Introduction

A major theme in the subject of quantum groups is the philosophy that in the passage from a classical coordinate ring to a quantized analog, the classical geometry is replaced by structures that should be treated as ‘noncommutative geometry’. Indeed, much work has been invested into the development of theories of noncommutative differential geometry and noncommutative algebraic geometry. We would like to pose the question whether these theories are entirely noncommutative, or whether traces of classical geometry are to be found in the noncommutative geometry. This rather vague question can, of course, be focused in any number of different directions. We discuss one particular direction here, which was developed in joint work with E. S. Letzter [6]; it concerns situations in which quantized analogs of classical varieties contain certain quotients of these varieties.