quo82.tex August 26, 1998

QUANTUM n-SPACE AS A QUOTIENT OF CLASSICAL n-SPACE

K. R. Goodearl and E. S. Letzter

Abstract. The prime and primitive spectra of O_{q}(k^{n}), the multiparameter quantized coordinate ring of affine n-space over an algebraically closed field k, are shown to be topological quotients of the corresponding classical spectra, spec O(k^{n}) and max O(k^{n}) ≈ k^{n}, provided the multiplicative group generated by the entries of q avoids −1.

Introduction

In the representation theory of a noncommutative ring A, the natural analog of the maximal spectrum of a commutative ring is prim A, the set of primitive ideals, equipped with the Zariski (Jacobson) topology. Thus, if A is a quantization of a classical coordinate ring O(V) over an algebraically closed field, one can view prim A as a quantization of the variety V. The question then naturally arises, how are prim A and V related? Some “piecewise” relations are known in many cases. For instance, if G is a connected, semisimple, complex algebraic group and H is a maximal torus of G, then various (generic) quantizations A of O(G) exhibit the following properties: H acts on A via automorphisms; there are only finitely many H-orbits in prim A, and they are locally closed; each H-orbit in prim A is homeomorphic to a torus; and each H-orbit in prim A is a set-theoretic quotient of a locally closed subset of G that is stable under translation by H (see [8, 9, 11, 12, 10]). Similar pictures have been observed to hold for quantized coordinate rings of affine spaces (see [4, 15, 3, 6]). One is therefore led to conjecture that in the above situations, prim A as a whole is a topological quotient of V. Similarly, spec A should be a topological quotient of spec O(V). In the “smallest” cases, such as A = O_{q}(SL_{2}(k)) and A = O_{q}(k^{2}), these conjectures can easily be verified by direct calculation. In this paper we establish these conjectures for multiparameter quantum affine spaces, when −1 cannot be written as a product of the defining parameters.

To provide some detail, first let k be an algebraically closed field. By a quantum kaffine n-space (i.e., a quantization of the coordinate ring of k^{n}), we mean an algebra with generators x_{1}, x_{2}, …, x_{n} subject only to the commutator relations x_{i}x_{j} = q_{ij}x_{j}x_{i}, for a chosen set of nonzero scalars q_{ij} ∈ k. (To avoid degeneracies, one assumes that q = (q_{ij}) is a multiplicatively antisymmetric n×n matrix.) We denote this algebra as O_{q}(k^{n}). Letting H: = (k^{×})^{n} be an algebraic n-torus, it is not hard to verify that the natural action of H on the vector space spanned by x_{1}, …, x_{n} extends to an action by k-algebra automorphisms on O_{q}(k^{n}). Consequently, there are induced H-actions on the prime and primitive spectra of O_{q}(k^{n}).

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16D30, 16D60, 16P40, 16S36, 17B37, 81R50.

The research of the first author was partially supported by NSF grant DMS-9622876, and the research of the second author was partially supported by NSF grant DMS-9623579.

Our methods now require us to make the following technical assumption: Either the subgroup of k^{×} generated by the q_{ij} does not contain −1, or the characteristic of k is 2. (While our present techniques do not appear to work in the absence of this hypothesis, we do not know if our main results will remain valid without it.) Now set A = O_{q}(k^{n}), and let R: = O(k^{n}) be a commutative polynomial ring in n variables, also equipped with the natural action of H by automorphisms. In the main result (4.11), we construct Hequivariant topological quotient maps

spec R → spec A and k^{n} ≈ max R → prim A.

Moreover, a description of the fibers of the second map is provided. Roughly speaking, we show that prim A is equal to k^{n} modulo the actions of a compatible system of groups acting separately on each H-orbit, and that spec A can be obtained from spec R in an analogous fashion.

Our analysis involves two separate steps:

(I) The spaces spec A and prim A are homeomorphic to certain sets of ideals of R, denoted G -spec R and G -max R, which are equipped with Zariski topologies.

(II) We obtain topological quotient maps spec R → G -spec R and max R → G -max R, by sending each prime (respectively, maximal) ideal P of R to the largest member of G -spec R contained within P.

Step II works very generally, and so we begin with that. Section 1 contains an axiomatic treatment that produces topological quotients of prime and primitive spectra in a noetherian ring R, using certain families of ideals of R. In Section 2, we restrict to the case of a commutative polynomial ring R over k and a family of ideals stable under a suitably compatible family of groups of automorphisms of R. This last setup yields the topological quotients of spec R and max R that will appear in our main theorem. Step I is developed in Sections 3 and 4. Section 3 is devoted to the case of quantum tori, which yields the necessary information about individual “strata” of prime and primitive ideals in A; in Section 4, we glue these results together over the whole prime and primitive spectra of A, and obtain the main theorem. Section 5 contains explicit calculations of the resulting topological quotient maps k^{n} → prim A for the case where A is a single parameter quantum affine space, and Section 6 applies the main theorem to twists of more general graded commutative algebras.

We remind the reader that a map ϕ : X → Y between topological spaces is a topological quotient map provided ϕ is surjective and the topology on Y coincides with the quotient topology induced by ϕ, that is, a subset W ⊆ Y is closed in Y precisely when ϕ^{−1}(W) is closed in X. When such a map exists, Y is completely determined (up to homeomorphism) by X and the fibers of ϕ.

1. Quotients of prime spectra

QUANTUM n-SPACE 3

We begin the work of this paper by specifying sufficient conditions under which a collection of ideals in a noetherian ring R, equipped with a Zariski topology, will form a topological quotient of the prime spectrum of R. These conditions are modelled on standard properties of the collection of those ideals of R which are prime relative to the ideals invariant under some set of operators, such as the collection of G-prime ideals where G is a group of automorphisms of R, or the collection of Δ-prime ideals where Δ is a set of derivations on R. Conditions for a subcollection to be a topological quotient of the primitive or the maximal spectrum of R are also given. At the end of the section, we specialize to the case of G-prime ideals, this being the case we shall need for application to quantum tori.

1.1. Let R be a noetherian ring, and let X -spec R be a nonempty set of ideals of R. The elements of X -spec R will be referred to as X-prime ideals, and intersections of collections of X-prime ideals will be termed X-semiprime ideals. We will further assume, throughout this section, that

(a) For each prime ideal P of R there exists a (unique) X-prime ideal, denoted (P : X), that is maximum among X-semiprime ideals contained in P. In other words, (P : X) is an X-prime ideal contained in P and containing all X-semiprime ideals which are contained in P.

(b) If P is a prime ideal of R minimal over an X-prime ideal Q, then Q = (P : X). In particular, for each X-prime ideal Q there exists a prime ideal P such that Q = (P : X).

1.2. If I is an ideal of R then V (I) will denote the set of prime ideals containing I. The set of prime ideals of R will be denoted spec R and will be equipped with the standard Zariski topology: The closed subsets are precisely those of the form V (I), for ideals I of R. The sets of (left) primitive and maximal ideals of R will be denoted prim R and max R respectively, and each of these sets will be given the relative topology from spec R.

1.3 Lemma. Let Q be an X-prime ideal in R, and let I and J be X-semiprime ideals of R such that IJ ⊆ Q. Then I or J is contained in Q.

Proof. Choose a prime ideal P such that Q = (P : X). Then IJ ⊆ P, and so one of I or J is contained in P, say I ⊆ P. But Q is the unique maximum X-semiprime ideal contained in P, and so I ⊆ Q. □

1.4. If I is an ideal of R, define V_{X}(I) to be the set of X-prime ideals of R containing I; note that V_{X}(I) = V_{X}(J), where J is the intersection of the X-prime ideals of R containing I (with the convention that the intersection of an empty collection of ideals is equal to R itself). By (1.3), if J_{1}, …, J_{t} are X-semiprime ideals of R, then V_{X}(J_{1}) ∪ ⋯ ∪ V_{X}(J_{t}) = V_{X}(J_{1} ∩ ⋯ ∩ J_{t}). Therefore, the standard Zariski topology generalizes to a topology on X -spec R: The closed sets are those subsets of the form V_{X}(I), for ideals I of R.

1.5. By our assumptions in (1.1), there is a surjection

π : spec R → X -spec R

sending each prime ideal P to (P : X). We first record two trivial observations concerning the behavior of π with respect to closed sets.

Lemma. Let J be an X-semiprime ideal of R.

(a) π(V (J)) = V_{X}(J).

(b) π^{−1}(V_{X}(J)) = V (J). □

1.6. A subset U of spec R will be termed π-stable provided U is a union of fibers of π, or equivalently, provided U = π^{−1}(π(U)).

Lemma. Let U be a closed π-stable subset of spec R. Then there exists an X-semiprime ideal J of R such that U = V (J).

Proof. Choose an ideal I of R such that U = V (I), and let P_{1}, …, P_{t} be the prime ideals minimal over I. Then P_{1}, …, P_{t} are the minimal elements of U. For 1 ≤ i ≤ t, set Q_{i} = (P_{i} : X), and set J = Q_{1} ∩ ⋯ ∩ Q_{t}. Observe that U ⊆ V (J). Next, let P be a prime ideal minimal over J. Note that P is minimal over Q_{i}, for some 1 ≤ i ≤ t, and so (P : X) = Q_{i} = (P_{i} : X). Therefore, since U is π-stable, P ∈ U. Consequently, V (J) ⊆ U, and hence U = V (J). □

1.7 Proposition. (a) π is continuous.

(b) π maps π-stable closed subsets of spec R to closed subsets of X -spec R.

