On Weil like functors on flag vector bundles with given length

Abstract

The complete description is given of the product preserving gauge bundle functors  $F$ on the category $\mathcal{F_\kappaVB}$ of flag vector bundles $K=(K;K_1,...,K_\kappa)$ of length $\kappa$ in terms of systems $I=(I_1,...,I_{\kappa-1})$ of $A$-module homomorphisms $I_i:V_{i+1}\to V_i$ for Weil algebras $A$ and finite dimensional (over $\mathbf{R}$)  A$-modules $V_1,...,V_\kappa$. The so called iteration problem is investigated. The natural affinors on $FK$ are classified. The gauge natural operators $C$ lifting $\kappa$-lag-linear vector fields on $K$ to vector fields $C(X)$ on $FK$ are completely described. The concept of the complete lift $\mathcal{F}\varphi$ of a $\kappa$-flag-linear semi-basic tangentvalued $p$-form $\varphi$ on $K$ is introduced.That the complete lifting $\mathcal{F}\varphi$ preserves the Frolicher-Nijenhuis bracket is observed. The obtained results are applied to study prolongation and torsion of $\kappa$-flag-linear connections.