PRIME IDEALS IN CERTAIN

QUANTUM DETERMINANTAL RINGS

K. R. Goodearl and T. H. Lenagan

Abstract. The ideal I_{1} generated by the 2 × 2 quantum minors in the coordinate algebra of quantum matrices, O_{q}(M_{m,n}(k)), is investigated. Analogues of the First and Second Fundamental Theorems of Invariant Theory are proved. In particular, it is shown that I_{1} is a completely prime ideal, that is, O_{q}(M_{m,n}(k))/I_{1} is an integral domain, and that O_{q}(M_{m,n}(k))/I_{1} is the ring of coinvariants of a coaction of k[x, x^{−1}] on O_{q}(k^{m}) ⊗ O_{q}(k^{n}), a tensor product of two quantum affine spaces. There is a natural torus action on O_{q}(M_{m,n}(k))/I_{1} induced by an (m + n)-torus action on O_{q}(M_{m,n}(k)). We identify the invariant prime ideals for this action and deduce consequences for the prime spectrum of O_{q}(M_{m,n}(k))/I_{1}.

Introduction

Let k be a field and let q ∈ k^{×}. The coordinate ring of quantum m × n matrices, A := O_{q}(M_{m,n}(k)), is a deformation of the classical coordinate ring of m × n matrices, O(M_{m,n}(k)). As such it is a k-algebra generated by mn indeterminates X_{ij}, for 1 ≤ i ≤ m and 1 ≤ j ≤ n, subject to the relations

X_{ij}X_{lj} = qX_{lj}X_{ij}_{X_{ij}X_{is} = qX_{is}X_{ij}}_{X_{is}X_{lj} = X_{lj}X_{is}}_{X_{ij}X_{ls} − X_{ls}X_{ij} = (q − q^{−1})X_{is}X_{lj}} when i < l;

when j < s;
when i < l and j < s;
when i < l and j < s.

In some references (e.g., [6, §3.5]), q is replaced by q^{−1}. When q = 1 we recover O(M_{m,n}(k)), which is the commutative polynomial algebra k[X_{ij}].

This research was partially supported by National Science Foundation research grant DMS-9622876 and NATO Collaborative Research Grant 960250.

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When m = n, the algebra A possesses a special element, the quantum determinant, D_{q}, defined by

D_{q} := ∑_{σ∈S_{n}}(−q)^{l(σ)}X_{1,σ(1)}X_{2,σ(2)} ⋯ X_{n,σ(n)},

where l(σ) denotes the number of inversions in the permutation σ. The quantum determinant D_{q} is a central element of A (see, for example, [6, Theorem 4.6.1]), and the localization A[D_{q}^{−1}] is the coordinate ring of the quantum general linear group, denoted O_{q}(GL_{n}(k)).

If I ⊆ {1, …, m} and J ⊆ {1, …, n} with |I| = |J| = t, let D(I, J) denote the t × t quantum minor obtained as the quantum determinant of the subalgebra of A obtained by deleting generators X_{ij} from the rows outside I and from the columns outside J. We write I_{t} for the ideal generated by the (t + 1) × (t + 1) quantum minors of A. In [3] it is proved that A/I_{t} is an integral domain, for each 1 ≤ t ≤ min{m, n}. Independently, Rigal [7] has shown that A/I_{1} is a domain; he also shows that A/I_{1} is a maximal order in its division ring of fractions.

There is an action of the torus H := (k^{×})^{m} × (k^{×})^{n} by k-algebra automorphisms on A such that

(α_{1}, …, α_{m}, β_{1}, …, β_{n}) · X_{ij} := α_{i}β_{j}X_{ij}

for all i, j. The ideals I_{t} are easily seen to be invariant under H; so there is an induced action of H on the factor algebras A/I_{t}. In this paper, we study the prime ideal structure in the algebra A/I_{1}, paying particular attention to the H-invariant prime ideals.

1. Complete primeness of I_{1}

We give a direct derivation of the fact that A/I_{1} is a domain. Although this is already established in both [3] and [7], the proof we give here is so much simpler and more transparent than either of the previous proofs that we think it will be useful to have it in a published form.

The coordinate ring of quantum affine n-space, denoted O_{q}(k^{n}), is defined to be the k-algebra generated by elements y_{1}, …, y_{n} subject to the relations y_{i}y_{j} = qy_{j}y_{i} for each 1 ≤ i < j ≤ n. It is well known that O_{q}(k^{n}) is an iterated Ore extension, and thus, in particular, O_{q}(k^{n}) is a domain. Our strategy is to produce a homomorphism of A into O_{q}(k^{m}) ⊗ O_{q}(k^{n}). This latter algebra can also be presented as an iterated Ore extension and thus is a domain. We show that I_{1} is the kernel of this map and so A/I_{1} is a domain.

