Algebraic Constructions in the Category of Lie Algebroids

TL;DR

This paper introduces a generalized framework for Lie algebroids, extending classical connection theory and providing new formulas and properties for these generalized structures, including special cases like Levi-Civita connections.

Contribution

It presents a novel generalized notion of Lie algebroids and develops a new perspective on linear connections within this framework, unifying and extending classical results.

Findings

Generalized tangent bundle for Lie algebroids constructed

New formulas of Ricci and Bianchi types derived

Classical Lie algebroid connections recovered as special cases

Abstract

A generalized notion of a Lie algebroid is presented. Using this, the Lie algebroid generalized tangent bundle is obtained. A new point of view over (linear) connections theory on a fiber bundle is presented. These connections are characterized by o horizontal distribution of the Lie algebroid generalized tangent bundle. Some basic properties of these generalized connections are investigated. Special attention to the class of linear connections is paid. The recently studied Lie algebroids connections can be recovered as special cases within this more general framework. In particular, all results are similar with the classical results. Formulas of Ricci and Bianchi type and linear connections of Levi-Civita type are presented.