# Poisson structures on Weil bundles

## (*Lobachevskii Journal of Mathematics, Volume XVII*)

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 algebra *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 *^{A}M and that *w*^{C} is a linear combination of some "basic" Poisson structures on *T *^{A}M induced by *w*. 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.