Local and global integrability of Lie brackets

Integrability of Lie Brackets

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-mode...

Lectures on Integrability of Lie Brackets

Integrability of Poisson Brackets

We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express the obstructions in terms of variations of symplectic areas. As an application ...

Prequantization and Lie Brackets

We start by describing the relationship between the classical prequantization condition and the integrability of a certain Lie algebroid associated to the problem and use this to give a global construction of the prequantizing bundle in terms of path spaces (Introduction), then we rephrase the problem in terms of groupoids (second section), and then we study the more general problem of prequant...

Lie Algebras and Lie Brackets of Lie Groups–matrix Groups

The goal of this paper is to study Lie groups, specifically matrix groups. We will begin by introducing two examples: GLn(R) and SLn(R). Then in each section we will prove basic results about our two examples and then generalize these results to general matrix groups.