A Poisson bracket on the space of Poisson structures
Thomas Machon

TL;DR
This paper introduces a Poisson bracket on the space of Poisson structures on a manifold, explores its Hamiltonian flow, and defines invariants that detect properties like unimodularity, connecting to symplectic cohomology.
Contribution
It constructs a new Poisson bracket on the space of Poisson structures depending on a volume form and introduces invariants related to flow fixed points and unimodularity.
Findings
Invariant detects unimodularity in regular Poisson 3-manifolds.
Flow acts by volume-preserving diffeomorphisms.
Invariant relates to symplectic cohomology groups.
Abstract
Let be a smooth closed orientable manifold and the space of Poisson structures on . We construct a Poisson bracket on depending on a choice of volume form. The Hamiltonian flow of the bracket acts on by volume-preserving diffeomorphism of . We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the and symplectic cohomology groups defined by Tseng and Yau.
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Taxonomy
TopicsAdvanced Differential Geometry Research · Advanced Topics in Algebra · Advanced Differential Equations and Dynamical Systems
