PRIME IDEALS INVARIANT UNDER WINDING

AUTOMORPHISMS IN QUANTUM MATRICES

K. R. Goodearl and T. H. Lenagan

Abstract. The main goal of the paper is to establish the existence of tensor product decompositions for those prime ideals P of the algebra A = O_{q}(M_{n}(k)) of quantum n × n matrices which are invariant under winding automorphisms of A, in the generic case (q not a root of unity). More specifically, every such P is the kernel of a map of the form

A ⟶ A ⊗ A ⟶ A^{+} ⊗ A^{−} ⟶ (A^{+}/P^{+}) ⊗ (A^{−}/P^{−})

where A → A ⊗ A is the comultiplication, A^{+} and A^{−} are suitable localized factor algebras of A, and P^{±} is a prime ideal of A^{±} invariant under winding automorphisms. Further, the algebras A^{±}, which vary with P, can be chosen so that the correspondence (P^{+}, P^{−}) ↦ P is a bijection. The main theorem is applied, in a sequel to this paper, to completely determine the winding-invariant prime ideals in the generic quantum 3 × 3 matrix algebra.

Introduction

This paper represents part of an ongoing project to determine the prime and primitive spectra of the generic quantized coordinate ring of n × n matrices, O_{q}(M_{n}(k)). Here k is an arbitrary field and q ∈ k^{×} is a non-root of unity. The current intermediate goal is to determine the prime ideals of O_{q}(M_{n}(k)) invariant under all winding automorphisms. (See below for a discussion of the relations between these winding-invariant primes and the full prime spectrum of O_{q}(M_{n}(k)).) Our main result exhibits a bijection between these primes and pairs of winding-invariant primes from certain ‘localized step-triangular factors’ of O_{q}(M_{n}(k)), namely the algebras

\begin{aligned}R_{r}^{+} & = (O_{q}(M_{n}(k))/⟨X_{ij} | j > t or i < r_{j}⟩)[\overline{X}_{r_{1}1}^{−1}, …, \overline{X}_{r_{t}t}^{−1}] \\ R_{c}^{−} & = (O_{q}(M_{n}(k))/⟨X_{ij} | i > t or j < c_{i}⟩)[\overline{X}_{1c_{1}}^{−1}, …, \overline{X}_{tc_{t}}^{−1}] \end{aligned}

This research was partially supported by NSF research grants DMS-9622876 and DMS-9970159 and by NATO Collaborative Research Grant CRG.960250. Some of the research was done while both authors were visiting the Mathematical Sciences Research Institute in Berkeley during the winter of 2000, and they thank MSRI for its support.

Typeset by AMS-TEX

where r = (r_{1}, …, r_{t}) and c = (c_{1}, …, c_{t}) are strictly increasing sequences of integers in the range 1, 2, …, n. In particular, since each R_{r}^{+} and R_{c}^{−} can be presented as a skew-Laurent extension of a localized factor algebra of O_{q}(M_{n−1}(k)), the above bijection can be used to obtain descriptions (as pullbacks of primes in the algebras R_{r}^{+} ⊗ R_{c}^{−}) of the winding-invariant primes of O_{q}(M_{n}(k)) from those of O_{q}(M_{n−1}(k)). In a sequel [7] to this paper, we follow the route just sketched to develop a complete list, with sets of generators, of the winding-invariant primes in O_{q}(M_{3}(k)).

The theorem indicated above depends on some detailed structural results concerning O_{q}(M_{n}(k)) and on some general work with primes in tensor product algebras invariant under group actions. First, we construct a partition of spec O_{q}(M_{n}(k)) indexed by pairs (r, c) as above, together with localized factor algebras A_{r,c} of O_{q}(M_{n}(k)), such that the portion of spec O_{q}(M_{n}(k)) indexed by (r, c) is Zariski-homeomorphic to spec A_{r,c}. We next prove that A_{r,c} is isomorphic to a subalgebra B_{r,c} of R_{r}^{+} ⊗ R_{c}^{−}, identify the structure of B_{r,c}, and show that R_{r}^{+} ⊗R_{c}^{−} is a skew-Laurent extension of B_{r,c}. Finally, with the help of some general work on tensor products, we prove that each winding-invariant prime of B_{r,c} extends uniquely to a winding-invariant prime of R_{r}^{+} ⊗R_{c}^{−}, and that the latter primes can be uniquely expressed in the form (P^{+} ⊗ R_{c}^{−}) + (R_{r}^{+} ⊗ P^{−}) where P^{+} (respectively, P^{−}) is a winding-invariant prime in R_{r}^{+} (respectively, R_{c}^{−}). We thus conclude that every winding-invariant prime of O_{q}(M_{n}(k)) can be uniquely expressed as the kernel of a map

O_{q}(M_{n}(k)) ⟶ O_{q}(M_{n}(k)) ⊗ O_{q}(M_{n}(k)) ⟶ R_{r}^{+} ⊗ R_{c}^{−} ⟶ (R_{r}^{+}/P^{+}) ⊗ (R_{c}^{−}/P^{−}),

where the first arrow is comultiplication and the others are tensor products of localization or quotient maps.

Algebraic background. The algebra O_{q}(M_{n}(k)) has standard generators X_{ij} for i, j = 1, …, n and relations which we recall in (5.1)(a), along with the bialgebra structure of this algebra. The latter structure allows us to define left and right winding automorphisms corresponding to those characters (k-algebra homomorphisms O_{q}(M_{n}(k)) → k) which are invertible in O_{q}(M_{n}(k))^{∗} with respect to the convolution product (cf. [1, (I.9.25)] or [12, (1.3.4)] for the Hopf algebra case). It is well known that the collection of left (respectively, right) winding automorphisms of O_{q}(M_{n}(k)) forms a group isomorphic to the diagonal subgroup of GL_{n}(k), whose action on the matrix of generators (X_{ij}) is given by left (respectively, right) multiplication. We combine these actions to obtain an action of the group H = (k^{×})^{n} × (k^{×})^{n} on O_{q}(M_{n}(k)) by k-algebra automorphisms satisfying the rule

(∗)  (α_{1}, …, α_{n}, β_{1}, …, β_{n}).X_{ij} = α_{i}β_{j}X_{ij}.

One indication of the extent of the symmetry given by this action is the fact that there are only finitely many (actually, at most 2^{n^{2}}) primes of O_{q}(M_{n}(k)) invariant under H [9, (5.7)(i)]. The quoted result also shows that all H-primes of this algebra are prime, and so the H-primes coincide with the winding-invariant primes in O_{q}(M_{n}(k)). (Recall that the definition of an H-prime ideal is obtained from the standard ideal-theoretic definition of a prime ideal by restricting to H-invariant ideals.)

In [9, Theorem 6.6], Letzter and the first author showed that the overall picture of the prime spectrum of an algebra with certain basic features like those of O_{q}(M_{n}(k)) is determined to a great extent by the primes invariant under a suitable group action. We quote the improved version of this picture presented in [1, Theorem II.2.13]. Let A be a noetherian algebra over an infinite field k, and let H = (k^{×})^{r} be (the group of k-points of) an algebraic torus acting rationally on A by k-algebra automorphisms. Each H-prime of A is a prime ideal, and spec A is the disjoint union of the sets

spec_{J} A := {P ∈ spec A | ⋂_{h∈H} h(P) = J}

as J ranges over the H-primes of A. Further:

(1) Let E_{J} denote the set of all regular H-eigenvectors in A/J. Then E_{J} is a denominator set, and the localization A_{J} = (A/J)[E_{J}^{−1}] is an H-simple ring.

(2) spec_{J} A is homeomorphic to spec A_{J} via localization and contraction.

(3) spec A_{J} is homeomorphic to spec Z(A_{J}) via contraction and extension.

(4) Z(A_{J}) is a Laurent polynomial ring, in at most r indeterminates, over the fixed

field Z(Fract A/J)^{H}. Under some additional hypotheses, satisfied by O_{q}(M_{n}(k)), we also have:

(5) The primitive ideals of A are exactly the maximal elements of the sets spec_{J} A.

(6) If k is algebraically closed, then the primitive ideals within each spec_{J} A are per-

muted transitively by H. Statement (5) was proved in [9, Corollary 6.9] (see also [1, Theorem II.8.4]). Statement (6) is a consequence of general transitivity theorems for algebraic group actions due to Moeglin-Rentschler [14, Théorème 2.12(ii)] and Vonessen [17, Theorem 2.2], but the case where the acting group is a torus is much easier (see [9, Theorem 6.8] or [1, Theorem II.8.14]).

The above results indicate that to draw a complete picture of spec O_{q}(M_{n}(k)), we need to determine the H-primes. That is easy to do in case n = 2; the result is recorded, for instance, in [5, (3.6)] (see [1, Example II.2.14(d)] for more detail). In general, we make the following

Conjecture. Every H-prime of O_{q}(M_{n}(k)) can be generated by a set of quantum minors.

This conjecture is easily checked in case n = 2 using the information above, and we verify it for the case n = 3 in [7]. Further supporting evidence is provided by recent work of Cauchon, who showed that distinct, comparable H-primes in O_{q}(M_{n}(k)) can be distinguished by the quantum minors they contain [2, Proposition 6.2.2 and Théorème 6.2.1]. Another source of support for the conjecture is the work of Hodges and Levasseur [10, 11], from which one can deduce that, up to certain localizations, the winding-invariant primes of O_{q}(SL_{n}(k)) are generated by quantum minors (cf. [1, Corollary II.4.12]). Since O_{q}(GL_{n}(k)) is isomorphic to a Laurent polynomial ring over O_{q}(SL_{n}(k)) [13, Proposition], the above statement also holds in O_{q}(GL_{n}(k)). In particular, those H-primes of O_{q}(M_{n}(k)) which do not contain the quantum determinant can be generated, up to suitable localizations, by quantum minors.

Geometric background. The classical origins of our main theorem, especially as it applies to winding-invariant primes of O_{q}(M_{n}(k)) not containing the quantum determinant, lie in the geometry of ‘LU-decompositions’ of invertible matrices. For this part of the introduction, let us assume (to avoid complications) that k is algebraically closed. An LUdecomposition of a matrix X ∈ GL_{n}(k) is any expression X = LU where L (respectively, U) is a lower (respectively, upper) triangular invertible matrix. It is well known that X has such a decomposition if and only if the principal minors of X (those indexed by rows and columns from an initial segment of {1, …, n}) are all nonzero. The LU-decomposable matrices thus form a dense open subvariety of GL_{n}(k), known as the big cell. We may write the big cell in the form B^{+}B^{−} where B^{+} (respectively, B^{−}) is the subgroup of lower (respectively, upper) triangular matrices in GL_{n}(k), and we have

O(B^{+}B^{−}) = O(GL_{n}(k))[D^{−1}]

where D is the multiplicative subset of O(GL_{n}(k)) generated by the principal minors. The comorphism of the multiplication map B^{+} × B^{−} → GL_{n}(k) provides an embedding

β : O(GL_{n}(k))[D^{−1}] ⟶ O(B^{+}) ⊗ O(B^{−});

the restriction of β to O(GL_{n}(k)) is just the composition of the comultiplication map O(GL_{n}(k)) → O(GL_{n}(k)) ⊗ O(GL_{n}(k)) with the tensor product of the restriction maps O(GL_{n}(k)) → O(B^{±}). The structure of the image of β is easy to determine, since B^{+}∩B^{−} is the diagonal subgroup of GL_{n}(k) and the subgroups B^{±} are semidirect products of their unipotent subgroups with this diagonal subgroup. Namely, the image of β is the subalgebra of O(B^{+}) ⊗ O(B^{−}) generated by (the cosets of) the elements

(†)  X_{ij}X_{jj}^{−1} ⊗ 1  1 ⊗ X_{ij}X_{ii}^{−1}  (X_{ii} ⊗ X_{ii})^{±1}.

Further, O(B^{+}) ⊗ O(B^{−}) is a Laurent polynomial ring over the image of β with respect to indeterminates 1 ⊗ X_{ii}^{±1}.

Quantum analogs of the above facts are known, but we have not been able to locate complete statements in the literature. To formulate them, let us write O_{q}(B^{+}) and O_{q}(B^{−}) for the respective quotients of O_{q}(GL_{n}(k)) modulo the ideals ⟨X_{ij} | i < j⟩ and ⟨X_{ij} | i > j⟩. Then:

(a) The composition of the comultiplication map on O_{q}(GL_{n}(k)) with the tensor product of the quotient maps O_{q}(GL_{n}(k)) → O_{q}(B^{±}) yields an embedding β : O_{q}(GL_{n}(k)) → O_{q}(B^{+}) ⊗ O_{q}(B^{−}) [16, Theorem 8.1.1].

(b) The multiplicative subset D of O_{q}(GL_{n}(k)) generated by the principal quantum minors is a denominator set.

(c) β extends to O_{q}(GL_{n}(k))[D^{−1}], and the image of this extension is the subalgebra B of O_{q}(B^{+}) ⊗ O_{q}(B^{−}) generated by (the cosets of) the elements (†).

(d) O_{q}(B^{+}) ⊗ O_{q}(B^{−}) is a skew-Laurent extension of B with respect to the variables 1 ⊗ X_{ii}^{±1}.

These facts will be proved as part of the case r = c = (1, …, n) of our work below. To obtain them via existing results in the literature, one first transfers the problem to O_{q}(SL_{n}(k)) using the isomorphism O_{q}(GL_{n}(k)) ≅ O_{q}(SL_{n}(k))[z^{±1}] established in [13, Proposition]; the desired conditions then hold in the generality of O_{q}(G), where G is an arbitrary semisimple algebraic group. The isomorphism of the appropriate localization of O_{q}(G) with the analog of B is given in [3, Theorem 4.6] and [12, Proposition 9.2.14]. It is easy to see that O_{q}(B^{+}) ⊗ O_{q}(B^{−}) is a skew-Laurent extension of B with appropriate variables; this is mentioned for the case where q is a root of unity in [4, (4.6)].

The above facts concerning O_{q}(GL_{n}(k)) immediately carry over to O_{q}(M_{n}(k)), since D is also a denominator set in that algebra and O_{q}(M_{n}(k))[D^{−1}] = O_{q}(GL_{n}(k))[D^{−1}]. Whereas in O_{q}(GL_{n}(k)) all prime ideals are disjoint from D, that no longer holds in O_{q}(M_{n}(k)), and a sequence of modified versions of (a)–(d) are needed to yield information about those H-primes of O_{q}(M_{n}(k)) that meet D. These modifications, involving maps

β_{r,c} : O_{q}(M_{n}(k)) ⟶ O_{q}(M_{n}(k)) ⊗ O_{q}(M_{n}(k)) ⟶ R_{r}^{+} ⊗ R_{c}^{−},

concern quantum analogs of what can be viewed as LU-decompositions for certain locally closed subsets of M_{n}(k) of the form B^{+}w^{+}ew^{−}B^{−} where w^{±} are permutation matrices and e = e_{11} + ⋯ + e_{tt} is a diagonal idempotent matrix. We leave the formulations of these geometric facts to the interested reader.

Some notation and conventions. Throughout the paper, let A = O_{q}(M_{n}(k)), where we fix a base field k, a positive integer n, and a nonzero scalar q ∈ k^{×}. See (5.1)(a) for the basic relations satisfied by the standard generators X_{ij} of O_{q}(M_{n}(k)). As mentioned in (5.2), our relations for A differ from those in [16] by an interchange of q and q^{−1}. On the other hand, they agree with the relations used in [2, 15]. To match the relations in [10, 11], replace q by q^{2}.

While our main interest is in the case that q is not a root of unity, some of our results do not require this assumption, and others require only that q ≠ ±1. Thus, we impose no blanket hypotheses on q. To simplify some formulas, write q̂ = q − q^{−1}. We also fix the torus H = (k^{×})^{n} × (k^{×})^{n} and its action on A by winding automorphisms as described in (∗) above. The algebra A is graded in a natural way by ℤ^{2n} = ℤ^{n} ×ℤ^{n}, each generator X_{ij} having degree (ϵ_{i}, ϵ_{j}) where ϵ_{1}, …, ϵ_{n} is the standard basis for ℤ^{n}. We refer to this grading as the standard grading on A. (As long as k is infinite, the homogeneous components of A for this grading coincide with the eigenspaces for the action of H.)

