Prime spectra of quantized coordinate rings

TL;DR

This paper surveys the structure and prime spectra of quantized coordinate rings, proposing a unifying framework and conjecture that these algebras share common properties, supported by stratification and axiomatic approaches.

Contribution

It introduces an axiomatic basis for understanding the prime spectra of quantized coordinate rings, advancing the conjecture of their shared structural properties.

Findings

Stratifications of prime spectra by torus actions

Support for the conjecture of common properties among these algebras

Identification of axioms involving normal elements

Abstract

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 joint work with E. S. Letzter in [The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras (to appear in Trans. Amer. Math. Soc.)], 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.