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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Prime spectra of quantum semisimple groups

Authors: K. A. Brown and K. R. Goodearl
Journal: Trans. Amer. Math. Soc. 348 (1996), 2465-2502
MSC (1991): Primary 16D30, 16D60, 16P40, 17B37
MathSciNet review: 1348148
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Abstract: We study the prime ideal spaces of the quantized function algebras $R_{q}[G]$, for $G$ a semisimple Lie group and $q$ an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras $R_{q}[G]$ satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar's strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature -- it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of $R_{q}[G]$. In the final section the results are specialized to the case $G= SL_{n}(\mathbb {C})$, where detailed calculations can be used to illustrate the general results. As a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine $n$-space satisfy our axiom scheme when the group generated by the parameters is torsionfree.

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Additional Information

K. A. Brown
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106

K. R. Goodearl
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland

Received by editor(s): November 4, 1994
Received by editor(s) in revised form: September 5, 1995
Additional Notes: The research of the second author was partially supported by a grant from the National Science Foundation (USA). Part of the work was carried out while he visited the University of Glasgow Mathematics Department during October 1993, supported by the Edinburgh and London Mathematical Societies. Work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the Department of Mathematics of the University of Washington, whom both thank for its hospitality. The travel costs of the first author were in part covered by a grant from the Carnegie Trust for the Universities of Scotland.
Article copyright: © Copyright 1996 American Mathematical Society