Some results on the Mackenzie obstruction for transitive Lie algebroids

TL;DR

This paper investigates the Mackenzie obstruction in the context of transitive Lie algebroids, providing conditions under which this obstruction is trivial, thereby advancing understanding of their structural properties.

Contribution

It proves that the Mackenzie obstruction is trivial for transitive Lie algebroids when the Lie algebra is a direct sum of its center and a subalgebra without the center.

Findings

Mackenzie obstruction is trivial for certain Lie algebras.

The paper characterizes conditions for the existence of transitive Lie algebroids.

Provides theoretical results on the structure of Lie algebroids.

Abstract

The preprint is prepared as description of results that were obtained during joint scientific project No: 71NC /2015/VNCCCT on the VIASM (Vietnam Institute for Advanced Study in Mathematics) from 08.12.2015 to 06.02.2016. The problem was formulated how to calculate so called the Mackenzie obstruction for existing of transitive Lie algebroid for given coupling between a finite dimensional Lie algebra and the tangent bundle of a smooth manifold. It is proved that the Mackenzie obstruction for transitive Lie algebroids is trivial for the finite dimensional Lie algebra which is the direct sum of the center and the subalgebra without the center.