(c) π is a topological quotient map.

Proof. (a) Let V be a closed subset of X -spec R. As seen in (1.4), there exists an Xsemiprime ideal J in R such that V = V_{X}(J). By (1.5b), π^{−1}(V) = V (J).

(b) Let U be a π-stable closed subset of spec R. It follows from (1.6) that there exists an X-semiprime ideal J in R such that U = V (J). By (1.5a), π(U) = V_{X}(J).

(c) Assume that V is a subset of X -spec R whose inverse image U = π^{−1}(V) is closed in spec R. It only remains to show that V is closed. However, U is clearly π-stable, and so V is closed by (b). □

1.8. Let us say that a subset U of prim R (or of max R) is relatively π-stable provided U = V ∩ prim R (or U = V ∩ max R) for some π-stable subset V of spec R.

Proposition. (a) Suppose that ⋂ U is X-semiprime for every relatively π-stable subset U of prim R. Then the restriction of π to the surjection

prim R ⟶ π(prim R)

is a topological quotient map (with respect to the relative topologies).

(b) Similarly, if the intersection of any relatively π-stable subset of max R is X-semi-prime, then π restricts to a topological quotient map

max R ⟶ π(max R).

Proof. (a) Let π_{prim} : prim R → π(prim R) denote the restriction of π. It is immediate from the continuity of π that π_{prim} is continuous. Now let W be a subset of π(prim R) whose inverse image U = π_{prim}^{−1}(W) is closed in prim R. Since U is relatively π-stable, its intersection is by hypothesis an X-semiprime ideal J. Moreover, U = V (J) ∩ prim R because U is closed in prim R. Then W = π(U) ⊆ π(V (J)) = V_{X}(J). If I ∈ V_{X}(J) ∩ π(prim R), there exists P ∈ prim R such that π(P) = I ⊇ J. Since P ⊇ π(P), we have P ∈ V (J) ∩ prim R = U, and so I ∈ W. Thus W = V_{X}(J) ∩ π(prim R), a relatively closed subset of π(prim R).

(b) Use the proof above, with “prim” replaced by “max” everywhere. □

The preceding observations apply easily to group actions, as follows.

1.9. For the remainder of this section, let G be a group acting by automorphisms on R. An ideal I of R, such that g(I) = I for all g ∈ G, is termed a G-ideal. Note that the set of G-ideals of R is closed under products, sums, and intersections. A (set-theoretically) proper G-ideal Q of R is G-prime provided IJ ⊈ Q for all G-ideals I and J not contained in Q. An intersection of G-prime ideals will be termed G-semiprime. For any ideal I of R, set (I : G) = ⋂_{g∈G} g(I). It is easy to see that (I : G) is the unique maximum G-ideal contained in I.

1.10 Lemma. (cf. [5, Remarks 4^{∗}, 5^{∗}]) (a) Let P be a prime ideal of R. Then (P : G) is a G-prime ideal.

(b) Let Q be a G-prime ideal of R, and let P be a prime ideal of R minimal over Q. Then Q = (P : G), and every prime ideal minimal over Q is in the G-orbit of P. In particular, Q is semiprime.

Proof. Part (a) follows from an argument mimicking the proof of (1.3).

(b) Let P_{1}, …, P_{t} be the prime ideals of R minimal over Q, and set N_{i} = (P_{i} : G) for each 1 ≤ i ≤ t. Note that N_{1}, …, N_{t} are G-ideals, all containing Q, and that some product of them is contained within Q. Therefore, Q = N_{i} for some i, because Q is G-prime. Now consider an arbitrary prime ideal P_{j} minimal over Q. Since P_{j} ⊇ Q = N_{i}, we see that P_{j} contains a product of prime ideals in the G-orbit of P_{i}, and so P_{j} ⊇ g(P_{i}) ⊇ Q for some g ∈ G. Therefore, P_{j} = g(P_{i}), and Q = (P_{i} : G) = (P_{j} : G). Part (b) follows. □

1.11. Let G -spec R denote the set of G-prime ideals of R. It follows from (1.9) and (1.10) that G -spec R satisfies the axioms for X -spec R specified in (1.1). In particular, if V_{G}(I) denotes the set of G-prime ideals of R that contain a given ideal I of R, then by (1.4) there is a Zariski topology on G -spec R: The closed sets are those of the form V_{G}(I), for ideals I of R. With respect to this topology, we can use (1.7c) to deduce that the assignment

P ↦ (P : G)

is a topological quotient map from spec R onto G -spec R.

Let G -max R denote the set of maximal proper G-ideals. Observe that G -max R is a subset of G -spec R and that G -max R is comprised of the maximal members of G -spec R. Equip G -max R with the relative topology. While each member of G -max R must be equal to (M : G) for some M ∈ max R, the converse fails in general. For example, let R be a polynomial ring k[x] over an infinite field k, and let G = k^{×} act on R by the rule g.f(x) = f(gx). Then (⟨x − α⟩ : G) = 0 for all nonzero α ∈ k, whereas G -max R consists of the single maximal ideal ⟨x⟩.

In the present setting, the hypothesis of (1.8b) is easily verified, as follows. Suppose U is a relatively π-stable subset of max R, and set J = ⋂ U. Since max R is stable under the action of G (within spec R), so is U, and hence J = ⋂_{M∈U}(M : G). Thus by (1.10a), J is G-semiprime, as desired. We therefore conclude from (1.8b) that the assignment

M ↦ (M : G)

is a topological quotient map from max R onto π(max R).

1.12. Suppose that G has no proper subgroups of finite index. Then all finite G-orbits of ideals of R are singletons, and so all G-prime ideals of R are prime, by (1.10b). We see, in this case, that G -spec R is a subset of spec R and that the topology on G -spec R described in (1.11) is the relative topology.

1.13. We conclude this section by considering a particular case of the group action setting that will be needed later.

Let k be an algebraically closed field, R = k[y_{1}^{±1}, …, y_{n}^{±1}] a Laurent polynomial ring over k, and H = (k^{×})^{n} the standard algebraic k-torus of rank n. Let H act on R by k-algebra automorphisms in the natural manner; namely,

(h_{1}, …, h_{n}).f(y_{1}, …, y_{n}) = f(h_{1}y_{1}, …, h_{n}y_{n}),

for (h_{1}, …, h_{n}) ∈ H and f(y_{1}, …, y_{n}) ∈ R. Since k is algebraically closed, the induced action of H on max R is transitive.

Proposition. Let G be a subgroup of H.

(a) The set {(M : G) | M ∈ max R} coincides with G -max R.

(b) If G is a closed subgroup of H, the fibers of the map M ↦ (M : G), from max R to G -max R, are precisely the G-orbits in max R.

Proof. (a) Given Q ∈ G -max R, choose a maximal ideal N ⊇ Q. Then (N : G) ⊇ Q, and so Q = (N : G) by the maximality of Q.

Conversely, consider Q = (M : G) where M ∈ max R. Choose Q′ ∈ G -max R containing Q. By the preceding paragraph, there exists a maximal ideal M′ in max R for which Q′ = (M′ : G). Since H acts transitively on max R, there exists an h ∈ H such that M = h(M′). Since H is abelian, Q = (M : G) = (h(M′) : G) = h(Q′). Hence Q′ ⊇ h(Q′), and so Q′ = h(Q′) = Q. Thus Q ∈ G -max R.

(b) Let M, M′ ∈ max R. If M and M′ are in the same G-orbit, then (M : G) = (M′ : G). Conversely, assume that (M : G) = (M′ : G). On intersecting with the ring of G-invariants R^{G}, we thus obtain

M ∩ R^{G} = (M : G) ∩ R^{G} = (M′ : G) ∩ R^{G} = M′ ∩ R^{G}.

Next, we may view R as the coordinate algebra of H and identify H with max R. By [14, Theorem 48, p. 220], for example, there is an identification of H/G with max R^{G} that produces the following commutative diagram:

H ⟶^{quotient} H/G
∥  ∥
max R ⟶^{restriction} max R^{G}

Letting H act on itself and on H/G by right translation (i.e., g ∈ H acts on h ∈ H by g.h = hg^{−1}), the diagram above is H-equivariant.

Since the cosets of G in H are the G-orbits with respect to right translation, it follows that the fibers of the restriction map max R → max R^{G} are the G-orbits in max R. Therefore M and M′ lie in the same G-orbit, as desired. □

2. Quotients of affine space

Throughout this section, k denotes a field, and R denotes a commutative polynomial ring k[y_{1}, …, y_{n}]. We study a “piecewise” action on R by a compatible system of groups, together with a collection of ideals which we show satisfies the axioms of the previous section. The resulting quotients of the prime and maximal spectra of R will play a crucial role in our main theorem. Namely, as we shall prove in Section 4, the quotients of spec R and max R with respect to a suitable system of groups turn out to be homeomorphic to the prime and primitive spectra, respectively, of a quantum affine n-space.

The setup we develop here amounts to patching together finitely many quotients by group actions, as follows. We first partition spec R into the 2^{n} locally closed sets determined by which subsets of {y_{1}, …, y_{n}} are contained in given prime ideals. These subsets are homeomorphic, via localization, to the prime spectra of the Laurent polynomial rings obtained by factoring some of the y_{i} out of R and inverting the remainder. Our “piecewise group action”, finally, amounts to compatible choices of groups acting as automorphisms on the above localizations.

