Journal of Symplectic Geometry

Volume 16 (2018)

Number 5

Integrability of central extensions of the Poisson Lie algebra via prequantization

Pages: 1351 – 1375

DOI: https://dx.doi.org/10.4310/JSG.2018.v16.n5.a4

Authors

Bas Janssens (Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands)

Cornelia Vizman (Department of Mathematics, West University of Timişoara, Romania)

Abstract

We present a geometric construction of central $S^1$-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this to find nontrivial central $S^1$-extensions of the universal cover of the group of Hamiltonian diffeomorphisms. In the process, we obtain central $S^1$-extensions of Lie groups that act by exact strict contact transformations.

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The first author was supported by the NWO grant 613.001.214 “Generalized Lie algebra sheaves” and the NWO VIDI grant 639.032.734 “Cohomology and representation theory of infinite dimensional Lie group s”.

The second author was supported by the grant PN-II-ID-PCE-2011-3-0921 of the Romanian National Authority for Scientific Research.

Received 11 May 2016

Accepted 7 February 2018

Published 26 February 2019