As in [6], we use the notations [I | J] and [i_{1} ⋯ i_{s} | j_{1} ⋯ j_{s}] for quantum minors in O_{q}(M_{n}(k)), where I (respectively, i_{1}, …, i_{s}) records the set (respectively, a list) of the corresponding row indices, and similarly for column indices. Recall that [I | J] corresponds to the quantum determinant in a subalgebra of O_{q}(M_{n}(k)) isomorphic to O_{q}(M_{s}(k)); we write O_{q}(M_{I,J}(k)) for that subalgebra (see (5.1)(c)). We allow the index sets I and J to be empty, following the convention that [∅ | ∅] = 1.

The symbols ⊂ and ⊆ will be reserved for proper and arbitrary inclusions, respectively. We write ⊔ to denote a disjoint union.

1. A partition of spec A

We begin by investigating sets spec_{r,c} A of prime ideals defined by certain ‘stepwise patterns’ of quantum minors corresponding to strictly increasing sequences r and c of row and column indices. Our aim in this section is to show that these sets partition spec A when q is not a root of unity. In fact, as long as q ≠ ±1, these sets at least partition the collection of completely prime ideals of A. (Recall from [8, Theorem 3.2] that when q is not a root of unity, all primes of A are completely prime.) We proceed without imposing special hypotheses on q until needed.

1.1. We introduce the following partial ordering ≤ on index sets I, I′ ⊆ {1, …, n} of the same cardinality. Write I = {i_{1} < ⋯ < i_{l}} and I′ = {i′_{1} < ⋯ < i′_{l}}; then I ≤ I′ if and only if i_{s} ≤ i′_{s} for s = 1, … l. (This is the same as the ‘column ordering’ ≤_{c} used in [6].) All order relations among index sets in this paper will refer to the above partial ordering. This includes statements that an index set with a particular property is minimal among index sets with the same cardinality satisfying that property (e.g., the sets Ĩ and J̃ in Theorem 1.9).

1.2. Let RC denote the set of all pairs (r, c) where r and c are strictly increasing sequences in {1, …, n} of the same length, that is, r = (r_{1}, …, r_{t}) and c = (c_{1}, …, c_{t}) in ℕ^{t} with 1 ≤ r_{1} < r_{2} < ⋯ < r_{t} ≤ n and 1 ≤ c_{1} < c_{2} < ⋯ < c_{t} ≤ n. We allow t = 0, in which case r and c are empty sequences, denoted either ( ) or ∅. When referring to the length t of r and c, we write (r, c) ∈ RC_{t}.

For (r, c) ∈ RC_{t}, let K_{r,c} be the ideal of A generated by the following set of quantum minors:

{[I | J] | |I| > t} ∪ {[I | J] | |I| = l ≤ t and I ≱ {r_{1}, …, r_{l}}}
∪ {[I | J] | |I| = l ≤ t and J ≱ {c_{1}, …, c_{l}}}.

In particular, K_{∅,∅} = ⟨X_{ij} | i, j ∈ {1, …, n}⟩.

Now set d_{l}^{r,c} = [r_{1} ⋯ r_{l} | c_{1} ⋯ c_{l}] for l ≤ t, and observe that these quantum minors commute with each other (cf. (5.1)(c)). Let D_{r,c} denote the multiplicative subset of A generated by d_{1}^{r,c}, …, d_{t}^{r,c}. (In particular, D_{∅,∅} = {1}.) Set d̃_{l}^{r,c} = d_{l}^{r,c} + K_{r,c} ∈ A/K_{r,c} for l ≤ t, and let D̃_{r,c} denote the image of D_{r,c} in A/K_{r,c}.

We prove in this section that D̃_{r,c} is a denominator set in A/K_{r,c}, and that when q is generic, spec A is partitioned by the subsets

spec_{r,c} A := {P ∈ spec A | K_{r,c} ⊆ P and P ∩ D_{r,c} = ∅}
≈ spec (A/K_{r,c})[D̃_{r,c}^{−1}]

as (r, c) ranges over RC.

1.3. Lemma. Let I, J ⊆ {1, …, n} with |I| = |J|, and set

L = ⟨ [I′ | J′] | |I′| = |I|, and I′ < I or J′ < J ⟩.

If r, c ∈ {1, …, n} with r ≤ max(I) or c ≤ max(J), then

(∗)  [I | J]X_{rc} − q^{2−δ(r,I)−δ(c,J)}X_{rc}[I | J] ∈ L.

Proof. If r ∈ I and c ∈ J, then [I | J] commutes with X_{rc} (cf. Lemma 5.2(a)), and so (*) holds.

Next, suppose that r ∈ I and c ∉ J. If j ∈ J and j > c, then J ⊔ {c} \ {j} < J, whence [I | J ⊔ {c} \ {j}] ∈ L by definition of L. It follows from Lemma 5.2(b2) that [I | J]X_{rc} − qX_{rc}[I | J] ∈ L. Thus (*) holds in this case. The case where r ∉ I and c ∈ J is proved similarly, using Lemma 5.2(c2).

Finally, suppose that r ∉ I and c ∉ J, and note that either r < max(I) or c < max(J). (This part of the proof is similar to that of Lemma 5.7.) There is no loss of generality in assuming that

I ⊔ {r} = J ⊔ {c} = {1, …, n},

whence either r < n or c < n. In the notation of [16, (4.3)], [I | J] = A(r c). Note that {1, …, n} \ {i} < I when i > r, while {1, …, n} \ {j} < J when j > c. Hence, A(i j) ∈ L whenever either i > r or j > c. Set D_{q} = [1 ⋯ n | 1 ⋯ n]. The basic q-Laplace relations (Corollary 5.5) imply that D_{q} lies in the ideal generated by all the A(n j), and in the ideal generated by all the A(i n). Since either n > r or n > c, it follows that D_{q} ∈ L.

We now use the basic q-Laplace relations in the form given in [16, Corollary 4.4.4]. The first two relations yield

(1)  ∑_{j=1}^{n}(−q)^{j−r}X_{rj}A(r j) = ∑_{j=1}^{n}(−q)^{r−j}A(r j)X_{rj} = D_{q} ∈ L.

Since A(r j) ∈ L for j > c, we obtain the following congruences, after multiplying the two sums in (1) by (−q)^{r−c} and (−q)^{c−r}, respectively:

(2)  X_{rc}A(r c) ≡ − ∑_{j<c}(−q)^{j−c}X_{rj}A(r j)  (mod L)
(3)  A(r c)X_{rc} ≡ − ∑_{j<c}(−q)^{c−j}A(r j)X_{rj}  (mod L).

For any j, the third relation of [16, Lemma 5.1.2] implies that

(4)  X_{rj}A(r j) ≡ A(r j)X_{rj} + (1 − q^{−2}) ∑_{l<j}(−q)^{j−l}A(r l)X_{rl}  (mod L),

since X_{sj}A(s j) ∈ L for s > r. Substituting (4) into (2) for all j < c, we obtain

(5)  \begin{aligned}X_{rc}A(r c) & ≡ − ∑_{j<c}(−q)^{j−c}A(r j)X_{rj} \\ & − (1 − q^{−2}) ∑_{j<c} ∑_{l<j}(−q)^{2j−l−c}A(r l)X_{rl}  (mod L) \\ & = − ∑_{l<c}[(−q)^{l−c} + (1 − q^{−2}) ∑_{l<j<c}(−q)^{2j−l−c}]A(r l)X_{rl}. \end{aligned}

The expression in square brackets can be simplified as follows:

(6) \begin{aligned}(−q)^{l−c} + (1 − q^{−2}) ∑_{l<j<c}(−q)^{2j−l−c} & = (−q)^{l−c}[1 + (1 − q^{−2}) ∑_{0<m<c−l}(−q)^{2m}] \\ & = (−q)^{c−l−2}. \end{aligned}

Substituting (6) into (5) and replacing l by j, we obtain

(7)  X_{rc}A(r c) ≡ − ∑_{j<c}(−q)^{c−j−2}A(r j)X_{rj}  (mod L).

Finally, combining (3) with (7), we conclude that

[I | J]X_{rc} = A(r c)X_{rc} ≡ q^{2}X_{rc}A(r c) = q^{2}X_{rc}[I | J] (mod L),

as desired. □

1.4. Corollary. Let I, J ⊆ {1, …, n} with |I| = |J|, and set

L = ⟨ [I′ | J′] | |I′| = |I|, and I′ < I or J′ < J ⟩.

Then the coset d = [I | J] + L generates a denominator set in A/L.

Proof. Set B = A/L, and set x_{ij} = X_{ij} + L for all i, j. Lemma 1.3 says that

(1)  dx_{ij} = q^{2−δ(i,I)−δ(j,J)}x_{ij}d

whenever i ≤ max(I) or j ≤ max(J). Hence, in this case we have d^{r}x_{ij} ∈ Bd^{r} for all r ≥ 0.

When i > max(I) and j > max(J), Lemma 5.7 says that

(2)  dx_{ij} − q^{2}x_{ij}d = e := (1 − q^{2})[I ⊔ {i} | J ⊔ {j}] + L.

Observe that d and e commute. Hence, it follows from (2) by an easy induction that

(3)  \begin{aligned}d^{r}x_{ij} & = q^{2r}x_{ij}d^{r} + (q^{2r−2} + ⋯ + q^{2} + 1)ed^{r−1} \\ & = [q^{2r}x_{ij}d + (q^{2r−2} + ⋯ + q^{2} + 1)e]d^{r−1} \end{aligned}

for all r > 0. Combining (1) and (3), we see that

(4)  d^{r}x_{ij} ∈ Bd^{r−1}  (i, j = 1, …, n; r = 1, 2, …).

Since B is spanned by products of the x_{ij}, it follows from (4) that D := {d^{r} | r ≥ 0} is a left Ore set in B. Similarly, D is right Ore, and therefore D is a denominator set because B is noetherian. □

1.5. Proposition. Let (r, c) ∈ RC_{t}, and set

L = ⟨ [I | J] | |I| = l ≤ t, and I < {r_{1}, …, r_{l}} or J < {c_{1}, …, c_{l}} ⟩.

Then the image of D_{r,c} in A/L is a denominator set. Consequently, D̃_{r,c} is a denominator set in A/K_{r,c}.

Proof. For each l = 1, …, t, Corollary 1.4 shows that d_{l}^{r,c} + L generates a denominator set in A/L. The proposition follows. □

1.6. In view of Proposition 1.5, we can form Ore localizations

A_{r,c} = (A/K_{r,c})[D̃_{r,c}^{−1}]

for (r, c) ∈ RC. (It will follow from Lemma 2.5 that A_{r,c} ≠ 0.) The localization maps A → A/K_{r,c} → A_{r,c} induce Zariski homeomorphisms

spec_{r,c} A ⟶ spec A_{r,c}.

Lemma 1.7. Let I, J ⊆ {1, …, n} with |I| = |J|, and let P be an ideal of A. (a) Fix J_{1} ⊆ J. If [I_{1} | J_{1}] ∈ P for all I_{1} ⊆ I with |I_{1}| = |J_{1}|, then [I | J] ∈ P.

(b) Fix I_{1} ⊆ I. If [I_{1} | J_{1}] ∈ P for all J_{1} ⊆ J with |J_{1}| = |I_{1}|, then [I | J] ∈ P.

Proof. By symmetry (see (5.1)(b)), we need only prove (a). Set J_{2} = J \ J_{1}. Then Lemma 5.4(a) provides a relation of the form

∑_{I_{1}⊔I_{2}=I} ±q^{•}[I_{1} | J_{1}][I_{2} | J_{2}] = ±q^{•}[I | J],

where ± is an unspecified sign and q^{•} stands for an unspecified power of q. Since all the [I_{1} | J_{1}] ∈ P by assumption, it follows that [I | J] ∈ P. □

Lemma 1.8. Let I_{1}, I_{2}, J_{1}, J_{2} ⊆ {1, …, n} with |I_{1}| = |J_{1}| and |I_{2}| = |J_{2}|, and let P be a completely prime ideal of A. Assume that one of the following conditions (a), (b), or (c) holds:

(a) (1) |I_{1} ∩ I_{2}| > |J_{1} ∩ J_{2}|, and (2) [I_{1} | J′] ∈ P whenever J_{1} ∩ J_{2} ⊆ J′ ⊆ J_{1} ∪ J_{2} with |J′| = |J_{1}| but J′ ≠ J_{1}.

(b) (1) |I_{1} ∩ I_{2}| < |J_{1} ∩ J_{2}|, and (2) [I′ | J_{1}] ∈ P whenever I_{1} ∩ I_{2} ⊆ I′ ⊆ I_{1} ∪ I_{2} with |I′| = |I_{1}| but I′ ≠ I_{1}.

(c) (1) |I_{1} ∩ I_{2}| = |J_{1} ∩ J_{2}| and [I_{1} ∪ I_{2} | J_{1} ∪ J_{2}] ∈ P, and (2) Either (a)(2) or (b)(2) holds.

Then either [I_{1} | J_{1}] ∈ P or [I_{2} | J_{2}] ∈ P.

Proof. By symmetry, it suffices to prove cases (a) and (c).

(a) Set V = J_{1} ∪ J_{2} = J ⊔ L where J = J_{1} ∩ J_{2} and L = V \ J. Since |I_{1} ∩ I_{2}| > |J|, we have |I_{1} ∪ I_{2}| < |V |, and so there exists U ⊆ {1, …, n} such that I_{1} ∪ I_{2} ⊂ U and |U| = |V |. Also, we have |I_{1}| + |I_{2}| = |J_{1}| + |J_{2}| = 2|J| + |L|. Thus, Lemma 5.6(b) yields a relation of the form

(†)  ∑_{L=L′⊔L′′} ±q^{•}[I_{1} | J ⊔ L′][I_{2} | J ⊔ L′′] = 0.

Now for each term in this sum, J_{1} ∩J_{2} ⊆ J ⊔L′ ⊆ J_{1} ∪J_{2} with |J ⊔L′| = |J_{1}|. If L′ ≠ J_{1} \J, then J ⊔ L′ ≠ J_{1}, whence [I_{1} | J ⊔ L′] ∈ P by hypothesis. Therefore the remaining term in (†) must lie in P. This is the term with L′ = J_{1} \J, whence L′′ = J_{2} \J, and so J ⊔L′ = J_{1} and J ⊔ L′′ = J_{2}. Thus

±q^{•}[I_{1} | J_{1}][I_{2} | J_{2}] ∈ P.

Since P is completely prime, either [I_{1} | J_{1}] ∈ P or [I_{2} | J_{2}] ∈ P.

(c) By symmetry, we may assume that (a)(2) holds. Again, set V = J_{1} ∪ J_{2} = J ⊔ L where J = J_{1} ∩ J_{2} and L = V \ J. Set U = I_{1} ∪ I_{2}, and observe that |I_{1}| + |I_{2}| = 2|J| + |L|. This time, Lemma 5.6(b) provides a relation of the form

(‡)  ∑_{L=L′⊔L′′} ±q^{•}[I_{1} | J ⊔ L′][I_{2} | J ⊔ L′′] = ±q^{•}[I_{1} ∩ I_{2} | J][U | V].

Since [U | V] ∈ P by hypothesis, the right hand side of (‡) lies in P. Therefore we can proceed as in the proof above. □

1.9. Theorem. Assume that q ≠ ±1. Let P be a completely prime ideal of A, and let t ≤ n be maximal such that P does not contain all t × t quantum minors. Choose

Ĩ = {r_{1} < ⋯ < r_{t}} ⊆ {1, …, n}  and  J̃ = {c_{1} < ⋯ < c_{t}} ⊆ {1, …, n}

with Ĩ minimal such that some [Ĩ | ∗] ∉ P, and J̃ minimal such that some [∗ | J̃] ∉ P. Then

(a) [r_{1} ⋯ r_{s} | c_{1} ⋯ c_{s}] ∉ P for s = 1, …, t. In particular, [Ĩ | J̃] ∉ P.

(b) [I | J] ∈ P whenever |I| = s ≤ t and either I ≱ {r_{1}, …, r_{s}} or J ≱ {c_{1}, …, c_{s}}. In particular, it follows that Ĩ and J̃ are unique.

Proof. Since the theorem holds trivially when t = 0, we may assume that t > 0. By assumption, there exists J_{0} such that [Ĩ | J_{0}] ∉ P. We first claim that

(1) [I | J] ∈ P whenever |I| = s ≤ t and I < {r_{1}, …, r_{s}}. Suppose not, so that some [I | J] ∉ P where |I| = s ≤ t and I < {r_{1}, …, r_{s}}. We may assume that s is minimal for this, and that with I fixed, |J ∩ J_{0}| is maximal. Note from the minimality of Ĩ that s < t.

