Integrating central extensions of Lie algebras via Lie 2-groups

Abstract

The purpose of this paper is to show how central extensions of (possibly infinitedimensional) Lie algebras integrate to central extensions ofétale Lie 2-groups in the sense of [Get09,Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π 2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finitedimensional Lie algebra.In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial π 2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions ofétale Lie 2-groups. As an application, we obtain a generalization of Lie's Third Theorem to infinite-dimensional Lie algebras.

String Principal Bundles and Courant Algebroids

“…This is a classical result. See [48,Proposition 2.4] for a concrete statement. See [22] and [48,Sect.…”

“…To build up a Lie 2-group, we first need to have an underlying Lie groupoid which presents the stack String p (G), and then establish a group structure "up to homotopy" on top of it. Here we follow the convention in [48,Section 2].…”

“…See [48,Proposition 2.4] for a concrete statement. See [22] and [48,Sect. 2] and references therein for definition and properties of the sheaf cohomology for simplicial objects.…”

Higher extensions of Lie algebroids

“…We have the following proposition which can be viewed as the global version of Lemma 4.2: (however, we shall not expect a classification result as in Theorem 4.5 with our current version of groupoid cohomology because even in the case of groups this version needs to be refined for the classification result to hold. See [WZ,Section 2] and [SP]).…”

Principal actions of stacky Lie groupoids

“…The reader can find more on stacky Lie groups e.g. in [7] (see also [2,22], and [55] for infinite dimensional examples arising from central extensions).…”