Associativity and Integrability

TL;DR

This paper extends classical results on Lie groups to Lie groupoids, linking the integrability of Lie algebroids with associativity failures in local integrations, and provides a construction for such extensions.

Contribution

It generalizes Mal'cev's theorem to groupoids, describes a method to obtain local Lie groupoids with integrable algebroids, and relates integrability to associativity failure.

Findings

Mal'cev's theorem extends to Lie groupoids

Construction method for local Lie groupoids with integrable algebroids

Associativity failure corresponds to monodromy groups in local integrations

Abstract

We provide a complete solution to the problem of extending a local Lie groupoid to a global Lie groupoid. First, we show that the classical Mal'cev's theorem, which characterizes local Lie groups that can be extended to global Lie groups, also holds in the groupoid setting. Next, we describe a construction that can be used to obtain any local Lie groupoid with integrable algebroid. Last, our main result establishes a precise relationship between the integrability of a Lie algebroid and the failure in associativity of a local integration. We give a simplicial interpretation of this result showing that the monodromy groups of a Lie algebroid manifest themselves combinatorially in a local integration, as a lack of associativity.