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

TL;DR

This paper explicitly describes and calculates the dimension of the space of linear natural liftings of differential forms from manifolds to Weil bundles associated with Weil algebras, generalizing previous results.

Contribution

It provides a comprehensive explicit description and dimension calculation for linear natural liftings of forms to Weil bundles with Weil algebras, covering all relevant parameters.

Findings

Derived explicit formulas for the dimension of the space of liftings.

Identified the structure of these liftings for various parameters.

Extended previous work to a broader class of Weil algebras.

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 $q=0$.