Integrability of Lie brackets

2003/03/01 by Marius Crainic, Rui Loja Fernandes · 15 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra #Algebraic structures and combinatorial models

paper · pdf · doi:10.4007/annals.2003.157.575

openalex publication_date 2003/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications.

Citations

• Lie Groupoids and Lie Algebroids in Differential Geometry
• Espaces fibr�s en alg�bres de Lie et en groupes
• Integration of Lie bialgebroids
• On integrability of infinitesimal actions
• Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes

Cited by

• Homotopy of representations up to homotopy
• Lagrangian Mechanics and Reduction on Fibered Manifolds
• General Theory of Lie Groupoids and Lie Algebroids
• Integrating Lie algebroids via stacks
• Symplectic algebroids, groupoid Toeplitz operators and deformation quantization
• An Explicit Construction of \mathbbS¹-Gerbes over the Stack [G/G]
• Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions
• Lie algebroid fibrations
• GROUPOIDS AND POISSON SIGMA MODELS WITH BOUNDARY
• On the integration of Poisson homogeneous spaces
• General Yang–Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical frameworkorFrom Bianchi identities to twisted Courant algebroids
• Poisson manifolds and their associated stacks
• Singular spaces, groupoids and metrics of positive scalar curvature
• Almost regular Poisson manifolds and their holonomy groupoids
• Lie algebroid [wikipedia]