Prolongation of Poisson $2$-form on Weil bundles

Norbert Mahoungou Moukala and Basile Guy Richard Bossoto

Address: Marien Ngouabi University, BP:69, Brazzaville, Congo

Institut de Recherche en Sciences Exactes et Naturelles (IRSEN), Brazzaville, Congo

E-mail:
bossotob@yahoo.fr
nmahomouk@yahoo.fr

Abstract: In this paper, $M$ denotes a smooth manifold of dimension $n$, $A$ a Weil algebra and $M^{A}$ the associated Weil bundle. When $(M,\omega _{M})$ is a Poisson manifold with $2$-form $\omega _{M}$, we construct the $2$-Poisson form $\omega _{M^{A}}^{A}$, prolongation on $M^{A}$ of the $2$-Poisson form $\omega _{M}$. We give a necessary and sufficient condition for that $M^{A}$ be an $A$-Poisson manifold.

AMSclassification: primary 58A20; secondary 58A32, 17D63, 53D17, 53D05.

Keywords: Weil bundle, Weil algebra, Poisson manifold, Lie derivative, Poisson 2-form.

DOI: 10.5817/AM2016-2-91