Central extensions of infinite-dimensional Lie groups
type: Item
instance of: scholarly article
title: Central extensions of infinite-dimensional Lie groups
zbMATH Open document ID: 1019.22012
publication date: 2003-03-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116014
review text: The central result of this extensive paper is the following exact sequence for a connected Lie group \(G\), its universal covering group \(\widetilde G\), the discrete central subgroup \(\pi_1(G) \subseteq \widetilde G\), and an abelian Lie group \(Z\) which can be written as \(Z=z/ \Gamma\), where \(\Gamma \subseteq z\) is a discrete subgroup: \[ \begin{multlined} \Hom(G,Z) \hookrightarrow \Hom(\widetilde G,Z)\to \Hom\bigl(\pi_1(G), Z\bigr)\to\text{Ext}_{\text{Lie}} (G,Z)\to\\ \to H^2_c(g,z) \to\Hom\bigl( \pi_2(G),Z\bigr) \times \Hom\bigl(\pi_1(G),\text{Lin} \bigl(g,z)\bigr),\end{multlined} \tag{1} \] where \(\text{Lin} (g,z)\) denotes the space of continuous linear maps \(g\to z\). The author describes the course of the paper as follows: ``In Section 2 we collect the necessary results on central extensions of topological groups and in Section 3 we provide some results on infinite-dimensional manifolds and Lie groups which are well-known in the finite-dimensional case. In Section 4 we explain how the setting for abstract, resp., topological groups has to be modified to deal with central extensions of Lie groups with smooth local sections. Section 5 is dedicated to the construction of the period homomorphism \(\text{per}_\omega: \pi_2 (G)\to z\). Section 6, which is the heart of the paper, contains the construction of a global group cocycle \(f:G\times G\to Z\) for simply connected groups \(G\) and any Lie algebra cocycle \(\omega\), where \(Z\) can be defined as \(z/ \Pi_\omega\). The so defined group \(Z\) is a Lie group iff \(\Pi_\omega\) is discrete, so that we obtain a Lie group extension iff \(\Pi_\omega\) is discrete. In Section 7 we eventually put all pieces together to prove the exactness of (1). The central result of Section 8 is Theorem 8.8 which gives a version of the exact sequence (1) for central Lie group extensions with smooth global sections. Section 9 is a collection of examples displaying various aspects in the description of the group \(\text{Ext…
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 58B05
zbMATH DE Number: 1880701
zbMATH Keywords: infinite-dimensional Lie group
zbMATH Keywords: invariant form
zbMATH Keywords: central extension
zbMATH Keywords: period map
zbMATH Keywords: Lie group cocycle
zbMATH Keywords: homotopy group
zbMATH Keywords: local cocycle
zbMATH Keywords: diffeomorphism group
author: Karl-Hermann Neeb
reviewed by: A. Klíč
MaRDI profile type: Publication
OpenAlex ID: W1863645054
Wikidata QID: Q115159047
cites work: Q3136738
cites work: Q3125206
cites work: Loop spaces, characteristic classes and geometric quantization
cites work: Sur les extensions des groupes topologiques
cites work: Q5842420
cites work: Espaces fibres en algèbres de Lie et en groupes
cites work: Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space
cites work: Q5732711
cites work: Q3804940
cites work: Relations between Homology and Homotopy Groups
cites work: Cohomology theory in abstract groups. II
cites work: Q5514215
cites work: Q3845399
cites work: Q3797348
cites work: Q5609457
cites work: Q4788984
cites work: Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups
cites work: Algebras whose groups of units are Lie groups
cites work: The construction of a simply connected Lie group with a given Lie algebra
cites work: Q5622916
cites work: The inverse function theorem of Nash and Moser
cites work: Principal bundles and group extensions with applications to Hopf algebras
cites work: Differential Topology
cites work: Group extensions of Lie groups. I, II
cites work: The structure of compact groups. A primer for the student -- a handbook for the expert
cites work: Homotopical cohomology and Čech cohomology
cites work: Q4360299
cites work: Q4226699
cites work: Q4788985
cites work: Les ensembles boreliens et les extensions des groupes
cites work: Convex Structures and Continuous Selections
cites work: Q3724586
cites work: Central extensions of current groups
cites work: Q4796254
cites work: Q2734524
cites work: Universal central extensions of Lie groups
cites work: Q4332327
cites work: Holomorphic highest weight representations of infinite dimensional complex classical groups
cites work: Q4331753
cites work: On the homotopy type of certain groups of operators
cites work: Homotopy theory of infinite dimensional manifolds
cites work: Q3754832
cites work: Central extensions of infinite-dimensional Lie algebras and Lie groups, Virasoro algebra and generalizations
cites work: Q5628962
cites work: Unitary representations of some infinite dimensional groups
cites work: Group extensions of compact Lie groups
cites work: Q4112906
cites work: Q5522742
cites work: Continuous cohomology of groups and classifying spaces
cites work: Q5825483
cites work: Q3998776
cites work: Q4743009
cites work: Integrating unitary representations of infinite-dimensional Lie groups
cites work: Central extensions and physics
cites work: Q3702812

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cites work: Q3664070

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cites work: Q3874497

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cites work: Q4863380

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Central extensions of infinite-dimensional Lie groups (DOI)
DOI: 10.5802/AIF.1921

Central extensions of infinite-dimensional Lie groups (published in)
published in: Annales de l’institut Fourier