Algebraic constructions in the category of vector bundles

TL;DR

This paper develops algebraic and differential calculus tools for the category of generalized Lie algebroids, extending classical results and introducing new Maurer-Cartan type theorems and Cartan-type theorems.

Contribution

It introduces a generalized Lie algebroid framework with an exterior calculus, Maurer-Cartan type results, and Cartan-type theorems for interior differential systems.

Findings

Established an exterior differential calculus for generalized Lie algebroids

Derived a Maurer-Cartan type theorem in this context

Extended Cartan and Bianchi identities to generalized Lie algebroids

Abstract

The category of generalized Lie algebroids is presented. We obtain an exterior differential calculus for generalized Lie algebroids. In particular, we obtain similar results with the classical and modern results for Lie algebroids. So, a new result of Maurer-Cartan type is presented. Supposing that any vector subbundle of the pullback vector bundle of a generalized Lie algebroid is called interior differential system (IDS) for that generalized Lie algebroid, a theorem of Cartan type is obtained. Extending the classical notion of exterior differential system (EDS) to generalized Lie algebroids, a theorem of Cartan type is obtained. Using the theory of linear connections of Ehresmann type presented in the paper [1], the identities of Cartan and Bianchi type are presented.