Prime spectra of quantum semisimple groups |
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| Authors: | K. A. Brown K. R. Goodearl |
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| Affiliation: | Department of Mathematics, University of California, Santa Barbara, California 93106 ; Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland |
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| Abstract: | We study the prime ideal spaces of the quantized function algebras ![$R_{q}[G]$](http://www.ams.org/tran/1996-348-06/S0002-9947-96-01597-8/gif-abstract/img1.gif) , for  a semisimple Lie group and  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 ![$R_{q}[G]$](http://www.ams.org/tran/1996-348-06/S0002-9947-96-01597-8/gif-abstract/img4.gif) 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 ![$R_{q}[G]$](http://www.ams.org/tran/1996-348-06/S0002-9947-96-01597-8/gif-abstract/img5.gif) . In the final section the results are specialized to the case  , 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  -space satisfy our axiom scheme when the group generated by the parameters is torsionfree. |
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