Linear natural liftings of forms to Weil bundles with Weil algebras ${\mathbb D}^r_k$

Abstract

We give an explicit description and calculate the  dimension of the vector space of linear natural liftings of  $p$-forms on $n$-dimensional manifolds $M$ to $q$-forms on  $T^{{\mathbb D}^r_k}M$, where ${\mathbb D}^r_k$ is the Weil algebra of  $r$-jets at $0$ of smooth functions ${\mathbb R}^k\longrightarrow{\mathbb R}$,  for all non-negative integers $n$, $p$, $q$, $r$, $k$ except the case $p=n$  and $q=0$.