Lifts of Poisson structures to Weil bundles

TL;DR

This paper explores how Poisson structures on a manifold can be lifted to its Weil bundle, demonstrating that these lifts preserve Poisson properties and affect cohomology and modular classes.

Contribution

It introduces methods for lifting Poisson tensors to Weil bundles and analyzes their properties and cohomological implications.

Findings

Complete and vertical lifts of Poisson tensors are themselves Poisson tensors.

Lifts induce homomorphisms in Poisson cohomology.

Modular classes of the lifted structures are computed.

Abstract

In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold $M$ to its Weil bundle $T^A M$ defined by a Frobenius Weil algebra $A$. For a Poisson manifold $(M,w)$, we show that the complete lift $w^C$ and the vertical lift $w^V$ of the Poisson tensor $w$ are Poisson tensors on $T^A M$ and establish their properties. We prove that the complete and the vertical lifts induce homomorphisms of the Poisson cohomology spaces. We compute the modular classes of the lifts of Poisson structures.