Quantized coordinate rings and related noetherian algebras

Abstract

This paper contains a survey of some ring-theoretic aspects of quantized coordinate rings, with primary focus on the prime and primitive spectra. For these algebras, the overall structure of the prime spectrum is governed by a partition into strata determined by the action of a suitable group of automorphisms of the algebra. We discuss this stratification in detail, as well as its use in determining the primitive spectrum -- under suitable conditions, the primitive ideals are precisely those prime ideals which are maximal within their strata. The discussion then turns to the global structure of the primitive spectra of quantized coordinate rings, and to the conjecture that these spectra are topological quotients of the corresponding classical affine varieties. We describe the solution to the conjecture for quantized coordinate rings of full affine spaces and (somewhat more generally) affine toric varieties. The final part of the paper is devoted to the quantized coordinate ring of $n\times n$ matrices. We mention parallels between this algebra and the classical coordinate ring, such as the primeness of quantum analogs of determinantal ideals. Finally, we describe recent work which determined, for the $3\times3$ case, all prime ideals invariant under the group of winding automorphisms governing the stratification mentioned above.