Integrable lifts for transitive Lie algebroids

Abstract

Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid, is the quotient of a finite dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an "Almeida-Molino" integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a "de Rham" integrable lift for any given transitive Abelian Lie algebroid. 1

On the integration of transitive Lie algebroids

“…It was shown by Tseng and Zhu [32] that the topological Lie groupoid G ⇒ M associated to any Lie algebroid has the structure of a 'stacky Lie groupoid'. In another direction, Androulidakis-Antonini [2] show that any transitive Lie groupoid with non-discrete monodromy groups admits a canonical lift to a transitive Lie algebroid over new manifold, in such a way that the monodromy groups do become discrete.…”

“…′ #A is defined by placing the squares for A ′ , A next to each other (horizontally), and concatenating.Let A ⇒ I 2 be a transitive Lie algebroid with a framing near ∂I 2 . It represents a trivial element of Tran k (I 2 , ∂I2 …”

Genus Integration, Abelianization, and Extended Monodromy

“…The notion of extended monodromy was introduced recently in [9], formalizing ideas from [10,15], as a mean to find obstructions to the existence of proper integrations of Poisson manifolds. It has also been used in [1]. We consider here the general case of a Lie algebroid and show what exactly these groups obstruct.…”