Exercises for Symplectic Geometry I
LMU M¨unchen, Summer term 09 Frederik Witt / Vincent Humili`ere Problem set 9, due for 09/07/09
Moment maps
1. An example
We consider the natural action ofU(n) onCn endowed with its canonical symplectic form.
(a) Show that this action is symplectic.
(b) The Lie algebra u(n) is the set of skew-hermitian matrices X = V +iW where Vt = −V and Wt=W. Show that the infinitesimal action is generated by the Hamiltonian function
µX(z) =−1
2hx, W xi+hy, V xi −1
2hy, W yi, wherez=x+iy ∈Cn.
(c) Show thatµX(z) = 12i z∗Xz= 12itrace(zz∗X).
(d) Show that the action ofU(n) onCn is Hamiltonian with moment map
µ:Cn→u(n), z7→ i 2zz∗,
where the Lie algebrau(n) is identified with its dual via the inner product (A, B) = trace(A∗B).
2. Cohomology of a Lie algebra and moment maps.
Let g be a Lie algebra. For any integer k, we denote by Ck the space of alternating k-linear maps g×. . .×g→R. We consider the mapδ:Ck→Ck+1 defined by
δc(X0, . . . , Xk) =X
i<j
c([Xi, Xj], X0, . . . ,Xˆi, . . . ,Xˆj, . . . , Xk).
It can be checked thatδ2= 0 so that we can define cohomology groups
Hk(g,R) = kerδ:Ck→Ck+1 imδ:Ck−1→Ck.
(a) IfG is a compact connected Lie group with Lie algebrag, thenHk(g,R) =HdeRahmk (G). Do you see why? (do not write the details!)
Let us suppose that H1(g,R) =H2(g,R) = 0. The goal of this exercise is to show that if Gis any Lie group with Lie algebragthen any symplectic action ofGis Hamiltonian.
(b) Show that the conditionH1(g,R) = 0 implies the equalityg= [g,g], where [g,g] denotes the set of linear combinations of elements of the form [X, Y], forX, Y ∈g
(c) Let (M, ω) be a symplectic manifold with a symplectic action ofG. Prove the equalityι[X,Y]ω = d(ω(X, Y)) and show that the Lie bracket of two symplectic vector fields is an Hamiltonian vector field. Deduce that the infinitesimal action of anyX ∈gis Hamiltonian.
(d) For any vectorXi of a basis ofg, we choose an Hamiltonian functionτXi ∈C∞(M) for the action ofXi and extend it by linearity to a well defined mapτ :g→C∞(M), X7→τX. We then consider the antisymmetric bilinear map c(X, Y) = τ[X,Y] − {τX, τY}, where {·,·} denotes the Poisson bracket. Show that c is a cocycle, i.e. δc = 0. Deduce that there exists some b ∈ C1 such that c(X, Y) =−b([X, Y]).
(e) Show thatµ∗ :g→C∞(M), X7→τX+b(X) is a comoment map.
This is finished, congratulations!