1.1. Theorem. The algebra O_{q}(M_{m,n}(k))/I_{1} is isomorphic to a subalgebra of the tensor product O_{q}(k^{m}) ⊗ O_{q}(k^{n}). In particular, I_{1} is a completely prime ideal of O_{q}(M_{m,n}(k)).

Proof. Let O_{q}(k^{m}) = k[y_{1}, …, y_{m}] and O_{q}(k^{n}) = k[z_{1}, …, z_{n}] be the coordinate rings of quantum affine m-space and n-space, respectively. We define an algebra homomorphism θ : A → O_{q}(k^{m})⊗O_{q}(k^{n}) such that θ(X_{ij}) = y_{i} ⊗z_{j} for all i, j. In order that this does extend to a well-defined algebra homomorphism, we must check that the elements y_{i} ⊗ z_{j} satisfy at least the relations defining A. These are routine verifications; for example, if i < l and j < s then

(y_{i} ⊗ z_{j})(y_{l} ⊗ z_{s}) = y_{i}y_{l} ⊗ z_{j}z_{s} = y_{i}y_{l} ⊗ qz_{s}z_{j} = q(y_{i} ⊗ z_{s})(y_{l} ⊗ z_{j}),

while

(y_{l} ⊗ z_{s})(y_{i} ⊗ z_{j}) = y_{l}y_{i} ⊗ z_{s}z_{j} = q^{−1}y_{i}y_{l} ⊗ z_{s}z_{j} = q^{−1}(y_{i} ⊗ z_{s})(y_{l} ⊗ z_{j}).

Thus,

(y_{i} ⊗ z_{j})(y_{l} ⊗ z_{s}) − (y_{l} ⊗ z_{s})(y_{i} ⊗ z_{j}) = (q − q^{−1})(y_{i} ⊗ z_{s})(y_{l} ⊗ z_{j}),

so that the fourth relation of the introduction holds. One can also obtain θ as the composition of the comultiplication A → A ⊗ A with the tensor product of the quotient maps from A to A/⟨X_{ij} | i > 1⟩ and A/⟨X_{ij} | j > 1⟩. We shall pursue the latter point of view in the next section.

Thus, there exists a unique k-algebra homomorphism

θ : A → O_{q}(k^{m}) ⊗ O_{q}(k^{n})

such that θ(X_{ij}) = y_{i} ⊗ z_{j} for all i, j. If i < l and j < s then the above calculations also show that θ(X_{ij}X_{ls} − qX_{is}X_{lj}) = 0; thus I_{1} ⊆ ker(θ).

Now, O_{q}(k^{m}) ⊗ O_{q}(k^{n}) is a domain, since it can be viewed as a (multiparameter) quantum affine (m + n)-space with respect to the generators y_{1} ⊗ 1, …, y_{m} ⊗ 1, 1 ⊗ z_{1}, …, 1 ⊗ z_{n}. Hence, ker(θ) is a completely prime ideal of A. We show that I_{1} = ker(θ), so that I_{1} is completely prime. It remains to show that the induced map \overline{θ} : A/I_{1} → O_{q}(k^{m}) ⊗ O_{q}(k^{n}) is injective. Let S denote the set of monomials X_{i_{1}j_{1}}X_{i_{2}j_{2}} … X_{i_{l}j_{l}} in A such that i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{l} and j_{1} ≤ j_{2} ≤ ⋯ ≤ j_{l}. (We allow the monomial to be equal to 1 when l = 0.) We claim that the set \overline{S} of images forms a spanning set of A/I_{1}.

It suffices to show that an arbitrary monomial C in A is congruent modulo I_{1} to a linear combination of monomials from S. We proceed by induction on the index sets, where row index sequences (i_{1}, i_{2}, …, i_{l}) are ordered lexicographically with respect to ≥, column index sequences (j_{1}, j_{2}, …, j_{l}) are ordered lexicographically with respect to ≤, and pairs of sequences are ordered lexicographically.

If the claim fails, then it fails for a monomial C = X_{i_{1}j_{1}}X_{i_{2}j_{2}} … X_{i_{l}j_{l}} whose index set is minimal with respect to the ordering given in the previous paragraph. In particular, C ∉ S. Let r be the first subindex such that either i_{r} < i_{r+1} or j_{r} > j_{r+1}.