2.1. Set H = (k^{×})^{n}, the algebraic k-torus of rank n, and equip H with the Zariski topology. As in (1.13), let H act on R by the automorphisms

(h_{1}, …, h_{n}).f(y_{1}, …, y_{n}) = f(h_{1}y_{1}, …, h_{n}y_{n}).

We will also consider the induced actions of H on spec R and max R. Next, let W denote the set of subsets of {1, …, n}. For each w ∈ W, let spec_{w} R, equipped with the relative topology, denote the set of prime ideals P of R such that

P ∩ {y_{1}, …, y_{n}} = {y_{i} | i ∈ w}.

Note that spec R = ⨆_{w∈W} spec_{w} R. Also note that spec_{w} R is stable under H, and that there is an obvious H-equivariant homeomorphism

spec_{w} R ⟶^{≅} spec k[y_{i}^{±1} | i ∉ w].

2.2. Let G = {G_{w} | w ∈ W } be a family of subgroups of H, indexed by W, subject to the following compatibility hypothesis: (P : G_{v}) ⊆ (P : G_{w}) for all v ⊆ w and every P ∈ spec_{w} R. We do not exclude the possibility that the G_{w}, for w ∈ W, are identically the same. Set

G -spec_{w} R = {(P : G_{w}) | P ∈ spec_{w} R} ,

for w ∈ W. Note, since the y_{i} are G_{w}-eigenvectors, that Q ∩ {y_{1}, …, y_{n}} = {y_{i} | i ∈ w} for all Q ∈ G -spec_{w} R. Consequently, the sets G -spec_{w} R for w ∈ W are pairwise disjoint, and we set

G -spec R = ⨆_{w∈W} G -spec_{w} R.

The members of G -spec R will be termed G-prime ideals, the intersections of collections of G-prime ideals will be called G-semiprime ideals, and the set of G-prime ideals containing an ideal I of R will be denoted V_{G}(I). Note from (1.10b) that all G-semiprime ideals are semiprime.

2.3 Lemma. G -spec R satisfies the conditions in (1.1), with (P : G) = (P : G_{w}) for P ∈ spec_{w} R.

Proof. To verify (1.1a), let P ∈ spec_{w} R, let Q = (P : G_{w}), and let Q′ be an arbitrary G-semiprime ideal of R contained in P. We will show that Q′ ⊆ Q. To start, write

Q′ = ⋂_{v∈W} N_{v},

where each N_{v} is an intersection of ideals from G -spec_{v} R, and where each N_{v} contains Q′. Since Q′ ⊆ P, some N_{v} ⊆ P. Then since y_{i} ∈ N_{v} for all i ∈ v, we must have v ⊆ w. Note that N_{v} is G_{v}-stable and so is contained in (P : G_{v}). By the compatibility hypothesis, N_{v} ⊆ Q, whence Q′ ⊆ Q, and (1.1a) is established.

To check (1.1b), let Q ∈ G -spec_{w} R and let P be a prime ideal of R minimal over Q. Observe first that P ∈ spec_{v} R for some v ⊇ w. Next, it follows from (1.10a) that Q is G_{w}-prime, and then from (1.10b) that Q = (P : G_{w}). Moreover, since each of the variables y_{i} is a G_{w}-eigenvector, it now follows that v = w, whence Q = (P : G) and (1.1b) is satisfied. □

2.4. Following (1.4), we equip G -spec R with the Zariski topology where the closed subsets are the V_{G}(I), for ideals I of R. In case none of the G_{w} has proper subgroups of finite index, it follows from (1.12) that G -spec R ⊆ spec R, and then the topology on G -spec R coincides with the relative topology.

Note, for each w ∈ W, that spec_{w} R is a union of H-orbits. Therefore, since H is abelian, we see that h(P : G) = (h(P) : G) for all h ∈ H and P ∈ spec R. From (1.7) we now obtain:

Proposition. The assignment P ↦ (P : G) produces an H-equivariant topological quotient map from spec R onto G -spec R. □

2.5. For w ∈ W, set max_{w} R = max R ∩ spec_{w} R. Note that max R = ⨆_{w∈W} max_{w} R. Moreover, max_{w} R is equal to the set of maximal members of spec_{w} R. Next, set

G -max_{w} R = {(M : G_{w}) | M ∈ max_{w} R}  (for w ∈ W);
G -max R = ⨆_{w∈W} G -max_{w} R = {(M : G) | M ∈ max R}.

Note that the definition of G -max_{w} R above and the definition of G_{w} -max R in (1.11) follow different patterns. However, we shall see in (2.6) that when k is algebraically closed, G -max_{w} R coincides with (G_{w} -max R) ∩ (G -spec_{w} R).

Proposition. The assignment M ↦ (M : G) produces an H-equivariant topological quotient map from max R onto G -max R.

Proof. By (1.8b), it suffices to show that if U is any relatively π-stable subset of max R and J = ⋂ U, then J is G-semiprime.

To start, set U_{w} = U ∩ max_{w} R for w ∈ W, and note that U_{w} is stable under G_{w} since (g(M) : G) = (M : G) for all M ∈ max_{w} R and g ∈ G_{w}. Hence,

⋂ U_{w} = ⋂_{M∈U_{w}}(M : G_{w}) = ⋂_{M∈U_{w}}(M : G),

which is G-semiprime. Therefore J = ⋂_{w∈W} (⋂ U_{w}) is a G-semiprime ideal, as required. □

2.6 Proposition. Assume that k is algebraically closed. For w ∈ W, the set G -max_{w} R coincides with the set of maximal members of G -spec_{w} R.

Proof. Set m = n−|w| and R_{w} = (R/⟨y_{i} | i ∈ w⟩)[y_{j}^{−1} | j ∉ w]. Note that R_{w} is a Laurent polynomial ring in the (images of the) m indeterminates y_{j}, for j ∉ w. We thereby obtain an action of the m-torus (k^{×})^{m} on R_{w}, following (1.13). Note that the actions on R_{w} by G_{w} and H, induced from their actions on R, factor through the (k^{×})^{m}-action. Moreover, localization provides H-equivariant bijections

max_{w} R → max R_{w}  and  G -spec_{w} R → G_{w} -spec R_{w}.

The proposition therefore follows from (1.13a). □

2.7. We close this section by considering the fibers of the map in (2.5) more closely. First note that each max_{w} R is a disjoint union of fibers. Next, on max_{w} R the map is given by the rule M ↦ (M : G_{w}), and so the fibers within max_{w} R are unions of G_{w}-orbits. Our interest is in conditions under which the fibers within max_{w} R coincide with the G_{w}-orbits.

The maximal ideals in max_{w} R have the form

⟨y_{i} | i ∈ w⟩ + ⟨f_{1}, …, f_{t}⟩

where f_{1}, …, f_{t} ∈ k[y_{j} | j ∉ w]. These ideals are all fixed by the group

H_{w} = {(h_{1}, …, h_{n}) ∈ H | h_{j} = 1 for all j ∉ w};

thus (M : G_{w}) = (M : G_{w}H_{w}) for all M ∈ max_{w} R. Hence, there is no loss of generality in assuming that G_{w} ⊇ H_{w}.

Proposition. Fix w ∈ W, and assume that k is algebraically closed. Assume that G_{w} ⊇ H_{w} and that G_{w}/H_{w} is a closed subgroup of H/H_{w}. Then the fibers of the map M ↦ (M : G), from max_{w} R to G -max_{w} R, are precisely the G_{w}-orbits in max_{w} R.

Proof. We shift everything to the localization R_{w}, as in the proof of (2.6), and we recall the (k^{×})^{m}-action on R_{w} described therein, where m = n − |w|. Now observe that the induced action of H/H_{w} on R_{w} is identical to the (k^{×})^{m}-action on R_{w}. Therefore, the proposition follows from (1.13b). □

3. Quantum tori

As in [6], much of our analysis of quantum affine spaces can be reduced, via localization, to quantum tori. This portion of our analysis is carried out in the present section. It leads, in particular, to a version of our main theorem for an arbitrary multiparameter quantum torus O_{q}((k^{×})^{n}) over an algebraically closed field k, which establishes that spec O_{q}((k^{×})^{n}) and prim O_{q}((k^{×})^{n}) can be presented as topological quotients of spec O((k^{×})^{n}) and max O((k^{×})^{n}), respectively. No restriction on the parameter matrix q is needed here; it is only in patching quantum tori together to cover a quantum affine space, as in the following section, that we will need to avoid −1.

Let k be a field. From (3.7) onward, we will assume that k is algebraically closed.

3.1. Let q = (q_{ij}) be a multiplicatively antisymmetric n×n matrix over k; that is, q_{ii} = 1 and q_{ji} = q_{ij}^{−1}, for all i, j. Let A = O_{q}((k^{×})^{n}) be the corresponding multiparameter quantized coordinate ring of the torus (k^{×})^{n}, that is, the k-algebra generated by elements x_{1}^{±1}, …, x_{n}^{±1} subject only to the relations x_{i}x_{j} = q_{ij}x_{j}x_{i} for all i, j. This algebra is also known as a McConnell-Pettit algebra, after [13]; in the notation of that paper, A = P (q).

