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

Moukala, Norbert Mahoungou; Bossoto, Basile Guy Richard

Arch. Math., Brno 52, No. 2, 91-111 (2016).

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.

Reviewer: Josef Janyška (Brno)

MSC:

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

Keywords:

Weil bundle; Weil algebra; Poisson manifold; Lie derivative; Poisson 2-form