Poisson-Jacobi reduction of homogeneous tensors

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold $M$, homogeneous with respect to a vector field $\mathrm{\Delta }$ on $M$, and first-order polydifferential operators on a closed submanifold $N$ of codimension 1 such that $\mathrm{\Delta }$ is transversal to $N$. This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on $M$ to the Schouten-Jacobi bracket of first-order polydifferential operators on $N$ and generalizes the Poissonization of Jacobi manifolds. Actually, it can be viewed as a super-Poissonization. This procedure of passing from a homogeneous multivector field to a first-order polydifferential operator can be also understood as a sort of reduction; in the standard case – a half of a Poisson reduction. A dual version of the above correspondence yields in particular the correspondence between $\mathrm{\Delta }$-homogeneous symplectic structures on $M$ and contact structures on $N$.