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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)

Abstract

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

Keywords

Symplectic Manifold Poisson Structure Morse Theory Momentum Mapping Coadjoint Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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