We introduce the concept of modified vertical Weil functors on the category \mathcal{F}\mathcal{M}_m of fibred manifolds with m-dimensional bases and their fibred maps with embeddings as base maps. Then we describe all fiber product preserving bundle functors on \mathcal{F}\mathcal{M}_m in terms of modified vertical Weil functors. The construction of modified vertical Weil functors is an (almost direct) generalization of the usual vertical Weil functor. Namely, in the construction of the usual vertical Weil functors, we replace the usual Weil functors T^{a} corresponding to Weil algebras A by the so called modified Weil functors T^{a} corresponding to Weil algebra bundle functors A on the category \mathcal{F}\mathcal{M}_m of m-dimensional manifolds and their embeddings., Włodzimierz M. Mikulski., and Obsahuje seznam literatury
We define equivariant tensors for every non-negative integer $p$ and every Weil algebra $A$ and establish a one-to-one correspondence between the equivariant tensors and linear natural operators lifting skew-symmetric tensor fields of type $(p,0)$ on an $n$-dimensional manifold $M$ to tensor fields of type $(p,0)$ on $T^AM$ if $1\le p\le n$. Moreover, we determine explicitly the equivariant tensors for the Weil algebras ${\mathbb D}^r_k$, where $k$ and $r$ are non-negative integers.
Let $A$ be a Weil algebra. The bijection between all natural operators lifting vector fields from $m$-manifolds to the bundle functor $K^A$ of Weil contact elements and the subalgebra of fixed elements $SA$ of the Weil algebra $A$ is determined and the bijection between all natural affinors on $K^A$ and $SA$ is deduced. Furthermore, the rigidity of the functor $K^A$ is proved. Requisite results about the structure of $SA$ are obtained by a purely algebraic approach, namely the existence of nontrivial $SA$ is discussed.
We characterize Weilian prolongations of natural bundles from the viewpoint of certain recent general results. First we describe the iteration $F(EM)$ of two natural bundles $E$ and $F$. Then we discuss the Weilian prolongation of an arbitrary associated bundle. These two auxiliary results enables us to solve our original problem.
We establish a formula for the Schouten-Nijenhuis bracket of linear liftings of skew-symmetric tensor fields to any Weil bundle. As a result we obtain a construction of some liftings of Poisson structures to Weil bundles.