Write I = {i_{1} < ⋯ < i_{s}}. There is some b ≤ s such that i_{l} = r_{l} for l < b while i_{b} < r_{b}, whence {i_{1}, …, i_{b}} < {r_{1}, …, r_{b}}. Since [I | J] ∉ P, Lemma 1.7(b) implies that some [i_{1} ⋯ i_{b} | ∗] ∉ P. Hence, the minimality of s implies that b = s. Thus, we have i_{l} = r_{l} for l < s while i_{s} < r_{s}. In particular, r_{s−1} = i_{s−1} < i_{s} < r_{s}.

Assume that |J ∩ J_{0}| ≥ s − 1. In this case, we will apply Lemma 1.8 with

I_{1} = Ĩ  I_{2} = I  J_{1} = J_{0}  J_{2} = J.

Note that |I_{1} ∩ I_{2}| = s − 1 and [I_{1} | J_{1}], [I_{2} | J_{2}] ∉ P. When I_{1} ∩ I_{2} ⊆ I′ ⊆ I_{1} ∪ I_{2} with |I′| = t and I′ ≠ I_{1}, we have I′ = Ĩ ∪ {i_{s}} \ {r_{l}} for some l ≥ s. Then since r_{s−1} < i_{s} < r_{s} ≤ r_{l}, we have I′ < Ĩ, and so [I′ | J_{1}] ∈ P by the minimality of Ĩ. Further, if |J_{1} ∩ J_{2}| = s − 1, then [I_{1} ∪ I_{2} | J_{1} ∪ J_{2}] ∈ P because P contains all (t + 1) × (t + 1) quantum minors. Therefore by Lemma 1.8(b), if |J ∩ J_{0}| ≥ s, or by Lemma 1.8(c), if |J ∩ J_{0}| = s − 1, we have either [I_{1} | J_{1}] ∈ P or [I_{2} | J_{2}] ∈ P, giving us a contradiction. Therefore |J ∩ J_{0}| < s − 1.

Next, we will apply Lemma 1.8 with the roles of I_{1}, I_{2} and J_{1}, J_{2} reversed, that is, with

I_{1} = I  I_{2} = Ĩ  J_{1} = J  J_{2} = J_{0}.

When J_{1} ∩J_{2} ⊆ J′ ⊆ J_{1} ∪J_{2} with |J′| = s and J′ ≠ J_{1}, we must have |J′ ∩J_{0}| = |J′ ∩J_{2}| > |J ∩ J_{0}|. By the maximality of |J ∩ J_{0}|, we obtain [I | J′] ∈ P in this case. But then Lemma 1.8(a) leads to the same contradiction.

Therefore (1) holds. By symmetry, we must also have

(2) [I | J] ∈ P whenever |I| = s ≤ t and J < {c_{1}, …, c_{s}}. We now proceed by induction on s = 1, …, t to verify the following properties:

(P_{s}) [r_{1} ⋯ r_{s} | c_{1} ⋯ c_{s}] ∉ P; (Q_{s}) [I | J] ∈ P whenever |I| = s and either I ≱ {r_{1}, …, r_{s}} or J ≱ {c_{1}, …, c_{s}}.

The theorem will then be established.

To start, note that we cannot have all [r_{1} | ∗] ∈ P or all [∗ | c_{1}] ∈ P, by Lemma 1.7. Choose i, j such that [r_{1} | j], [i | c_{1}] ∉ P. If j = c_{1} or i = r_{1}, then [r_{1} | c_{1}] ∉ P. Otherwise, in view of (1) and (2) we must have j > c_{1} and i > r_{1}. Hence, because of the assumption that q ≠ ±1, we have

[r_{1} | c_{1}][i | j] − [i | j][r_{1} | c_{1}] = (q − q^{−1})[r_{1} | j][i | c_{1}] ∉ P,

which implies that [r_{1} | c_{1}] ∉ P. Therefore (P_{1}) holds. Property (Q_{1}) is immediate from (1) and (2).

Now let 1 < s ≤ t and assume that (P_{a}) and (Q_{a}) hold for all a < s. By Lemma 1.7, there exist j_{1} < ⋯ < j_{s} such that

[r_{1} ⋯ r_{s} | j_{1} ⋯ j_{s}] ∉ P,

and we may assume that {j_{1}, …, j_{s}} is minimal for this. Likewise, there exist i_{1} < ⋯ < i_{s} such that [i_{1} ⋯ i_{s} | c_{1} ⋯ c_{s}] ∉ P and such that {i_{1}, …, i_{s}} is minimal for this. We cannot have {i_{1}, …, i_{s−1}} ≱ {r_{1}, …, r_{s−1}}, since then (Q_{s−1}) would imply that all [i_{1} ⋯ i_{s−1} | ∗] ∈ P, whence Lemma 1.7(b) would imply that all [i_{1} ⋯ i_{s} | ∗] ∈ P. Therefore {i_{1}, …, i_{s−1}} ≥ {r_{1}, …, r_{s−1}}, and similarly {j_{1}, …, j_{s−1}} ≥ {c_{1}, …, c_{s−1}}.

Suppose there exists b < s such that j_{l} = c_{l} for l < b while j_{b} > c_{b}. We will apply Lemma 1.8 with

I_{1} = {r_{1}, …, r_{s}}  I_{2} = {r_{1}, …, r_{b}}
J_{1} = {j_{1}, …, j_{s}}  J_{2} = {c_{1}, …, c_{b}}.

Observe that |I_{1} ∩ I_{2}| = b > |J_{1} ∩ J_{2}|. If J_{1} ∩ J_{2} ⊆ J′ ⊆ J_{1} ∪ J_{2} with |J′| = s and J′ ≠ J_{1}, then J′ = J_{1} ∪ {c_{b}} \ {j_{d}} for some d ≥ b. In this case, J′ < {j_{1}, …, j_{s}}, whence [I_{1} | J′] ∈ P by the minimality of {j_{1}, …, j_{s}}. Therefore Lemma 1.8(a) implies that either [I_{1} | J_{1}] ∈ P or [I_{2} | J_{2}] ∈ P. But [I_{1} | J_{1}] ∉ P by choice of J_{1}, and [I_{2} | J_{2}] ∉ P by (P_{b}), so we have a contradiction. Therefore j_{l} = c_{l} for all l < s. Similarly, i_{l} = r_{l} for all l < s.

If j_{s} = c_{s}, then (P_{s}) holds. If j_{s} < c_{s}, then {j_{1}, …, j_{s}} < {c_{1}, …, c_{s}}, which would imply [r_{1} ⋯ r_{s} | j_{1} ⋯ j_{s}] ∈ P by (2), contradicting our assumptions. Therefore we may assume that j_{s} > c_{s}. Likewise, we may assume that i_{s} > r_{s}.

Set U = {r_{1}, …, r_{s}, i_{s}} and V = {c_{1}, …, c_{s}, j_{s}}. By Lemma 5.3(d), we have

[i_{1} ⋯ i_{s} | j_{1} ⋯ j_{s}][r_{1} ⋯ r_{s} | c_{1} ⋯ c_{s}] − [r_{1} ⋯ r_{s} | c_{1} ⋯ c_{s}][i_{1} ⋯ i_{s} | j_{1} ⋯ j_{s}]

= (q^{−1} − q)[r_{1} ⋯ r_{s} | j_{1} ⋯ j_{s}][i_{1} ⋯ i_{s} | c_{1} ⋯ c_{s}].

Since neither of the factors [r_{1} ⋯ r_{s} | j_{1} ⋯ j_{s}] and [i_{1} ⋯ i_{s} | c_{1} ⋯ c_{s}] is in P, it follows that [r_{1} ⋯ r_{s} | c_{1} ⋯ c_{s}] cannot be in P. This establishes property (P_{s}).

Finally, suppose that (Q_{s}) fails. By symmetry, we may assume that [I | J] ∉ P for some I, J with |I| = s and I ≱ {r_{1}, …, r_{s}}. We may also assume that I is minimal for this, and that with I fixed, J is minimal.

Write I = {i_{1} < ⋯ < i_{s}} and J = {j_{1} < ⋯ < j_{s}}. If {i_{1}, …, i_{s−1}} ≱ {r_{1}, …, r_{s−1}}, then by (Q_{s−1}) we would have all [i_{1} ⋯ i_{s−1} | ∗] ∈ P, whence Lemma 1.7(b) would imply that all [I | ∗] ∈ P, contradicting our choice of I. Thus {i_{1}, …, i_{s−1}} ≥ {r_{1}, …, r_{s−1}}, and similarly {j_{1}, …, j_{s−1}} ≥ {c_{1}, …, c_{s−1}}. Since I ≱ {r_{1}, …, r_{s}}, we must also have i_{s} < r_{s}. Note that r_{s−1} ≤ i_{s−1} < i_{s} < r_{s}, and so i_{s} ∉ {r_{1}, …, r_{s−1}}. Further, {r_{1}, …, r_{s−1}, i_{s}} < {r_{1}, …, r_{s}}, and so all [r_{1} ⋯ r_{s−1}i_{s} | ∗] ∈ P by (1), whence I ≠ {r_{1}, …, r_{s−1}, i_{s}}. Therefore there is some b < s such that i_{l} = r_{l} for l < b while i_{b} > r_{b}.

Suppose there exists d ≤ b such that j_{m} = c_{m} for m < d while j_{d} > c_{d}. We will apply Lemma 1.8 with

I_{1} = I = {i_{1}, …, i_{s}}  I_{2} = {r_{1}, …, r_{d}}
J_{1} = J = {j_{1}, …, j_{s}}  J_{2} = {c_{1}, …, c_{d}}.

Note that |I_{1} ∩ I_{2}| ≥ d − 1 = |J_{1} ∩ J_{2}|, and that |I_{1} ∩ I_{2}| = |J_{1} ∩ J_{2}| only when d = b. Since r_{b−1} < r_{b} < i_{b}, we have {r_{1}, …, r_{b}, i_{b}, …, i_{s−1}} < I, and so all [r_{1} ⋯ r_{b}i_{b} ⋯ i_{s−1} | ∗] ∈ P by the minimality of I. Then Lemma 1.7(b) implies that all [r_{1} ⋯ r_{b}i_{b} ⋯ i_{s} | ∗] ∈ P. In particular, when d = b we find that [I_{1} ∪ I_{2} | J_{1} ∪ J_{2}] ∈ P.

If J_{1} ∩ J_{2} ⊆ J′ ⊆ J_{1} ∪ J_{2} with |J′| = s and J′ ≠ J_{1}, then J′ = J ∪ {c_{d}} \ {j_{p}} for some p ≥ d. In this case, J′ < J, and so [I_{1} | J′] ∈ P by the minimality of J. Hence, case (a) of Lemma 1.8 (if d > b) or case (c) (if d = b) implies that either [I_{1} | J_{1}] ∈ P or [I_{2} | J_{2}] ∈ P. But [I_{1} | J_{1}] ∉ P by assumption, and [I_{2} | J_{2}] ∉ P by (P_{d}), so we have a contradiction. Therefore j_{m} = c_{m} for all m ≤ b.

We will conclude by applying Lemma 1.8 with

I_{1} = I = {i_{1}, …, i_{s}}  I_{2} = {r_{1}, …, r_{b}}
J_{1} = J = {j_{1}, …, j_{s}}  J_{2} = {c_{1}, …, c_{b}}.

Note that i_{b−1} = r_{b−1} < r_{b} < i_{b} implies r_{b} ∉ I, and so |I_{1} ∩ I_{2}| < |J_{1} ∩ J_{2}|. If I_{1} ∩ I_{2} ⊆ I′ ⊆ I_{1} ∪ I_{2} with |I′| = s and I′ ≠ I_{1}, then I′ = I ∪ {r_{b}} \ {i_{p}} for some p ≥ b. In this case, I′ < I, whence [I′ | J_{1}] ∈ P by the minimality of I. Thus Lemma 1.8(b) implies that either [I_{1} | J_{1}] ∈ P or [I_{2} | J_{2}] ∈ P, and again we have reached a contradiction.

Therefore (Q_{s}) must hold, which establishes our induction step. □

1.10. Corollary. Assume that q ≠ ±1. Given any completely prime ideal P ∈ spec A, there is a unique pair (r, c) ∈ RC such that K_{r,c} ⊆ P and P ∩ D_{r,c} = ∅.

Thus, if q is not a root of unity,

spec A = ⨆_{(r,c)∈RC} spec_{r,c} A.

Proof. Let t ≤ n be maximal such that P does not contain all t × t quantum minors, let {r_{1} < ⋯ < r_{t}} and {c_{1} < ⋯ < c_{t}} be as in Theorem 1.9, and set r = (r_{1}, …, r_{t}) and c = (c_{1}, …, c_{t}). The theorem implies that K_{r,c} ⊆ P and that d_{s}^{r,c} ∉ P for s = 1, …, t. Since P is completely prime, it follows that P ∩ D_{r,c} = ∅.

Now suppose that we also have (r′, c′) ∈ RC_{t′} for some t′ such that K_{r′,c′} ⊆ P and P ∩ D_{r′,c′} = ∅. Then P contains all (t′ + 1) × (t′ + 1) quantum minors but not all t′ × t′ quantum minors, whence t′ = t. Moreover, we have d_{t}^{r,c} ∉ K_{r′,c′} and d_{t}^{r′,c′} ∉ K_{r,c}. The first relation implies that r ≥ r′ and c ≥ c′, and the second relation yields the reverse inequalities. (Here we have transferred the relation ≤ in (1.1) from index sets to sequences in the obvious manner.) Therefore r′ = r and c′ = c. □

2. Structure of A_{r,c}

The purpose of this section is to develop a structure theorem for the localizations A_{r,c}. We introduce localized factor algebras R_{r}^{+} and R_{c}^{−} of A (patterned after quantized coordinate rings of groups of triangular matrices) together with subalgebras B_{r}^{+} ⊂ R_{r}^{+} and B_{c}^{−} ⊂ R_{c}^{−} (patterned after quantized coordinate rings of unipotent groups of triangular matrices), and we show that A_{r,c} is isomorphic to an algebra B_{r,c} trapped between B_{r}^{+} ⊗ B_{c}^{−} and R_{r}^{+} ⊗ R_{c}^{−}. More precisely, we prove that B_{r,c} is a skew-Laurent extension of B_{r}^{+} ⊗ B_{c}^{−} (this is a key ingredient in establishing that A_{r,c} ≅ B_{r,c}), and that R_{r}^{+} ⊗ R_{c}^{−} is a skew-Laurent extension of B_{r,c}.

2.1. Fix t ∈ {0, 1, …, n} and (r, c) ∈ RC_{t} throughout the section. Set

R_{r,0}^{+} = A/⟨X_{ij} | j > t or i < r_{j}⟩  and  R_{c,0}^{−} = A/⟨X_{ij} | i > t or j < c_{i}⟩.

Write Y_{ij} and Z_{ij} for the images of X_{ij} in R_{r,0}^{+} and R_{c,0}^{−}, respectively. Note that these algebras are iterated skew polynomial extensions of k, hence noetherian domains, the natural indeterminates for these iterated skew polynomial structures being those Y_{ij} and Z_{ij} which are nonzero. These indeterminates, when recorded within an n × n matrix, display ‘stairstep’ patterns – for example, if n = 4 and r = (1, 2, 4), the Y_{ij} may be displayed as follows:

[ \begin{matrix}Y_{11} & 0 & 0 & 0 \\ Y_{21} & Y_{22} & 0 & 0 \\ Y_{31} & Y_{32} & 0 & 0 \\ Y_{41} & Y_{42} & Y_{43} & 0 \end{matrix} ] .

Since all the information is recorded in the placement of the zero and nonzero positions within this matrix, a convenient abbreviation for this example is to write

R_{(1,2,4),0}^{+} = k [ \begin{matrix}+ & 0 & 0 & 0 \\ + & + & 0 & 0 \\ + & + & 0 & 0 \\ + & + & + & 0 \end{matrix} ] .

