PRIME SPECTRA OF QUANTUM SEMISIMPLE GROUPS

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 hypo theses, and then to demonstrate, using work of Joseph, Hedges and Leva sseur, that the algebras R(q)[G] satisfy this list of assumptions. Rin gs satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar's strong second layer condition. For such ri ngs 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. Hom ogeneity 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)(C), where detailed calculations can be used to i llustrate the general results. As a preliminary set of examples we sho w also that the multiparameter quantum coordinate rings of affine n-sp ace satisfy our axiom scheme when the group generated by the parameter s is torsionfree.