Summary: QUANTUM DETERMINANTAL IDEALS
K. R. Goodearl and T. H. Lenagan
Abstract. Quantum determinantal ideals in coordinate algebras of quantum matrices are
investigated. The ideal I t generated by all (t + 1) \Theta (t + 1) quantum minors in Oq (Mm;n (k))
is shown to be a completely prime ideal, that is, Oq (Mm;n (k))=I t is an integral domain.
The corresponding result is then obtained for the multiparameter quantum matrix algebra
O –;p (Mm;n (k)). The main idea involved in the proof is the construction of a preferred basis
for Oq (Mn (k)) in terms of certain products of quantum minors, together with rewriting rules
for expressing elements of this algebra in terms of the preferred basis.
Fix a base field k. The quantized coordinate ring of n \Theta n matrices over k, denoted
O q (M n (k)), is a deformation of the classical coordinate ring of n \Theta n matrices, O(M n (k)).
As such it is a kalgebra generated by n 2 indeterminates X ij , for 1 ź i; j ź n, subject to
relations which we recall in (1.1). Here q is a nonzero element of the field k. When q = 1,
we recover O(M n (k)), which is the commutative polynomial algebra k[X ij ]. The algebra
O q (M n (k)) has a distinguished element D q , the quantum determinant , which is a central
element. Two important algebras O q (GL n (k)) and O q (SL n (k)) are formed by inverting
D q and setting D q = 1, respectively.
The structures of the primitive and prime ideal spectra of the algebras O q (GL n (k)) and
O q (SL n (k)) have been investigated recently; see, for example, ,  and . Results