Stratified Symplectic Spaces and Reduction

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 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 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 −1 (0). Finally we give a detailed description of the stratification of M 0 and prove the existence of a connected open dense stratum.

Poisson geometry of flat connections for SU(2)-bundles on surfaces

“…While the space N is known to have a structure of a normal projective variety [19], it is unclear whether and how the stratification (0.1) relates to the corresponding complex analytic one; I am indebted to M. S. Narasimhan for this comment. In another earlier paper [11], we proved that the space N inherits a structure of a stratified symplectic space in the sense of Sjamaar-Lerman [21]; in particular, we constructed a Poisson algebra (C ∞ N, {·, ·}) of continuous functions on N which, on each stratum, restricts to a symplectic Poisson algebra of smooth functions in the ordinary sense. In the present paper we describe the strata locally explicitly in terms of certain related classical constrained systems; in particular, for the special case of genus two, we describe the Poisson algebra and resulting Poisson geometry of the space N explicitly.…”

Singular reduction of implicit Hamiltonian systems

“…In this case, the Hamiltonian dynamics defined by (5.5) is exactly the singular reduced Hamiltonian dynamics as defined in [10,9,23,24,27].…”

“…The manifold M can be decomposed into submanifolds as follows [9,24,27]. Let K be a compact subgroup of G and define M (K) to be the set of points in M whose stabilizer group G x = {g ∈ G | φ(g, x) = x} is conjugate to K, i.e., M (K) = {x ∈ M | ∃g ∈ G such that gG x g −1 = K}.…”

“…Thus the bracket in (6.7) is a well defined generalized Poisson bracket {·, ·} (K) on C ∞ ((M 0 ) (K) ). The associated generalized Poisson structure J (K) is the inverse of the nondegenerate two-form ω (K) defined by the condition that the pullback of ω (K) to P −1 (0) ∩ M (K) equals the restriction of ω to P −1 (0) ∩ M (K) , see also [9,24,27].…”

“…Already in simple examples like the spherical pendulum moving with angular momentum zero (i.e., moving in a plane) and the Lagrange top (reducing the gravitational S 1 symmetry after regular reduction of the internal S 1 symmetry corresponding to the homogeneous mass distribution), one recognizes the need for the investigation of these special cases. This task has been taken up in [3,4,10,11,23,24,27] leading to the theory of so-called singular reduction of (symplectic and Poisson) Hamiltonian systems. See [9] for a nice overview and some worked out examples (including the spherical pendulum and the Lagrange top).…”