9 Prime Spectra of Quantized Coordinate Rings

This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common properties, there must be a common basis on which to treat them. The first part of the paper is expository. We survey a number of classes of quantized coordinate rings, as well as some related algebras that share common properties, and we record some of the basic properties known to occur for many of these algebras, culminating in stratifications of the prime spectra by the actions of tori of automorphisms. As our main interest is in the generic case, we assume various parameters are not roots of unity whenever convenient. In the second part of the paper, which is based on [20], we offer some support for the conjecture above, in the form of an axiomatic basis for the observed stratifications and their properties. At present, the existence of a suitable supply of normal elements is taken as one of the axioms; the search for better axioms that yield such normal elements is left as an open problem. This part of the paper is an expository account of the prime ideal structure of algebras on the " quantized coordinate ring " side of the theory of quantum groups – quantizations of the coordinate rings of affine spaces, matrices, semisimple groups, symplectic and Euclidean spaces, as well as a few related algebras – quantized enveloping algebras of Borel and nilpotent subalgebras of semisimple Lie algebras, and quantized Weyl algebras. These algebras occur widely throughout the quantum groups literature, and different papers