 
Summary: QUANTIZED COORDINATE RINGS AND
RELATED NOETHERIAN ALGEBRAS
K. R. Goodearl
Abstract. This paper contains a survey of some ringtheoretic aspects of quantized co
ordinate 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 deter
mined 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 spec
tra are topological quotients of the corresponding classical a#ne varieties. We describe
the solution to the conjecture for quantized coordinate rings of full a#ne spaces and
(somewhat more generally) a#ne toric varieties. The final part of the paper is devoted
to the quantized coordinate ring of n × 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 × 3
case, all prime ideals invariant under the group of winding automorphisms governing the
stratification mentioned above.
Introduction
