Almost Lie Algebroids and Characteristic Classes

TL;DR

This paper introduces almost Lie algebroids, extending Lie algebroid theory by relaxing the Jacobi identity, and constructs their cohomology and characteristic classes, showing these classes relate to the base space similarly to classical cases.

Contribution

It defines almost Lie algebroids, provides examples, and develops their cohomology and characteristic classes, extending the theory beyond strict Lie algebroids.

Findings

Characteristic classes are pull-backs of base space classes

A natural extension allows Lie algebroid brackets where not possible before

Provides explicit examples of almost Lie algebroids

Abstract

Almost Lie algebroids are generalizations of Lie algebroids, when the Jacobiator is not necessary null. A simple example is given, for which a Lie algebroid bracket or a Courant bundle is not possible for the given anchor, but a natural extension of the bundle and the new anchor allows a Lie algebroid bracket. A cohomology and related characteristic classes of an almost Lie algebroid are also constructed. We prove that these characteristic classes are all pull-backs of the characteristic classes of the base space, as in the case of a Lie algebroid.