August 22, 2025Open Access

Universal central extension of the Lie algebra of exact divergence-free vector fields

Key Points

The universal central extension of the Lie algebra was successfully constructed, validating Claude Roger's 1995 conjecture.
Key finding shows the link between divergence-free vector fields and Leibniz algebras in mathematical structures.
The approach involves an analysis of underlying mathematical frameworks relevant to infinite-dimensional Lie groups.
Implications extend to the construction of diffeomorphisms on a compact 3-dimensional manifold based on the findings.

Abstract

Abstract 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.

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