6 1. Geometry of Coadjoint Orbits
2.1. The first (original) approach.
We use the explicit formula (19) in Appendix II.2.3 for the differential
of a 2-form:
dσ(ξ, η, ζ) = ξσ(η, ζ)− σ([ξ, η], ζ)
where the sign denotes the summation over cyclic permutations of ξ, η, ζ.
Let ξ, η, ζ be the vector fields on Ω which correspond to elements X, Y ,
Z of the Lie algebra
Then ξ(F ) = K∗(X)F, η(F ) = K∗(Y )F, ζ(F ) =
K∗(Z)F , and we obtain
σ(η, ζ) = F, [Y, Z] , ξσ(η, ζ) = K∗(X)F, [Y, Z] = −F, [X, [Y, Z]] ,
[ξ, η] = −K∗([X, Y ])F, σ([ξ, η], ζ) = −F, [[X, Y ], Z] .
dσ(ξ, η, ζ) = 2 F, [X, [Y, Z]] = 0 via the Jacobi identity.
Since G acts transitively on Ω, the vectors K∗(X)F, X ∈ g , span the whole
tangent space TF Ω. Thus, dσ = 0.
This proof of Theorem 1, being short enough, can be however not quite
satisfactory for a geometric-minded reader. So, we give a variant of it which
is based on more geometric observations. This variant of the proof is also
in accordance with the general metamathematical homology principle men-
tioned in Remark 1 in Appendix I.3.
Consider again the fibration pF : G → Ω and introduce the form ΣF :=
(σ) on the group G. By the very construction, ΣF is a left-invariant 2-
form on G with initial value ΣF (e) = BF . We intend to show that this form
is not only closed but exact.
To see this, we shall use the so-called Maurer-Cartan form Θ. By
definition, it is a g-valued left-invariant 1-form on G defined by the condition
Θ(e)(X) = X. Since the left action of G on itself is simply transitive
(i.e. there is exactly one left shift which sends a given point g1 to another
given point g2), to define a left-invariant form on G we only need to specify
arbitrarily its value at one point. The explicit formula for Θ in matrix
(8) Θ(g)(X) =
· X for X ∈ TgG.
Often, especially in physics papers, this form is denoted by
cause for any smooth curve g = g(t) we have g
˙(t)dt g =
that for left G-manifolds the vector field ξ corresponding to X ∈ g is defined by
(exp tX) · x
. The map X → ξ is an antihomomorphism of g to Vect M.