Annals of Global Analysis and Geometry

, Volume 36, Issue 4, pp 381–418

Central extensions of groups of sections

Original Paper

Abstract

If K is a Lie group and q : PM is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra \({\mathfrak{k}}\) of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact 1-forms. In this article, we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components, we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by the specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context, we provide sufficient conditions for integrability in terms of data related only to the group K.

Keywords

Gauge group Gauge algebra Central extension Lie group extension Integrable Lie algebra Lie group bundle Lie algebra bundle 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Mathematisches InstitutGeorg-August-Universität GöttingenGöttingenGermany

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