Claude Roger made in [2] two conjectures on universal central extensions of Lie algebras of vector fields. In [1] we proof the conjecture about the universal central extension of the Lie algebra of Hamiltonian vector fields. In this talk I will explain how we solved the second one: the universal central extension of the Lie algebra of exact divergence free vector fields. Unlike the Hamiltonian setting, the divergence free setting needs a detour in the realm of Leibniz algebras.
(joint with Bas Janssens and Leonid Ryvkin)
References:
[1] B. Janssens, C. Vizman, Universal central extension of the Lie algebra of Hamiltonian vector fields, IMRN, 2016.16 (2016) 4996-5047.
[2] C. Roger, Extensions centrales d’algebres et de groupes de Lie de dimensioninfinie, algebre de Virasoro et generalisations, Rep. Math. Phys. 35 (1995) 225-266.