§2. Symplectic structure on coadjoint orbits 7

Proposition 1. The 2-form ΣF = pF

∗

(σ) on G is the exterior derivative of

the left-invariant real-valued 1-form θF given by

(9) θF = − F, Θ .

Proof. We shall use the formula for the exterior derivative of a 1-form (see

(19) from Appendix II.2.3):

dθ(ξ, η) = ξθ(η) − ηθ(ξ) − θ([ξ, η]).

Let X and Y be left-invariant vector fields on G (see the fourth definition

of a Lie algebra in Appendix III.1.3). Putting θ = θF , ξ = X, η = Y , we

get

dθF (X, Y ) = XθF (Y ) − Y θF (X) − θF ([X, Y ]).

The first and second terms in the right-hand side vanish because θF (X)

and θF (Y ) are constant functions. We can rewrite the last term as

−θF ([X, Y ]) = −θF ([X, Y ]) = F, [X, Y ] = pF

∗

(σ)(X, Y ).

Now we return to the form σ. Since pF is a submersion, the linear map

(pF )∗ is surjective. Therefore, the dual map pF

∗

is injective. But pF

∗

dσ =

dpF

∗

(σ) = d ΣF =

d2θF

= 0. Hence, σ is closed.

Note that in general θF cannot be written as pF

∗

(φ) for some 1-form φ

on Ω, so we cannot claim that σ is exact (and actually it is not in general).

2.2. The second (Poisson) approach.

We now discuss another way to introduce the canonical symplectic struc-

ture on coadjoint orbits. It is based on the notion of Poisson manifold (see

Appendix II.3.2).

Consider the real n-dimensional vector space V and, making an excep-

tion to the general rules, denote the coordinates (X1, . . . , Xn) on V using

lower indices. Let c be a bivector field on V with linear coeﬃcients:

(10) c = cijXk

k ∂i

⊗

∂j

where

∂i

= ∂/∂Xi ,

∂j

= ∂/∂Xj , and cij

k

=

−cji.k

Lemma 3. The bivector (10) defines a Poisson structure on V if and only

if the coeﬃcients cij

k

form a collection of structure constants for some Lie

algebra g.