 
Summary: WINDINGINVARIANT PRIME IDEALS
IN QUANTUM 3 \Theta 3 MATRICES
K. R. Goodearl and T. H. Lenagan
Abstract. A complete determination of the prime ideals invariant under winding auto
morphisms in the generic 3 \Theta 3 quantum matrix algebra Oq (M3 (k)) is obtained. Explicit
generating sets consisting of quantum minors are given for all of these primes, thus verifying
a general conjecture in the 3 \Theta 3 case. The result relies heavily on certain tensor product
decompositions for windinginvariant prime ideals, developed in an accompanying paper. In
addition, new methods are developed here, which show that certain sets of quantum minors,
not previously manageable, generate prime ideals in Oq (Mn (k)).
Introduction
Although the quantized coordinate ring of n \Theta n matrices, O q (M n (k)), appears to have
a very straightforward structure  for instance, it is an iterated skew polynomial ring over
the base field  many basic questions about this algebra remain unanswered. In particular,
the structure of the prime spectrum of O q (M n (k)) is only partially understood, even in
the generic case (that is, when q is not a root of unity), where there are far fewer prime
ideals than in the root of unity case. The 2 \Theta 2 situation is relatively easy and has long
since been dealt with; here we complete the picture for the far more complicated 3 \Theta 3
quantum matrix algebra.
First, consider the algebra A = O q (M n (k)), with arbitrary matrix size n and arbitrary
