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Integrating Lie algebroids via stacks and applications to Jacobi manifolds

Chenchang Zhu

TL;DR

This work addresses the long-standing problem of integrating Lie algebroids by replacing the classical target—Lie groupoids—with Weinstein groupoids, differentiable stacks equipped with groupoid-like laws that hold up to 2-morphisms. It proves a stacky version of Lie's third theorem: every Weinstein groupoid has a Lie algebroid and every Lie algebroid integrates to a Weinstein groupoid, via two canonical constructions G(A) and H(A) that reflect monodromy and holonomy data. The framework is then applied to Jacobi manifolds by constructing contact groupoids and relating Jacobi data to Poissonization, enabling a unified treatment of integrability and prequantization for Poisson and Jacobi structures. The results illuminate when Poisson bivectors and Jacobi structures admit global groupoid integration, and they connect integrability to the exactness of symplectic forms and period groups, with concrete examples and a clear path toward quantization via prequantization of symplectic groupoids.

Abstract

Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid, and every Lie algebroid can be integrated into a Weinstein groupoid. Furthermore, we apply this general result to Jacobi manifolds and construct contact groupoids for Jacobi manifolds. There are further applications in prequantization and integrability of Poisson bivectors.

Integrating Lie algebroids via stacks and applications to Jacobi manifolds

TL;DR

This work addresses the long-standing problem of integrating Lie algebroids by replacing the classical target—Lie groupoids—with Weinstein groupoids, differentiable stacks equipped with groupoid-like laws that hold up to 2-morphisms. It proves a stacky version of Lie's third theorem: every Weinstein groupoid has a Lie algebroid and every Lie algebroid integrates to a Weinstein groupoid, via two canonical constructions G(A) and H(A) that reflect monodromy and holonomy data. The framework is then applied to Jacobi manifolds by constructing contact groupoids and relating Jacobi data to Poissonization, enabling a unified treatment of integrability and prequantization for Poisson and Jacobi structures. The results illuminate when Poisson bivectors and Jacobi structures admit global groupoid integration, and they connect integrability to the exactness of symplectic forms and period groups, with concrete examples and a clear path toward quantization via prequantization of symplectic groupoids.

Abstract

Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid, and every Lie algebroid can be integrated into a Weinstein groupoid. Furthermore, we apply this general result to Jacobi manifolds and construct contact groupoids for Jacobi manifolds. There are further applications in prequantization and integrability of Poisson bivectors.

Paper Structure

This paper contains 42 sections, 53 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Differentiable stacks
  3. Stacks over the category of differentiable (or Banach) manifolds
  4. The definitions
  5. Representability
  6. Differentiable (Banach) stacks
  7. Morphisms and 2-morphisms
  8. Lie groupoids and differentiable stacks
  9. From stacks to groupoids
  10. From groupoids to stacks
  11. Morita equivalence
  12. Morphisms and 2-morphisms---in the world of groupoids
  13. Morphisms
  14. 2-morphisms
  15. Invariant maps
  16. ...and 27 more sections

Key Result

Theorem 1.0.2

To each Weinstein groupoid one can associate a Lie algebroid. For every Lie algebroid $A$, there are naturally two Weinstein groupoids $\mathcal{G}(A)$ and $\mathcal{H}(A)$ with Lie algebroid $A$.

Theorems & Definitions (175)

  • Definition 1.0.1: Weinstein groupoid
  • Theorem 1.0.2: Lie's third theorem
  • Conjecture 1.0.3
  • Conjecture 1.0.4: Lie's second theorem
  • Theorem 1.0.5
  • Theorem 1.0.6
  • Theorem 1.0.7
  • Theorem 1.0.8
  • Theorem 1.0.9
  • Theorem 1.0.10
  • ...and 165 more