33 research outputs found
Equivalences of Smooth and Continuous Principal Bundles with Infinite-Dimensional Structure Group
Let K be a a Lie group, modeled on a locally convex space, and M a
finite-dimensional paracompact manifold with corners. We show that each
continuous principal K-bundle over M is continuously equivalent to a smooth one
and that two smooth principal K-bundles over M which are continuously
equivalent are also smoothly equivalent. In the concluding section, we relate
our results to neighboring topics.Comment: 18 pages, final versio
Integrating central extensions of Lie algebras via Lie 2-groups
The purpose of this paper is to show how central extensions of (possibly
infinite-dimensional) Lie algebras integrate to central extensions of \'etale
Lie 2-groups. In finite dimensions, central extensions of Lie algebras
integrate to central extensions of Lie groups, a fact which is due to the
vanishing of \pi_2 for each finite-dimensional Lie group. This fact was used by
Cartan (in a slightly other guise) to construct the simply connected Lie group
associated to each finite-dimensional Lie algebra. In infinite dimensions,
there is an obstruction for a central extension of Lie algebras to integrate to
a central extension of Lie groups. This obstruction comes from non-trivial
\pi_2 for general Lie groups. We show that this obstruction may be overcome by
integrating central extensions of Lie algebras not to Lie groups but to central
extensions of \'etale Lie 2-groups. As an application, we obtain a
generalization of Lie's Third Theorem to infinite-dimensional Lie algebras.Comment: 51 pages, 3 figures, submitted version, to appear in Journal of the
European Mathematical Societ
Principal 2-bundles and their gauge 2-groups
In this paper we introduce principal 2-bundles and show how they are
classified by non-abelian Cech cohomology. Moreover, we show that their gauge
2-groups can be described by 2-group-valued functors, much like in classical
bundle theory. Using this, we show that, under some mild requirements, these
gauge 2-groups possess a natural smooth structure. In the last section we
provide some explicit examples.Comment: 40 pages; v3: completely revised and extended, classification
corrected, name changed, to appear in Forum Mat