A Poisson bracket on the space of Poisson structures (Q6042869)

Let \(M\) be a smooth, closed and orientable manifold. The author considers the set of all Poisson structures on \(M\), denoted \(\mathcal{P}(M)\) and shows that \(\mathcal{P}(M)\) has itself a family of Poisson structures \(\{\,,\}_{\mu}\), depending on a choice of a volume form \(\mu\). The motivation for this work comes from ideal fluid dynamics. The aim is to extend the Poisson bracket on the space of Poisson structures and to define a Poisson bracket on the space of admissible functions. The bracket is explicitly given and the corresponding proofs are detailed. The author considers also the space of symplectic manifolds with a symplectic volume form. In this case, he constructs a further and related Poisson bracket. He studies the induced flow and gives a description in terms of the symplectic cohomology groups introduced by \textit{L.-S. Tseng} and \textit{S.-T. Yau} [J. Differ. Geom. 91, No. 3, 383--416 (2012; Zbl 1275.53079)].