 
Summary: 4 December 99
QUANTIZED PRIMITIVE IDEAL SPACES AS
QUOTIENTS OF AFFINE ALGEBRAIC VARIETIES
K. R. Goodearl
Abstract. Given an affine algebraic variety V and a quantization Oq (V ) of its coordinate
ring, it is conjectured that the primitive ideal space of Oq (V ) can be expressed as a topolog
ical quotient of V . Evidence in favor of this conjecture is discussed, and positive solutions
for several types of varieties (obtained in joint work with E. S. Letzter) are described. In
particular, explicit topological quotient maps are given in the case of quantum toric varieties.
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.
