Integrating central extensions of Lie algebras via Lie 2-groups

Wockel, Christoph, and Zhu, Chenchang. "Integrating central extensions of Lie algebras via Lie 2-groups." Journal of the European Mathematical Society 018.6 (2016): 1273-1320. <http://eudml.org/doc/277426>.

@article{Wockel2016,
abstract = {The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale 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 $\pi _2$ 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$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 étale Lie 2-groups. As an application, we obtain a generalization of Lie’s Third Theorem to infinite-dimensional Lie algebras.},
author = {Wockel, Christoph, Zhu, Chenchang},
journal = {Journal of the European Mathematical Society},
keywords = {infinite-dimensional Lie group; central extension; smooth group cohomology; group stack; Lie 2-group; integration of cocycles; Lie’s Third Theorem; 2-connected cover; infinite-dimensional Lie group; central extension; smooth group cohomology; group stack; Lie 2-group; integration of cocycles; Lie's third theorem; 2-connected cover},
language = {eng},
number = {6},
pages = {1273-1320},
publisher = {European Mathematical Society Publishing House},
title = {Integrating central extensions of Lie algebras via Lie 2-groups},
url = {http://eudml.org/doc/277426},
volume = {018},
year = {2016},
}

TY - JOUR
AU - Wockel, Christoph
AU - Zhu, Chenchang
TI - Integrating central extensions of Lie algebras via Lie 2-groups
JO - Journal of the European Mathematical Society
PY - 2016
PB - European Mathematical Society Publishing House
VL - 018
IS - 6
SP - 1273
EP - 1320
AB - The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale 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 $\pi _2$ 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$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 étale Lie 2-groups. As an application, we obtain a generalization of Lie’s Third Theorem to infinite-dimensional Lie algebras.
LA - eng
KW - infinite-dimensional Lie group; central extension; smooth group cohomology; group stack; Lie 2-group; integration of cocycles; Lie’s Third Theorem; 2-connected cover; infinite-dimensional Lie group; central extension; smooth group cohomology; group stack; Lie 2-group; integration of cocycles; Lie's third theorem; 2-connected cover
UR - http://eudml.org/doc/277426
ER -