Let us first express A in terms of ordered monomials, using standard multi-index notation. Thus A has a k-basis of monomials, x^{α} = x_{1}^{α_{1}}x_{2}^{α_{2}} ⋯ x_{n}^{α_{n}}, for n-tuples α = (α_{1}, …, α_{n}) from the group Γ := ℤ^{n}. Define σ : Γ × Γ → k^{×} by

σ(α, β) = ∏_{i,j=1}^{n} q_{ij}^{α_{i}β_{j}}.

Then σ determines the commutation rules in A; namely, x^{α}x^{β} = σ(α, β)x^{β}x^{α} for α, β ∈ Γ. Moreover, σ is an alternating bicharacter on Γ, because

σ(α, α) = 1,  σ(β, α) = σ(α, β)^{−1},  and  σ(α, β + β′) = σ(α, β)σ(α, β′),

for α, β, β′ ∈ Γ.

3.2. It is convenient to view A as a twisted group algebra of Γ. One way to do this is to use the function d : Γ × Γ → k^{×} such that x^{α}x^{β} = d(α, β)x^{α+β} for all α, β ∈ Γ. It follows from the associativity of multiplication in A that d must be a 2-cocycle on Γ. Thus by writing A in terms of the k-basis {x^{α} | α ∈ Γ}, we have expressed A as a twisted group algebra k^{d}Γ.

Note that σ(α, β) = d(α, β)d(β, α)^{−1} for α, β ∈ Γ. Conversely, if c is any 2-cocycle on Γ such that c(α, β)c(β, α)^{−1} = σ(α, β) for α, β ∈ Γ, then k^{c}Γ ≅ A. Namely, k^{c}Γ has a basis {x_{α} | α ∈ Γ} such that x_{α}x_{β} = c(α, β)x_{α+β} for α, β ∈ Γ. If ϵ_{1}, …, ϵ_{n} denotes the standard basis for Γ, then k^{c}Γ is generated by the elements x_{ϵ_{1}}^{±1}, …, x_{ϵ_{n}}^{±1}, which satisfy the same commutation rules as x_{1}, …, x_{n}, namely

x_{ϵ_{i}}x_{ϵ_{j}} = c(ϵ_{i}, ϵ_{j})x_{ϵ_{i}+ϵ_{j}} = c(ϵ_{i}, ϵ_{j})c(ϵ_{j}, ϵ_{i})^{−1}x_{ϵ_{j}}x_{ϵ_{i}} = σ(ϵ_{i}, ϵ_{j})x_{ϵ_{j}}x_{ϵ_{i}} = q_{ij}x_{ϵ_{j}}x_{ϵ_{i}}

for all i, j. Therefore there exists a k-algebra isomorphism k^{c}Γ → A such that x_{ϵ_{i}} ↦ x_{i} for all i. Note that this isomorphism sends each x_{α} to a scalar multiple of x^{α}.

While the cocycle d is useful for some purposes, it has some drawbacks; for example, it is neither symmetric nor antisymmetric in general. We therefore leave the choice of a particular cocycle until later (see (3.5) and (3.12)).

3.3. Let c be any 2-cocycle on Γ such that c(α, β)c(β, α)^{−1} = σ(α, β) for α, β ∈ Γ, and identify A with k^{c}Γ. This allows us to write A in terms of a k-basis {x_{α} | α ∈ Γ}, with multiplication given by

x_{α}x_{β} = c(α, β)x_{α+β}

for α, β ∈ Γ. We shall also need the ordinary group algebra kΓ. Let us express this algebra in terms of a k-basis {y_{α} | α ∈ Γ}, with y_{α}y_{β} = y_{α+β} for α, β ∈ Γ.

3.4. Set H = Hom(Γ, k^{×}), an algebraic torus of rank n (which will henceforth be equipped with the Zariski topology). Write the application of H to Γ in terms of a pairing ⟨−, −⟩ : H × Γ → k^{×}. Define actions of H on both A and kΓ via k-algebra automorphisms such that

h.x_{α} = ⟨h, α⟩x_{α} and h.y_{α} = ⟨h, α⟩y_{α},

for h ∈ H and α ∈ Γ.

3.5. Let Φ_{c} : A → kΓ be the k-linear isomorphism such that Φ_{c}(x_{α}) = y_{α} for α ∈ Γ. Observe that Φ_{c} is H-equivariant. The extent to which Φ_{c} fails to preserve products can be expressed in terms of c: since x_{α}x_{β} = c(α, β)x_{α+β}, we have

Φ_{c}(x_{α}x_{β}) = c(α, β)y_{α+β} = c(α, β)Φ_{c}(x_{α})Φ_{c}(x_{β})

for α, β ∈ Γ.

As in [6, 1.2], set S = rad(σ) = {α ∈ Γ | σ(α, −) ≡ 1}, a subgroup of Γ; then

Z(A) = k[x_{α} | α ∈ S].

If we identify the group algebra kS with the k-linear span of {y_{α} | α ∈ S} inside kΓ, then Φ_{c}(Z(A)) = kS. In fact:

Lemma. Assume that c ≡ 1 on S × Γ ⊆ Γ × Γ. Then the map Φ_{c} restricts to a kalgebra isomorphism of Z(A) onto kS. With respect to this isomorphism, Φ_{c} itself gives a semilinear isomorphism of A as Z(A)-module onto kΓ as kS-module: Φ_{c}(za) = Φ_{c}(z)Φ_{c}(a) for all z ∈ Z(A) and a ∈ A.

Proof. By assumption, Φ_{c}(x_{α}x_{β}) = Φ_{c}(x_{α})Φ_{c}(x_{β}) for α ∈ S and β ∈ Γ. Consequently, Φ_{c}(za) = Φ_{c}(z)Φ_{c}(a) for all z ∈ Z(A) and a ∈ A, and the lemma follows. □

We shall need the following fact to show that S equals the intersection of the kernels of some homomorphisms from H (see (3.7)).

3.6 Lemma. The order of the torsion subgroup of Γ/S is not divisible by char k.

Proof. The torsion subgroup of Γ/S is a finite abelian group, hence a finite direct sum of finite cyclic groups, and so the order of this subgroup equals the product of the orders of certain of its elements. Thus it suffices to show that the order of each torsion element of Γ/S is not divisible by char k.

Let α + S be a torsion element of Γ/S, and choose a basis ϵ_{1}, …, ϵ_{n} for Γ. For each positive integer m, we have

mα ∈ S ⟺ σ(mα, γ_{i}) = 1 for all i ⟺ σ(α, ϵ_{i})^{m} = 1 for all i.

Hence, the order of α + S equals the order of the element (σ(α, ϵ_{1}), …, σ(α, ϵ_{n})) in (k^{×})^{n}, and the latter order is clearly not divisible by char k. □

3.7. From now until the end of the section, assume that k is algebraically closed.

We use ^{⊥} to denote orthogonals of subsets of H and Γ, relative to the pairing ⟨−, −⟩. In particular,

S^{⊥} = {h ∈ H | ⟨h, α⟩ = 1 for α ∈ S} = {h ∈ H | ker h ⊇ S},

a closed subgroup of H, and

S^{⊥⊥} = {α ∈ Γ | ⟨h, α⟩ = 1 for h ∈ S^{⊥}} = ⋂_{h∈S^{⊥}} ker h,

a subgroup of Γ.

Lemma. S^{⊥⊥} = S.

Proof. Consider α ∈ Γ\S. The order of the coset α+S in Γ/S is either infinite, or – if finite – not divisible by char k, in view of (3.6). In either case, the subgroup ℤ(α + S) ⊆ Γ/S supports a nontrivial homomorphism to k^{×}. Since k^{×} is divisible, any such homomorphism extends to a homomorphism Γ/S → k^{×}. Thus, there exists a homomorphism h : Γ → k^{×} (i.e., h ∈ H) such that S ⊆ ker h and h(α) ≠ 1, which verifies that α ∉ S^{⊥⊥}. □

3.8 Proposition. (a) A^{S^{⊥}} = Z(A) and (kΓ)^{S^{⊥}} = kS.

(b) Every S^{⊥}-stable ideal of kΓ is generated by its contraction to kS.

(c) Every ideal of A is generated by its contraction to Z(A).

(d) Assume that c ≡ 1 on S × Γ. The map I ↦ Φ_{c}(I), for ideals I of A, provides a product preserving, H-equivariant lattice isomorphism from the lattice of ideals of A onto the lattice of S^{⊥}-ideals of kΓ.

Proof. (a) It follows from (3.7) that A^{S^{⊥}} is spanned by the x_{α} for α ∈ S, and that (kΓ)^{S^{⊥}} is spanned by the y_{α} for α ∈ S. The second part of (a) is thus clear, and the first follows from (3.5).

(b) Consider an S^{⊥}-stable ideal I in kΓ. Since S^{⊥} acts semisimply on kΓ, this ideal must be spanned by S^{⊥}-eigenvectors. Let a ∈ I be an S^{⊥}-eigenvector, and choose an element γ from the support of a. Then x_{γ}^{−1}a is also an S^{⊥}-eigenvector and contains 1 in its support. Hence S^{⊥} fixes x_{γ}^{−1}a. Thus x_{γ}^{−1}a ∈ (kΓ)^{S^{⊥}} = kS, and so a ∈ (kΓ)(I ∩ kS). Therefore I = (kΓ)(I ∩ kS).

(c) See, for example, [6, 1.4].

(d) To start, note that U ↦ Φ_{c}(U) produces an H-equivariant, bijective lattice isomorphism from the lattice of k-subspaces U of A onto the lattice of k-subspaces of kΓ.

