On the prime and primitive spectra of the algebra of functions on a quantum group.

*(English)*Zbl 0814.17013Let \({\mathfrak g}\) be a complex semisimple Lie algebra, \(U:= U_ q ({\mathfrak g})\) its quantized enveloping algebra, and \(R:= R_ q [G]\) the algebra of functions on its (as yet undefined) “quantum group” \(G\); here \(R\) may be defined as the span of the matrix coefficients of finite- dimensional representations of \(U\), or essentially just the Hopf dual of \(U\). The purpose of this article is to classify the prime and primitive spectrum of \(R\), proving thereby a conjectural description of T. J. Hodges and T. Levasseur [Commun. Math. Phys. 156, 581-605 (1993; Zbl 0801.17012)], who had already proved it for \({\mathfrak g}= {\mathfrak {sl}} (n)\) [J. Algebra 168, No. 2, 455-468 (1994; Zbl 0814.17012) see the preceding review]. It turns out that, up to tensoring with one- dimensional representations, the primitive and prime spectra \(\text{Prim }R\), \(\text{Spec }R\) are parametrized by elements of \(W\times W\), where \(W\) is the Weyl group of \({\mathfrak g}\), and that every prime ideal is completely prime. Many of the ideas in the classification proof come from a fundamental paper of Ya. Soibelman [Leningr. Math. J. 2, 161-178 (1991); translation from Algebra Anal. 2, 190-212 (1990; Zbl 0708.46029)], while the techniques in the proof largely come from the author and G. Letzter’s paper [J. Algebra 153, 289-318 (1992; Zbl 0779.17012)]. There is another exposition of most of the proofs in A. Joseph’s recently published book entitled “Quantum groups and their primitive ideals” [Ergebnisse der Mathematik, 3. Folge, Band 29, Springer-Verlag (1995; Zbl 0808.17004)].

Reviewer: W.M.McGovern (Seattle)

##### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |