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Non-Abelian convexity by symplectic cuts. (English) Zbl 0913.58023
Providing further support for the view that in symplectic geometry the smooth category should include orbifolds, this paper extends key results about the properties of moment maps on symplectic manifolds to the case of symplectic orbifolds. In particular, the authors establish the convexity of the moment map image and the connectedness of its fibers. They also extend to orbifolds a theorem of R. Sjamaar [Adv. Math. 138, 46-91 (1998)] on the openness of the map from the quotient \(M/G\) to the image of the moment map \(\Phi:M\to \mathfrak g^*\) (here \(G\) is the Lie group which acts symplectically on the symplectic manifold \(M\), and \(\mathfrak g\) is its Lie algebra). The proofs apply to moment maps for non-Abelian (compact) groups. They use Lerman’s technique of symplectic cutting in a slightly generalized version which accommodates Hamiltonian torus actions (rather than Hamiltonian circle actions).

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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