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Stratified Symplectic Spaces and Reduction

Reyer Sjamaar and Eugene Lerman
Annals of Mathematics
Second Series, Vol. 134, No. 2 (Sep., 1991), pp. 375-422
Published by: Annals of Mathematics
DOI: 10.2307/2944350
Stable URL: http://www.jstor.org/stable/2944350
Page Count: 48
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Stratified Symplectic Spaces and Reduction
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Abstract

Let (M, ω) be a Hamiltonian G-space with proper momentum map J: M → g*. It is well-known that if zero is a regular value of J and G acts freely on the level set J-1(0), then the reduced space M0 := J-1(0)/G is a symplectic manifold. We show that if the regularity assumptions are dropped, the space M0 is a union of symplectic manifolds; i.e., it is a stratified symplectic space. Arms et al. [2] proved that M0 possesses a natural Poisson bracket. Using their result, we study Hamiltonian dynamics on the reduced space. In particular we show that Hamiltonian flows are strata-preserving and give a recipe for lifting a reduced Hamiltonian flow to the level set J-1(0). Finally we give a detailed description of the stratification of M0 and prove the existence of a connected open dense stratum.

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