If i_{r} < i_{r+1} and j_{r} ≥ j_{r+1} then C = λC′, where λ is either 1 or q and C′ is obtained from C by switching X_{i_{r}j_{r}} and X_{i_{r+1}j_{r+1}}. However,

(i_{1}, , …, i_{r−1}, i_{r+1}, i_{r}, i_{r+2}, , …, i_{l}) < (i_{1}, i_{2}, …, i_{l})

in our ordering, so C′ is congruent modulo I_{1} to a linear combination of elements of S. Then C is congruent to such a linear combination, contradicting our assumptions. A similar contradiction occurs if i_{r} ≤ i_{r+1} and j_{r} > j_{r+1}: this time, the row indices might not change, but

(j_{1}, …, j_{r−1}, j_{r+1}, j_{r}, j_{r+2}, …, j_{l}) < (j_{1}, …, j_{l}),

so again we have a contradiction. Therefore, we must either have i_{r} < i_{r+1} and j_{r} < j_{r+1} or i_{r} > i_{r+1} and j_{r} > j_{r+1}.

Suppose that i_{r} < i_{r+1} and j_{r} < j_{r+1}. In this case, we have

X_{i_{r}j_{r}}X_{i_{r+1}j_{r+1}} − qX_{i_{r+1}j_{r}}X_{i_{r}j_{r+1}} ∈ I_{1},

so that C − qC′ ∈ I_{1}, where

C′ = X_{i_{1}j_{1}} ⋯ X_{i_{r−1}j_{r−1}}X_{i_{r+1}j_{r}}X_{i_{r}j_{r+1}}X_{i_{r+2}j_{r+2}} ⋯ X_{i_{l}j_{l}}.

We obtain a contradiction as above.

The final case is i_{r} > i_{r+1} and j_{r} > j_{r+1}, where we have

X_{i_{r}j_{r}}X_{i_{r+1}j_{r+1}} − q^{−1}X_{i_{r}j_{r+1}}X_{i_{r+1}j_{r}} ∈ I_{1}.

Thus, C − q^{−1}C′ ∈ I_{1}, where

C′ = X_{i_{1}j_{1}} ⋯ X_{i_{r−1}j_{r−1}}X_{i_{r}j_{r+1}}X_{i_{r+1}j_{r}}X_{i_{r+2}j_{r+2}} ⋯ X_{i_{l}j_{l}},

and once again we reach a contradiction. This finishes the proof of the claim and establishes that \overline{S} spans A/I_{1}.

Now, observe that in O_{q}(k^{m}) ⊗ O_{q}(k^{n}) we have

θ(X_{i_{1}j_{1}}X_{i_{2}j_{2}} … X_{i_{l}j_{l}}) = y_{i_{1}}y_{i_{2}} … y_{i_{l}} ⊗ z_{j_{1}}z_{j_{2}} … z_{j_{l}}.

The monomials y_{i_{1}}y_{i_{2}} … y_{i_{l}} with i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{l} are linearly independent over k, and, likewise, the monomials z_{j_{1}}z_{j_{2}} … z_{j_{l}} with j_{1} ≤ j_{2} ≤ ⋯ ≤ j_{l} are linearly independent over k. Hence, θ maps S bijectively to a linearly independent set in O_{q}(k^{m}) ⊗ O_{q}(k^{n}), so that \overline{S} is a linearly independent set in A/I_{1}. Therefore, the map \overline{θ} : A/I_{1} → O_{q}(k^{m}) ⊗ O_{q}(k^{n}) maps the k-basis \overline{S} bijectively onto a linearly independent set, so that \overline{θ} is injective. □

2. Coinvariants

Theorem 1.1 has an invariant theoretic interpretation, which we discuss in this section. First, we outline what happens in the classical (q = 1) case.

2.1. Let M_{u,v}(k) denote the algebraic variety of u × v matrices over k. Fix positive integers m, n and t ≤ min{m, n}. The general linear group GL_{t}(k) acts on M_{m,t}(k) × M_{t,n}(k) via

g · (A, B) := (Ag^{−1}, gB).

Matrix multiplication yields a map

μ : M_{m,t}(k) × M_{t,n}(k) → M_{m,n}(k),

the image of which is the variety of m × n matrices with rank at most t, and there is an induced map

μ_{∗} : O(M_{m,n}(k)) → O(M_{m,t}(k) × M_{t,n}(k)) = O(M_{m,t}(k)) ⊗ O(M_{t,n}(k)).

The First Fundamental Theorem of invariant theory identifies the fixed ring of the coordinate ring O(M_{m,t}(k) × M_{t,n}(k)) under the induced action of GL_{t}(k) as precisely the image of μ_{∗}. The Second Fundamental Theorem states that the kernel of μ_{∗} is I_{t}, the ideal generated by the (t + 1) × (t + 1) minors of O(M_{m,n}(k)), so that the coordinate ring of the variety of m × n matrices of rank at most t is O(M_{m,n}(k))/I_{t}. As a consequence, since this variety is irreducible, the ideal I_{t} is a prime ideal of O(M_{m,n}(k)).