Observe that the Y_{r_{s}s} are regular normal elements in R_{r,0}^{+}, and that the Z_{sc_{s}} are regular normal elements in R_{c,0}^{−}. More precisely,

Y_{r_{s}s}Y_{ij} = \{ \begin{matrix}q^{−1}Y_{ij}Y_{r_{s}s} & (i = r_{s}, j ≠ s) \\ qY_{ij}Y_{r_{s}s} & (i ≠ r_{s}, j = s) \\ Y_{ij}Y_{r_{s}s} & (i ≠ r_{s}, j ≠ s) \end{matrix}

Z_{sc_{s}}Z_{lm} = \{ \begin{matrix}qZ_{lm}Z_{sc_{s}} & (l = s, m ≠ c_{s}) \\ q^{−1}Z_{lm}Z_{sc_{s}} & (l ≠ s, m = c_{s}) \\ Z_{lm}Z_{sc_{s}} & (l ≠ s, m ≠ c_{s}). \end{matrix}

(For instance, the first relation above holds when j > s because Y_{ij} = 0 in that case. To verify the third relation, observe that Y_{r_{s}j} = 0 if j > s, while Y_{is} = 0 if i < r_{s}.) In particular, the Y_{r_{s}s} commute with each other, and the Z_{sc_{s}} commute with each other.

Due to the normality of the Y_{r_{s}s} and the Z_{sc_{s}}, we can form Ore localizations

R_{r}^{+} = R_{r,0}^{+}[Y_{r_{1}1}^{−1}, Y_{r_{2}2}^{−1}, …, Y_{r_{t}t}^{−1}]  and  R_{c}^{−} = R_{c,0}^{−}[Z_{1c_{1}}^{−1}, Z_{2c_{2}}^{−1}, …, Z_{tc_{t}}^{−1}].

These algebras are noetherian domains, and they may be viewed as quantized coordinate rings of certain locally closed subvarieties of M_{n}(k). Extending the abbreviated description given for the example above, we display the following abbreviation for R_{r}^{+} in that case:

R_{(1,2,4)}^{+} = k [ \begin{matrix}± & 0 & 0 & 0 \\ + & ± & 0 & 0 \\ + & + & 0 & 0 \\ + & + & ± & 0 \end{matrix} ] .

Observe that the standard ℤ^{2n}-grading on A induces ℤ^{2n}-gradings on R_{r}^{+} and R_{c}^{−}, which we also refer to as standard.

2.2. Let π_{r,0}^{+} : A → R_{r,0}^{+} and π_{c,0}^{−} : A → R_{c,0}^{−} be the quotient maps, and define

β_{r,c} : A ⟶^{Δ} A ⊗ A ⟶^{π_{r,0}^{+}⊗π_{c,0}^{−}} R_{r,0}^{+} ⊗ R_{c,0}^{−} ⟶^{⊆} R_{r}^{+} ⊗ R_{c}^{−}.

Observe that

β_{r,c}(X_{ij}) = ∑_{l≤t, r_{l}≤i, c_{l}≤j} Y_{il} ⊗ Z_{lj}

for all i, j. In particular, β_{r,c}(X_{ij}) = 0 when i < r_{1} or j < c_{1}.

2.3. Lemma. K_{r,c} ⊆ ker(β_{r,c}).

Remark. We conjecture that ker(β_{r,c}) = K_{r,c}.

Proof. Since X_{ij} ∈ ker(π_{r,0}^{+}) for j > t, all (t + 1) × (t + 1) and larger quantum minors lie in ker(π_{r,0}^{+}). In view of the rule for comultiplication of quantum minors (see (5.1)(d)), it follows that all (t + 1) × (t + 1) and larger quantum minors lie in ker(β_{r,c}).

Consider an index set I with |I| = l ≤ t and I ≱ {r_{1}, …, r_{l}}. Write I = {i_{1} < ⋯ < i_{l}}; then i_{m} < r_{m} for some m ≤ l. Hence, Y_{i_{s}j} = 0 for all s ≤ m and j ≥ m. This implies that π_{r,0}^{+}[i_{1} ⋯ i_{m} | M] = 0 for all M with |M| = m, whence π_{r,0}^{+}[I | K] = 0 for all K with |K| = l (cf. Lemmas 1.7 or 5.4). Therefore β_{r,c}[I | J] = 0 for all J with |J| = l.

Likewise, β_{r,c}[I | J] = 0 whenever |I| = l ≤ t and J ≱ {c_{1}, …, c_{l}}. □

2.4. Let B_{r,c} denote the k-subalgebra of R_{r}^{+} ⊗ R_{c}^{−} generated by the set

{Y_{il} ⊗ Z_{lj} | l ≤ t, i ≥ r_{l}, j ≥ c_{l}} ∪ {Y_{r_{l}l}^{−1} ⊗ Z_{lc_{l}}^{−1} | l ≤ t}.

We may also express B_{r,c} as the subalgebra of R_{r}^{+} ⊗ R_{c}^{−} generated by

{Y_{il}Y_{r_{l}l}^{−1} ⊗ 1 | l ≤ t, i > r_{l}} ∪ {1 ⊗ Z_{lj}Z_{lc_{l}}^{−1} | l ≤ t, j > c_{l}} ∪ {(Y_{r_{l}l} ⊗ Z_{lc_{l}})^{±1} | l ≤ t}.

Note that β_{r,c}(X_{ij}) ∈ B_{r,c} for all i, j, so that β_{r,c}(A) ⊆ B_{r,c}.

Let l ≤ t. Since Y_{r_{s}j} = 0 for s ≤ l and j > s, we have

π_{r,0}^{+}[r_{1} ⋯ r_{l} | K] = \{ \begin{matrix}Y_{r_{1}1}Y_{r_{2}2} ⋯ Y_{r_{l}l} & (K = \{1, …, l\}) \\ 0 & (K ≠ \{1, …, l\}). \end{matrix}

Similarly, π_{c,0}^{−}[1 ⋯ l | c_{1} ⋯ c_{l}] = Z_{1c_{1}}Z_{2c_{2}} ⋯ Z_{lc_{l}}, and therefore

β_{r,c}(d_{l}^{r,c}) = (Y_{r_{1}1} ⊗ Z_{1c_{1}})(Y_{r_{2}2} ⊗ Z_{2c_{2}}) ⋯ (Y_{r_{l}l} ⊗ Z_{lc_{l}}).

In particular, β_{r,c}(d_{l}^{r,c}) is invertible in B_{r,c}.

2.5. Lemma. The map β_{r,c} induces a surjective k-algebra homomorphism

β̃_{r,c} : A_{r,c} ⟶ B_{r,c}.

Remark. We shall prove later (Theorem 2.11) that β̃_{r,c} is an isomorphism. Note that the lemma already implies that A_{r,c} is nonzero.

Proof. We have K_{r,c} ⊆ ker(β_{r,c}) by Lemma 2.3 and β_{r,c}(A) ⊆ B_{r,c} by (2.4), and so β_{r,c} induces a homomorphism β′_{r,c} : A/K_{r,c} → B_{r,c}. It also follows from (2.4) that β′_{r,c} sends the elements of D̃_{r,c} to units of B_{r,c}, and therefore β′_{r,c} does induce a homomorphism β̃_{r,c} : A_{r,c} → B_{r,c}.

Set E = β̃_{r,c}(A_{r,c}); we must show that E = B_{r,c}. Note that

(Y_{r_{l}l} ⊗ Z_{lc_{l}})^{±1} = β_{r,c}(d_{l}^{r,c})^{±1}β_{r,c}(d_{l−1}^{r,c})^{∓1} ∈ E

for l ≤ t, where d_{0}^{r,c} = 1. It remains to show that Y_{il} ⊗ Z_{lj} ∈ E for all i, l, j.

As in (2.4), we see that

β_{r,c}([r_{1} ⋯ r_{l−1}i | c_{1} ⋯ c_{l}]) = (Y_{r_{1}1} ⊗ Z_{1c_{1}}) ⋯ (Y_{r_{l−1},l−1} ⊗ Z_{l−1,c_{l−1}})(Y_{il} ⊗ Z_{lc_{l}})
= β_{r,c}(d_{l−1}^{r,c})(Y_{il} ⊗ Z_{lc_{l}})

for l ≤ t and i ≥ r_{l}, whence Y_{il} ⊗ Z_{lc_{l}} ∈ E. Similarly, Y_{r_{l}l} ⊗ Z_{lj} ∈ E for j ≥ c_{l}, and therefore

Y_{il} ⊗ Z_{lj} = q^{1−δ(j,c_{l})}(Y_{r_{l}l}^{−1} ⊗ Z_{lc_{l}}^{−1})(Y_{r_{l}l} ⊗ Z_{lj})(Y_{il} ⊗ Z_{lc_{l}}) ∈ E,

as desired. □

2.6. Lemma. R_{r}^{+} ⊗ R_{c}^{−} is a skew-Laurent extension of B_{r,c} of the form

R_{r}^{+} ⊗ R_{c}^{−} = B_{r,c}[1 ⊗ Z_{1c_{1}}^{±1}, …, 1 ⊗ Z_{tc_{t}}^{±1}; τ_{1}, …, τ_{t}]

for some τ_{1}, …, τ_{t} ∈ {1}^{2n} × H.

Proof. First observe that there exist σ_{1}, …, σ_{t} ∈ H such that Z_{lc_{l}}r = σ_{l}(r)Z_{lc_{l}} for l ≤ t and r ∈ R_{c,0}^{−}. This relation extends to r ∈ R_{c}^{−}, and so

(1 ⊗ Z_{lc_{l}})w = (1 × σ_{l})(w)(1 ⊗ Z_{lc_{l}})

for l ≤ t and w ∈ R_{r}^{+} ⊗ R_{c}^{−}. In particular, if τ_{l} = (1, …, 1, σ_{l}) then 1 ⊗ Z_{lc_{l}} is τ_{l}-normal with respect to B_{r,c}.

The standard ℤ^{2n}-gradings on R_{r}^{+} and R_{c}^{−} (cf. (2.1)) induce a ℤ^{4n}-grading on R_{r}^{+} ⊗R_{c}^{−}. With respect to this grading, B_{r,c} is a homogeneous subalgebra of R_{r}^{+} ⊗ R_{c}^{−}, and its homogeneous components have degrees of the form (∗, b, b, ∗). On the other hand, 1 ⊗ Z_{lc_{l}} has degree (0, 0, ϵ_{l}, ∗), so the monomials

(1 ⊗ Z_{1c_{1}})^{m_{1}}(1 ⊗ Z_{2c_{2}})^{m_{2}} ⋯ (1 ⊗ Z_{tc_{t}})^{m_{t}}

have degrees (0, 0, m_{1}ϵ_{1} + ⋯ + m_{t}ϵ_{t}, ∗). It follows that these monomials are left (or right) linearly independent over B_{r,c}. Hence, the subalgebra

C = ∑_{m_{1},...,m_{t}∈ℤ} B_{r,c}(1 ⊗ Z_{1c_{1}})^{m_{1}}(1 ⊗ Z_{2c_{2}})^{m_{2}} ⋯ (1 ⊗ Z_{tc_{t}})^{m_{t}}

of R_{r}^{+} ⊗ R_{c}^{−} is a skew-Laurent extension of B_{r,c} of the desired form.

It remains to show that C = R_{r}^{+}⊗R_{c}^{−}. First note that Y_{il}⊗1 = (Y_{il}⊗Z_{lc_{l}})(1⊗Z_{lc_{l}}^{−1}) ∈ C for l ≤ t and i ≥ r_{l}, and that

Y_{r_{l}l}^{−1} ⊗ 1 = (Y_{r_{l}l}^{−1} ⊗ Z_{lc_{l}}^{−1})(1 ⊗ Z_{lc_{l}}) ∈ C

for l ≤ t. On the other hand,

1 ⊗ Z_{lj} = (Y_{r_{l}l} ⊗ Z_{lj})(Y_{r_{l}l}^{−1} ⊗ Z_{lc_{l}}^{−1})(1 ⊗ Z_{lc_{l}}) ∈ C

for l ≤ t and j ≥ c_{l}, and 1 ⊗ Z_{lc_{l}}^{−1} ∈ C for l ≤ t. Therefore C = R_{r}^{+} ⊗ R_{c}^{−}. □

2.7. Let B_{r}^{+} and B_{c}^{−} denote the subalgebras of R_{r}^{+} and R_{c}^{−} generated by the respective subsets

{Y_{il}Y_{r_{l}l}^{−1} | l ≤ t, i > r_{l}}  and  {Z_{lj}Z_{lc_{l}}^{−1} | l ≤ t, j > c_{l}}.

The algebra B_{r}^{+}, for instance, may be viewed as a quantized coordinate ring of the variety

{(a_{ij}) ∈ M_{n}(k) | a_{ij} = 0 when j > t or i < r_{j}, and a_{r_{j}j} = 1 for j ≤ t}.

(The factor algebra R_{r}^{+}/⟨Y_{r_{1}1} − 1, …, Y_{r_{t}t} − 1⟩, which one might expect to appear in the above role, is inappropriate because it collapses to k[Y_{r_{1}1}^{±1}, …, Y_{r_{t}t}^{±1}] when q ≠ 1.)

As noted in (2.4), B_{r,c} is generated by its subalgebra B_{r}^{+} ⊗ B_{c}^{−} together with the elements (Y_{r_{l}l} ⊗ Z_{lc_{l}})^{±1} for l ≤ t. In fact:

Lemma. B_{r,c} is a skew-Laurent extension of B_{r}^{+} ⊗ B_{c}^{−} of the form

B_{r,c} = (B_{r}^{+} ⊗ B_{c}^{−})[(Y_{r_{1}1} ⊗ Z_{1c_{1}})^{±1}, …, (Y_{r_{t}t} ⊗ Z_{tc_{t}})^{±1}; η_{1}, …, η_{t}]

for some η_{1}, …, η_{t} ∈ H × H.

Proof. This is proved in the same manner as Lemma 2.6. □

2.8. We use the above structure of B_{r,c} in constructing a homomorphism B_{r,c} → A_{r,c} which will be the inverse of β̃_{r,c}. To begin the construction, we will define suitable homomorphisms from B_{r}^{+} and B_{c}^{−} to A_{r,c} whose images centralize each other. For that purpose, we need to know the defining relations for B_{r}^{+} and B_{c}^{−}, as well as certain commutation relations in A_{r,c}.

Set y_{ij} = Y_{ij}Y_{r_{j}j}^{−1} for j ≤ t and i > r_{j}. In view of the basic commutation relations satisfied by the Y_{lm}, it is easily checked that the y_{ij} satisfy the following relations:

(1)  \begin{aligned}y_{ij}y_{im} & = qy_{im}y_{ij} & (j < m) \\ y_{ij}y_{lj} & = qy_{lj}y_{ij} & (i < l) \\ y_{ij}y_{lm} & = y_{lm}y_{ij} & (i < l, j > m) \\ y_{ij}y_{lm} & = \{ \begin{matrix}y_{lm}y_{ij} & (i < r_{m}) \\ q^{−1}y_{lm}y_{ij} + q̂y_{lj} & (i = r_{m}) \\ y_{lm}y_{ij} + q̂y_{im}y_{lj} & (i > r_{m}) \end{matrix} & (i < l, j < m). \end{aligned}

Lemma. The relations (1) are defining relations for the elements y_{ij} generating the algebra B_{r}^{+}.

Proof. Let S be the k-algebra presented by generators s_{ij} for j ≤ t and i > r_{j} satisfying the analogs of (1). Then there is a k-algebra homomorphism ϕ : S → B_{r}^{+} such that ϕ(s_{ij}) = y_{ij} for all i, j.

In R_{r}^{+}, list the Y_{ij} lexicographically, and observe that the ordered monomials in the Y_{ij} are linearly independent. This remains true for ordered monomials in which we allow the Y_{r_{j}j} to have negative exponents. Since the Y_{r_{j}j} commute up to scalars with the Y_{lm} (recall (2.1)), it follows that the ordered monomials in the y_{ij} are linearly independent, and so these monomials form a basis for B_{r}^{+}. On the other hand, there are sufficient commutation relations for the s_{ij} to show that the ordered monomials in the s_{ij} span S. Hence, ϕ maps a spanning subset of S to a basis for B_{r}^{+}. Therefore ϕ is an isomorphism, and the lemma is proved. □

2.9. Lemma. Let M = [I ⊔ {a} | J] and N = [I′ | J′] be quantum minors in A with I ⊂ I′ and J ⊆ J′. Assume that a ∉ I′ and that b = max(I′) ∉ I.

