§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
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
(σ) = d ΣF =
= 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
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
= ∂/∂Xi ,
= ∂/∂Xj , and cij
Lemma 3. The bivector (10) defines a Poisson structure on V if and only
if the coeﬃcients cij
form a collection of structure constants for some Lie