Poisson structures on Weil bundles

In the present paper, we construct complete lifts of covariant and contravariant tensor fields from the smooth manifold $M$ to its Weil bundle $T^{\mathbf A}M$ for the case of a Frobenius Weil algebra $\mathbf A$. For a Poisson manifold $(M,w)$ we show that the complete lift $w^C$ of a Poisson tensor $w$ is again a Poisson tensor on $T^{\mathbf A}M$ and that $w^C$ is a linear combination of some “basic” Poisson structures on $T^{\mathbf A}M$ induced by $w$. Finally, we introduce the notion of a weakly symmetric Frobenius Weil algebra $\mathbf A$ and we compute the modular class of $(T^{\mathbf A}M,w^C)$ for such algebras.