Exercises for Symplectic Geometry I
LMU M¨unchen, Summer term 09 Frederik Witt / Vincent Humili`ere Problem set 7, due for 25/06/09
Co-adjoints orbits
LetGbe a Lie group. Remember that its Lie algebrag, with its canonical Lie bracket [·,·], can be seen either as the tangent space to Gateor as the space of left-invariant vector fields onG. We are going to construct a canonical symplectic form on some submanifolds of the dualg∗ ofg.
The groupGacts on itself by conjugation:
ψg:G→G, a7→gag−1.
The associated linearized action is the so called adjoint representation ofGong:
Adg:g→g, X 7→dψg(e)·X.
Consider the canonical pairing h·,·ibetweeng∗ andg. Theco-adjoint representation ofGong∗ is Ad∗g :g∗→g∗, ξ,7→Ad∗g(ξ),
where Ad∗g(ξ) is defined by hAd∗g(ξ), Xi=hξ, Adg(X)ifor allX ∈g.
(a) Show the formula
d
dtAdexp(tX)(Y)
¯¯
¯¯
t=0
= [X, Y].
(b) Show the formula ¿
d
dtAd∗exp(tX)(ξ)
¯¯
¯¯
t=0
, Y À
=hξ,[X, Y]i.
For anyξ∈g∗ we define the following skew-symmetric bilinear form ong:
ωξ(X, Y) =hξ,[X, Y]i.
(c) Show that the kernel ofωξ is the Lie algebra of the stabilizer ofξfor the co-adjoint representation.
(d) Show thatωξdefines a non-degenerate 2-form on the tangent space atξto co-adjoint orbit through ξ.
(e) Show thatωξ defines a closed 2-form on the orbit of ξing∗.
Hint: This will follow from the Jacobi identity ong.