2.2. We now proceed to explain the connection between Theorem 1.1 and the above invariant theoretic point of view.

The analog of μ_{∗} is the k-algebra homomorphism

θ_{t} : O_{q}(M_{m,n}(k)) → O_{q}(M_{m,t}(k)) ⊗ O_{q}(M_{t,n}(k))

induced from the comultiplication on O_{q}(M_{m,n}(k)), that is,

θ_{t}(X_{ij}) = ∑_{l=1}^{t} X_{il} ⊗ X_{lj}

for 1 ≤ i ≤ m and 1 ≤ j ≤ n. The comultiplications on O_{q}(M_{m,t}(k)) and O_{q}(M_{t,n}(k)) yield k-algebra homomorphisms

ρ_{t} : O_{q}(M_{m,t}(k)) → O_{q}(M_{m,t}(k)) ⊗ O_{q}(M_{t}(k))
→ O_{q}(M_{m,t}(k)) ⊗ O_{q}(GL_{t}(k))
λ_{t} : O_{q}(M_{t,n}(k)) → O_{q}(M_{t}(k)) ⊗ O_{q}(M_{t,n}(k))
→ O_{q}(GL_{t}(k)) ⊗ O_{q}(M_{t,n}(k))

which make O_{q}(M_{m,t}(k)) into a right O_{q}(GL_{t}(k))-comodule and O_{q}(M_{t,n}(k)) into a left O_{q}(GL_{t}(k))-comodule. Since O_{q}(GL_{t}(k)) is a Hopf algebra, the right comodule O_{q}(M_{m,t}(k)) becomes a \underline{left} O_{q}(GL_{t}(k))-comodule on composing ρ_{t} with 1 ⊗ S followed by the flip (where S denotes the antipode). Finally, the tensor product of the two left O_{q}(GL_{t}(k))-comodules O_{q}(M_{m,t}(k)) and O_{q}(M_{t,n}(k)) becomes a left O_{q}(GL_{t}(k))-comodule via the multiplication map on O_{q}(GL_{t}(k)). This comodule structure map,

γ_{t} : O_{q}(M_{m,t}(k))⊗O_{q}(M_{t,n}(k)) → O_{q}(GL_{t}(k))⊗O_{q}(M_{m,t}(k))⊗O_{q}(M_{t,n}(k)),

can be described (using the Sweedler summation notation) as follows:

γ_{t}(a ⊗ b) = ∑_{(a)} ∑_{(b)} S(a_{1})b_{−1} ⊗ a_{0} ⊗ b_{0}

where ρ_{t}(a) = ∑_{(a)} a_{0} ⊗ a_{1} and λ_{t}(b) = ∑_{(b)} b_{−1} ⊗ b_{0} for a ∈ O_{q}(M_{m,t}(k)) and b ∈ O_{q}(M_{t,n}(k)). Note that for t > 1, the map γ_{t} is not an algebra homomorphism, since neither the antipode nor the multiplication map on O_{q}(GL_{t}(k)) is an algebra homomorphism. On the other hand, γ_{1} is a k-al-gebra homomorphism.

Recall that the coinvariants of the coaction γ_{t} are the elements x in the tensor product O_{q}(M_{m,t}(k)) ⊗ O_{q}(M_{t,n}(k)) such that γ_{t}(x) = 1 ⊗ x. Quantum analogs of the First and Second Fundamental Theorems would be the following:

Conjecture 1. The set of coinvariants of γ_{t} equals the image of θ_{t}.

Conjecture 2. The kernel of θ_{t} is the ideal I_{t}. We have proved Conjecture 2 in [3, Proposition 2.4] (essentially; the cited result covers the case m = n, and the general case follows easily by the method of [3, Corollary 2.6]). However, Conjecture 1 is open at present. Here we shall establish it in the case t = 1.

