Prolongation of Poisson 2-form on Weil bundles

TL;DR

This paper extends the concept of Poisson 2-forms to Weil bundles, providing conditions under which the prolongation preserves the Poisson structure on these bundles.

Contribution

It introduces a method to prolong Poisson 2-forms to Weil bundles and establishes necessary and sufficient conditions for the resulting bundle to be an A-Poisson manifold.

Findings

Prolongation of Poisson 2-forms to Weil bundles is possible under specific conditions.

Necessary and sufficient conditions for Weil bundles to be A-Poisson manifolds are identified.

The construction generalizes Poisson structures to a broader geometric context.

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,_{M}) is a Poisson manifold with 2-form _{M}, we construct the 2-Poisson form _{M^{A}}^{A}, prolongation on M^{A} of the 2-Poisson form _{M}. We give a necessary and sufficent condition for that M^{A} be an A-Poisson manifold.