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(2010 / vol.12 / no.1)
A Convexity theorem for three tangled Hamiltonian torus actions
Hiraku Abe
Pages. 115-119   



The convexity theorem for hamiltonian torus actions states that the
image of a moment map of a hamiltonian torus action on a compact, connected
symplectic manifold is a convex polytope (cite{A}, cite{G-S}).
Kirwan generalized this
theorem to the case of any compact, connected Lie group, which also gives us a
convex polytope (cite{K}).
On the other hand, if a torus which has half the dimension
of the manifold acts effectively in a hamiltonian fashion, then its moment map
provides a completely integrable system. Many important completely integrable
systems are super-integrable systems in which each trajectory is linear in a
smaller torus than the Liouville torus: the harmonic oscillators, the Kepler system, the
Toda lattice, etc. In this note, motivated by the structure of the super-integrable
system on the Toda lattice (cite{A-D-S}), we shall give a new generalization of the convexity theorem for hamiltonian torus actions.



1. Backgrounds and Results
2. An Example