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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@misc{Janssens-Universal-central-extension-2024,
 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.},
 arxiv = {arXiv:2409.05182},
 author = {Janssens, Bas and Ryvkin, Leonid and Vizman, Cornelia},
 eprint = {2409.05182},
 howpublished = {Preprint, {arXiv}:2409.05182 [math.{DG}] (2024)},
 keywords = {17B56,58D05,17B66},
 title = {Universal central extension of the {Lie} algebra of exact divergence-free vector fields},
 url = {https://arxiv.org/abs/2409.05182},
 year = {2024}
}