(a) If a > b, then MN = q^{−1}NM.

(b) If a < b, then

MN − q^{−1}NM ≡ q̂(−q)^{|(I′\I)∩(a,b)|}[I ⊔ {b} | J][I′ ⊔ {a} \ {b} | J′]

modulo the ideal L := ⟨ [I′ ⊔ {a} \ {i′} | J′] | i′ ∈ I′ ∩ (a, b) ⟩.

Proof. (a) Expand M using the q-Laplace relation of Corollary 5.5(b2) with r = a. Since (a, n] ∩ I = ∅, we get

(1)  M = ∑_{j∈J}(−q)^{|(j,n]∩J|}[I | J \ {j}]X_{aj}.

Note that all the [I | J \ {j}] commute with N. Lemma 5.2(c1) implies that X_{aj}N = q^{−1}NX_{aj} for all j ∈ J, and thus it follows from (1) that MN = q^{−1}NM.

(b) Set α = |(a, n] ∩ I|, and again expand M using Corollary 5.5(b2) with r = a. This time, we get

(2)  (−q)^{α}M = ∑_{j∈J}(−q)^{|(j,n]∩J|}[I | J \ {j}]X_{aj}.

For all j ∈ J, Lemma 5.2(c2) implies that

NX_{aj} − qX_{aj}N = q̂ ∑_{\begin{matrix}i′∈I′ \\ i′>a \end{matrix}}(−q)^{|I′∩[a,i′]|}X_{i′j}[I′ ⊔ \{a\} \ \{i′\} | J′],

whence

(3)  X_{aj}N − q^{−1}NX_{aj} ≡ q̂(−q)^{β}X_{bj}[I′ ⊔ {a} \ {b} | J′]  (mod L),

where β = |I′ ∩ [a, b]| − 1 = |I′ ∩ (a, b)|. Note that β − α = |(I′ \ I) ∩ (a, b)|.

Combining (2) and (3), we obtain

(4) MN ≡ ∑_{j∈J}(−q)^{|(j,n]∩J|−α}[I | J \ {j}](q^{−1}NX_{aj} + q̂(−q)^{β}X_{bj}[I′ ⊔ {a} \ {b} | J′])

modulo L. Since all the [I | J \ {j}] commute with N, it follows from (2) that

(5)  ∑_{j∈J}(−q)^{|(j,n]∩J|−α}[I | J \ {j}]q^{−1}NX_{aj} = q^{−1}NM.

Further, an application of Corollary 5.5(b2) with r = b yields

(6)  ∑_{j∈J}(−q)^{|(j,n]∩J|}[I | J \ {j}]X_{bj} = [I ⊔ {b} | J].

Combining (4), (5), and (6), we complete the proof. □

2.10. Corollary. Let M = [I | J ⊔ {a}] and N = [I′ | J′] be quantum minors in A with I ⊆ I′ and J ⊂ J′. Assume that a ∉ J′ and that b = max(J′) ∉ J.

(a) If a > b, then MN = q^{−1}NM.

(b) If a < b, then

MN − q^{−1}NM ≡ q̂(−q)^{|(J′\J)∩(a,b)|}[I | J ⊔ {b}][I′ | J′ ⊔ {a} \ {b}]

modulo the ideal L := ⟨ [I′ | J′ ⊔ {a} \ {j′}] | j′ ∈ J′ ∩ (a, b) ⟩.

Proof. This follows from Lemma 2.9 by symmetry. □

2.11. Theorem. The map β̃_{r,c} : A_{r,c} ⟶ B_{r,c} is an isomorphism.

Proof. Recall the notation d̃_{l}^{r,c} = d_{l}^{r,c} + K_{r,c} from (1.2). Similarly, we shall use tildes to denote other cosets in A/K_{r,c}. To abbreviate the relation of congruence modulo K_{r,c}, we adopt the notation ≡_{r,c}. We shall use the same symbols for elements of A/K_{r,c} and their images in the localization A_{r,c}, which does not cause problems as long as we only transfer equations from A/K_{r,c} to A_{r,c} and not in the reverse direction.

We proceed to construct, in several steps, a k-algebra homomorphism ϕ : B_{r,c} → A_{r,c} that will be an inverse for β_{r,c}. The construction of ϕ is based on the skew-Laurent structure of B_{r,c} given in Lemma 2.7. The first ingredient will be a homomorphism from B_{r}^{+} to A_{r,c}. To describe it, set

u_{ij} = [r_{1} ⋯ r_{j−1}i | c_{1} ⋯ c_{j}] ∈ A  (j ≤ t, i ≥ r_{j})
v_{ij} = ũ_{ij}(d̃_{j}^{r,c})^{−1} ∈ A_{r,c}  (j ≤ t, i > r_{j})
y_{ij} = Y_{ij}Y_{r_{j}j}^{−1} ∈ R_{r}^{+}  (as in (2.8))  (j ≤ t, i > r_{j}).

(While the given expressions for v_{ij} and y_{ij} also make sense for j ≤ t and i = r_{j}, they yield v_{r_{j}j} = 1 ∈ A_{r,c} and y_{r_{j}j} = 1 ∈ R_{r}^{+}. It is more convenient for the proof to exclude these possibilities.) Recall that the y_{ij} generate B_{r}^{+}.

Claim 1: There exists a homomorphism ϕ^{+} : B_{r}^{+} → A_{r,c} such that ϕ^{+}(y_{ij}) = v_{ij} for j ≤ t and i > r_{j}.

To prove this, we must show that the v_{ij} satisfy the analogs of the relations (2.8)(1), i.e., the corresponding equations with all y’s replaced by v’s. We first check that the u_{ij} satisfy the relations (1) below. The first relation follows from Lemma 2.9(a); for the second, observe that {r_{1}, …, r_{j−1}, i} ⊂ {r_{1}, …, r_{m−1}, l} and {c_{1}, …, c_{j}} ⊂ {c_{1}, …, c_{m}} in that case.

(1)  u_{lm}u_{ij} = \{ \begin{matrix}q^{−1}u_{ij}u_{lm} & (i < l, j ≥ m) \\ u_{ij}u_{lm} & (i ≤ l, j < m, i ∈ \{r_{j}, …, r_{m−1}, l\}). \end{matrix}

Furthermore, when i < l and j < m but i ∉ {r_{j}, …, r_{m−1}}, Lemma 2.9(b) implies that

(2)  u_{ij}u_{lm} − q^{−1}u_{lm}u_{ij} ≡_{r,c} \{ \begin{matrix}q̂u_{lj}u_{im} & (i ≥ r_{m}) \\ 0 & (i < r_{m}). \end{matrix}

Since u_{r_{l}l} = d_{l}^{r,c} for l ≤ t, the relations (1) and (2) yield commutation relations for the d_{l}^{r,c} and u_{ij}, which we combine in the following form:

(3) d_{l}^{r,c}u_{ij} = u_{ij}d_{l}^{r,c}  \{ \begin{matrix}(l < j), or \\ (l ≥ j and i ∈ \{r_{j}, …, r_{l}\}) \end{matrix}

d_{l}^{r,c}u_{ij} ≡_{r,c} qu_{ij}d_{l}^{r,c}  (l ≥ j and i ∉ {r_{j}, …, r_{l}}).

(Note that when l < j, we have i ≥ r_{j} > r_{l}.)

It follows from (1), (2), and (3) that the elements v_{ij} ∈ A_{r,c} indeed satisfy the analogs of the relations (2.8)(1), as desired. This establishes Claim 1.

Next, set

w_{lm} = [r_{1} ⋯ r_{l} | c_{1} ⋯ c_{l−1}m] ∈ A  (l ≤ t, m ≥ c_{l})
t_{lm} = w̃_{lm}(d̃_{l}^{r,c})^{−1} ∈ A_{r,c}  (l ≤ t, m > c_{l})
z_{lm} = Z_{lm}Z_{lc_{l}}^{−1} ∈ R_{c}^{−}  (l ≤ t, m > c_{l}),

and recall that the z_{lm} generate B_{c}^{−}.

Claim 2. There exists a homomorphism ϕ^{−} : B_{c}^{−} → A_{r,c} such that ϕ^{−}(z_{lm}) = t_{lm} for l ≤ t and m > c_{l}.

This follows from Claim 1 by symmetry.

Claim 3. Each v_{ij} commutes with each t_{lm}.

We first collect the following commutation relations between the u_{ij} and the w_{lm}:

(4)

u_{ij}w_{lm} = w_{lm}u_{ij}  \{ \begin{matrix}(j = l), or \\ (j < l and i ∈ \{r_{j+1}, …, r_{l}\}), or \\ (j > l and m ∈ \{c_{l+1}, …, c_{j}\}) \end{matrix}

u_{ij}w_{lm} ≡_{r,c} \{ \begin{matrix}q^{−1}w_{lm}u_{ij} & (j < l and i ∉ \{r_{j+1}, …, r_{l}\}) \\ qw_{lm}u_{ij} & (j > l and m ∉ \{c_{l+1}, …, c_{j}\}). \end{matrix}

The first equation in (4) follows from Lemma 5.3(c). The next two equations hold because {r_{1}, …, r_{j−1}, i} ⊂ {r_{1}, …, r_{l}} and {c_{1}, …, c_{j}} ⊂ {c_{1}, …, c_{l−1}, m} in the first case, while those inclusions are reversed in the second case. Finally, the last two relations follow from Lemma 2.9 and Corollary 2.10.

Commutation relations between the d_{l}^{r,c} and the u_{ij} are given in (3), and, by symmetry, we also have

(5) d_{j}^{r,c}w_{lm} = w_{lm}d_{j}^{r,c}  \{ \begin{matrix}(j < l), or \\ (j ≥ l and m ∈ \{c_{l}, …, c_{j}\}) \end{matrix}

d_{j}^{r,c}w_{lm} ≡_{r,c} qw_{lm}d_{j}^{r,c}  (j ≥ l and m ∉ {c_{l}, …, c_{j}}).

It follows from (3), (4), and (5) that v_{ij} indeed commutes with t_{lm}, and Claim 3 is proved. Combining Claims 1, 2, and 3, we see that there exists a homomorphism

ϕ : B_{r}^{+} ⊗ B_{c}^{−} → A_{r,c}

such that ϕ(y_{ij} ⊗ 1) = v_{ij} and ϕ(1 ⊗ z_{lm}) = t_{lm} for all i, j, l, m.

Claim 4. ϕ extends to a homomorphism B_{r,c} → A_{r,c} such that ϕ(Y_{r_{s}s} ⊗ Z_{sc_{s}}) = d̃_{s}^{r,c}(d̃_{s−1}^{r,c})^{−1} for s ≤ t.

In view of the relations given in (2.1), we see that

(6)  \begin{aligned}Y_{r_{s}s}y_{ij}Y_{r_{s}s}^{−1} & = \{ \begin{matrix}q^{−1}y_{ij} & (i = r_{s}) \\ qy_{ij} & (j = s) \\ y_{ij} & (i ≠ r_{s}, j ≠ s) \end{matrix} \\ Z_{sc_{s}}z_{lm}Z_{sc_{s}}^{−1} & = \{ \begin{matrix}qz_{lm} & (l = s) \\ q^{−1}z_{lm} & (m = c_{s}) \\ z_{lm} & (l ≠ s, m ≠ c_{s}). \end{matrix} \end{aligned}

On the other hand, it follows from (3) and (5) that the units d̃_{s}^{r,c}(d̃_{s−1}^{r,c})^{−1} in A_{r,c} normalize the elements v_{ij} and t_{lm} in exactly the same way, that is,

(7)  \begin{aligned}d̃_{s}^{r,c}(d̃_{s−1}^{r,c})^{−1}v_{ij}d̃_{s−1}^{r,c}(d̃_{s}^{r,c})^{−1} & = \{ \begin{matrix}q^{−1}v_{ij} & (i = r_{s}) \\ qv_{ij} & (j = s) \\ v_{ij} & (i ≠ r_{s}, j ≠ s) \end{matrix} \\ d̃_{s}^{r,c}(d̃_{s−1}^{r,c})^{−1}t_{lm}d̃_{s−1}^{r,c}(d̃_{s}^{r,c})^{−1} & = \{ \begin{matrix}qt_{lm} & (l = s) \\ q^{−1}t_{lm} & (m = c_{s}) \\ t_{lm} & (l ≠ s, m ≠ c_{s}). \end{matrix} \end{aligned}

Claim 4 follows from (6), (7), and Lemma 2.7.

Claim 5. ϕ = β̃_{r,c}^{−1}.

Since β̃_{r,c} is surjective (Lemma 2.5), it is enough to show that ϕβ̃_{r,c} is the identity on A_{r,c}. First note, using (2.4), that

ϕβ̃_{r,c}(d̃_{s}^{r,c}(d̃_{s−1}^{r,c})^{−1}) = ϕ(Y_{r_{s}s} ⊗ Z_{sc_{s}}) = d̃_{s}^{r,c}(d̃_{s−1}^{r,c})^{−1}

for all s, whence ϕβ̃_{r,c}(d̃_{s}^{r,c}) = d̃_{s}^{r,c} for all s. Next, for j ≤ t and i > r_{j} we have

ϕβ̃_{r,c}(ũ_{ij}) = ϕ((Y_{r_{1}1} ⊗ Z_{1c_{1}}) ⋯ (Y_{r_{j−1},j−1} ⊗ Z_{j−1,c_{j−1}})(Y_{ij} ⊗ Z_{jc_{j}}))
= ϕ((y_{ij} ⊗ 1)(Y_{r_{1}1} ⊗ Z_{1c_{1}}) ⋯ (Y_{r_{j}j} ⊗ Z_{jc_{j}}))
= ϕ((y_{ij} ⊗ 1)β_{r,c}(d̃_{j}^{r,c})) = v_{ij}d̃_{j}^{r,c} = ũ_{ij}.

By symmetry, ϕβ̃_{r,c}(w̃_{lm}) = w̃_{lm} for l ≤ t and m > c_{l}. Therefore ϕβ̃_{r,c} at least equals the identity on the subalgebra C of A_{r,c} generated by (the image of) the set

{(d̃_{s}^{r,c})^{±1} | s ≤ t} ∪ {ũ_{ij} | j ≤ t, i > r_{j}} ∪ {w̃_{lm} | l ≤ t, m > c_{l}}.

To finish the proof, we just need to show that C = A_{r,c}, that is, that X̃_{ij} ∈ C for all i, j. This is clear in case i < r_{1} or j < c_{1}, since in those cases X̃_{ij} = 0. We also have X̃_{ic_{1}} = ũ_{i1} ∈ C for i ≥ r_{1} and X̃_{r_{1}j} = w̃_{1j} ∈ C for j ≥ c_{1}. Hence, X̃_{ij} ∈ C whenever i ≤ r_{1} or j ≤ c_{1}.

Now let 1 < l ≤ t and assume that X̃_{ij} ∈ C whenever i ≤ r_{l−1} or j ≤ c_{l−1}. For r_{l−1} < i ≤ t and c_{l−1} < j ≤ c_{l}, it follows from Corollary 5.5(b1) that

(8) (−q)^{l−1}[r_{1} ⋯ r_{l−1}i | c_{1} ⋯ c_{l−1}j] = (−q)^{l−1}X_{ij}[r_{1} ⋯ r_{l−1} | c_{1} ⋯ c_{l−1}]

+ ∑_{s=1}^{l−1}(−q)^{s−1}X_{ic_{s}}[r_{1} ⋯ r_{l−1} | c_{1} ⋯ ĉ_{s} ⋯ c_{l−1}j].

Most of the cosets of the factors in (8) can be seen to lie in C right away. For instance, the coset of the left hand side is zero if either i < r_{l} or j < c_{l}, and it equals ũ_{il} if i ≥ r_{l} and j = c_{l}. For s < l, we have X̃_{ic_{s}} ∈ C by the inductive hypothesis, and similarly the coset [r_{1} ⋯ r_{l−1} | c_{1} ⋯ ĉ_{s} ⋯ c_{l−1}j] + K_{r,c} is in C since it is a linear combination of products of elements X̃_{ab} with a ≤ r_{l−1}. Consequently, we obtain from (8) that

X_{ij}[r_{1} ⋯ r_{l−1} | c_{1} ⋯ c_{l−1}] + K_{r,c} ∈ C.

