A Poisson bracket on the space of Poisson structures

Abstract

Let $M$ be a smooth closed orientable manifold and $\mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $\mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on $\mathcal{P}(M)$ by volume-preserving diffeomorphism of $M$. 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 $d{d}^{\Lambda }$ and $d+d^{\Lambda }$ symplectic cohomology groups defined by Tseng and Yau.

18 pages. v2: Many updates and corrections

References in corpus (1)

• Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Cited by in corpus (1)

• Some Morse-type inequalities for symplectic manifolds