Quantization of the Dixmier localization of \(U(sl_{n+1}(\mathbb C))\)
In the classical case, Dixmier proved that, up to localization, the universal enveloping algebra of \(sl_{n+1}({\mathbb C})\) is a central polynomial extension of the enveloping algebra of some parabolic subalgebra. The latter algebra has a semiinvariant element by which it is sufficient to localize. This result was used by \textit{C. Moeglin} [J. Algebra 106, 287-366 (1987; Zbl 0637.17005)] to obtain the description of the completely prime ideals of the enveloping algebra of \(sl_{n+1}({\mathbb C})\). In the paper under review the author continues the project started in her paper [Bull. Sci. Math. 122, No. 7, 495-519 (1998; Zbl 0938.17013)]. The main purpose is to extend to the quantum case the result of Dixmier. The methods used here are quite different from those in the classical case because the powers of the ``quantum'' indeterminate \(q\) would make the calculations in the spirit of those of Dixmier too complicated. Instead, the author uses the quantum analogue of the Killing form due to [\textit{M. Rosso}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 23, No. 3, 445-467 (1990; Zbl 0721.17012)] and works in the Hopf dual of the quantum enveloping algebra \(\check U_q(sl_{n+1}({\mathbb C}))\) which significantly simplifies the considerations. As an application of the main result the author obtains that the ring of fractions of \(\check U_q(sl_{n+1}({\mathbb C}))\) is isomorphic to the ring of fractions of a certain quantum Weyl algebra and proves the conjecture of \textit{I. M. Gelfand} and \textit{A. A. Kirillov} [Publ. Math., Inst. Hautes Etud. Sci. 31, 506-523 (1966; Zbl 0144.02104)] in the case of \(\check U_q(sl_{n+1}({\mathbb C}))\).
universal enveloping algebras of simple Lie algebras
scientific article; zbMATH DE number 1340573
scientific article