Lie algebroid cohomology and Lie algebroid extensions

TL;DR

This paper investigates the extension problem for Lie algebroids over schemes, identifying obstructions and classifying extensions using Lie algebroid hypercohomology, thus advancing the understanding of their structure and cohomological properties.

Contribution

It provides a new cohomological framework for classifying Lie algebroid extensions over schemes, including explicit obstruction criteria.

Findings

Obstruction to extension is characterized by a specific hypercohomology class.

Extensions are classified via Lie algebroid hypercohomology groups.

Preliminary results on free Lie algebroids support the main classification theory.

Abstract

We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.