2.3. Note that O_{q}(M_{m,1}(k)) and O_{q}(M_{1,n}(k)) are quantum affine spaces on generators X_{11}, X_{21}, …, X_{m1} and X_{11}, X_{12}, …, X_{1n}, respectively. In studying the case t = 1, it is convenient to replace O_{q}(M_{m,1}(k)) and O_{q}(M_{1,n}(k)) by O_{q}(k^{m}) = k[y_{1}, …, y_{m}] and O_{q}(k^{n}) = k[z_{1}, …, z_{n}], respectively. Then θ_{1} becomes the k-algebra homomorphism

θ : O_{q}(M_{m,n}(k)) → O_{q}(k^{m}) ⊗ O_{q}(k^{n}), X_{ij} ↦ y_{i} ⊗ z_{j}

used in the proof of Theorem 1.1. Next, the (quantum) coordinate ring of 1 × 1 matrices is just a polynomial ring k[x], and the (quantum) coordinate ring of the 1 × 1 general linear group is the localization k[x, x^{−1}]. Thus, in the present case the coaction γ_{1} becomes the k-algebra homomorphism

γ : O_{q}(k^{m}) ⊗ O_{q}(k^{n}) → k[x^{±1}] ⊗ O_{q}(k^{m}) ⊗ O_{q}(k^{n}),
y_{i} ⊗ 1 ↦ x^{−1} ⊗ y_{i} ⊗ 1, 1 ⊗ z_{j} ↦ x ⊗ 1 ⊗ z_{j}.

2.4. Theorem. The set of coinvariants of γ is exactly the image of the algebra O_{q}(M_{m,n}(k)) in O_{q}(k^{m}) ⊗ O_{q}(k^{n}) under θ.

Proof. Clearly γθ(X_{ij}) = 1 ⊗ y_{i} ⊗ z_{j} = 1 ⊗ θ(X_{ij}) for all i, j. Since θ and γ are k-algebra homomorphisms, it follows that the image of θ is contained in the coinvariants of γ.

The algebra O_{q}(k^{m})⊗O_{q}(k^{n}) has a basis consisting of pure tensors Y ⊗Z where Y is an ordered monomial in the y_{i} and Z is an ordered monomial in the z_{j}. Note that γ(Y ⊗ Z) = x^{s−r} ⊗ Y ⊗ Z where r and s are the total degrees of Y and Z, respectively. Hence, the images γ(Y ⊗ Z) are k-linearly independent, and a linear combination ∑_{l=1}^{d} α_{l}Y_{l} ⊗ Z_{l} of distinct monomial tensors is a coinvariant for γ if and only if each Y_{l} ⊗ Z_{l} is a coinvariant.

Thus, we need only consider a single term

Y ⊗ Z = y_{i_{1}}y_{i_{2}} ⋯ y_{i_{r}} ⊗ z_{j_{1}}z_{j_{2}} ⋯ z_{j_{s}}.

If Y ⊗ Z is a coinvariant, then because γ(Y ⊗ Z) = x^{s−r} ⊗ Y ⊗ Z we must have r = s. Therefore

Y ⊗ Z = θ(X_{i_{1}j_{1}}X_{i_{2}j_{2}} ⋯ X_{i_{r}j_{r}}),

which shows that Y ⊗ Z is in the image of θ, as desired. □

3. H-invariant prime ideals of O_{q}(M_{m,n}(k))/I_{1}

Under the mild assumption that our ground field k is infinite, we identify the H-invariant prime ideals of the domain A/I_{1} = O_{q}(M_{m,n}(k))/I_{1}. (Recall that H denotes the torus (k^{×})^{m} × (k^{×})^{n}, acting on A as described in the introduction.) This identifies the minimal elements in a stratification of spec A/I_{1}, and yields a description of this prime spectrum as a finite disjoint union of commutative schemes corresponding to Laurent polynomial rings.

3.1. Let H be a group acting as automorphisms on a ring A. We refer the reader to [1] for the definition of the H-stratification of spec A, and here recall only that the H-stratum of spec A corresponding to an H-prime ideal J is the set

spec_{J} A := {P ∈ spec A | (P : H) = J}.

In the case of the algebra A/I_{1}, we shall (assuming k infinite) identify the H-prime ideals – they turn out to be the same as the H-invariant primes – and thus pin down the minimum elements of the H-strata. Further, we shall see that each H-stratum of spec A/I_{1} is homeomorphic to the spectrum of a Laurent polynomial ring over an algebraic extension of k. This pattern is also known to hold for spec A itself (at least when q is not a root of unity), but there the H-prime ideals have not yet been completely identified.

3.2. It turns out that if a generator X_{ij} lies in an H-prime ideal P of A containing I_{1}, then either all the generators from the same row, or all the generators from the same column must also lie in P. This leads us to make the following definition.

For subsets I ⊆ {1, …, m} and J ⊆ {1, …, n}, set

P (I, J) := I_{1} + ⟨X_{ij} | i ∈ I⟩ + ⟨X_{ij} | j ∈ J⟩.

Obviously, P (I, J) is an H-invariant ideal of A. We shall show that P (I, J) is (completely) prime, and hence H-prime.

