Primitive ideals of \(\mathbb{C}_ q[SL(n)]\)
type: Item
instance of: scholarly article
title: Primitive ideals of \(\mathbb{C}_ q[SL(n)]\)
zbMATH Open document ID: 0814.17012
author: Timothy J. Hodges
author: Thierry Levasseur
published in: Journal of Algebra
publication date: 1994-09-27T00:00:00Z
review text: Let \(q\) be a nonzero complex number which is not a root of unity. The authors classify the primitive spectrum of the quantum group \(\mathbb{C}_ q [\text{SL} (n)]\), showing that there is a natural bijection between primitive ideals in this ring and symplectic leaves of the associated Poisson group \(\text{SL} (n, \mathbb{C})\). They also show that orbits of primitive ideals under a natural \(n-1\)-torus of automorphisms are parametrized by \(W\times W\), where \(W\) is the associated Weyl group, isomorphic to the symmetric group on \(n\) letters. This paper generalizes an earlier one of the authors which dealt with the case \(n=3\) [Commun. Math. Phys. 156, 581-605 (1993; Zbl 0801.17012)]. Many of the ideas and methods come from a paper of \textit{Ya. Soibelman} [Leningr. Math. J. 2, 161-178 (1991); translation from Algebra Anal. 2, 190-212 (1990; Zbl 0708.46029)]. The results have been generalized to quantum groups attached to arbitrary simply connected semisimple complex Lie groups by \textit{A. Joseph} [J. Algebra 169, 441-511 (1994; Zbl 0814.17013) see the following review; see also C. R. Acad. Sci., Paris, Sér. I 316, 1139-1142 (1993; Zbl 0784.17021)].
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 645090
zbMATH Keywords: primitive spectrum
zbMATH Keywords: quantum group
zbMATH Keywords: primitive ideals
zbMATH Keywords: symplectic leaves
zbMATH Keywords: Poisson group
zbMATH Keywords: Weyl group
reviewed by: William M. McGovern
MaRDI profile type: Publication
full work available at URL: jabr.1994.1239
OpenAlex ID: W2059811707
DOI: 10.1006/JABR.1994.1239
description: scientific article; zbMATH DE number 645090