Some linear Jacobi structures on vector bundles

We study Jacubi structures on the dual bundle A* to a vector bundle A such that the Jacobi bracket of linear functions is again linear and the Jacobi bracket of a linear function and the constant function 1 is a basic functio n. We prove that a Lie algebroid structure on A and a 1-cocycle phi is an e lement of Gamma(A*) induce a Jacobi structure on A* satisfying the above co nditions. Moreover, we show that this correspondence is a bijection. Finall y, we discuss some examples and applications. (C) 2000 Academie des science s/Editions scientifiques et medicales Elsevier SAS.