ar X iv : m at h / 03 10 24 6 v 1 [ m at h . D G ] 1 6 O ct 2 00 3 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 to which ∆ is transversal. This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on M to the Schouten-Jacobi bracket of first-order polydifferential operators on N . A particular case is the well-known correspondence: Jacobi structures vs their poissonization. Mathematics Subject Classification (2000): 53D17, 53D10