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.