Next, let I be an ideal of A. It follows from (c) that I = I′A for some ideal I′ of Z(A). By (3.5), Φ_{c}(I′) is an ideal of kS, and Φ_{c}(I) = Φ_{c}(I′)Φ_{c}(A) = Φ_{c}(I′)kΓ is an ideal of kΓ. Furthermore, since Φ_{c}(I′) is an ideal of kS, and since kS is fixed pointwise by S^{⊥}, we see that Φ_{c}(I′)kΓ is stable under S^{⊥}. Similarly, it follows from (b) and (3.5) that Φ_{c}^{−1} maps each S^{⊥}-ideal of kΓ to an ideal of A. Therefore the set maps Φ_{c} and Φ_{c}^{−1} provide mutually inverse bijections between the lattices of ideals of A and S^{⊥}-ideals of kΓ. It follows easily from the preceding paragraph that these bijections are H-equivariant lattice isomorphisms.

Now let I and J be ideals of A. It only remains to show that Φ_{c}(IJ) = Φ_{c}(I)Φ_{c}(J). As above, write I = I′A and J = J′A, for ideals I′ and J′ of Z(A). Then IJ = I′J′A, and so by (3.5), Φ_{c}(IJ) = Φ_{c}(I′)Φ_{c}(J′)kΓ = Φ_{c}(I)Φ_{c}(J). □

3.9. Let S^{⊥}- spec kΓ and S^{⊥}- max kΓ denote the sets of S^{⊥}-prime and maximal proper S^{⊥}-ideals of kΓ, respectively. As noted in (1.11), S^{⊥}- max kΓ ⊆ S^{⊥}- spec kΓ. Equip these sets with their respective Zariski topologies, again following (1.11).

Proposition. Assume that c ≡ 1 on S × Γ. The set maps Φ_{c} and Φ_{c}^{−1} provide pairs of mutually inverse, H-equivariant, homeomorphisms

spec A ⟷ S^{⊥}- spec kΓ  and  max A ⟷ S^{⊥}- max kΓ.

Proof. This follows from (3.8d). □

3.10 Proposition. The assignment P ↦ (P : S^{⊥}), for prime ideals P of kΓ, produces H-equivariant topological quotient maps

spec kΓ → S^{⊥}- spec kΓ  and  max kΓ → S^{⊥}- max kΓ.

Proof. By (1.13a), S^{⊥}- max kΓ = {(M : S^{⊥}) | M ∈ max kΓ}, and so the given assignment does map max kΓ onto S^{⊥}- max kΓ. The given maps are H-equivariant because H is abelian. The remainder of the proposition follows from (1.11). □

The two propositions above provide the information necessary to exhibit spec A and prim A as topological quotients of their classical analogs, as in the following theorem. A cocycle c with the required condition is constructed in (3.12).

3.11 Theorem. Assume that c ≡ 1 on S ×Γ. Then the rule P ↦ Φ_{c}^{−1}(P : S^{⊥}), for prime ideals P of kΓ, defines H-equivariant topological quotient maps

spec kΓ → spec A  and  max kΓ → max A.

The fibers of the second map are exactly the S^{⊥}-orbits in max kΓ.

Proof. The first part follows from (3.9) and (3.10). The final part is a consequence of (1.13b). □

3.12. In order to apply (3.11), we must be able to choose the cocycle c to be trivial on S × Γ. This can be done as follows:

Lemma. There exists a 2-cocycle c on Γ such that c ≡ 1 on S × Γ and c(α, β)c(β, α)^{−1} = σ(α, β) for α, β ∈ Γ.

Proof. As in [6, 1.6], Γ has a basis γ_{1}, …, γ_{n} such that S is generated by m_{1}γ_{1}, …, m_{t}γ_{t}, for some t ≤ n and some positive integers m_{i}. Set p_{ij} = σ(γ_{i}, γ_{j}) for all i, j, and note that the matrix (p_{ij}) is multiplicatively antisymmetric. Given α, β ∈ Γ, write α = ∑_{i} a_{i}γ_{i} and β = ∑_{j} b_{j}γ_{j} for some a_{i}, b_{j} ∈ ℤ; then

σ(α, β) = ∏_{i,j} p_{ij}^{a_{i}b_{j}}.

Now define

c(α, β) = ∏_{i<j} p_{ij}^{a_{i}b_{j}}

for α = ∑_{i} a_{i}γ_{i} and β = ∑_{j} b_{j}γ_{j} in Γ. Then c is a bicharacter on Γ, and hence a 2-cocycle. It is easily checked that c(α, β)c(β, α)^{−1} = σ(α, β) for α, β ∈ Γ. Since m_{i}γ_{i} ∈ S, we obtain

1 = σ(m_{i}γ_{i}, γ_{j}) = p_{ij}^{m_{i}}

for all i, j. Hence, if α = ∑_{i} a_{i}γ_{i} ∈ S, then p_{ij}^{a_{i}} = 1 for all i, j, and therefore c(α, β) = 1 for all β ∈ Γ. □

3.13 Remark. It would result in a stronger theorem if we could conclude in (3.11) that the given maps were closed. However, this conclusion is not possible in general. For instance, take n = 3, choose p ∈ k^{×} not a root of unity, and take

q = ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & p^{2} \\ 1 & p^{−2} & 1 \end{matrix} ) .

Then σ is given by the rule σ(s, t) = p^{2(s_{2}t_{3}−s_{3}t_{2})}, and we may choose c to be given by

c(s, t) = p^{s_{2}t_{3}−s_{3}t_{2}}.

We compute that S = ℤ(1, 0, 0) and S^{⊥} = {(h_{1}, h_{2}, h_{3}) ∈ H | h_{1} = 1}.

Now write kΓ = k[y_{1}^{±1}, y_{2}^{±1}, y_{3}^{±1}], where y_{1}, y_{2}, y_{3} correspond to the standard basis for Γ, and let

V = {⟨y_{1} − α_{1}, y_{2} − α_{2}, y_{3} − α_{3}⟩ | α_{i} ∈ k^{×} and (α_{1} − 1)α_{2} = 1},

a closed subset of max kΓ. The map M ↦ (M : S^{⊥}) sends V to the set

{⟨y_{1} − α_{1}⟩ | α_{1} ∈ k^{×} and α_{1} ≠ 1},

a non-closed subset of S^{⊥}- max kΓ, and thus M ↦ Φ_{c}^{−1}(M : S^{⊥}) sends V to a non-closed subset of max A.

4. Quantum affine spaces

We now present and prove the main results of this paper. In essence, we partition the prime and primitive spectra of a quantum affine space into corresponding spectra of quantum tori, apply the results of the previous section to these spectra, and then patch everything together. It is the patching process that requires a more careful choice of cocycle than previously.

We continue to assume that the base field k is algebraically closed.

4.1. Again let q = (q_{ij}) be a multiplicatively antisymmetric n × n matrix over k. Now, however, let A = O_{q}(k^{n}) be the corresponding multiparameter quantized coordinate ring of affine n-space. In other words, A is the k-algebra generated by elements x_{1}, …, x_{n}, subject only to the relations x_{i}x_{j} = q_{ij}x_{j}x_{i}. We will treat A as a twisted semigroup algebra, as follows.

Set Γ = ℤ^{n} and Γ^{+} = (ℤ^{+})^{n}, and define an alternating bicharacter σ on Γ by the rule

σ(α, β) = ∏_{i,j=1}^{n} q_{ij}^{α_{i}β_{j}}

as in (3.1). If c is any 2-cocycle on Γ such that c(α, β)c(β, α)^{−1} = σ(α, β) for α, β ∈ Γ, we can write A as the twisted semigroup algebra k^{c}Γ^{+}, viewed in the obvious way as a subalgebra of the twisted group algebra k^{c}Γ.

We will need to apply the results of Section 3 to the localizations A_{w} = (A/⟨x_{i} | i ∈ w⟩)[x_{j}^{−1} | j ∉ w]

as w ranges over the subsets of {1, …, n}. Thus, we require a cocycle c whose restrictions to the subgroups of Γ generated by subsets of the standard basis all satisfy the hypothesis of (3.11). The cocycle obtained in (3.12) is not sufficient for this purpose. With a minor technical assumption, we can construct a suitable cocycle as follows.

4.2. Let ⟨q_{ij}⟩ denote the subgroup of k^{×} generated by the q_{ij}. Assume from now on that either −1 ∉ ⟨q_{ij}⟩ or char k = 2.

Lemma. There exists an alternating bicharacter c on Γ such that c^{2} = σ and such that c(α, β) = 1 whenever σ(α, β) = 1.

Proof. Since k is algebraically closed, the abelian group k^{×} is divisible, and so it contains a divisible hull D for the subgroup ⟨q_{ij}⟩. In view of our hypotheses, ⟨q_{ij}⟩ contains no elements of order 2, whence the same holds for D. Now each q_{ij} has a unique square root in D, say p_{ij}, and it follows from the uniqueness of the choices that (p_{ij}) is a multiplicatively antisymmetric matrix over k. Thus, the rule

c(α, β) = ∏_{i,j=1}^{n} p_{ij}^{α_{i}β_{j}}

defines an alternating bicharacter c on Γ such that c^{2} = σ. For any α, β ∈ Γ such that σ(α, β) = 1, we have c(α, β)^{2} = 1 and therefore c(α, β) = 1, because D has no elements of order 2. □

4.3. Fix an alternating bicharacter c as in (4.2). Then c is a 2-cocycle on Γ and

c(α, β)c(β, α)^{−1} = c(α, β)^{2} = σ(α, β)

for α, β ∈ Γ. As in (3.3), we write k^{c}Γ in terms of a basis {x_{α} | α ∈ Γ} such that x_{α}x_{β} = c(α, β)x_{α+β} for α, β ∈ Γ. Thus A = k^{c}Γ^{+} equals the subspace of k^{c}Γ spanned by the x_{α} for α ∈ Γ^{+}.

