Central extensions of Lie groups preserving a differential form
Bas Janssens (TU Delft - Analysis)
T. Diez (TU Delft - Analysis)
Karl Hermann Neeb (Friedrich-Alexander-Universität Erlangen-Nürnberg)
Cornelia Vizman (West University of Timisoara (UVT))
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Abstract
Let M be a manifold with a closed, integral (k+1)-form ω, and let G be a Fréchet–Lie group acting on (M,ω). As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of g by R, indexed by Hk−1(M,R)∗. We show that the image of Hk−1(M,Z) in Hk−1(M,R)∗ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of G by the circle group T. The idea is to represent a class in Hk−1(M,Z) by a weighted submanifold (S,β), where β is a closed, integral form on S. We use transgression of differential characters from S and M to the mapping space C∞(S,M) and apply the Kostant–Souriau construction on C∞(S,M).