On differential operators generating higher brackets

On supermanifolds, a Poisson structure can be either even, corresponding to a Poisson bivector, or odd, corresponding to an odd Hamiltonian quadratic in momenta. An odd Poisson bracket can also be defined by an odd second-order differential operator that squares to zero, known as a "BV-type" operator.
A higher analog, $P_\infty$ or $S_\infty$, is a series of brackets of alternating parities or all odd, respectively, that satisfy relations that are higher homotopy analogs of the Jacobi identity. These brackets are generated by arbitrary multivector fields or Hamiltonians. However, generating an $S_\infty$-structure by a higher-order differential operator is not straightforward, as this would violate the Leibniz identities. Kravchenko and others studied these structures, and Voronov addressed the Leibniz identity issue by introducing formal $\hbar$-differential operators.
In this talk, we revisit the construction of an $\hbar$-differential operator that generates higher Koszul brackets on differential forms on a $P_\infty$-manifold.
It is well known that a chain map between the de Rham and Poisson complexes on a Poisson manifold at the same time maps the Koszul bracket of differential forms to the Schouten bracket of multivector fields. In the $P_\infty$-case, however, the chain map is also known, but it does not connect the corresponding bracket structures. An $L_\infty$-morphism from the higher Koszul brackets to the Schouten bracket has been constructed recently, using Voronov's thick morphism technique. In this talk, we will show how to lift this morphism to the level of operators.
The talk is partly based on joint work with Yagmur Yilmaz.