Symmetry Groups of Principal Bundles Over Non-Compact Bases

Abstract

In this work, we describe how to obtain the structure of an infinite-dimensional Lie group on the group of compactly carried bundle automorphisms Autc(P) for a locally convex prinicpal bundle P over a finite-dimensional smooth sigma-compact base M. This is a generalization of previous work by Wockel, where the base M was compact. We first consider the Lie group structure on the group of compactly carried vertical bundle morphisms Gauc(P) (in both cases "compactly carried" refers to being compactly carried on the base in a certain sense). We then introduce the Lie group structure on Autc(P) as an extension of a certain open Lie subgroup of the compactly carried diffeomorphisms Diffc(M) by the gauge group Gauc(P). We find an explicit condition on P ensuring that Gauc(P) can be equipped with a Lie group structure enabling the extension just mentioned and show that this condition is satisfied by selected classes of bundles.