We construct the universal central extension of the Lie algebra of exact divergence- free vector fields, proving a conjecture by Claude Roger from 1995. The proof relies on the analysis of a Leibniz algebra that underlies these vector fields. As an application, we construct the universal central extension of the (infinite-dimensional) Lie group of exact divergence- free diffeomorphisms of a compact 3-dimensional manifold. De Gruyter 2025.

We construct an L1-algebra on the truncated canonical homology complex of a symplectic manifold, which naturally projects to the universal central extension of the Lie algebra of Hamiltonian vector fields.