Integrating central extensions of Lie algebras via Lie 2-groups

TL;DR

This paper demonstrates how central extensions of infinite-dimensional Lie algebras can be integrated into central extensions of étale Lie 2-groups, overcoming obstructions present in traditional Lie group integration.

Contribution

It introduces a method to integrate central extensions of Lie algebras into étale Lie 2-groups, generalizing Lie's Third Theorem to infinite dimensions.

Findings

Central extensions of Lie algebras can be integrated into étale Lie 2-groups.

Overcomes obstructions caused by non-trivial  in infinite-dimensional cases.

Provides a framework for extending Lie theory beyond finite dimensions.

Abstract

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of \'etale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of \pi_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 finite-dimensional 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 \pi_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 of \'etale Lie 2-groups. As an application, we obtain a generalization of Lie's Third Theorem to infinite-dimensional Lie algebras.