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Prolongation of Poisson \(2\)-form on Weil bundles. (English) Zbl 1389.58001

Let \(M\) be an \(n\)-dimensional smooth manifold and \(A\) be a Weil algebra. The associated Weil bundle is denoted as \(\pi_{M}:M^A\to M\). Geometrical objects on \(M\) can be lifted to geometrical object on \(M^A\). In the paper the authors study lifts of 2-forms \(\omega_M\) on \(M\) to \(A\)-valued 2-forms \(\omega^A_{M^A}\). If \((M,\omega_M)\) is a Poisson manifold then a necessary and sufficient condition for \((M^A,\omega^A_{M^A})\) to be an \(A\)-Poisson manifold are found.

MSC:

58A20 Jets in global analysis
58A32 Natural bundles
17B63 Poisson algebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
53D05 Symplectic manifolds (general theory)
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