In other words, X̃_{ij}d̃_{l−1}^{r,c} ∈ C, whence X̃_{ij} ∈ C. Thus, X̃_{ij} ∈ C whenever j ≤ c_{l}. By symmetry, X̃_{ij} ∈ C whenever i ≤ r_{l}.

The above induction proves that X̃_{ij} ∈ C whenever i ≤ r_{t} or j ≤ c_{t}. If there exist indices i > r_{t} and j > c_{t}, we have

(9) (−q)^{t}[r_{1} ⋯ r_{t}i | c_{1} ⋯ c_{t}j] = (−q)^{t}X_{ij}[r_{1} ⋯ r_{t} | c_{1} ⋯ c_{t}]

+ ∑_{s=1}^{t}(−q)^{s−1}X_{ic_{s}}[r_{1} ⋯ r_{t} | c_{1} ⋯ ĉ_{s} ⋯ c_{t}j]

by Corollary 5.5(b1), from which we see as above that X̃_{ij} ∈ C. (Note that the left hand side of (9) necessarily lies in K_{r,c} because it involves a (t + 1) × (t + 1) quantum minor.) Therefore all X̃_{ij} ∈ C, and the proof is complete. □

3. Tensor product decompositions of H-primes

Throughout this section, we assume that q is not a root of unity; we shall place reminders of this hypothesis in the relevant results. Thus, by [8, Theorem 3.2], all primes of A are completely prime. Since this property survives in factors and localizations, all primes in the algebras A_{r,c}, R_{r}^{+}, and R_{c}^{−} are completely prime, and also in B_{r,c} because of Theorem 2.11. We have already observed that the algebras R_{r,0}^{+} and R_{c,0}^{−} are iterated skew polynomial algebras over k, and so is their tensor product. The iterated skew polynomial structure of R_{r,0}^{+} ⊗ R_{c,0}^{−} is easily seen to satisfy the hypotheses of [8, Theorem 2.3], and thus all its primes are completely prime. Consequently, all primes in the localizations R_{r}^{+} ⊗ R_{c}^{−} are completely prime.

3.1. In order to deal with H-primes in tensor products, we need the following rationality property. Suppose that S is a noetherian k-algebra and that G is a group acting on S by k-algebra automorphisms. We say that a G-prime P of S is strongly G-rational provided the algebra Z(Fract S/P)^{G}, the fixed ring of the center of the Goldie quotient ring of S/P under the induced G-action, equals k.

By [9, (5.7)(i)] (cf. [1, Theorem II.5.14]) and [1, Corollary II.6.5], A has only finitely many H-primes, and they are all completely prime and strongly H-rational. These properties carry over to A_{r,c}, R_{r}^{+}, and R_{c}^{−}. Analogous results [1, Theorems II.5.12 and II.6.4] imply that R_{r,0}^{+} ⊗ R_{c,0}^{−} has only finitely many (H × H)-primes, and they are all completely prime and strongly (H × H)-rational. These properties now carry over to R_{r}^{+} ⊗ R_{c}^{−}.

Identify H with the subgroup

H̃ = ((k^{×})^{n} × {1}^{n}) × ({1}^{n} × (k^{×})^{n}) ◁ H × H

in the obvious way. With this identification, β_{r,c} and β̃_{r,c} are H-equivariant. In particular, it follows that B_{r,c} has only finitely many H̃-primes, and they are all completely prime and strongly H̃-rational.

3.2. Lemma. For i = 1, 2, let A_{i} be a k-algebra, H_{i} a group acting on A_{i} by k-algebra automorphisms, and P_{i} an H_{i}-prime ideal of A_{i}. Set P = (P_{1} ⊗ A_{2}) + (A_{1} ⊗ P_{2}), and let H_{1} × H_{2} act on A_{1} ⊗ A_{2} in the natural manner.

(a) If each A_{i}/P_{i} is H_{i}-simple and Z(A_{1}/P_{1})^{H_{1}} = k, then (A_{1} ⊗ A_{2})/P is (H_{1} × H_{2})- simple.

(b) If each A_{i} is noetherian and P_{1} is strongly H_{1}-rational, then P is an (H_{1}×H_{2})-prime ideal of A_{1} ⊗ A_{2}. Moreover, P is the only (H_{1} × H_{2})-prime ideal of A_{1} ⊗ A_{2} that contracts to P_{1} ⊗ 1 in A_{1} ⊗ 1 and to 1 ⊗ P_{2} in 1 ⊗ A_{2}.

(c) If each A_{i} is noetherian and each P_{i} is strongly H_{i}-rational, then P is strongly (H_{1} × H_{2})-rational.

Proof. Since (A_{1} ⊗ A_{2})/P ≅ (A_{1}/P_{1}) ⊗ (A_{2}/P_{2}), there is no loss of generality in assuming that each P_{i} = 0.

(a) This is a standard shortest length argument. Let I be a nonzero (H_{1} × H_{2})-ideal of A_{1} ⊗ A_{2}, and let m be the shortest length for nonzero elements of I (as sums of pure tensors). Choose a nonzero element

x = b_{1} ⊗ c_{1} + ⋯ + b_{m} ⊗ c_{m} ∈ I

of length m, where the b_{j} ∈ A_{1} and c_{j} ∈ A_{2}, and note that the c_{j} are linearly independent over k. Now the set

{b ∈ A_{1} | (b ⊗ c_{1} + A_{1} ⊗ c_{2} + ⋯ + A_{1} ⊗ c_{m}) ∩ I ≠ ∅}

is a nonzero H_{1}-ideal of A_{1}, and so it equals A_{1}. Thus, without loss of generality, b_{1} = 1. For any a ∈ A_{1}, we now have x(a ⊗ 1) − (a ⊗ 1)x in I with length less than m, whence x(a ⊗ 1) − (a ⊗ 1)x = 0 and so b_{j}a = ab_{j} for all j. For any h ∈ H_{1}, we have (h, 1)(x) − x in I with length less than m, whence (h, 1)(x) − x = 0 and so h(b_{j}) = b_{j} for all j. Therefore all b_{j} lie in Z(A_{1})^{H_{1}} = k. It follows that x ∈ 1 ⊗ A_{2}, whence m = 1 and x = 1 ⊗ c_{1}. Consequently, the set {c ∈ A_{2} | 1 ⊗ c ∈ I} is a nonzero H_{2}-ideal of A_{2}, and so it equals A_{2}. Therefore 1 ⊗ 1 ∈ I, proving that I = A_{1} ⊗ A_{2}.

(b) Each A_{i} is an H_{i}-prime noetherian ring, and so is semiprime. Let C_{i} be the set of regular elements in A_{i}, and note that the set

C = {c_{1} ⊗ c_{2} | c_{i} ∈ C_{i}}

is an (H_{1} × H_{2})-stable denominator set in A_{1} ⊗ A_{2}, consisting of regular elements. Each A_{i}C_{i}^{−1} is H_{i}-simple artinian, and Z(A_{1}C_{1}^{−1})^{H_{1}} = k by our hypothesis on P_{1}. Thus by part (a), the localization

(A_{1} ⊗ A_{2})C^{−1} = A_{1}C_{1}^{−1} ⊗ A_{2}C_{2}^{−1}

is (H_{1} × H_{2})-simple. It follows that each nonzero (H_{1} × H_{2})-ideal of A_{1} ⊗ A_{2} meets C; in particular, A_{1} ⊗ A_{2} is an (H_{1} × H_{2})-prime ring.

Now let Q be any (H_{1} × H_{2})-prime ideal of A_{1} ⊗ A_{2} that contracts to zero in both A_{1} ⊗ 1 and 1 ⊗ A_{2}. Then Q is a semiprime ideal, disjoint from both C_{1} ⊗ 1 and 1 ⊗ C_{2}. If some prime ideal Q_{0} minimal over Q meets C_{1} ⊗ 1, then h(Q_{0}) meets C_{1} ⊗ 1 for all h ∈ H_{1} × H_{2}. But since Q is a finite intersection of some of the h(Q_{0}), it would follow that Q meets C_{1} ⊗ 1, a contradiction. Therefore C_{1} ⊗ 1 is disjoint from all primes minimal over Q, whence C_{1} ⊗ 1 is regular modulo Q. Likewise, 1 ⊗ C_{2} is regular modulo Q. It follows that C is disjoint from Q, and therefore Q = 0.

(c) After localization, we can assume that each A_{i} is H_{i}-simple artinian. By part (a), A_{1} ⊗ A_{2} is now (H_{1} × H_{2})-simple. Consider an element u in Z(Fract(A_{1} ⊗ A_{2}))^{H_{1}×H_{2}}. The set {a ∈ A_{1} ⊗ A_{2} | au ∈ A_{1} ⊗ A_{2}} is a nonzero (H_{1} × H_{2})-ideal of A_{1} ⊗ A_{2}, and so it equals A_{1} ⊗ A_{2}. Therefore u ∈ A_{1} ⊗ A_{2}. Now write u = v_{1} ⊗ w_{1} + ⋯ + v_{t} ⊗ w_{t} for some v_{j} ∈ A_{1} and some linearly independent w_{j} ∈ A_{2}. Since u is fixed by H_{1} × 1 and commutes with A_{1} ⊗ 1, we see that all v_{j} ∈ Z(A_{1})^{H_{1}} = k. Hence, u = 1 ⊗ w for some w ∈ A_{2}. But then w ∈ Z(A_{2})^{H_{2}} = k, and therefore u ∈ k. □

3.3. Proposition. For i = 1, 2, let A_{i} be a noetherian k-algebra and H_{i} a group acting on A_{i} by k-algebra automorphisms. Assume that all H_{1}-primes of A_{1} are strongly H_{1}-rational. Then the rule (P_{1}, P_{2}) ↦ (P_{1} ⊗ A_{2}) + (A_{1} ⊗ P_{2}) provides a bijection

(H_{1} -spec A_{1}) × (H_{2} -spec A_{2}) ⟶ (H_{1} × H_{2}) -spec(A_{1} ⊗ A_{2}).

Proof. Lemma 3.2(b) shows that the given rule maps (H_{1} -spec A_{1}) × (H_{2} -spec A_{2}) to (H_{1} × H_{2}) -spec(A_{1} ⊗ A_{2}).

Now consider an (H_{1} × H_{2})-prime P in A_{1} ⊗ A_{2}. Let P_{1} and P_{2} be the inverse images of P under the natural maps A_{i} → A_{1} ⊗ A_{2}. Then each P_{i} is an H_{i}-ideal of A_{i}, and (P_{1} ⊗ A_{2}) + (A_{1} ⊗ P_{2}) ⊆ P. There are H_{1}-primes Q_{1}, …, Q_{t} in A_{1}, containing P_{1}, such that Q_{1}Q_{2} ⋯ Q_{t} ⊆ P_{1}. Then the Q_{i} ⊗ A_{2} are (H_{1} × H_{2})-ideals of A_{1} ⊗ A_{2} such that

(Q_{1} ⊗ A_{2})(Q_{2} ⊗ A_{2}) ⋯ (Q_{t} ⊗ A_{2}) ⊆ P_{1} ⊗ A_{2} ⊆ P.

Consequently, some Q_{j} ⊗ A_{2} ⊆ P, whence Q_{j} ⊆ P_{1}, and so Q_{j} = P_{1}. This shows that P_{1} is H_{1}-prime. Similarly, P_{2} is H_{2}-prime.

By Lemma 3.2(b), (P_{1} ⊗ A_{2}) + (A_{1} ⊗ P_{2}) is an (H_{1} × H_{2})-prime of A_{1} ⊗ A_{2}, and it is the only (H_{1} ×H_{2})-prime of A_{1} ⊗A_{2} that contracts to P_{1} ⊗1 in A_{1} ⊗1 and to 1⊗P_{2} in 1⊗A_{2}. Therefore P = (P_{1} ⊗A_{2})+(A_{1} ⊗P_{2}). It is clear that P_{1} and P_{2} are unique, since P_{i} equals the inverse image of (P_{1} ⊗ A_{2}) + (A_{1} ⊗ P_{2}) under the natural map A_{i} → A_{1} ⊗ A_{2}. □

3.4. Lemma. [q not a root of unity] Let (r, c) ∈ RC. If P̃ is an H̃-prime of B_{r,c}, then there exists a unique (H × H)-prime Q of R_{r}^{+} ⊗ R_{c}^{−} such that Q ∩ B_{r,c} = P̃.

Proof. Since B_{r,c} has only finitely many H̃-primes and H×H just permutes them, the (H× H)-orbit of P̃ in spec B_{r,c} is finite. Since P̃ is prime, it now follows from [1, Proposition II.2.9] that P̃ must be invariant under H × H. In view of Lemma 2.6, Q = P̃(R_{r}^{+} ⊗ R_{c}^{−}) is an (H × H)-invariant prime of R_{r}^{+} ⊗ R_{c}^{−} such that Q ∩ B_{r,c} = P̃. It remains to show that if Q′ is any (H × H)-prime of R_{r}^{+} ⊗ R_{c}^{−} that contracts to P̃, then Q′ = Q. Note that Q′ ⊇ Q by definition of Q.

Set G = (k^{×})^{n}, let ϕ : G → ({1}^{n} × (k^{×})^{n}) × ((k^{×})^{n} × {1}^{n}) ⊂ H × H be the homomorphism given by the rule

ϕ(α_{1}, …, α_{n}) = (1, …, 1, α_{1}^{−1}, …, α_{n}^{−1}, α_{1}, …, α_{n}, 1, …, 1),

and use ϕ to pull back the action of H × H on R_{r}^{+} ⊗ R_{c}^{−} to an action of G. With respect to this G-action, B_{r,c} is generated by fixed elements, and each of the elements 1 ⊗ Z_{sc_{s}} is a G-eigenvector with eigenvalue equal to the projection (α_{1}, …, α_{n}) ↦ α_{s}. In view of Lemma 2.6 and the fact that k is infinite, it follows that the G-eigenspaces of R_{r}^{+} ⊗ R_{c}^{−} are the subspaces B_{r,c}(1 ⊗ (Z_{1c_{1}}^{m_{1}}Z_{2c_{2}}^{m_{2}} ⋯ Z_{tc_{1}}^{m_{t}})) for (m_{1}, …, m_{t}) ∈ ℤ^{t}. Consequently, any G-eigenvector in R_{r}^{+} ⊗R_{c}^{−} has the form du where d ∈ B_{r,c} and u is a unit. If du ∈ Q′, then d ∈ Q′ ∩ B_{r,c} = P̃, whence du ∈ Q. Since Q′ is G-invariant, we conclude that Q′ = Q, as desired. □

3.5. Set H- spec_{r,c} A = (H- spec A) ∩ (spec_{r,c} A) for (r, c) ∈ RC. These sets partition H- spec A because of Corollary 1.10.

Theorem. [q not a root of unity] For each (r, c) ∈ RC, there is a bijection

(H- spec R_{r}^{+}) × (H- spec R_{c}^{−}) ⟶ H- spec_{r,c} A

given by the rule (Q^{+}, Q^{−}) ↦ β_{r,c}^{−1}((Q^{+} ⊗ R_{c}^{−}) + (R_{r}^{+} ⊗ Q^{−})).

Proof. If Q^{+} ∈ H- spec R_{r}^{+} and Q^{−} ∈ H- spec R_{c}^{−}, then Proposition 3.3 shows that the ideal Q = (Q^{+} ⊗ R_{c}^{−}) + (R_{r}^{+} ⊗ Q^{−}) is an (H × H)-prime of R_{r}^{+} ⊗ R_{c}^{−}. In particular, Q is completely prime, and so Q ∩ B_{r,c} is an H̃-prime of B_{r,c}, whence β_{r,c}^{−1}(Q) = β_{r,c}^{−1}(Q ∩ B_{r,c}) is an H-prime of A lying in spec_{r,c} A. This shows that the given rule does define a map from (H- spec R_{r}^{+}) × (H- spec R_{c}^{−}) to H- spec_{r,c} A.

Now consider an arbitrary H-prime P in H- spec_{r,c} A. Then P induces an H-prime in A_{r,c} that contracts to P under the localization map. In view of Theorem 2.11, it follows that β_{r,c}(P) induces an H̃-prime P̃ of B_{r,c} such that β_{r,c}^{−1}(P̃) = P. By Lemma 3.4, there is a unique (H × H)-prime Q of R_{r}^{+} ⊗ R_{c}^{−} such that Q ∩ B_{r,c} = P̃. Then Proposition 3.3 implies that Q = (Q^{+} ⊗ R_{c}^{−}) + (R_{r}^{+} ⊗ Q^{−}) for some H-primes Q^{+} in R_{r}^{+} and Q^{−} in R_{c}^{−}. Thus,

β_{r,c}^{−1}((Q^{+} ⊗ R_{c}^{−}) + (R_{r}^{+} ⊗ Q^{−})) = β_{r,c}^{−1}(Q) = β_{r,c}^{−1}(Q ∩ B_{r,c}) = β_{r,c}^{−1}(P̃) = P.