Similarly, we write the group algebra kΓ in terms of a basis {y_{α} | α ∈ Γ}, with y_{α}y_{β} = y_{α+β} for α, β ∈ Γ, and we identify the semigroup algebra R := kΓ^{+} with the subspace of kΓ spanned by the y_{α} for α ∈ Γ^{+}. We also view R as a polynomial ring k[y_{1}, …, y_{n}], as in Section 2, where y_{i} = y_{ϵ_{i}} and ϵ_{i} is the i-th standard basis element of Γ.

4.4. Following (3.4), set H = Hom(Γ, k^{×}) and write the application of H to Γ in terms of a pairing ⟨−, −⟩ : H × Γ → k^{×}. We again have actions of H on k^{c}Γ and kΓ via k-algebra automorphisms such that h.x_{α} = ⟨h, α⟩x_{α} and h.y_{α} = ⟨h, α⟩y_{α}, for h ∈ H and α ∈ Γ. The subalgebras A and R are stable under these actions.

4.5. Let W denote the set of subsets of {1, …, n}. For w ∈ W, let spec_{w} A be the set, equipped with the relative Zariski topology, of those prime ideals P in A such that

P ∩ {x_{1}, …, x_{n}} = {x_{i} | i ∈ w}.

Then spec A is the disjoint union of the sets spec_{w} A. Likewise, prim A is the disjoint union of the sets prim_{w} A = prim A ∩ spec_{w} A, each of which is also endowed with the relative topology. Note that the sets spec_{w} A and prim_{w} A are invariant under the action of H. Also, it was proved in [6, 2.3], for w ∈ W, that the set prim_{w} A coincides with the set of maximal members of spec_{w} A. Define spec_{w} R and max_{w} R similarly, as in (2.1) and (2.5). Recall that max_{w} R equals the set of maximal elements of spec_{w} R.

It is convenient to label the localizations

A_{w} = (A/⟨x_{i} | i ∈ w⟩)[x_{j}^{−1} | j ∉ w]  and  R_{w} = (R/⟨y_{i} | i ∈ w⟩)[y_{j}^{−1} | j ∉ w],

and to identify A_{w} (respectively, R_{w}) with the k-subalgebra of k^{c}Γ (respectively, kΓ) spanned by the x_{α} (respectively, y_{α}) for those α ∈ Γ such that α_{i} = 0 for i ∈ w. In view of the preceding paragraph and [7, 9.20], for example, we see that localization induces H-equivariant homeomorphisms

spec_{w} A ⟶ spec A_{w}  spec_{w} R ⟶ spec R_{w}
prim_{w} A ⟶ max A_{w}  max_{w} R ⟶ max R_{w}.

4.6. As in (3.5), we have an H-equivariant k-linear isomorphism Φ = Φ_{c} : k^{c}Γ → kΓ such that Φ(x_{α}) = y_{α} for α ∈ Γ. This map restricts to an H-equivariant k-linear isomorphism from A onto R. Note that Φ also restricts to H-equivariant k-linear isomorphisms Φ_{w} : A_{w} → R_{w} for w ∈ W.

4.7. Retain ϵ_{1}, …, ϵ_{n} as the standard basis for Γ, and fix w ∈ W. Let Γ_{w} be the subgroup of Γ generated by {ϵ_{i} | i ∉ w}, and let σ_{w} and c_{w} denote the restrictions of σ and c to Γ_{w}. With this notation, the identifications made in (4.5) are A_{w} = k^{c_{w}}Γ_{w} and R_{w} = kΓ_{w}.

4.8. Set S_{w} = rad(σ_{w}) = {α ∈ Γ_{w} | σ(α, −) ≡ 1 on Γ_{w}}, for w ∈ W. As in (3.7), set

S_{w}^{⊥} = {h ∈ H | ⟨h, α⟩ = 1 for α ∈ S_{w}} and Γ_{w}^{⊥} = {h ∈ H | ⟨h, α⟩ = 1 for α ∈ Γ_{w}} .

Because σ ≡ 1 on S_{w} × Γ_{w}, it follows from (4.2) that c ≡ 1 on S_{w} × Γ_{w}.

Lemma. Let v ⊆ w in W.

(a) S_{w}^{⊥} ⊆ S_{v}^{⊥}Γ_{w}^{⊥}.

(b) (P : S_{v}^{⊥}) ⊆ (P : S_{w}^{⊥}) for all P ∈ spec_{w} R.

Proof. (a) Observe that Γ_{w} ⊆ Γ_{v} and that S_{v} ∩Γ_{w} ⊆ S_{w}. Any h ∈ S_{w}^{⊥} is a homomorphism Γ → k^{×} such that ker h ⊇ S_{w} ⊇ S_{v} ∩ Γ_{w}. Hence, h|_{Γ_{w}} extends to a homomorphism h_{1} : Γ_{w} +S_{v} → k^{×} such that S_{v} ⊆ ker h_{1}. Since k^{×} is divisible, h_{1} extends to a homomorphism h_{2} : Γ → k^{×}. Now h_{2} is a homomorphism in H such that h_{2} = h on Γ_{w} and h_{2} = 1 on S_{v}. Thus h_{2} ∈ S_{v}^{⊥} and h_{2}^{−1}h ∈ Γ_{w}^{⊥}, whence h ∈ S_{v}^{⊥}Γ_{w}^{⊥} as desired.

(b) Observe that Γ_{w}^{⊥} acts trivially on R/⟨y_{i} | i ∈ w⟩. Hence, P is stable under Γ_{w}^{⊥}, and so (P : S_{v}^{⊥}) = (P : S_{v}^{⊥}Γ_{w}^{⊥}). In view of part (a), (P : S_{v}^{⊥}Γ_{w}^{⊥}) ⊆ (P : S_{w}^{⊥}), and we are done. □

4.9. Set G = {S_{w}^{⊥} | w ∈ W }, and note by (4.8) that G satisfies the hypotheses in (2.2), as applied to R. Define G -spec R as in (2.2), and define (P : G) to be (P : S_{w}^{⊥}) for P ∈ spec_{w} R, as in (2.3). Apply the Zariski topology to G -spec R, following (2.4). We may also use (2.4) to conclude that the assignment P ↦ (P : G) is a topological quotient map from spec R onto G -spec R.

Next, define G -max R as in (2.5). It follows from (2.6), for each w ∈ W, that G -max_{w} R is the set of maximal members of G -spec_{w} R. From (2.5) it follows that the assignment M ↦ (M : G), for M ∈ max R, produces an H-equivariant topological quotient map from max R onto G -max R. Moreover, by (2.7), the fibers in max R over points in G -max_{w} R are precisely the S_{w}^{⊥}-orbits in max_{w} R.

It remains to relate spec A to G -spec R and prim A to G -max R.

4.10 Lemma. The set function I ↦ Φ(I), for ideals I of R, produces an H-equivariant homeomorphism from spec A onto G -spec R that restricts to a homeomorphism from prim A onto G -max R.

Proof. Given w ∈ W, there exists an H-equivariant commutative diagram

A ⟶^{Φ} R
λ_{w}↓  ↓^{μ_{w}}
A_{w} ⟶^{Φ_{w}} R_{w}

where λ_{w} and μ_{w} are the respective localization maps. Let Q ∈ G -spec_{w} R, and write Q = (P : S_{w}^{⊥}) for some P ∈ spec_{w} R. Then P = μ_{w}^{−1}(P′) for some P′ ∈ spec R_{w}; the ideal Q′ = (P′ : S_{w}^{⊥}) lies in S_{w}^{⊥}- spec R_{w}; and Q = μ_{w}^{−1}(Q′). By (3.9), Φ_{w}^{−1}(Q′) ∈ spec A_{w}, and so λ_{w}^{−1}Φ_{w}^{−1}(Q′) ∈ spec_{w} A. Since λ_{w}^{−1}Φ_{w}^{−1}(Q′) = Φ^{−1}μ_{w}^{−1}(Q′) = Φ^{−1}(Q), we thus see that the set Φ^{−1}(Q) is a prime ideal of A, lying in spec_{w} A. Similarly, Φ^{−1}(M) ∈ prim_{w} A for all M ∈ G -max_{w} R.

Since Φ and Φ_{w} are bijections, analogous arguments show that Φ(Q) ∈ G -spec_{w} R, for all Q ∈ spec_{w} A, and Φ(P) ∈ G -max_{w} R, for all P ∈ prim_{w} A.

The results above, taken over all w ∈ W, show that the set maps I ↦ Φ(I) and J ↦ Φ^{−1}(J) yield pairs of mutually inverse bijections

spec A ⟷ G -spec R  and  prim A ⟷ G -max R.

