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Some linear Jacobi structures on vector bundles. (English. Abridged French version) Zbl 0983.53055

Summary: We study Jacobi 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 function. We prove that a Lie algebroid structure on \(A\) and a 1-cocycle \(\varphi\in\Gamma(A^*)\) induce a Jacobi structure on \(A^*\) satisfying the above conditions. Moreover, we show that this correspondence is a bijection. Finally, we discuss some examples and applications.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
17B66 Lie algebras of vector fields and related (super) algebras
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