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.

Using a nonlinear version of the tautological bundle over Graßmannians, we construct a transgression map for differential characters from M to the nonlinear Graßmannian Gr^S(M) of submanifolds of M of a fixed type S. In particular, we obtain prequantum circle bundles of the nonlinear Graßmannian endowed with the Marsden–Weinstein symplectic form. The associated Kostant–Souriau prequantum extension yields central Lie group extensions of a group of volume-preserving diffeomorphisms integrating Lichnerowicz cocycles.

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.

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)⁠.

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 S1-extensions of Lie groups that act by exact strict contact transformations.