Universality of the central Lie algebra extension for exact divergence-free vector fields on a 3-manifold

Establish that the central extension of Lie algebras 0 → H^1_dR(M) → Ω^1(M)/dΩ^0(M) → X_ex(M,µ) → 0, associated to the Lie algebra X_ex(M,µ) of exact divergence-free vector fields on a compact, connected, orientable 3-manifold M with volume form µ, is universal among continuous, linearly split central extensions of X_ex(M,µ).

Background

Section 5 develops a central Lie group extension for the exact volume-preserving diffeomorphism group of a closed, orientable 3-manifold M with volume form µ. At the Lie algebra level, the corresponding central extension is H^1_dR(M) → Ω^1(M)/dΩ^0(M) → X_ex(M,µ), where Ω^1(M)/dΩ^0(M) carries a bracket induced by contraction with µ and projects onto the Lie algebra X_ex(M,µ) of exact divergence-free vector fields. This extension is classical and has been studied in the literature.

The paper notes that Roger [Ro95] conjectured the universality of this central Lie algebra extension. If this conjecture holds, Neeb’s Recognition Theorem would provide conditions under which a simply connected cover of the constructed Lie group extension is universal. The authors indicate ongoing work toward proving the conjecture, highlighting the significance of the universality question for integrating the Lie algebra extension to a universal central extension at the group level.