These maps are H-equivariant because Φ and Φ^{−1} are H-equivariant. Finally, note that the topologies on all of the spaces considered in this section can be defined using subsets of A and R instead of ideals – for instance, the closed subsets of spec A are precisely those of the form {Q ∈ spec A | Q ⊇ X}, for arbitrary subsets X ⊆ A. It therefore follows that the above maps are homeomorphisms, as desired. □

Combining (4.9) and (4.10), we obtain the main theorem of the paper, which we state as follows, using the notation developed in (4.2)–(4.8).

4.11 Theorem. Suppose that k is an algebraically closed field, that q = (q_{ij}) is a multiplicatively antisymmetric n × n matrix over k, that A = O_{q}(k^{n}), and that R = O(k^{n}). Further assume that −1 ∉ ⟨q_{ij}⟩ or that char k = 2. Let H = (k^{×})^{n} act by k-algebra automorphisms on A and on R in the standard manner. Then there exist H-equivariant topological quotient maps

spec R → spec A  and  max R → prim A

given by P ↦ Φ^{−1}(P : S_{w}^{⊥}), for P ∈ spec_{w} R. The fibers of the second map, over points in prim_{w} A, consist precisely of the S_{w}^{⊥}-orbits within max_{w} R. □

5. Some examples

We illustrate (4.11) by calculating the explicit form of the maps from max R to prim A in the cases of the standard single parameter quantum affine spaces, and we comment on some more general cases related to bilinear forms and Poisson brackets. Let k, q, A, and R be as in (4.11), and let

Ψ : max R ⟶ prim A

denote the topological quotient map given by the theorem therein. Composing Ψ with the natural isomorphism of affine n-space onto max R, we obtain a similar topological quotient map

ψ : k^{n} ⟶ prim A.

For simplicity of notation, we describe ψ rather than Ψ.

5.1. First, let A be the standard one-parameter quantization O_{q}(k^{n}), for some q ∈ k^{×}. Then A is generated by x_{1}, …, x_{n} such that x_{i}x_{j} = qx_{j}x_{i} for all i < j. The alternating bicharacter σ as in (4.1) can be expressed by

σ(α, β) = q^{b(α,β)},

where b : Γ × Γ → ℤ is the alternating bilinear form

b(α, β) = ∑_{i<j} α_{i}β_{j} − ∑_{i>j} α_{i}β_{j}.

To complete the hypotheses of (4.11), we must assume that q is either not a root of unity or an odd root of unity. In either case, q has a square root p ∈ k^{×} such that −1 ∉ ⟨p⟩ (if char k ≠ 2), and the rule

c(α, β) = p^{b(α,β)}

defines an alternating bicharacter c on Γ satisfying the conclusions of (4.2). Hence, we can identify A with k^{c}Γ^{+} for this c.

5.2. Continue with (5.1), and suppose that q is not a root of unity. We calculate ψ(λ) for points λ = (λ_{1}, …, λ_{n}) in k^{n}. This depends on the position of λ, with respect to the stratification of k^{n} corresponding to the subsets max_{w} R. Set

(k^{n})_{w} = {λ ∈ k^{n} | λ_{i} = 0 for i ∈ w and λ_{j} ≠ 0 for j ∉ w}

for w ∈ W.

If n − |w| is even, we calculate that S_{w} = 0. In this case,

ψ(λ) = ⟨x_{i} | i ∈ w⟩

for all λ ∈ (k^{n})_{w}.

Now suppose that n−|w| is odd. List the elements of the complement of w in ascending order, say

{1, …, n} \ w = {w_{1} < w_{2} < ⋯ < w_{2m+1}},

and then set

w_{+} = {w_{1}, w_{3}, …, w_{2m+1}}  and  w_{−} = {w_{2}, w_{4}, …, w_{2m}}.

Finally, write γ_{±}(w) = ∑_{i∈w_{±}} ϵ_{i} ∈ Γ, where ϵ_{1}, …, ϵ_{n} is, as usual, the standard basis for Γ. We calculate that S_{w} = ℤ(γ_{+}(w) − γ_{−}(w)), and that

ψ(λ) = ⟨x_{i} | i ∈ w⟩ + ⟨( ∏_{j∈w_{−}} λ_{j})x_{γ_{+}(w)} − ( ∏_{j∈w_{+}} λ_{j})x_{γ_{−}(w)}⟩

for λ ∈ (k^{n})_{w}.

If one wishes to write the last formula in terms of ordinary monomials in the x_{i}, additional scalar factors are introduced. For instance, suppose that n = 3 and w = ∅, so that w_{+} = {1, 3} and w_{−} = {2}. Since x_{1}x_{3} = c(ϵ_{1}, ϵ_{3})x_{(1,0,1)} = px_{γ_{+}(w)}, we have

ψ(λ_{1}, λ_{2}, λ_{3}) = ⟨p^{−1}λ_{2}x_{1}x_{3} − λ_{1}λ_{3}x_{2}⟩

for (λ_{1}, λ_{2}, λ_{3}) ∈ (k^{3})_{w} = (k^{×})^{3}.

5.3. Continue with (5.1), but suppose now that q is a primitive t-th root of unity, for some odd t > 1. In this case, the correct choice for p is q^{(t+1)/2}. The form of ψ(λ) for λ ∈ (k^{n})_{w} again depends on the parity of n − |w|.

If n − |w| is even, we calculate that S_{w} = tΓ_{w}, and that

ψ(λ) = ⟨x_{i} | i ∈ w⟩ + ⟨x_{j}^{t} − λ_{j}^{t} | j ∉ w⟩

for λ ∈ (k^{n})_{w}.

If n − |w| is odd, we calculate that S_{w} = tΓ_{w} + ℤ(γ_{+}(w) − γ_{−}(w)), and that

ψ(λ) = ⟨x_{i} | i ∈ w⟩ + ⟨x_{j}^{t} − λ_{j}^{t} | j ∉ w⟩ + ⟨( ∏_{j∈w_{−}} λ_{j})x_{γ_{+}(w)} − ( ∏_{j∈w_{+}} λ_{j})x_{γ_{−}(w)}⟩

for λ ∈ (k^{n})_{w}.

5.4. Let b : Γ × Γ → ℤ be an arbitrary alternating bilinear form, let q ∈ k^{×} be a non-root of unity, and set q_{ij} = q^{b(ϵ_{i},ϵ_{j})} for all i, j. The alternating bicharacter σ associated with A as in (4.1) then takes the form σ(α, β) = q^{b(α,β)}. To obtain a bicharacter c as in (4.2), take a square root p for q, and define c(α, β) = p^{b(α,β)}. Observe that since q is not a root of unity, S_{w} = rad(σ|_{Γ_{w}}) = rad(b|_{Γ_{w}}) for w ∈ W.

Now suppose that char k = 0. There is a Poisson bracket on R such that {y_{α}, y_{β}} = b(α, β)y_{α}y_{β} for α, β ∈ Γ. The analysis in [15, 1.4] (with ℂ replaced by k) shows that the S_{w}^{⊥}-orbits in (k^{n})_{w} coincide with the minimal Poisson subvarieties relative to the Poisson structure just defined. Therefore the fibers of ψ in this case coincide with the minimal Poisson subvarieties of k^{n}.

When k = ℂ, these minimal Poisson subvarieties are not necessarily the same as the symplectic leaves in k^{n}. See [15, 1.4] for necessary and sufficient conditions.

5.5. The relation σ = q^{b} in (5.4) is stronger than necessary for the analysis given – it would suffice to have an alternating bilinear form b : Γ × Γ → k such that rad(σ|_{Γ_{w}}) = rad(b|_{Γ_{w}}) for all w ∈ W. To find further examples where such b exist, and some situations in which the fibers of a map analogous to ψ coincide with symplectic leaves, see [15, Section 3].

5.6. To test whether the hypothesis −1 ∉ ⟨q_{ij}⟩ in (4.11) is necessary, the natural examples to investigate are the algebras O_{−1}(k^{n}) in characteristic different from 2. Our method works precisely when O_{−1}(k^{n}) can be expressed in the form k^{c}Γ^{+} for some 2-cocycle c such that c ≡ 1 on S_{w} × Γ_{w} for all w ∈ W. It is easy to find such a c in case n = 2, and one also exists in case n = 3. However, it can be shown that no such c exists when n ≥ 4. Thus, our method cannot handle O_{q}(k^{n}) for arbitrary q and n. We leave it as an open question whether prim O_{q}(k^{n}) is always an H-equivariant topological quotient of max O(k^{n}).

6. Twists of graded commutative algebras

To conclude the paper, we apply the main theorem (4.11) to a class of twists of finitely generated commutative graded algebras. Assume throughout that k is a field, that R is a kalgebra graded by an abelian group G (which we write additively), and that c : G×G → k^{×} is a 2-cocycle.

6.1. There exists a G-graded k-algebra R′ equipped with a G-graded k-linear isomorphism

\begin{matrix}R & ⟶ & R′ \\ r & ⟼ & r′ \end{matrix}

such that, for all α, β ∈ G, the multiplication of arbitrary homogeneous elements r′ ∈ R′_{α} and s′ ∈ R′_{β} is given by the rule r′s′ = c(α, β)(rs)′. Up to a G-graded k-algebra isomorphism, R′ depends only on the cohomology class of c [2, p. 888]. We call R′ the twist of R by c (cf. [2, Section 3]), and we refer to the above function R → R′ as the twist map (associated to c). If U is a subset of R, we will use U′ to denote the image of U under the twist map.

Note, for computational purposes, that if I and J are (G-)homogeneous ideals of R then I′J′ = (IJ)′.

