  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.
