Integrating Lie algebroids via stacks

Abstract

Lie algebroids cannot always be integrated into Lie groupoids. We introduce a new structure, 'Weinstein groupoid', which may be viewed as stacky groupoids. We use this structure to present a solution to 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 such a groupoid.

Group-like objects in Poisson geometry and algebra

“…The best theory of such generalized morphisms comes about when M is acted upon by groupoids over X and Y . 1 The result is the theory of stacks, which we will describe in its smooth version [2] [3] [18]. (See [14] for the topological case.…”

Kan Replacement of Simplicial Manifolds

“…Thus, we can define an order relation on the collapsible simplicial sets: we say that S is not greater than T , and we write S ≺ T , if T = S t and S = S s with s ≤ t in Equation (11). By convention, the notation S ≺ T also indicates the inclusion map S → T .…”

“…A classical example is the case of Lie groups and Lie algebras, However, when the symmetries become more complicated, such as those of L ∞ -algebras, or L ∞ -algebroids, the integration and differentiation both become harder. The following problems have been solved for these higher symmetries: integration of nilpotent L ∞ -algebras by Getzler [5], integration of general L ∞ -algebras by Henriques [6], differentiation of L ∞ -groupoids byŠevera [13], both directions for Lie 1-algebroids by Cattaneo and Felder [2], Crainic and Fernandes [3], and from a higher viewpoint by Tseng and Zhu [11]. Here, the author wants to emphasize a middle step of local symmetries missing in the above correspondence, Indeed, to obtain infinitesimal symmetries by differentiation, we only need local symmetries.…”

The Volume of a Differentiable Stack

“…Conversely, if G 0 is any Poisson manifold for which the associated Lie algebroid structure on T * G 0 is integrable to a groupoid, then the integrating groupoid with simply-connected source fibres is a symplectic groupoid with underlying Poisson manifold G 0 . If we allow integration by stacks rather than just manifolds, then there is a bijective (up to natural isomorphisms) correspondence between Poisson manifolds and symplectic groupoids with simply-connected source fibres [15].…”