A Universal Reduction Procedure for Hamiltonian Group Actions

Arms, Judith M.; Cushman, Richard; Gotay, Mark J.

doi:10.1007/978-1-4613-9725-0_4

Cited by 105 publications

(180 citation statements)

References 24 publications

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“…Moreover, cf. [1], the symplectic Poisson structure on H 1 (π, g φ ) passes to a Poisson structure {·, ·} φ on C ∞ (H φ ). The space H φ together with the Poisson algebra (C ∞ (H φ ), {·, ·} φ ) is our local model for the moduli space N near the point represented by φ, as a stratified symplectic space.…”

Section: The Local Modelmentioning

confidence: 99%

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

Huebschmann

1996

Math Z

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Abstract. In earlier work we have shown that the moduli space N of flat connections for the (trivial) SU(2)-bundle on a closed surface of genus ℓ ≥ 2 is a stratified symplectic space so that, in particular, the data give rise to a Poisson algebra (C ∞ N, {·, ·}) of continuous functions on N; furthermore, the strata are Kähler manifolds, and the stratification consists of an open connected and dense submanifold N Z of real dimension 6(ℓ − 1), a connected stratum N (T ) of real dimension 2ℓ, and 2 2ℓ isolated points. In this paper we show that, close to each point of N (T ) , the space N and Poisson algebra (C ∞ N, {·, ·}) look like a product of C ℓ endowed with the standard symplectic Poisson structure with the reduced space and Poisson algebra of the system of (ℓ − 1) particles in the plane with total angular momentum zero, while close to one of the isolated points, the Poisson algebra (C ∞ N, {·, ·}) looks like that of the reduced system of ℓ particles in R 3 with total angular momentum zero. Moreover, in the genus two case where the space N is known to be smooth we locally describe the Poisson algebra (C ∞ N, {·, ·}) and the various underlying symplectic structures on the strata and their mutual positions explicitly in terms of the Poisson structure.1991 Mathematics Subject Classification. 32G13, 32G15, 32S60, 58C27, 58D27, 58E15, 81T13.

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Section: The Local Modelmentioning

confidence: 99%

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

Huebschmann

1996

Math Z

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“…Then there exists a well defined (see [12]) skewsymmetric vector bundle map 10) wherev ∈ (Γ(x)) * is the restriction of some v ∈ T x M to Γ(x) ⊂ T * x M which satisfies the condition that (v, v * ) ∈ D(x). Notice that the kernel of J(x) is given by the codistribution Γ 0 with fibers defined by 3 Γ…”

Section: Implicit Hamiltonian Systemsmentioning

confidence: 99%

“…, corresponding to the bundle map J, we can use the theory in [3,9] to define a generalized Poisson bracket {·, ·} 0 :…”

Section: Singular Reductionmentioning

confidence: 99%

“…In principle, however, these results generalize immediately to the case of singular reduction of generalized Poisson brackets, as described above by (4.3). In particular, [3,9] show that (under the assumption that G acts properly) nondegeneracy of {·, ·} implies that of {·, ·} 0 . Also, from (4.3) it follows immediately that {·, ·} 0 satisfies the Jacobi identity if {·, ·} does.…”

Section: Singular Reductionmentioning

confidence: 99%

“…Furthermore, the set of smooth functions C ∞ (M 0 ) equals the set of smooth functions as defined by the differential structure on M 0 . Indeed, since N = P −1 (0) is closed in M , every smooth G-invariant function on N can be smoothly extended to a G-invariant function on M [3]. In that case the notion of a "projecting derivation" as defined above has the usual meaning of projection of a vector field on M to a vector field on the reduced space M 0 .…”

Section: Singular Reductionmentioning

confidence: 99%

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Singular reduction of implicit Hamiltonian systems

Blankenstein

Raţiu

2004

Reports on Mathematical Physics

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Geometric Aspects of Degenerate Modulation Equations

Bridges

1994

Stud Appl Math

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Geometrie aspects of degenerate modulation equations associated with spatially reversible systems are considered. Our primary observation is that stationary solutions of such equations always have a Poisson structure that is reminiscent of the equations governing the rigid body in mechanics. The Poisson structure is used to study the geometry of "spatial" phase space: A non trivial Casimir of the Poisson structure provides a foliation of the phase space, spatially periodie states are given by critieal points on level sets of the Casimir and stability type is given by the rate of change of the Casimir function. The bifurcation of spatially periodic states is then studied using singularity theory. The case where branches intersect transversely is treated in detail.

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A Universal Reduction Procedure for Hamiltonian Group Actions

Cited by 105 publications

References 24 publications

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

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

Singular reduction of implicit Hamiltonian systems

Geometric Aspects of Degenerate Modulation Equations

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