Lie groups of bundle automorphisms and their extensions

Abstract

We describe natural abelian extensions of the Lie algebra 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, k) 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 aut(P), and, under the assumption that TM is trivial, we also describe the space H2(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 aut(P) and integrate to global Lie group extensions.