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^A M for the case of a Frobenius Weil algebraA. 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^A M and that w^C is a linear combination of some "basic" Poisson structures on T^A M induced byw. Finally, we introduce the notion of a weakly symmetric Frobenius Weil algebra A and we compute the modular class of (T^A M, w^C) for such algebras.

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