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From Symplectic to Poisson. A Study of Reduction and a Proposal Towards Implosion

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arxiv 2202.05783 v1 pith:D3HYP6QX submitted 2022-02-11 math.SG math.DG

classification math.SGmath.DG
keywords symplecticpoissonimplosionreductiontheoremcross-sectionguillemin-sternbergmanifolds
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The imploded cross-section of a symplectic manifold is a stratified space allowing for an abelianization of its symplectic reduction. After recalling symplectic and Poisson reduction and reviewing the basics of symplectic implosion, we prove a cross-section theorem for Poisson manifolds, generalizing the Guillemin-Sternberg theorem for symplectic manifolds, which constitutes a first step towards Poisson implosion. On our way, we find and fix a mistake in the proof of Guillemin-Sternberg's theorem, and we identify Poisson transversals as the right analogue to symplectic submanifolds in this context.

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