It remains to show that Q^{+} and Q^{−} are unique. Consider any T^{+} ∈ H- spec R_{r}^{+} and T^{−} ∈ H- spec R_{c}^{−} such that P = β_{r,c}^{−1}(T) where T = (T^{+} ⊗ R_{c}^{−}) + (R_{r}^{+} ⊗ T^{−}). As in the first paragraph of the proof, T is an (H × H)-prime of R_{r}^{+} ⊗ R_{c}^{−} and T ∩ B_{r,c} is an H̃-prime of B_{r,c}. Since β_{r,c}^{−1}(T ∩ B_{r,c}) = P, we must have T ∩ B_{r,c} = P̃, whence T = Q by the uniqueness of Q. Therefore Proposition 3.3 shows that T^{+} = Q^{+} and T^{−} = Q^{−}, as desired. □

3.6. Fix t ∈ {0, 1, …, n}, and let H- spec^{[t]} A be the set of those H-primes of A which contain all (t+1)×(t+1) quantum minors but not all t×t quantum minors. By Corollary 1.10,

H- spec^{[t]} A = ⨆_{(r,c)∈RC_{t}} H- spec_{r,c} A,

and consequently Theorem 3.5 implies that

|H- spec^{[t]} A| = ∑_{(r,c)∈RC_{t}} |H- spec R_{r}^{+}| · |H- spec R_{c}^{−}|.

If R_{t} denotes the set of sequences (r_{1}, …, r_{t}) ∈ ℕ^{t} with 1 ≤ r_{1} < ⋯ < r_{t} ≤ n, then RC_{t} = R_{t} × R_{t}. For each r ∈ R_{t}, the automorphism τ of A discussed in (5.1) induces an isomorphism \overline{τ} : R_{r}^{+} → R_{r}^{−}. While \overline{τ} is not H-equivariant, there is an automorphism γ of H, given by (α_{1}, …, α_{n}, β_{1}, …, β_{n}) ↦ (β_{1}, …, β_{n}, α_{1}, …, α_{n}), such that the following diagram commutes:

H ⟶^{γ} H
↓  ↓
Aut R_{r}^{+} ⟶^{\overline{τ}^{∗}} Aut R_{r}^{−}

(Here the vertical arrows denote the standard actions of H on R_{r}^{+} and R_{r}^{−}.) Hence, \overline{τ} provides a bijection of H- spec R_{r}^{+} onto H- spec R_{r}^{−}. Therefore

|H- spec^{[t]} A| = ∑_{r,c∈R_{t}} |H- spec R_{r}^{+}| · |H- spec R_{c}^{+}| = ( ∑_{r∈R_{t}} |H- spec R_{r}^{+}|)^{2},

a perfect square. These numbers are known in three cases:

|H- spec^{[t]} A| = \{ \begin{matrix}1 & (t = 0) \\ (2^{n} − 1)^{2} & (t = 1) \\ (n!)^{2} & (t = n). \end{matrix}

The case when t = 0 is trivial, and the case when t = 1 is given by [5, Corollary 3.5]. For the remaining case, note first that H- spec^{[n]} A ≈ H- spec O_{q}(GL_{n}(k)). It can be checked that there is a bijection between H- spec O_{q}(GL_{n}(k)) and the set of winding-in-variant primes of O_{q}(SL_{n}(k)) (e.g., see [1, Lemma II.5.16]), and it follows from the work of Hodges and Levasseur [11] that the latter set is in bijection with the double Weyl group S_{n} × S_{n} (cf. [1, Corollary II.4.12]).

3.7. As a corollary of Theorem 3.5, we obtain the following less specific but more digestible result.

Corollary. [q not a root of unity] Set

R^{+} = A/⟨X_{ij} | i < j⟩  and  R^{−} = A/⟨X_{ij} | i > j⟩,

let π^{±} : A → R^{±} denote the quotient maps, and let β denote the composition

A ⟶^{Δ} A ⊗ A ⟶^{π^{+}⊗π^{−}} R^{+} ⊗ R^{−}.

Given any H-prime P in A, there exist H-primes P^{±} in R^{±} such that

P = β^{−1}((P^{+} ⊗ R^{−}) + (R^{+} ⊗ P^{−})).

Proof. By Corollary 1.10 and Theorem 3.5, P = β_{r,c}^{−1}((Q^{+} ⊗ R_{c}^{−}) + (R_{r}^{+} ⊗ Q^{−})) for some (r, c) ∈ RC and some H-primes Q^{+} in R_{r}^{+} and Q^{−} in R_{c}^{−}. Observe that X_{ij} ∈ ker π_{r,0}^{+} when i < j, and that X_{ij} ∈ ker π_{c,0}^{−} when i > j. Hence, there are surjective k-algebra homomorphisms τ^{+} : R^{+} → R_{r,0}^{+} and τ^{−} : R^{−} → R_{c,0}^{−} such that τ^{+}π^{+} = π_{r,0}^{+} and τ^{−}π^{−} = π_{c,0}^{−}. Consequently, (τ^{+} ⊗ τ^{−})β = β_{r,c} (with the obvious adjustment of codomains).

Next, observe that Q_{0}^{+} = Q^{+} ∩ R_{r,0}^{+} and Q_{0}^{−} = Q^{−} ∩ R_{c,0}^{−} are H-primes of R_{r,0}^{+} and R_{c,0}^{−}, respectively, whence the ideals P^{±} = (τ^{±})^{−1}(Q_{0}^{±}) are H-primes in R^{±}. Finally,

P = ((τ^{+} ⊗ τ^{−})β)^{−1}((Q_{0}^{+} ⊗ R_{c,0}^{−}) + (R_{r,0}^{+} ⊗ Q_{0}^{−})) = β^{−1}((P^{+} ⊗ R^{−}) + (R^{+} ⊗ P^{−})),

as desired. □

4. Illustration: O_{q}(M_{2}(k))

Theorem 3.5 opens a potential route to computing the H-primes of A in the generic case: If we can find all the H-primes in each R_{r}^{+} and R_{c}^{−}, we immediately obtain descriptions of all the H-primes in A. Since these descriptions would be in terms of pullbacks of H-primes from the algebras R_{r}^{+} ⊗ R_{c}^{−}, it would still remain to find generating sets for these ideals.

To illustrate the procedure, we sketch the case where n = 2, for which H- spec A is already known. In [7], we use the above process to compute H- spec A when n = 3.

4.1. Assume that q is not a root of unity, and fix n = 2. There are only four choices for r and c, namely ∅, (1), (2), and (1, 2). The corresponding algebras R_{r}^{+} and R_{c}^{−} are

R_{∅}^{+} = A/⟨X_{11}, X_{12}, X_{21}, X_{22}⟩ = k  R_{∅}^{−} = k
R_{(1)}^{+} = (A/⟨X_{12}, X_{22}⟩)[X_{11}^{−1}] = k⟨Y_{11}^{±1}, Y_{21}⟩  R_{(1)}^{−} = k⟨Z_{11}^{±1}, Z_{12}⟩
R_{(2)}^{+} = (A/⟨X_{11}, X_{12}, X_{22}⟩)[X_{21}^{−1}] = k[Y_{21}^{±1}]  R_{(2)}^{−} = k[Z_{12}^{±1}]
R_{(1,2)}^{+} = (A/⟨X_{12}⟩)[X_{11}^{−1}, X_{22}^{−1}] = k⟨Y_{11}^{±1}, Y_{21}, Y_{22}^{±1}⟩ R_{(1,2)}^{−} = k⟨Z_{11}^{±1}, Z_{12}, Z_{22}^{±1}⟩.

The H-primes in these algebras are easily computed:

H- spec R_{∅}^{+} = {⟨0⟩}  H- spec R_{∅}^{−} = {⟨0⟩}
H- spec R_{(1)}^{+} = {⟨0⟩, ⟨Y_{21}⟩}  H- spec R_{(1)}^{−} = {⟨0⟩, ⟨Z_{12}⟩}
H- spec R_{(2)}^{+} = {⟨0⟩}  H- spec R_{(2)}^{−} = {⟨0⟩}
H- spec R_{(1,2)}^{+} = {⟨0⟩, ⟨Y_{21}⟩}  H- spec R_{(1,2)}^{−} = {⟨0⟩, ⟨Z_{12}⟩}.

The only choice for (r, c) ∈ RC_{0} is r = c = ∅. In this case, the only H-primes in R_{r}^{+} and R_{c}^{−} are the zero ideals, and β_{∅,∅}^{−1}(⟨0⟩) = ⟨X_{11}, X_{12}, X_{21}, X_{22}⟩, the augmentation ideal of A. We record this H-prime using the symbol

•• ••

to denote the generating set {X_{11}, X_{12}, X_{21}, X_{22}}, the bullet in position (i, j) being a marker for the element X_{ij}.

Corresponding to the four pairs (r, c) ∈ RC_{1}, there are nine H-primes in A of the form

β_{r,c}^{−1}((Q^{+} ⊗ R_{c}^{−}) + (R_{r}^{+} ⊗ Q^{−}))

where Q^{+} is an H-prime in R_{r}^{+} and Q^{−} is an H-prime in R_{c}^{−}. We can record generating sets for these ideals as follows, continuing the notation introduced in the previous paragraph; here ∘ is a placeholder and □ denotes the 2 × 2 quantum determinant.

⟨0⟩ R_{(1)}^{−} ⟨Z_{12}⟩ ⟨0⟩ R_{(2)}^{−} ⟨0⟩ R_{(1)}^{+}  ∘ •  • ∘ ∘ •  • ∘ ⟨Y_{21}⟩  ∘ ∘ ∘ •  • ∘ • • • •  • • ⟨0⟩ R_{(2)}^{+}  • • • •  • • ∘ ∘ ∘ •  • ∘

(See, e.g., [5, Theorem 1.1] for a proof that the quantum determinant generates the kernel of β_{(1),(1)}. We leave it to the reader to check that the other H-primes are generated as indicated.)

Finally, the only choice for (r, c) ∈ RC_{2} is r = c = (1, 2), and there are four H-primes in A of the form

β_{(1,2),(1,2)}^{−1}((Q^{+} ⊗ R_{(1,2)}^{−}) + (R_{(1,2)}^{+} ⊗ Q^{−}))

with Q^{±} ∈ H- spec R_{(1,2)}^{±}. We record generating sets for these ideals as follows:

⟨0⟩ ⟨Z_{12}⟩ ⟨0⟩  ∘ ∘ ∘ • ∘ ∘ ∘ ∘ ⟨Y_{21}⟩  ∘ ∘ ∘ • • ∘ • ∘

We now conclude from Theorem 3.5 that we have found all the H-primes of A = O_{q}(M_{2}(k)). There are 14 in total, which we can display as follows:

• •  ∘ • • ∘ ∘ ∘ ∘ • • •  ∘ • • ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ • • ∘ ∘ ∘ ∘ • • • • • • • • ∘ • ∘ • • • • • • ∘ ∘ ∘ • • ∘

For a display showing the inclusions among these ideals, see [5, (3.6)].

5. Appendix. Relations in O_{q}(M_{n}(k))

The proofs in this paper rely on a number of relations among the generators and quantum minors in quantum matrix algebras. We record and/or derive those relations in this appendix. Throughout, let A = O_{q}(M_{n}(k)) with k an arbitrary field and q ∈ k^{×} an arbitrary nonzero scalar.

5.1. (a) We present the algebra A with generators X_{ij} for i, j = 1, …, n and relations

X_{ij}X_{lj} = qX_{lj}X_{ij}  (i < l)
X_{ij}X_{im} = qX_{im}X_{ij}  (j < m)
X_{ij}X_{lm} = X_{lm}X_{ij}  (i < l, j > m)
X_{ij}X_{lm} − X_{lm}X_{ij} = (q − q^{−1})X_{im}X_{lj}  (i < l, j < m).

As is well known, A is in fact a bialgebra, with comultiplication Δ : A → A⊗A and counit ϵ : A → k such that

Δ(X_{ij}) = ∑_{l=1}^{n} X_{il} ⊗ X_{lj}  and  ϵ(X_{ij}) = δ_{ij}

for all i, j.

(b) The algebra A possesses various symmetries. In particular, it supports a k-algebra automorphism τ such that τ (X_{ij}) = X_{ji} for all i, j [16, Proposition 3.7.1]. Pairs of results which imply each other just through applications of τ will be simply referred to as “symmetric”.

(c) Given U, V ⊆ {1, …, n} with |U| = |V | = t, let O_{q}(M_{U,V} (k)) denote the ksubalgebra of A generated by those X_{ij} with i ∈ U and j ∈ V. There is a natural isomorphism O_{q}(M_{t}(k)) → O_{q}(M_{U,V} (k)), which sends the quantum determinant of O_{q}(M_{t}(k)) to the quantum minor in A involving rows from U and columns from V. As in [6], we denote this quantum minor by [U | V], or in the form [u_{1} ⋯ u_{t} | v_{1} ⋯ v_{t}] if we wish to list the elements of U and V.

We shall also use the isomorphism O_{q}(M_{t}(k)) → O_{q}(M_{U,V} (k)) to simplify various proofs, since it will allow us to work in smaller quantum matrix algebras than A on occasion. For example, since the quantum determinant in O_{q}(M_{t}(k)) is central (e.g., [16, Theorem 4.6.1]), the quantum minor [U | V] commutes with X_{ij} for all i ∈ U and j ∈ V. Consequently,

[U | V][I | J] = [I | J][U | V]  (I ⊆ U, J ⊆ V).

(d) Recall from [15, Equation 1.9] the comultiplication rule for quantum minors:

Δ([I | J]) = ∑_{|K|=|I|}[I | K] ⊗ [K | J].

5.2. We next restate some identities from [16], given there for generators and maximal minors, in a form that applies to minors of arbitrary size. Note the difference between our choice of relations for A (see (5.1)(a)) and that in [16, p. 37]. Because of this, we must interchange q and q^{−1} whenever carrying over a formula from [16].

Lemma. Let r, c ∈ {1, …, n} and I, J ⊆ {1, …, n} with |I| = |J| ≥ 1.

(a) If r ∈ I and c ∈ J, then X_{rc}[I | J] = [I | J]X_{rc}.

(b) If r ∈ I and c ∉ J, set J^{+} = J ⊔ {c}. Then

(1)  X_{rc}[I | J] − q^{−1}[I | J]X_{rc} = (q^{−1} − q) ∑_{\begin{matrix}j∈J \\ j>c \end{matrix}}(−q)^{−|J∩[c,j]|}[I | J^{+} \ \{j\}]X_{rj}

(2)  [I | J]X_{rc} − qX_{rc}[I | J] = (q − q^{−1}) ∑_{\begin{matrix}j∈J \\ j>c \end{matrix}}(−q)^{|J∩[c,j]|}X_{rj}[I | J^{+} \ \{j\}]

(c) If r ∉ I and c ∈ J, set I^{+} = I ⊔ {r}. Then

(1)  X_{rc}[I | J] − q^{−1}[I | J]X_{rc} = (q^{−1} − q) ∑_{\begin{matrix}i∈I \\ i>r \end{matrix}}(−q)^{−|I∩[r,i]|}[I^{+} \ \{i\} | J]X_{ic}

(2)  [I | J]X_{rc} − qX_{rc}[I | J] = (q − q^{−1}) ∑_{\begin{matrix}i∈I \\ i>r \end{matrix}}(−q)^{|I∩[r,i]|}X_{ic}[I^{+} \ \{i\} | J]

Proof. Part (a) is clear. To obtain part (b), we may work in O_{q}(M_{I⊔{r′},J^{+}}(k)) for some r′ ∉ I, and so there is no loss of generality in assuming that I ⊂ J^{+} = {1, …, n}. The desired relations then follow from the second cases of parts (1) and (2) of [16, Lemma 4.5.1]. Part (c) is symmetric to (b). □

5.3. Lemma. Let U, V ⊆ {1, …, n} with |U| = |V |, and let u_{1}, u_{2} ∈ U and v_{1}, v_{2} ∈ V. Set U_{s} = U \ {u_{s}} and V_{s} = V \ {v_{s}} for s = 1, 2.