Lemma. The factor algebra A/P (I, J) is isomorphic to O_{q}(M_{m′,n′}(k))/I′_{1}, where m′ = m − |I| and n′ = n − |J|, and I′_{1} is the ideal generated by the 2 × 2 quantum minors of O_{q}(M_{m′,n′}(k)). Hence, P (I, J) is a completely prime ideal of A.

Proof. The second statement follows immediately from the first statement and Theorem 1.1.

Set I′ := {1, …, m}\I, and J′ := {1, …, n}\J, and let A′ be the ksubalgebra of A generated by the X_{ij} for i ∈ I′ and j ∈ J′. Note that A′ is isomorphic to O_{q}(M_{m′,n′}(k)). Let I′_{1} be the ideal of A′ generated by the 2×2 quantum minors of A′; that is, those for which both row indices are in I′ and both column indices are in J′. Obviously, I′_{1} ⊆ A′ ∩ I_{1}, so that the inclusion A′ ↪ A induces a k-algebra homomorphism f : A′/I′_{1} → A/P (I, J). It suffices to show that f is an isomorphism.

The factor A/P (I, J) is generated by the cosets of those X_{ij} with i ∈ I′ and j ∈ J′, since X_{ij} ∈ P (I, J) whenever i ∈ I or j ∈ J. These cosets are all in the image of f; so f is surjective.

Observe that there exists a k-algebra homomorphism g : A → A′ such that g(X_{ij}) = X_{ij} when i ∈ I′ and j ∈ J′, and g(X_{ij}) = 0 otherwise. To see this, note that the only problematic relations are those of the form X_{ij}X_{ls} − X_{ls}X_{ij} = (q − q^{−1})X_{is}X_{lj} for i < l and j < s. However, if i ∉ I′ then X_{ij} and X_{is} both map to zero, and the relation maps to 0 = 0. Likewise, this happens in all cases except when i, l ∈ I′ and j, s ∈ J′: in this case, the relation above maps to a relation in A′.

Consider a 2 × 2 quantum minor in A of the form D = X_{ij}X_{ls} − qX_{is}X_{lj} where i < l and j < s. If i ∉ I′ then both X_{ij} and X_{is} are in ker(g) , so that D ∈ ker(g). Likewise, g(D) = 0 when l ∉ I′, or j ∉ J′, or s ∉ J′. On the other hand, g(D) = D when i, l ∈ I′ and j, s ∈ J′. Further, g(X_{ij}) = 0 when i ∈ I or j ∈ J. Hence, g(P (I, J)) ⊆ I′_{1}.

Therefore, g induces a k-algebra homomorphism \overline{g} : A/P (I, J) → A′/I′_{1}. Both of these algebras are generated by the cosets corresponding to those X_{ij} such that i ∈ I′ and j ∈ J′. It follows that both f\overline{g} and \overline{g}f are identity maps, since both f and \overline{g} preserve these cosets. Hence, f is an isomorphism. □

Somewhat suprisingly, the P (I, J) turn out to be the only H-prime ideals of A that contain I_{1}. The following lemma will be helpful in establishing this fact.

3.3. Lemma. Let i, s ∈ {1, …, m} and j, t ∈ {1, …, n}. Then there exist scalars α ∈ {1, q^{±1}, q^{±2}} and β ∈ {1, q^{±1}} such that X_{ij}X_{st} − αX_{st}X_{ij} and X_{ij}X_{st} − βX_{it}X_{sj} lie in I_{1}. In particular, the cosets X_{ij} + I_{1} are all normal elements of A/I_{1}.

Proof. If i = s, then in view of the relations in A we can take α = β to be q, 1, or q^{−1} (depending on whether j < t or j = t or j > t). Similarly, if j = t, we can take α ∈ {1, q^{±1}} and β = 1.

If i < s and j > t, or if i > s and j < t, then X_{ij} and X_{st} commute, and we can take α = 1. On the other hand, one of X_{it}X_{sj} − q^{±1}X_{ij}X_{st} is a 2 × 2 quantum minor, and so we can take β to be q or q^{−1}.

Now suppose that i < s and j < t. Then X_{ij}X_{st} − qX_{it}X_{sj} is a quantum minor, and we can take β = q. But X_{st}X_{ij} − q^{−1}X_{it}X_{sj} is also a quantum minor, so we have X_{ij}X_{st} ≡ qX_{it}X_{sj} ≡ q^{2}X_{st}X_{ij} (mod I_{1}), and hence we can take α = q^{2}.

The remaining case follows from the previous one by exchanging (i, j) and (s, t), and then the final statement of the lemma is clear. □

3.4. Proposition. Assume that k is an infinite field. Then the H-prime ideals of O_{q}(M_{m,n}(k)) that contain I_{1} are precisely the ideals P (I, J).

