# Stratified symplectic spaces and reduction

**Reyer Sjamaar****Eugene Lerman**

CiteWeb id: 20160000100

CiteWeb score: 423

Let (M,!) 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 i1 (**0**), then the reduced space M0 := J i1 (**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 a lift of a reduced Hamiltonian flow to the level set J i1 (**0**). Finally we give a detailed description of the stratification of M0 and prove the existence of a connected open dense stratum. ⁄ Partially supported by a grant from the Netherlands Organization for Scientific Research (NWO)

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