Stratified symplectic spaces and reduction

authors: Reyer Sjamaar, Eugene Lerman

year: 1991

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Literature Notes

Section 1 - Stratified symplectic spaces

Defines

• stratified topological space (with a smooth structure)
• stratified symplectic space

Section 2 - Decomposition of symplectic reduction

Using the shifting trick, they look only at the level set of , that is .

The main theorem of this section proves that the orbit type stratification induces a decomposition on the reduced space, so the stratum of are manifolds. This proves the first condition in the definition of a stratified symplectic space.

Main Theorem

Let be a Hamiltonian G-space with moment map . The intersection of the stratum of orbit type with the zero level set is a manifold. That is, is a manifold. The orbit space

has a natural symplectic structure whose pullpack to coincides with the restriction to of the symplectic form on . Said another way,

Consequently, the stratification of by orbit types induces a decomposition of the reduced space into a disjoint union of symplectic manifolds

Proof (general ideas)

todo

Section 3 - Dynamics on reduced phase space

Section 3 is concerned with defining a Poisson structure on to prove the second and third condition for the definition of a stratified symplectic space.

Define a Poisson bracket on the smooth functions simply by using the symplectic forms of the pieces which is again defined using the symplectic form of .

Poison structure

The Poisson bracket of two functions on is a smooth function.

Proof (ideas)

For functions , let be two -invariant functions on with , and .

That means for , then the Poisson bracket would need to be

or written another way,

It would be enough to show this is true for an arbitrary point in .

Section 4 - Reduction in stages

Reduction in stages can be done for non-regular elements, in the exact same way as normal.

Section 5 - Local structure of the decomposition

Section 6 - Whitney embedding of a reduced phase space

Section 7 - Symplectic tubular neighborhood of a stratum