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

TL;DR

This paper constructs the universal central extension of the Lie algebra of exact divergence-free vector fields, confirming a longstanding conjecture and extending the understanding of the associated infinite-dimensional Lie groups.

Contribution

It provides the first explicit construction of the universal central extension for this class of Lie algebras, resolving a conjecture from 1995.

Findings

Confirmed the conjecture by Claude Roger from 1995.

Constructed the universal central extension of the Lie algebra.

Extended the construction to the Lie group of exact divergence-free diffeomorphisms.

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.