Quantum Determinantal Ideals

Introduction. Fix a base field k. The quantized coordinate ring of n×n matrices over k, denoted by q(Mn(k)), is a deformation of the classical coordinate ring of n×n matrices, (Mn(k)). As such, it is a k-algebra generated by n2 indeterminates Xij , for 1 ≤ i,j ≤ n, subject to relations which we state in (1.1). Here, q is a nonzero element of the field k. When q = 1, we recover (Mn(k)), which is the commutative polynomial algebra k[Xij ]. The algebra q(Mn(k)) has a distinguished element Dq , the quantum determinant, which is a central element. Two important algebras q(GLn(k)) and q(SLn(k)) are formed by invertingDq and settingDq = 1, respectively. The structures of the primitive and prime ideal spectra of the algebras q(GLn(k)) and q(SLn(k)) have been investigated recently (see, for example, [2], [7], and [10]). Results obtained in these investigations can be pulled back to partial results about the primitive and prime ideal spectra of q(Mn(k)). However, these techniques give no information about the closed subset of the spectrum determined by Dq . In this paper, we begin the study of this portion of the spectrum. In the classical commutative setting, much attention has been paid to determinantal ideals: that is, the ideals generated by the minors of a given size. In particular, these are special prime ideals of (Mn(k)) containing the determinant. Moreover, there are interesting geometrical and invariant theoretical reasons for the importance of these ideals (see, for example, [4]). In order to put our results into context, it may be useful to review some highlights of the commutative theory. Let Ml,m(k) denote the algebraic variety of l×m matrices over k. For t ≤ n, the general linear group GLt (k) acts on Mn,t (k)×Mt,n(k) via g ·(A,B) := (Ag−1,gB).