Prime spectra of quantum semisimple groups

Brown, K.; Goodearl, K.

doi:10.1090/S0002-9947-96-01597-8

Prime spectra of quantum semisimple groups
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by K. A. Brown and K. R. Goodearl PDF

Trans. Amer. Math. Soc. 348 (1996), 2465-2502 Request permission

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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