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 -