Équivalence rationnelle d'algèbres polynomiales classiques et quantiques

This article is devoted to rational equivalence for non-commutative polynomial algebras in a context including both the classical Gelfand-Kirillov problem and its quantum version. We introduce in this “mixed” context some reference algebras and define two new invariants allowing us to separate the fraction fields of these algebras up to isomorphism. The first one is linked to the notion of maximal simple quantum sub-torus, and the second one is a dimensional invariant measuring the classical character (in terms of Weyl algebras) of the skew-fields into consideration. As an application we obtain results concerning the rational equivalence of multiparametrized quantum Weyl algebras.