Flux homomorphisms and principal bundles over infinite dimensional manifolds
type: Item
instance of: scholarly article
title: Flux homomorphisms and principal bundles over infinite dimensional manifolds
zbMATH Open document ID: 1029.22027
published in: Monatshefte für Mathematik
publication date: 2003-09-23T00:00:00Z
review text: In the article under review, the concept of flux homomorphism for a closed differential form (see, e.g., [\textit{A. Banyaga}, The structure of classical diffeomorphism groups, Kluwer (1997; Zbl 0874.58005)]) is generalized to the case of a closed \(p\)-form \(\omega\) on a smooth manifold \(M\) modelled on a locally convex space, taking its values in a sequentially complete, locally convex space \({\mathfrak z}\). For \(p=2\) and connected \(M\), the flux homomorphism is used to analyze which automorphisms of \((M,\omega)\) can be lifted to automorphisms of a given principal \(Z\)-bundle \(P\to M\) with connection \(1\)-form \(\theta\) and curvature \(\omega\), where \(Z:={\mathfrak z}/\Gamma\) for a discrete subgroup of \({\mathfrak z}\). Generalizing a result of \textit{B. Konstant} in the finite-dimensional case [Lect.\ Notes Math. 170, 87-208 (1970; Zbl 0223.53028)], the latter are characterized as those automorphisms of \((M,\omega)\) which preserve the holonomy of piecewise smooth closed curves in \(M\). The results are then used to construct central extensions of infinite-dimensional Lie groups.
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 53D50
zbMATH DE Number: 1984997
zbMATH Keywords: flux homomorphism
zbMATH Keywords: symplectic manifold
zbMATH Keywords: holonomy group
zbMATH Keywords: symplectomorphism
zbMATH Keywords: central extension
author: Karl-Hermann Neeb
author: Cornelia Vizman
MaRDI profile type: Publication
full work available at URL: s00605-002-0001-6
OpenAlex ID: W2031808925
DOI: 10.1007/S00605-002-0001-6
description: scientific article; zbMATH DE number 1984997