Complete lifting of double-linear semi-basic tangent valued forms to Weil like functors on double vector bundles

Abstract

Let $F$ be a product preserving gauge bundle functor on double vector bundles. We introduce the complete lifting $\mathcal{F}\phi :FK\to {\wedge }^p{T}^{\ast }FM\otimes TFK$ of a double-linear semi-basic tangent valued $p$-form $\phi :K\to {\wedge }^p{T}^{\ast }M\otimes TK$ on a double vector bundle $K$ with base $M$. We prove that this complete lifting preserves the Frolicher-Nijenhuis bracket. We apply the results obtained to double-linear connections.