(a) If u_{1} < u_{2}, then [U_{1} | V_{1}][U_{2} | V_{1}] = q^{−1}[U_{2} | V_{1}][U_{1} | V_{1}].

(b) If v_{1} < v_{2}, then [U_{1} | V_{1}][U_{1} | V_{2}] = q^{−1}[U_{1} | V_{2}][U_{1} | V_{1}].

(c) If u_{1} < u_{2} and v_{1} > v_{2}, then [U_{1} | V_{1}][U_{2} | V_{2}] = [U_{2} | V_{2}][U_{1} | V_{1}].

(d) If u_{1} < u_{2} and v_{1} < v_{2}, then

[U_{1} | V_{1}][U_{2} | V_{2}] − [U_{2} | V_{2}][U_{1} | V_{1}] = (q^{−1} − q)[U_{2} | V_{1}][U_{1} | V_{2}].

Proof. Since we may work in O_{q}(M_{U,V} (k)), there is no loss of generality in assuming that U = V = {1, …, n}. The result then follows from [16, Theorem 5.2.1]. □

5.4. We also require the form of the q-Laplace relations given in [15]. For index sets I and J, set

ℓ(I; J) = |{(i, j) ∈ I × J | i > j}|.

Lemma. (q-Laplace relations) Let I, J ⊆ {1, …, n}.

(a) If J_{1}, J_{2} ⊆ {1, …, n} with |J_{1}| + |J_{2}| = |I|, then

∑_{\begin{matrix}I_{1}⊔I_{2}=I \\ |I_{ν}|=|J_{ν}| \end{matrix}}(−q)^{ℓ(I_{1};I_{2})}[I_{1} | J_{1}][I_{2} | J_{2}] = \{ \begin{matrix}(−q)^{ℓ(J_{1};J_{2})}[I | J_{1} ⊔ J_{2}] & (J_{1} ∩ J_{2} = ∅) \\ 0 & (J_{1} ∩ J_{2} ≠ ∅). \end{matrix}

(b) If I_{1}, I_{2} ⊆ {1, …, n} with |I_{1}| + |I_{2}| = |J|, then

∑_{\begin{matrix}J_{1}⊔J_{2}=J \\ |J_{ν}|=|I_{ν}| \end{matrix}}(−q)^{ℓ(J_{1};J_{2})}[I_{1} | J_{1}][I_{2} | J_{2}] = \{ \begin{matrix}(−q)^{ℓ(I_{1};I_{2})}[I_{1} ⊔ I_{2} | J] & (I_{1} ∩ I_{2} = ∅) \\ 0 & (I_{1} ∩ I_{2} ≠ ∅). \end{matrix}

Proof. [15, Proposition 1.1]. □

5.5. The q-Laplace relations simplify somewhat when one of the index sets is a singleton, as follows.

Corollary. Let r, c ∈ {1, …, n} and I, J ⊆ {1, …, n}.

(a) If |I| = |J| + 1, then

(1)  ∑_{i∈I}(−q)^{|[1,i)∩I|}X_{ic}[I \ \{i\} | J] = \{ \begin{matrix}(−q)^{|[1,c)∩J|}[I | J ⊔ \{c\}] & (c ∉ J) \\ 0 & (c ∈ J) \end{matrix}

(2)  ∑_{i∈I}(−q)^{|(i,n]∩I|}[I \ \{i\} | J]X_{ic} = \{ \begin{matrix}(−q)^{|(c,n]∩J|}[I | J ⊔ \{c\}] & (c ∉ J) \\ 0 & (c ∈ J). \end{matrix}

(b) If |J| = |I| + 1, then

(1)  ∑_{j∈J}(−q)^{|[1,j)∩J|}X_{rj}[I | J \ \{j\}] = \{ \begin{matrix}(−q)^{|[1,r)∩I|}[I ⊔ \{r\} | J] & (r ∉ I) \\ 0 & (r ∈ I) \end{matrix}

(2)  ∑_{j∈J}(−q)^{|(j,n]∩J|}[I | J \ \{j\}]X_{rj} = \{ \begin{matrix}(−q)^{|(r,n]∩I|}[I ⊔ \{r\} | J] & (r ∉ I) \\ 0 & (r ∈ I). \end{matrix}

Proof. (a) For the first case, fix J_{1} = {c} and J_{2} = J. We will use Lemma 5.4(a), which involves a sum over I_{1} ⊔ I_{2} = I with |I_{1}| = 1; thus I_{1} = {i} and I_{2} = I \ {i} for some i ∈ I. In that case, ℓ(I_{1}; I_{2}) = |[1, i) ∩ I| and ℓ(J_{1}; J_{2}) = |[1, c) ∩ J|. Thus, formula (1) follows directly from Lemma 5.4(a). Formula (2) follows similarly, where this time we fix J_{1} = J and J_{2} = {c}.

(b) These follow from (a) by symmetry. □

5.6. Lemma. Let U, V ⊆ {1, …, n} with |U| = |V |.

(a) Let U = I ⊔ K, and let J_{1}, J_{2} ⊆ V such that |J_{1}| + |J_{2}| = 2|I| + |K|. Then

∑_{\begin{matrix}K=K′⊔K′′ \\ |K′|=|J_{1}|−|I| \end{matrix}}(−q)^{ℓ(I;K′)+ℓ(K′;K′′⊔I)}[I ⊔ K′ | J_{1}][K′′ ⊔ I | J_{2}]

= \{ \begin{matrix}(−q)^{ℓ(J_{1}∩J_{2};J_{1}\J_{2})+ℓ(J_{1}\J_{2};J_{2})}[I | J_{1} ∩ J_{2}][U | V] & (|J_{1} ∩ J_{2}| = |I|) \\ 0 & (|J_{1} ∩ J_{2}| > |I|). \end{matrix}

(b) Let V = J ⊔ L, and let I_{1}, I_{2} ⊆ U such that |I_{1}| + |I_{2}| = 2|J| + |L|. Then

∑_{\begin{matrix}L=L′⊔L′′ \\ |L′|=|I_{1}|−|J| \end{matrix}}(−q)^{ℓ(J;L′)+ℓ(L′;L′′⊔J)}[I_{1} | J ⊔ L′][I_{2} | J ⊔ L′′]

= \{ \begin{matrix}(−q)^{ℓ(I_{1}∩I_{2};I_{1}\I_{2})+ℓ(I_{1}\I_{2};I_{2})}[I_{1} ∩ I_{2} | J][U | V] & (|I_{1} ∩ I_{2}| = |J|) \\ 0 & (|I_{1} ∩ I_{2}| > |J|). \end{matrix}

Proof. By symmetry, we need only prove (a). Note that

|J_{1}| + |J_{2}| = 2|I| + |K| = |I| + |U| ≥ |I| + |J_{1} ∪ J_{2}| = |I| + |J_{1}| + |J_{2}| − |J_{1} ∩ J_{2}|,

whence |J_{1} ∩ J_{2}| ≥ |I|. Let (1) denote the left hand side of the formula to be established, and (2) the first choice on the right hand side.

Expand each term (−q)^{ℓ(I;K′)}[I ⊔K′ | J_{1}] using Lemma 5.4(b) and insert into (1). Thus,

(1) equals

(3)  ∑_{\begin{matrix}K=K′⊔K′′ \\ J_{1}=J′_{1}⊔J′′_{1} \end{matrix}}(−q)^{ℓ(J′_{1};J′′_{1})+ℓ(K′;K′′⊔I)}[I | J′_{1}][K′ | J′′_{1}][K′′ ⊔ I | J_{2}].

We next claim that the sum

(4)  ∑_{\begin{matrix}U=I_{1}⊔I_{2} \\ J_{1}=J′_{1}⊔J′′_{1} \end{matrix}}(−q)^{ℓ(I_{1};I_{2})+ℓ(J′_{1};J′′_{1})}[I | J′_{1}][I_{1} | J′′_{1}][I_{2} | J_{2}]

equals (3). For I_{1} as in (4), we have |I| + |I_{1}| = |J′_{1}| + |J′′_{1}| = |J_{1}|, and so Lemma 5.4(b) gives

(5)  ∑_{J_{1}=J′_{1}⊔J′′_{1}}(−q)^{ℓ(J′_{1};J′′_{1})}[I | J′_{1}][I_{1} | J′′_{1}] = 0  (I_{1} ⊈ K),

because I ∩ I_{1} ≠ ∅ in this case. On the other hand, for fixed J′_{1}, J′′_{1} as in (4), we have

(6) ∑_{\begin{matrix}U=I_{1}⊔I_{2} \\ I_{1}⊆K \end{matrix}}(−q)^{ℓ(I_{1};I_{2})}[I_{1} | J′′_{1}][I_{2} | J_{2}]

= ∑_{K=K′⊔K′′}(−q)^{ℓ(K′;K′′⊔I)}[K′ | J′′_{1}][K′′ ⊔ I | J_{2}].

It follows from (5) and (6) that (4) = (3) as claimed, and thus (1) = (4).

For J′′_{1} as in (4), we have |J′′_{1}| = |K′| = |K| − |K′′| = |U| − |K′′ ⊔ I| = |V | − |J_{2}|. Hence, Lemma 5.4(a) says that

(7) ∑_{U=I_{1}⊔I_{2}}(−q)^{ℓ(I_{1};I_{2})}[I_{1} | J′′_{1}][I_{2} | J_{2}] = \{ \begin{matrix}(−q)^{ℓ(J′′_{1};J_{2})}[U | V] & (J′′_{1} ∩ J_{2} = ∅) \\ 0 & (J′′_{1} ∩ J_{2} ≠ ∅). \end{matrix}

Substituting (7) into (4), it follows that (1) is equal to the sum

(8)  ∑_{J_{1}=J′_{1}⊔J′′_{1}}(−q)^{ℓ(J′_{1};J′′_{1})+ℓ(J′′_{1};J_{2})}[I | J′_{1}]d(J′′_{1}),

where d(J′′_{1}) = [U | V] if J′′_{1} and J_{2} are disjoint, but d(J′′_{1}) = 0 otherwise.

If |J_{1} ∩ J_{2}| > |I|, then since any |J′_{1}| = |I|, we see that J_{1} ∩ J_{2} ⊈ J′_{1} and so J′′_{1} ∩ J_{2} ≠ ∅. Thus in this case all d(J′′_{1}) = 0, and so (1) = (8) = 0.

Finally, suppose that |J_{1} ∩ J_{2}| = |I|. Then the only time J′′_{1} and J_{2} can be disjoint is when J′_{1} = J_{1} ∩ J_{2}, and therefore (1) = (8) = (2) in this case. □

5.7. Lemma. Let r, c ∈ {1, …, n} and I, J ⊆ {1, …, n} with |I| = |J| ≥ 1. If r > max(I) and c > max(J), then

[I | J]X_{rc} − q^{2}X_{rc}[I | J] = (1 − q^{2})[I ⊔ {r} | J ⊔ {c}].

Proof. Since we may work in O_{q}(M_{I⊔{r},J⊔{c}}(k)), it suffices to consider the case that r = c = n and I = J = {1, …, n − 1}. Now [I | J] = A(n n) in the notation of [16, (4.3)]. Set D_{q} = [I ⊔ {r} | J ⊔ {c}] = [1 ⋯ n | 1 ⋯ n].

The first two q-Laplace relations in [16, Corollary 4.4.4] yield

(1)  ∑_{j=1}^{n}(−q)^{j−n}X_{nj}A(n j) = ∑_{j=1}^{n}(−q)^{n−j}A(n j)X_{nj} = D_{q}.

Solving for X_{nn}A(n n) and A(n n)X_{nn}, we obtain

(2)  X_{nn}A(n n) = D_{q} − ∑_{j<n}(−q)^{j−n}X_{nj}A(n j)
(3)  A(n n)X_{nn} = D_{q} − ∑_{j<n}(−q)^{n−j}A(n j)X_{nj}.

For any j, the third relation of [16, Lemma 5.1.2] implies that

(4)  X_{nj}A(n j) = A(n j)X_{nj} + (1 − q^{−2}) ∑_{l<j}(−q)^{j−l}A(n l)X_{nl}.

Substituting (4) into (2) for all j < n, we obtain

(5)  \begin{aligned}X_{nn}A(n n) & = D_{q} − ∑_{j<n}(−q)^{j−n}A(n j)X_{nj} \\ & − (1 − q^{−2}) ∑_{j<n} ∑_{l<j}(−q)^{2j−l−n}A(n l)X_{nl} \\ & = D_{q} − ∑_{l<n}[(−q)^{l−n} + (1 − q^{−2}) ∑_{l<j<n}(−q)^{2j−l−n}]A(n l)X_{nl}. \end{aligned}

The expression in square brackets can be simplified as follows:

(6) \begin{aligned}(−q)^{l−n} + (1 − q^{−2}) ∑_{l<j<n}(−q)^{2j−l−n} & = (−q)^{l−n}[1 + (1 − q^{−2}) ∑_{0<m<n−l}(−q)^{2m}] \\ & = (−q)^{n−l−2}. \end{aligned}

Substituting (6) into (5) and replacing l by j, we obtain

(7)  X_{nn}A(n n) = D_{q} − ∑_{j<n}(−q)^{n−j−2}A(n j)X_{nj}.

Finally, combining (3) with (7), we conclude that

A(n n)X_{nn} − q^{2}X_{nn}A(n n) = (1 − q^{2})D_{q},

as desired. □

References

1. K. A. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups, Centre de Recerca Matemàtica Advanced Course Series, Birkhäuser (to appear).

2. G. Cauchon, Spectre premier de O_{q}(M_{n}(k)). Image canonique et séparation normale (to appear).

3. C. De Concini and V. Lyubashenko, Quantum function algebra at roots of 1, Advances in Math. 108 (1994), 205-262.

4. C. De Concini and C. Procesi, Quantum Schubert cells and representations at roots of 1, in Algebraic Groups and Lie Groups (G.I. Lehrer, ed.), Austral. Math. Soc. Lecture Series 9, Cambridge Univ. Press, Cambridge, 1997, pp. 127-160.

5. K. R. Goodearl and T. H. Lenagan, Prime ideals in certain quantum determinantal rings, in Interactions between Ring Theory and Representations of Algebras (Murcia 1998) (F. Van Oystaeyen and M. Saorín, eds.), Dekker, New York, 2000, pp. 239-251.

6. , Quantum determinantal ideals, Duke Math. J. 103 (2000), 165-190.

7. , Winding-invariant prime ideals in quantum 3 × 3 matrices (to appear).

8. K. R. Goodearl and E. S. Letzter, Prime factor algebras of the coordinate ring of quantum matrices, Proc. Amer. Math. Soc. 121 (1994), 1017-1025.

9. , The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras, Trans. Amer. Math. Soc. 352 (2000), 1381-1403.

10. T. J. Hodges and T. Levasseur, Primitive ideals of C_{q}[SL(3)], Comm. Math. Phys. 156 (1993), 581-605.

11. , Primitive ideals of C_{q}[SL(n)], J. Algebra 168 (1994), 455-468.

12. A. Joseph, Quantum Groups and Their Primitive Ideals, Ergebnisse der Math. (3) 29, Springer-Verlag, Berlin, 1995.

13. T. Levasseur and J. T. Stafford, The quantum coordinate ring of the special linear group, J. Pure Applied Algebra 86 (1993), 181-186.

14. C. Moeglin and R. Rentschler, Orbites d’un groupe algébrique dans l’espace des idéaux rationnels d’une algèbre enveloppante, Bull. Soc. Math. France 109 (1981), 403–426.

15. M. Noumi, H. Yamada, and K. Mimachi, Finite dimensional representations of the quantum group GL_{q}(n; ℂ) and the zonal spherical functions on U_{q}(n − 1)\U_{q}(n), Japanese J. Math. 19 (1993), 31-80.

16. B. Parshall and J.-P. Wang, Quantum linear groups, Memoirs Amer. Math. Soc. 439 (1991).

17. N. Vonessen, Actions of algebraic groups on the spectrum of rational ideals, II, J. Algebra 208 (1998), 216-261.

Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

E-mail address: goodearl@math.ucsb.edu

Department of Mathematics, J.C.M.B., Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland

E-mail address: tom@maths.ed.ac.uk