Stratified symplectic spaces and reduction. (English) Zbl 0759.58019

Sjamaar, Reyer; Lerman, Eugene

Ann. Math. (2) 134, No. 2, 375-422 (1991).

This work deals with the reduction \(J^{-1}(0)/G\) of the singular momentum map, associated with a Hamiltonian \(G\)-space \((M,\omega)\), at a singular value 0. Several attempts have already been done in this direction [see J. M. Arms, M. J. Gotay and G. Jennings, Adv. Math. 79, No. 1, 43-103 (1990; Zbl 0721.53033)]. The originality of the work under review lies in the notion of a stratified symplectic space, where each stratum \(S\) is a symplectic manifold patched in such a way that the whole space possesses a Poisson structure extending the symplectic structure of the stratum \(S\). The authors also give a complete description of the local structure of the reduction: any stratum possesses a symplectic tubular neighborhood, the fiber being a linear reduced space. Although the space \(J^{-1}(0)/G\) is not a manifold, it is still possible to define the notion of Hamiltonian flow on \(J^{- 1}(0)/G\). The authors prove that the usual lifting techniques, in order to obtain the dynamics on \(M\), can be applied in this context.

Reviewer: M.Saralegi (Madrid)

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H05	Hamilton’s equations

Keywords:

momentum map; reduction; singular value; Poisson structure

Citations:

Zbl 0721.53033