Extensions essentielles privilégiées d'algèbres de Lie classiques de dimension infinite. (Priviliged essential extensions of classical infinite dimensional Lie algebras) (Q1098427)

[For an announcement of the results; see the second author's conference lecture (Salamanca/Spain 1985) in Lect. Notes Math. 1251, 265-283 (1987).] Let W be a connected \(C^{\infty}\) manifold (dim \(W\geq 2)\), L be the Lie algebra of vector fields on W and \(\Phi_ 2\) be the vector space of 2- forms on W on which L acts by the Lie derivative. M. Flato, A. Lichnerowicz, M. de Wilde and P. Lecomte proved that there exists a canonical cohomology class \(\beta \in H^ 2(L;\Phi_ 2)\), always non- zero. The corresponding extension of L is called the universal extension of L. Given a geometric structure on W (e.g. a unimodular structure or a Poisson structure or a general Jacobi conformal structure) one can consider the attached Lie subalgebra \(L^*\) of L. The aim of the paper under review is to prove that the restriction of the universal extension of L to \(L^*\) is an essential extension of \(L^*\) (i.e. the restriction of \(\beta\) to \(L^*\) is non-zero).