A Morse Theoretic Proof of Poisson Lie Convexity

  • Hermann Flaschka
  • Tudor Ratiu
Conference paper
Part of the Progress in Mathematics book series (PM, volume 145)


Let K be a connected compact Lie group that acts on the connected compact symplectic manifold (X, ω) preserving the symplectic form. If for every ξ in the Lie algebra p of K, the vector field ξx(x) = d/dt| t=o exp(tξ) · x is Hamiltonian relative to a function, say Hξ, then the map Φ : Xp* (to the dual p* of p) defined by
$$H_\xi (x) = \left\langle {\xi ,\Phi (x)} \right\rangle $$
is called momentum mapping for the action of K. If the momentum mapping is equivariant relative to the given action of K on X and the coadjoint action of K on p*, the action is called Hamiltonian. A remarkable property of this map was discovered by Guillemin-Sternberg [GS1,GS2] and Kirwan [Ki2]. It asserts that if T is a maximal torus of K and t + * is a positive Weyl chamber, then Φ(X) ∩ t + * is a convex polytope. This theorem was first proved as follows. The image Φ(X) ∩ t + * was shown to be a finite union of compact convex polytopes in [GS1], and a convex polytope for X a Kähler manifold [GS2]. From the partial result in [GS1], Kirwan [Ki2]deduced convexity by appealing to her Morse theory (developed in [Ki1]) for||Φ||2


Symplectic Manifold Poisson Structure Morse Theory Momentum Mapping Coadjoint Orbit 
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Copyright information

© Birkhäuser Boston 1997

Authors and Affiliations

  • Hermann Flaschka
    • 1
  • Tudor Ratiu
    • 2
  1. 1.Department of MathematicsThe University of ArizonaTucsonUSA
  2. 2.Department of MathematicsUniversity of California at Santa CruzSanta CruzUSA

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