar´ e conjecture claims that such a manifold is diffeomorphic to
S3
but it
is still unknown. We use only the equalities H2(M) = H1(M) = {0}.
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