8 1. Geometry of Coadjoint Orbits
Proof. Consider the bracket operation defined by c:
(11) {f1, f2} = cij
k
Xk
∂f1
∂Xi
∂f2
∂Xj
.
Since the bivector c has linear coefficients, the space V

of linear functions
on V is closed under this operation.
If c defines a Poisson structure on V , then V

is a Lie subalgebra in
C∞(V
) that we denote by g. Therefore, V itself can be identified with the
dual space
g∗,
which justifies the labelling of coordinates by lower indices.
In the natural basis in V

formed by coordinates (X1, . . . , Xn) the coef-
ficients cij
k
are precisely the structure constants of g:
{Xi, Xj} =
k
cijXk.k
Conversely, if cij
k
are structure constants of a Lie algebra g, then the brackets
(11) satisfy the Jacobi identity for any linear functions f1, f2, f3. But this
identity involves only the first partial derivatives of fi. Therefore, it is true
for all functions.
Remark 3. The existence of a Poisson bracket on
g∗
was already known
to Sophus Lie in 1890, as was pointed out recently by A. Weinstein. It seems
that Lie made no use of it. F. A. Berezin rediscovered this bracket in 1967
in connection with his study of universal enveloping algebras [Be1]. The
relation of this fact to coadjoint orbits was apparently first noted in [Ki4].

This relation can be formulated as follows.
Theorem 2. The symplectic leaves of the Poisson manifold
(g∗,
c) are ex-
actly the coadjoint orbits.
Proof. Let LF be the leaf that contains the point F
g∗.
The tangent
space to LF at F by definition (see Theorem 6 in Appendix II.3.2) is spanned
by vectors vi =
cijXk∂j. k
But vi is exactly the value at F of the vector field on
g∗
corresponding to the infinitesimal coadjoint action of Xi g. Therefore,
the coadjoint orbit ΩF and the leaf LF have the same tangent space. Since
it is true for every point F
g∗,
we get LF = ΩF .
Theorem 2 gives an alternative approach to the construction of the
canonical symplectic structure on coadjoint orbits.
Previous Page Next Page