For a connected symplectic manifold X, we determine the universal central extension of the Lie algebra ham(X) of hamiltonian vector fields. We classify the central extensions of ham(X), of the Lie algebra sp(X) of symplectic vector fields, of the Poisson Lie algebra C∞(X), and of its compactly supported version C∞c(X).
Janssens, B., & Vizman, C. (2015). Universal central extension of the Lie algebra of Hamiltonian vector fields. International Mathematics Research Notices, 2016(16), 4996-5047. https://doi.org/10.1093/imrn/rnv301