Integrability of Lie brackets

By M. Crainic, Rui Loja Fernandes. 2001

533 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/402f3717a3af5f7305b88e38e71037b8218a47e7

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Lectures on integrability of Lie brackets (2006, unknown): “The monodromy groups were introduced by us in [15].”
Geometric structures as deformed infinitesimal symmetries (2004, methodology): “By the groupoid version of Lie’s Second Fundamental Theorem (see, e.g., [5]), the Lie algebroid morphism ∇ lifts to a groupoid morphism Φ: M × M → GL( E ).”
PROPER GROUPOIDS AND MOMENTUM MAPS: LINEARIZATION, AFFINITY, AND CONVEXITY GROUPOPROPRES ET APPLICATIONS MOMENT: LIN ´ EARISATION, CARACT ` ERE AFFINE, ET CONVEXIT ´ E (2004, background): “3 and of CranicFernandes’ theorem about integrability of Lie algebroids [6] might lead to results about (partial) linearization of Lie algebroids and Poisson structures.”
Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model (2003, methodology): “Using the language of Lie algebroids, one may also give the following interpretation [9,19]: Elements of C(M) are precisely those bundle maps that are also morphisms of Lie algebroids, where the tangent bundles are given the canonical Lie algebroid structure and T ∗M the one induced from the Poisson”
On the universal enveloping algebra of a Lie algebroid (2010, background): “However, not every Lie algebroid is integrable; for a characterization of integrability, see [2].”
Universal lifting theorem and quasi-Poisson groupoids (2005, methodology): “The general case was due to Crainic-Fernandes [7].”
Interface and mixed boundary value problems on $n$-dimensional polyhedral domains (2010, unknown): “Documenta Mathematica 15 (2010) 687–745”
The Weil algebra and the Van Est isomorphism (2009, background): “The precise conditions for the integrability are described in [9].”
Multiplicative forms at the infinitesimal level (2010, methodology): “For an integrable Lie algebroid A, there is an explicit model for its integrating source-simply-connected Lie groupoid G(A) [12, 27]; namely, G(A) is the quotient PA/ ∼, where PA is the subspace of A-paths in the Banach manifold of all C1 paths from the interval I = [0, 1] into A with C2 projection ”
Differentiable stacks and gerbes (2006, background): “Remark It would be interesting to investigate how the integrability condition of CrainicFernandes [10] is related to the theorem above.”
Maximally homogeneous para-CR manifolds of semisimple type (2006, unknown): “[11] for the necessary and sufficient conditions for the integrability of Lie algebroids).”