Proof. By Lemma 3.2, we know that the ideals P (I, J) are H-prime. Consider an arbitrary H-prime ideal P of A that contains I_{1}. If all of the X_{ij} are in P then P must be the maximal ideal generated by the X_{ij}. In that case, P = P (I, J), where I = {1, …, m} and J = {1, …, n}. Hence, we may assume that not all X_{ij} are in P. Set

I = {i ∈ {1, …, m} | X_{ij} ∈ P for all j}
J = {j ∈ {1, …, n} | X_{ij} ∈ P for all i}.

We first show that X_{ij} ∈ P if and only if i ∈ I or j ∈ J. Certainly, if i ∈ I or j ∈ J then X_{ij} ∈ P, by the definition of I and J. Suppose that there exists an X_{ij} ∈ P such that i ∉ I and j ∉ J. Then there exists an index s ≠ i such that X_{sj} ∉ P and also there exists an index t ≠ j such that X_{it} ∉ P. By Lemma 3.3, there is a nonzero scalar β ∈ k such that X_{ij}X_{st} − βX_{it}X_{sj} ∈ P. Thus, X_{ij} ∈ P would imply that X_{it}X_{sj} ∈ P. However, X_{it} and X_{sj} are H-eigenvectors which, by Lemma 3.3, are normal modulo P. Hence, because P is H-prime, X_{it}X_{sj} ∈ P would imply X_{it} ∈ P or X_{sj} ∈ P, contradicting the choices of s and t. Thus, we have established that X_{ij} ∈ P if and only if i ∈ I or j ∈ J. Now P (I, J) ⊆ P, and we need to establish equality.

Set B := A/P (I, J) and \overline{P} = P/P (I, J), and note that B is a domain by Lemma 3.2. Write Y_{ij} for the image of X_{ij} in B. The claim just established implies that Y_{ij} ∉ \overline{P} if i ∉ I or j ∉ J. Recall from Lemma 3.3 that the Y_{ij} scalar-commute among themselves.

Now, I ≠ {1, …, m} and J ≠ {1, …, n}, since not all of the X_{ij} are in P. Let s ∈ {1, …, m} \ I and t ∈ {1, …, n} \ J be minimal, and consider the localization C := B[Y_{st}^{−1}]. Since Y_{st} ∉ \overline{P} there is an embedding of B into C, and \overline{P} C is an H-prime ideal of C such that \overline{P} C ∩ B = \overline{P}.

Note that Y_{ij} = 0 if i < s or j < t. If i > s and j > t, then we have Y_{st}Y_{ij} − qY_{sj}Y_{it} = 0, so that Y_{ij} = qY_{st}^{−1}Y_{sj}Y_{it} in C. Hence, C is generated as an algebra by Y_{st}^{±1} together with Y_{sj} for j > t and Y_{it} for i > s. Thus, C is a homomorphic image of a localized multiparameter quantum affine space O_{λ}(k^{r})[z_{1}^{−1}], for r = m − s + n − t + 1 and for a suitable parameter matrix λ.

The standard action of the torus H_{r} := (k^{×})^{r} on O_{λ}(k^{r}) has 1-dimensional eigenspaces generated by individual monomials (here, we use the fact that k is infinite). Therefore, the same holds for C. Hence, any nonzero H_{r}-in-variant ideal of C contains a monomial, and so any nonzero H_{r}-prime ideal of C must contain one of Y_{s+1,t}, …, Y_{mt}, Y_{s,t+1}, …, Y_{sn}. Since \overline{P} C contains none of these elements, to show that \overline{P} C = 0 it suffices to establish that \overline{P} C is H_{r}-prime. But \overline{P} C is already H-prime, so it is enough to see that the H_{r}-invariant ideals of C are the same as the H-invariant ideals. This will follow from showing that the images of H and H_{r} in aut C coincide.

Since the Y_{ij} are H-eigenvectors, it is clear that the image of H is contained in the image of H_{r}. The reverse inclusion amounts to the following statement:

(*) Given any α_{s}, …, α_{m}, β_{t+1}, …, β_{n} ∈ k^{×}, there exists h ∈ H such that h(Y_{it}) = α_{i}Y_{it} for i = s, …, m and h(Y_{sj}) = β_{j}Y_{sj} for j = t + 1, …, n.

Now, there exists h_{1} ∈ H such that h_{1}(X_{ij}) = X_{ij} for all i, j with i < s, and h_{1}(X_{ij}) = α_{i}X_{ij} for all i, j with i ≥ s. Also, there exists h_{2} ∈ H such that h_{2}(X_{ij}) = X_{ij} for all i, j with j ≤ t and h_{2}(X_{ij}) = α_{s}^{−1}β_{j}X_{ij} for all i, j with j > t. Setting h = h_{1}h_{2} gives the desired element of H, establishing (*).

