Stratified Symplectic Spaces and Reduction

Reyer Sjamaar, Eugene Lerman
1991 Annals of Mathematics  
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 M 0 := J −1 (0)/G is a symplectic manifold. We show that if the regularity assumptions are dropped the space M 0 is a union of symplectic manifolds, i.e., it is a stratified symplectic space. Arms et al., [2] , proved that M 0 possesses a natural Poisson bracket. Using their result we study
more » ... 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 −1 (0). Finally we give a detailed description of the stratification of M 0 and prove the existence of a connected open dense stratum.
doi:10.2307/2944350 fatcat:abflpeugmzfjhk5hi5xh2fup6m