Quantification de la Localisation de Dixmier de U(sln + 1(C))

Résumé In this article, we extend to the quantum case the classical result of Dixmier, according to which, up to localization, the enveloping algebra of sln + 1(C) is a central polynomial extension of the enveloping algebra of some parabolic subalgebra. In fact, the latter has a semiinvariant element d by which it is sufficient to localize to obtain the result. The methods we use here are quite different from those used by Dixmier. Indeed, in the quantum case, the powers of the indeterminate q would make calculations similar to those of Dixmier much too complicated. On the other hand, when we use the Rosso form, which allows us to work in the Hopf dual of q(sln + 1(C)), the calculations become much easier than they would be in q(sln + 1(C)).