[Paper Review] Higher Extensions of Lie Algebroids and Application to Courant Algebroids

Yunhe Sheng, Chenchang Zhu|arXiv (Cornell University)|Mar 30, 2011

Homotopy and Cohomology in Algebraic Topology16 citations

TL;DR

This paper introduces a framework for extending Lie algebroids via representations up to homotopy, constructing higher Lie algebroids; it applies this to build exact Courant algebroids and string Lie 2-algebras, and achieves a Lie 2-groupoid integration of an exact Courant algebroid through this extension mechanism.

ABSTRACT

We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obtain a Lie 2-groupoid integration of an exact Courant algebroid.

Motivation & Objective

• To develop a systematic theory of Lie algebroid extensions using representations up to homotopy.
• To generalize semidirect products of Lie algebroids to include representations up to homotopy.
• To construct higher Lie algebroids as extensions, particularly focusing on exact Courant algebroids and string Lie 2-algebras.
• To apply the extension framework to achieve a Lie 2-groupoid integration of an exact Courant algebroid.

Proposed method

• Utilizes representations up to homotopy as the data for extending a Lie algebroid.
• Constructs the extension as a higher Lie algebroid structure, generalizing classical Lie algebroid extensions.
• Applies the construction to exact Courant algebroids, showing they arise naturally as such extensions.
• Applies the same framework to string Lie 2-algebras, demonstrating their realization as higher extensions.
• Employs the theory of Lie 2-groupoids to integrate the resulting higher Lie algebroids.
• Relies on homotopy-theoretic and categorified structures to generalize classical integration theorems.

Experimental results

Research questions

• RQ1 How can Lie algebroid extensions be generalized beyond classical representations using homotopy-theoretic data?
• RQ2 In what way do representations up to homotopy lead to higher Lie algebroid structures?
• RQ3 Can exact Courant algebroids be systematically constructed as extensions of Lie algebroids via representations up to homotopy?
• RQ4 How do string Lie 2-algebras emerge as special cases of such higher extensions?
• RQ5 Is it possible to integrate an exact Courant algebroid into a Lie 2-groupoid using this extension framework?

Key findings

• The extension of a Lie algebroid by a representation up to homotopy yields a higher Lie algebroid structure.
• Exact Courant algebroids are realized as specific instances of such higher extensions.
• String Lie 2-algebras are shown to arise naturally as extensions of Lie algebroids via representations up to homotopy.
• The framework enables a Lie 2-groupoid integration of an exact Courant algebroid, extending classical integration results.
• The construction generalizes semidirect products to the setting of representations up to homotopy, enriching the categorical structure of Lie algebroid extensions.