ar X iv : m at h / 03 10 24 6 v 2 [ m at h . D G ] 1 5 Fe b 20 04 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 ∆ on M and first-order polydifferential operators on a closed submanifold N of codimension 1 such that ∆ 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 ∆-homogeneous symplectic structures on M and contact structures on N . Mathematics Subject Classification (2000): 53D17, 53D10