§2. Symplectic structure on coadjoint orbits 9
The third (symplectic reduction) approach.
We apply the symplectic reduction procedure described in Appendix
II.3.2 to the special case when the symplectic manifold M is the cotangent
over a Lie group G. This case has its peculiarities.
First, the bundle T
is trivial. Namely, we shall use the left action of
G on itself to make the identification T
In matrix notation
the covector g · F ∈ Tg−1
G corresponds to the pair (g, F ) ∈ G ×
Further, the set T
is itself a group with respect to the law
(g1, F1)(g2, F2) = (g1g2,
If we identify (g, 0) ∈ T
with g ∈ G and (e, F ) ∈ T
with F ∈
becomes a semidirect product G
So, we can write (g, F ) =
g · F both in matrix notation and in the sense of the group law in T
Note also the identity g · F ·
= K(g)F .
is a Lie group, the tangent bundle T (T
is also trivial.
We identify T(g,F )T
with Lie (T
using the left shift and obtain
× (g ⊕
Theorem 3. The canonical symplectic structure on T
in the trivial-
ization above is given by the bilinear form σ:
(12) σ(g,F )(X1 ⊕ F1, X2 ⊕ F2) = F1, X2 − F2, X1 − F, [X1, X2] .
Proof. Let us compute first the canonical 1-form θ on T
For a tangent
vector v = (g, F ; X, F ) ∈ T(g,F )T
the projection to TgG equals X ·
Therefore, θ(v) = g · F, X ·
= F, X .
Now we can compute σ as the exterior derivative of θ:
σ(ξ1, ξ2) = ξ1θ(ξ2) − ξ2θ(ξ1) − θ([ξ1, ξ2])
for any vector fields ξ1, ξ2 on T
We choose ξ1, ξ2 as the left-invariant fields on the Lie group T
ξ1(e, 0) = (X1, F1), ξ2(e, 0) = (X2, F2).
3Recall that g · F is defined by g · F, ξ = F, ξ · g−1 .