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

g.2

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] .

Therefore,

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 :=

pF

∗

(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

notations is

(8) Θ(g)(X) =

g−1

· X for X ∈ TgG.

Often, especially in physics papers, this form is denoted by

g−1dg

be-

cause for any smooth curve g = g(t) we have g

∗Θ

=

g(t)−1

˙(t)dt g =

g−1dg.

2Recall

that for left G-manifolds the vector field ξ corresponding to X ∈ g is defined by

ξ(x) =

d

dt

(exp tX) · x

t=0

. The map X → ξ is an antihomomorphism of g to Vect M.