Central extensions of Lie groups preserving a differential form

B. Janssens, T. Diez, Karl-Hermann Neeb, Cornelia Vizman

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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)⁠.
Original languageEnglish
Pages (from-to)3794-3821
Number of pages28
JournalInternational Mathematics Research Notices
Issue number5
Publication statusPublished - 2021


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