TY - JOUR

T1 - Central extensions of Lie groups preserving a differential form

AU - Janssens, B.

AU - Diez, T.

AU - Neeb, Karl-Hermann

AU - Vizman, Cornelia

PY - 2021

Y1 - 2021

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=85122443322&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnaa085

DO - 10.1093/imrn/rnaa085

M3 - Article

SN - 1073-7928

VL - 2021

SP - 3794

EP - 3821

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 5

ER -