The Schouten bracket is a natural binary operation on multi-vector fields, extended from the Lie bracket of vector fields.
One can study the algebra of multi-linear multi-differential operations on multi-vector fields, and insertion of operations into one another.
An insertion takes an $m$-ary operation and inserts its output into an input of an $n$-ary operation, to produce a $(n+m-1)$-ary operation.
Considering an abstract version of this (that maps to the concrete one), one is led to...
The graph complex introduced by Kontsevich is a differential graded Lie algebra with a basis consisting of isomorphism classes of undirected graphs. Here's an example of an element:
The Lie bracket of two graphs is the graded commutator with respect to an insertion operation: insert the right graph into a vertex of the left graph and again re-attach the incoming edges in all possible ways (sum over all possibilities).
The graph differential can be defined as the taking the bracket with the stick graph: $d(\gamma) = [\bullet\!\!\!-\!\!\!\bullet, \gamma]$.
It is known (from a combination of results by T. Willwacher and F. Brown) that cohomology classes $[\gamma_n]$ marked by $n$-wheels for odd $n \geqslant 3$ generate a free Lie algebra inside the degree-0 graph cohomology, and the Deligne–Drinfeld conjecture states that this is all there is.
From each graph cocycle $\gamma$ on $n$ vertices and $2n-2$ edges one obtains an $n$-ary operation $\operatorname{Op}(\gamma)$ on multi-vector fields.
For an arbitrary Poisson bi-vector field $P$ on $\mathbb{R}^d$ the formula $Q_\gamma(P) = \operatorname{Op}(\gamma)(P^{\otimes n})$ then defines a Poisson $2$-cocycle: $[P, Q_\gamma(P)] = 0$.
If $\gamma$ is a graph coboundary, then $Q_\gamma(P) = \operatorname{Op}(\gamma)(P^{\otimes n})$ is a Poisson coboundary.
Note: By construction, nonzero graph cohomology classes $\gamma$ act in such a way that $Q_\gamma(P)$ is not universally a Poisson $2$-coboundary.
However, it is a long open problem to actually find a non-trivial example of this action.
Open problem (Kontsevich, 1996): Find an example of a Poisson structure $P$ and a graph cocycle $\gamma$ such that $Q_\gamma(P)$ is not a Poisson $2$-coboundary, or prove that no such pair exists.
This website is dedicated to this open problem.