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Lie Algebroids

Eckhard Meinrenken

TL;DR

Lie algebroids provide a unifying infinitesimal framework for geometric structures by pairing a vector bundle $A\to M$ with a bracket on sections and an anchor $\mathsf a:A\to TM$ that satisfies a Leibniz-type rule. The paper outlines the core definitions, morphisms, pullbacks, and constructions, then develops the orbit foliation and splitting theorem to describe local models. It then connects Lie algebroids to Lie groupoids via the Lie functor, detailing integrability obstructions through monodromy groups and the Crainic–Fernandes integrability theorem, and finally presents the Lie algebroid complex, its contravariant viewpoint, and the Van Est map, linking algebroid cohomology to groupoid cohomology. The outlook highlights numerous directions, including bialgebroids, Dirac structures, and applications in Poisson and differential geometry. Overall, the work provides a consolidated perspective on infinitesimal-global correspondences and foundational tools for studying Lie algebroids and their integrations.

Abstract

This is an overview article on Lie algebroids, and their role as the infinitesimal counterparts of Lie groupoids.

Lie Algebroids

TL;DR

Lie algebroids provide a unifying infinitesimal framework for geometric structures by pairing a vector bundle with a bracket on sections and an anchor that satisfies a Leibniz-type rule. The paper outlines the core definitions, morphisms, pullbacks, and constructions, then develops the orbit foliation and splitting theorem to describe local models. It then connects Lie algebroids to Lie groupoids via the Lie functor, detailing integrability obstructions through monodromy groups and the Crainic–Fernandes integrability theorem, and finally presents the Lie algebroid complex, its contravariant viewpoint, and the Van Est map, linking algebroid cohomology to groupoid cohomology. The outlook highlights numerous directions, including bialgebroids, Dirac structures, and applications in Poisson and differential geometry. Overall, the work provides a consolidated perspective on infinitesimal-global correspondences and foundational tools for studying Lie algebroids and their integrations.

Abstract

This is an overview article on Lie algebroids, and their role as the infinitesimal counterparts of Lie groupoids.
Paper Structure (19 sections, 6 theorems, 25 equations)

This paper contains 19 sections, 6 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Definitions and basic properties
  3. Definition and examples
  4. Subalgebroids
  5. Morphisms and isomorphisms
  6. Pullbacks
  7. Other constructions
  8. Orbits, splitting theorem
  9. Orbits of a Lie algebroid
  10. Splitting theorem
  11. The correspondence between Lie groupoids and Lie algebroids
  12. The Lie functor
  13. Integrability
  14. Monodromy groups
  15. Crainic-Fernandes integrability theorem
  16. ...and 4 more sections

Key Result

Theorem 3.1

bur:spl There exists an open neighborhood $U\subseteq \nu(M,N)$ of the zero section and a tubular neighborhood embedding $\phi\colon U \to M$, lifting to a Lie algebroid isomorphism

Theorems & Definitions (7)

  • Definition 2.1: Pradines pra:th
  • Theorem 3.1: Linearization along transversals
  • Theorem 3.2: Splitting theorem
  • Theorem 4.1
  • Theorem 4.2: Crainic-Fernandes
  • Theorem 5.1
  • Theorem 5.2