 
Summary: PRIME IDEALS IN CERTAIN
QUANTUM DETERMINANTAL RINGS
K. R. Goodearl and T. H. Lenagan
Abstract. The ideal I1 generated by the 2 \Theta 2 quantum minors in the co
ordinate algebra of quantum matrices, Oq (Mm;n (k)), is investigated. Ana
logues of the First and Second Fundamental Theorems of Invariant Theory
are proved. In particular, it is shown that I1 is a completely prime ideal, that
is, Oq (Mm;n (k))=I1 is an integral domain, and that Oq (Mm;n (k))=I1 is the
ring of coinvariants of a coaction of k[x; x \Gamma1 ] on Oq (k m
)\Omega Oq (k n ), a tensor
product of two quantum affine spaces. There is a natural torus action on
Oq (Mm;n (k))=I1 induced by an (m + n)torus action on Oq (Mm;n (k)). We
identify the invariant prime ideals for this action and deduce consequences
for the prime spectrum of Oq (Mm;n (k))=I1 .
Introduction
Let k be a field and let q 2 k \Theta . The coordinate ring of quantum m \Theta n
matrices, A := O q (M m;n (k)), is a deformation of the classical coordinate
ring of m \Theta n matrices, O(M m;n (k)). As such it is a kalgebra generated
by mn indeterminates X ij , for 1 Ÿ i Ÿ m and 1 Ÿ j Ÿ n, subject to the
relations