Therefore, \overline{P} C = 0, and so \overline{P} = 0. This means that P = P (I, J). □

3.5. Corollary. If the field k is infinite, then O_{q}(M_{m,n}(k))/I_{1} has precisely (2^{m} − 1)(2^{n} − 1) + 1 distinct H-prime ideals, all of which are completely prime. Further, each H-stratum of spec O_{q}(M_{m,n}(k))/I_{1} is homeomorphic to the prime spectrum of a Laurent polynomial ring over an algebraic field extension of k.

Proof. The first statement is clear from Proposition 3.4. The second statement is not actually a corollary of the Proposition, but is included to fill in the picture. It may be obtained from [1, Theorems 5.3, 5.5] (all but the algebraicity of the coefficient fields also follows from [4, Theorem 6.6]). □

3.6. In particular, the corollary above explains why in the algebra O_{q}(M_{2}(k)) there are precisely 10 = (2^{2} − 1)^{2} + 1 distinct H-primes which contain the quantum determinant. This fact was known previously by direct enumeration of these primes. The remaining H-primes correspond to H-primes of O_{q}(GL_{2}(k)); there are 4 of these, as has long been known. We can display the lattice of H-prime ideals of O_{q}(M_{2}(k)) as in the diagram below, where the symbols • and ∘ stand for generators X_{ij} which are or are not included in a given prime, and □ stands for the 2 × 2 quantum determinant. For example, \binom{∘•}{•∘} stands for the ideal ⟨X_{12}, X_{21}⟩, and (□) stands for the ideal ⟨X_{11}X_{22} − qX_{12}X_{21}⟩.

[figure: \binom{••}{••}; _{\binom{••}{∘•}}; _{\binom{••}{•∘}}; _{\binom{∘•}{••}}; _{\binom{•∘}{••}}; _{\binom{••}{∘∘}}; _{\binom{∘•}{∘•}}; _{\binom{∘•}{•∘}}; _{\binom{•∘}{•∘}}; _{\binom{∘∘}{••}}; _{\binom{∘•}{∘∘}}; (□); _{\binom{∘∘}{•∘}}; _{\binom{∘∘}{∘∘}}; H-spec O_{q}(M_{2}(k))]

The corresponding H-strata in spec O_{q}(M_{2}(k)) can be easily calculated. For instance, if q is not a root of unity, the strata corresponding to \binom{∘∘}{∘∘} and \binom{∘•}{•∘} are 2-dimensional, the strata corresponding to \binom{••}{∘•}, \binom{••}{•∘}, \binom{∘•}{••}, and \binom{•∘}{••} are all 1-dimensional, and the remaining 8 strata are singletons.

3.7. We close with some remarks concerning catenarity. (Recall that the prime spectrum of a ring A is catenary provided that for any comparable primes P ⊂ Q in spec A, all saturated chains of primes from P to Q have the same length.) It is conjectured that spec O_{q}(M_{m,n}(k)) is catenary. In [2, Theorem 1.6], we showed that catenarity holds for any affine, noetherian, Auslander-Gorenstein, Cohen-Macaulay algebra A with finite Gelfand-Kirillov dimension, provided spec A has normal separation. All hypotheses but the last are known to hold for the algebra A = O_{q}(M_{m,n}(k)). We can, at least, say that the portion of spec A above I_{1} – that is, spec A/I_{1} – is catenary: In view of Lemma 3.3, A/I_{1} is a homomorphic image of a multiparameter quantum affine space O_{λ}(k^{n^{2}}), and spec O_{λ}(k^{n^{2}}) is catenary by [2, Theorem 2.6].

References

1. K. R. Goodearl, Prime spectra of quantized coordinate rings, This volume.

2. K. R. Goodearl and T. H. Lenagan, Catenarity in quantum algebras, J. Pure and Applied Algebra 111 (1996), 123-142.

3. K. R. Goodearl and T. H. Lenagan, Quantum determinantal ideals, Preprint, 1997.

4. K. R. Goodearl and E. S. Letzter, The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras, Trans. Amer. Math. Soc. (to appear).

5. J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley-Inter-science, Chichester-New York, 1987.

6. B. Parshall and J.-p. Wang, Quantum linear groups, Memoirs Amer. Math. Soc. 89 (1991).

7. L. Rigal, Normalité de certains anneaux déterminantiels quantiques, Proc. Edinburgh Math. Soc. (to appear).

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