[Paper Review] A construction of local Lie groupoids using Lie algebroid sprays

Alejandro Cabrera|arXiv (Cornell University)|Jan 1, 2017

Homotopy and Cohomology in Algebraic Topology9 references3 citations

TL;DR

This paper presents a direct, connection-based construction of a local Lie groupoid—called a spray groupoid—that integrates any given Lie algebroid, using Lie algebroid sprays. The method yields explicit formulas for integrating infinitesimal multiplicative objects and provides concrete integrations of geometric structures such as Poisson, Dirac, and Jacobi structures via corresponding local symplectic, presymplectic, or contact groupoids.

ABSTRACT

We give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a connection. We also give a complete account of local Lie theory based on these explicit constructions. On the resulting local Lie groupoid, called a spray groupoid, we obtain formulas for integrating infinitesimal multiplicative objects. These general results produce concrete integrations of several geometrical structures: (Nijenhuis-)Poisson, Dirac, Jacobi structures by local symplectic (Nijenhuis), presymplectic, contact groupoids, respectively. Joint work with A. Cabrera and I. Marcut.

Motivation & Objective

• To provide a self-contained, explicit construction of a local Lie groupoid integrating a given Lie algebroid.
• To eliminate dependence on abstract or indirect integration methods by relying solely on the choice of a connection.
• To establish a foundation for local Lie theory through concrete geometric constructions.
• To derive explicit formulas for integrating infinitesimal multiplicative objects on the resulting groupoid.
• To apply the framework to integrate specific geometric structures—Poisson, Dirac, Jacobi—into corresponding local groupoids.

Proposed method

• Utilizes Lie algebroid sprays as the central geometric tool to construct the local groupoid structure.
• Constructs the local groupoid via integral curves of the spray vector field associated with a chosen connection.
• Employs a connection on the Lie algebroid to define a canonical spray, ensuring the construction is explicit and independent of auxiliary choices.
• Applies the resulting groupoid structure to integrate infinitesimal multiplicative tensors and geometric objects.
• Derives explicit integration formulas by leveraging the spray's homogeneity and curvature properties.
• Applies the framework to known geometric structures, showing that the spray groupoid naturally induces symplectic, presymplectic, or contact groupoid structures.

Experimental results

Research questions

• RQ1 How can a local Lie groupoid be explicitly constructed from a Lie algebroid using only a connection?
• RQ2 What is the role of Lie algebroid sprays in enabling a direct integration of infinitesimal multiplicative objects?
• RQ3 How do the resulting spray groupoids integrate specific geometric structures such as Poisson or Jacobi structures?
• RQ4 Can the construction be made self-contained and independent of abstract or indirect integration techniques?
• RQ5 What are the explicit formulas for integrating multiplicative tensors on the resulting local groupoid?

Key findings

• A direct, explicit, and self-contained construction of a local Lie groupoid integrating any Lie algebroid is achieved using Lie algebroid sprays.
• The construction depends only on the choice of a connection, making it canonical and computationally accessible.
• The resulting groupoid, termed the spray groupoid, supports explicit formulas for integrating infinitesimal multiplicative objects.
• The framework provides concrete integrations of Nijenhuis-Poisson, Dirac, and Jacobi structures via local symplectic, presymplectic, and contact groupoids, respectively.
• The method establishes a complete local Lie theory based on these explicit constructions, unifying various geometric integration problems.
• The spray groupoid construction naturally extends to multiplicative structures, offering a systematic approach to integrating geometric objects in Poisson and Jacobi geometry.