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Lie Groups of Bundle Automorphisms and Their Extensions

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Developments and Trends in Infinite-Dimensional Lie Theory

Part of the book series: Progress in Mathematics ((PM,volume 288))

Summary

We describe natural abelian extensions of the Lie algebra \(\mathfrak{aut}(P)\) of infinitesimal automorphisms of a principal bundle over a compact manifold M and discuss their integrability to corresponding Lie group extensions. The case of a trivial bundle P = M * K is already quite interesting. In this case, we show that essentially all central extensions of the gauge algebra C ∞ (M T) can be obtained from three fundamental types of cocycles with values in one of the spaces ℨ := C ∞ (M V), Ω1 (M V) and Ω1 (M V)/ dC ∞ (M V).These cocycles extend to \(\mathfrak{aut}(P)\), and, under the assumption that T M is trivial, we also describe the space H 2 (v(M), ℨ) classifying the twists of these extensions. We then show that all fundamental types have natural generalizations to non-trivial bundles and explain under which conditions they extend to \(\mathfrak{aut}(P)\) and integrate to global Lie group extensions.

2000 Mathematics Subject Classifications: Primary 22E65. Secondary 22E67, 17B66.

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  1. Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054, Erlangen, Germany

    Karl-Hermann Neeb

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  1. Friedrich-Alexander-Universität Erlangen, Department of Mathematics, Bismarckstrasse 1 1/2, Erlangen, 91054, Germany

    Karl-Hermann Neeb

  2. University of Alberta, Department of Mathematical Sciences, Edmonton, T6G 2G1, Alberta, Canada

    Arturo Pianzola

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Neeb, KH. (2011). Lie Groups of Bundle Automorphisms and Their Extensions. In: Neeb, KH., Pianzola, A. (eds) Developments and Trends in Infinite-Dimensional Lie Theory. Progress in Mathematics, vol 288. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4741-4_9

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