Central extensions of current groups
type: Item
instance of: scholarly article
title: Central extensions of current groups
zbMATH Open document ID: 1029.22025
author: Peter Maier
author: Karl-Hermann Neeb
published in: Mathematische Annalen
publication date: 2003-09-16T00:00:00Z
review text: Let \(G\) be the identity component of the current group \(C^\infty(M,K)\), where \(M\) is a compact manifold and \(K\) a finite- or infinite-dimensional Lie group. The authors consider continuous Lie algebra \(2\)-cocycles \(\omega\) on the Lie algebra \({\mathfrak g}=C^\infty(M,{\mathfrak k})\) of \(G\) which are of ``product type'', meaning that \(\omega(\xi,\eta):=[\kappa(\xi,d\eta)]\in {\mathfrak z} :=\Omega^1(M,Y)/ dC^\infty(M,Y)\) for some continuous, symmetric, invariant bilinear map \(\kappa : {\mathfrak k}\times {\mathfrak k}\to Y\) on the Lie algebra \({\mathfrak k}\) of \(K\), with values in a sequentially complete, locally convex space \(Y\). The article is centered around the question of whether there exists a central Lie group extension \(\widehat{G}\) of \(G\) corresponding to \(\omega\). It is known from the general extension theory developed by the second author [Ann. Inst. Fourier 52, 1365-1442 (2002; Zbl 1019.22012)] that \(\widehat{G}\) exists if and only if the period group \(\Pi_{M,\kappa}\subseteq {\mathfrak z}\) (the image of the period homomorphism \(\text{per}_\omega : \pi_2(G)\to {\mathfrak z}\)) is discrete and the adjoint action of \({\mathfrak g}\) on \({\mathfrak g}\oplus_\omega {\mathfrak z}\) integrates to a smooth representation of \(G\). The second condition is always satisfied in the present context; the Lie algebra action even integrates to a smooth representation of \(C^\infty(M,K)\) (Proposition III.3). As the authors show, for fixed \(\kappa\), the first condition is satisfied for every \(M\) if and only if it is satisfied for \(M={\mathbb S}^1\), i.e., in the case of loop groups (Theorem I.6). Hence \(\widehat{G}\) exists for general \(M\) if and only if it does so for \(M={\mathbb S}^1\). The proof is based on an explicit calculation of \(\Pi_{M,\kappa}\) in terms of \(\Pi_{{\mathbb S}^1,\kappa}\) which is quite remarkable, because \(\pi_2(C^\infty(M,K))\) is not well accessible when \(\dim(M)>2\). The authors then turn to …
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 22E67
Mathematics Subject Classification ID: 57T10
zbMATH DE Number: 1982575
zbMATH Keywords: current group
zbMATH Keywords: loop group
zbMATH Keywords: mapping group
zbMATH Keywords: central extension
zbMATH Keywords: universal central extension
zbMATH Keywords: period group
zbMATH Keywords: period homomorphism
zbMATH Keywords: obstruction
zbMATH Keywords: cocycle
zbMATH Keywords: product type
zbMATH Keywords: infinite-dimensional Lie group
zbMATH Keywords: automorphic action
MaRDI profile type: Publication
full work available at URL: s00208-003-0425-x
OpenAlex ID: W2005821260
DOI: 10.1007/S00208-003-0425-X
description: scientific article; zbMATH DE number 1982575