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

Prove that the central Lie algebra extension H^1_dR(M) → Ω^1(M)/dΩ^0(M) → X_ex(M, μ) of 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.

Background

Let M be a compact, connected, orientable 3-manifold with volume form μ. The Lie algebra X_ex(M, μ) consists of exact divergence-free vector fields. The paper recalls the central Lie algebra extension 0 → H^1_dR(M) → Ω^1(M) → X_ex(M, μ) → 0 (with Ω^1(M) understood modulo exact forms) and details its bracket and projection. This extension has been studied in the literature.

The authors note that this extension is conjectured to be universal (in the sense of covering all continuous, linearly split central extensions uniquely) and indicate ongoing work toward a rigorous proof. Establishing universality would clarify the structural role of this extension within the broader theory of infinite-dimensional Lie algebras associated to volume-preserving dynamics.