Singular symplectic reduction - Imported Knowlpedia Content

Definition

Let a compact Lie group $G$ have a Hamiltonian action on a symplectic manifold $(M,\omega)$ with equivariant moment map $\mu:M\to\mathfrak g^*$ . The singular symplectic reduction at zero is the orbit space

M/\!/G=\mu^{-1}(0)/G

when $0$ need not be a regular value and the action on $\mu^{-1}(0)$ need not be free. It is generally not a manifold. Its decomposition by connected components of orbit-type sets gives it the structure of a stratified space whose strata are smooth symplectic manifolds. Reduction at a nonzero value is handled similarly using its coadjoint stabilizer, or by the shifting trick.

Orbit-type strata

For a closed subgroup $H\subseteq G$ , let $M_{(H)}$ denote the points whose stabilizers are conjugate to $H$ . The pieces

\bigl(\mu^{-1}(0)\cap M_{(H)}\bigr)/G

need not be connected; their connected components are the symplectic strata. On each component, the restricted form descends by the same kernel-removal mechanism as regular reduction. Different strata can have different dimensions and fit together according to stabilizer type.

Sjamaar and Lerman prove that, in this compact-group setting, the orbit-type pieces are symplectic and their decomposition is a stratification; the natural smooth functions carry a compatible Poisson bracket Sjamaar–Lerman, Theorems 2.1 and 6.11 and §3.

Relation to regular reduction

If $0$ is regular and the action on $\mu^{-1}(0)$ is free, there is only the principal orbit type on the level set, and the singular construction reduces to the Marsden–Weinstein–Meyer reduced manifold. With locally free rather than free action, the quotient is typically a symplectic orbifold. Stabilizers of different dimensions are what force genuinely stratified behavior.

As a basic model, a circle acting on a symplectic vector space with weights of both signs can have a zero level containing the fixed origin and free or locally free points away from it. Their images lie in strata of different dimensions, so no single manifold chart can describe the quotient near the image of the origin.

Conventions and scope

For noncompact groups, proper Hamiltonian actions admit broader singular-reduction theorems, but the precise hypotheses and stratification category require care. The compact-group formulation in the core is the classical Sjamaar–Lerman setting. Authors also use “singular reduction” for Poisson, presymplectic, and infinite-dimensional quotients; those are not included automatically here.

References

Reyer Sjamaar and Eugene Lerman, “Stratified Symplectic Spaces and Reduction,” Annals of Mathematics 134 (1991), 375–422. DOI record. Relevant: Theorem 2.1 on symplectic pieces, §3 on the reduced Poisson algebra, and Theorem 6.11 on stratification.
Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. DOI record. Relevant: Chapter 8, singular reduction and symplectic strata.