A Poisson bracket on the space of Poisson structures

Let $M$ be a smooth, closed, orientable manifold and $\mathcal{P}(M)$ the set of Poisson structures on $M$. We construct a Poisson bracket for a class of admissible functions on $\mathcal{P}(M)$, depending on a choice of volume form for $M$. The Hamiltonian flow of the bracket acts on $\mathcal{P}(M)$ by volume-preserving diffeomorphisms of $M$, corresponding to exact gauge transformations. Fixed points of the flow equation define a sub-algebra of Poisson vector fields, which are computed for Poisson structures on $2$ and $3$-manifolds. On the space of symplectic manifolds with a symplectic volume form (up to scaling) we define a further, related Poisson bracket and show that the behaviour of the induced flow on symplectic structures is described naturally in terms of the $d d^\Lambda$ and $d + d^\Lambda$ symplectic cohomology groups defined by Tseng and Yau[19].

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Received 10 May 2021

Accepted 15 December 2021

Published 24 April 2023