# Integrating central extensions of Lie algebras via Lie 2-groups

Christoph Wockel; Chenchang Zhu

Journal of the European Mathematical Society (2016)

- Volume: 018, Issue: 6, page 1273-1320
- ISSN: 1435-9855

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topWockel, 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 -

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