6.2. Recall that a proper homogeneous ideal K, in R or R′, is graded-prime provided it contains no product IJ of homogeneous ideals I and J not contained in K. Every ideal of R or R′ contains a unique maximum homogeneous ideal, and an argument similar to (1.3) demonstrates that the maximum homogeneous ideal within a prime ideal is graded-prime.

Lemma. Suppose that G is torsionfree and that R is noetherian. If P_{1}, …, P_{m} are the minimal prime ideals of R, then P_{1}, …, P_{m} are homogeneous, and P′_{1}, …, P′_{m} are the minimal prime ideals of R′.

Proof. Let P_{i}^{∘} denote the largest homogeneous ideal contained in P_{i}. Since G is torsionfree, the graded-prime ideal P_{i}^{∘} must actually be a prime ideal, by [1, Corollary 3.3]. Hence, P_{i} = P_{i}^{∘} by minimality.

Since the twist map preserves homogeneous ideals and their products, it follows that P′_{i} is a graded-prime ideal of R′. A second application of [1, Corollary 3.3] then shows that P′_{i} is a prime ideal of R′. Now some product of the P_{i} is equal to 0 in R, and by homogeneity the corresponding product of the P′_{i} must be 0 in R′. It follows that every minimal prime ideal of R′ must occur among the P′_{i}.

Let Q′_{1}, …, Q′_{n} be the minimal prime ideals of R′, and denote their respective preimages under the twist map by Q_{1}, …, Q_{n}. Arguing as above, we see that the Q_{j} are prime ideals of R, and that all minimal prime ideals of R occur among the Q_{j}. The lemma follows. □

6.3 Theorem. Let k be an algebraically closed field, let R be a commutative affine kalgebra graded by a torsionfree abelian group G, and let A be the twist of R by a 2-cocycle c : G × G → k^{×}. Assume either that −1 is not in the subgroup of k^{×} generated by the image of c, or that char k = 2. Then there exist topological quotient maps

spec R → spec A  and  max R → prim A.

Proof. Since R is affine, its support is contained in a finitely generated subgroup G^{fg} ⊆ G. Then R is also graded by G^{fg}, and A is equal to the twist of R by c|_{G^{fg}}. Hence, there is no loss of generality in assuming that G is finitely generated. Thus G is now a free abelian group of finite rank.

Set σ(α, β) = c(α, β)c(β, α)^{−1}, for α, β ∈ G. By [2, Proposition 1(ii), p. 888], σ is an alternating bicharacter on G. Since either −1 ∉ ⟨im σ⟩ or char k = 2, it follows from (4.2) that there exists an alternating bicharacter d on G with the following properties: d^{2} = σ, and d(α, β) = 1 whenever σ(α, β) = 1. By [2, Proposition 1(i), p. 888], d and c are in the same cohomology class, and hence A can also be written as the twist of R by d. Thus, there is no loss of generality in assuming that c is an alternating bicharacter and that c(α, β) = 1 whenever c(α, β)^{2} = 1.

The inverse of the twist map is a G-graded k-linear isomorphism Φ : A → R such that Φ(ab) = c(α, β)Φ(a)Φ(b) for a ∈ A_{α} and b ∈ A_{β}. Choose homogeneous elements r_{1}, …, r_{n} generating R as a k-algebra, and let δ_{1}, …, δ_{n} ∈ G be the degrees of these elements.

Set Γ = ℤ^{n} and Γ^{+} = (ℤ^{+})^{n}, and let ρ : Γ → G be the group homomorphism given by

ρ(α_{1}, …, α_{n}) = α_{1}δ_{1} + ⋯ + α_{n}δ_{n}.

Set c̃(α, β) = c(ρ(α), ρ(β)) for α, β ∈ Γ. Then c̃ is an alternating bicharacter on Γ such that c̃(α, β) = 1 whenever c̃(α, β)^{2} = 1. Now set r^{α} = r_{1}^{α_{1}}r_{2}^{α_{2}} ⋯ r_{n}^{α_{n}} ∈ R_{ρ(α)} and a_{α} = Φ^{−1}(r^{α}) ∈ A_{ρ(α)} for α ∈ Γ^{+}. Since R is commutative, r^{α}r^{β} = r^{α+β} for α, β ∈ Γ^{+}, whence

Φ(a_{α}a_{β}) = c(ρ(α), ρ(β))r^{α}r^{β} = c̃(α, β)r^{α+β} = c̃(α, β)Φ(a_{α+β}),

and so a_{α}a_{β} = c̃(α, β)a_{α+β} for α, β ∈ Γ^{+}.

Now set R̃ = kΓ^{+} and Ã = kc̃Γ^{+}. Write R̃ in terms of a k-basis {y_{α} | α ∈ Γ^{+}} such that y_{α}y_{β} = y_{α+β} for α, β ∈ Γ^{+}, and write Ã in terms of a k-basis {x_{α} | α ∈ Γ^{+}} such that x_{α}x_{β} = c̃(α, β)x_{α+β} for α, β ∈ Γ^{+}. Let Φ̃ : Ã → R̃ be the k-linear isomorphism such that Φ̃(x_{α}) = y_{α} for α ∈ Γ^{+}. There are k-algebra epimorphisms f : Ã → A and g : R̃ → R such that f(x_{α}) = a_{α} and g(y_{α}) = r^{α} for α ∈ Γ^{+}. We now obtain the following commutative diagram:

Ã ⟶^{Φ̃} R̃
f↓  ↓^{g}
A ⟶^{Φ} R

Thus if I = ker f, then Φ̃(I) = ker g.

Define G as in (4.9), with c̃, R̃, Ã playing the roles of c, R, A. By (4.11), the rule P ↦ Φ̃^{−1}(P : G) gives topological quotient maps

ϕ_{s} : spec R̃ → spec Ã and ϕ_{m} : max R̃ → prim Ã.

These maps restrict to topological quotient maps

ϕ_{s}^{−1}V (I) → V (I) ≈ spec A
ϕ_{m}^{−1}(V (I) ∩ prim Ã) → V (I) ∩ prim Ã ≈ prim A.

To finish the proof of the theorem, it suffices to show that ϕ_{s}^{−1}V (I) = V (Φ̃(I)). The inclusion ⊆ is easy: If P ∈ ϕ_{s}^{−1}V (I), then Φ̃^{−1}(P : G) ⊇ I, whence P ⊇ (P : G) ⊇ Φ̃(I).

Let P_{1}, …, P_{m} be the prime ideals of Ã minimal over I. In view of (6.2), the sets Φ̃(P_{1}), …, Φ̃(P_{m}) are the prime ideals of R̃ minimal over Φ̃(I). By (4.10), each Φ̃(P_{j}) ∈ G -spec R̃. Given any P ∈ V (Φ̃(I)), we have P ⊇ Φ̃(P_{j}) for some j. Since Φ̃(P_{j}) is a G-prime ideal, (P : G) ⊇ Φ̃(P_{j}) ⊇ Φ̃(I), and thus ϕ_{s}(P) ∈ V (I). Therefore ϕ_{s}^{−1}V (I) = V (Φ̃(I)), as desired. □

6.4 Remark. Retain the notation in the statement of the preceding theorem, and let r_{1}, …, r_{n} be homogeneous generators for R. As seen in the proof, we may assume without loss of generality that G = ℤ^{n} and that c is an alternating bicharacter on ℤ^{n}. In particular, it can be shown that the topological quotient maps in (6.3) are equivariant with respect to suitable actions by a subgroup of (k^{×})^{n}; details are left to the interested reader.

References

1. G. Abrams and J. Haefner, Primeness conditions for group graded rings, in Ring Theory, Proc. Biennial Ohio State – Denison Conf. 1992, World Scientific, Singapore, 1993, pp. 1-19.

2. M. Artin, W. Schelter, and J. Tate, Quantum deformations of GL_{n}, Communic. Pure Appl. Math. 44 (1991), 879-895.

3. K. A. Brown and K. R. Goodearl, Prime spectra of quantum semisimple groups, Trans. Amer. Math. Soc. 348 (1996), 2465–2502.

4. C. De Concini, V. Kac, and C. Procesi, Some remarkable degenerations of quantum groups, Comm. Math. Phys. 157 (1993), 405–427.

5. A. W. Goldie and G. O. Michler, Ore extensions and polycyclic group rings, J. London Math. Soc. (2) 9 (1974), 337–345.

6. K. R. Goodearl and E. S. Letzter, Prime and primitive spectra of multiparameter quantum affine spaces, in Trends in Ring Theory. Proc. Miskolc Conf. 1996 (V. Dlab and L. Márki, eds.), Canad. Math. Soc. Conf. Proc. Series 22 (1998), 39-58.

7. K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, London Math. Soc. Student Texts 16, Cambridge Univ. Press, Cambridge, 1989.

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

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

10. T. J. Hodges, T. Levasseur, and M. Toro, Algebraic structure of multi-parameter quantum groups, Advances in Math. 126 (1997), 52-92.

11. A. Joseph, On the prime and primitive spectra of the algebra of functions on a quantum group, J. Algebra 169 (1994), 441–511.

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

13. J. C. McConnell and J. J. Pettit, Crossed products and multiplicative analogs of Weyl algebras, J. London Math. Soc. (2) 38 (1988), 47-55.

14. D. G. Northcott, Affine Sets and Affine Groups, London Math. Soc. Lecture Note Series 39, Cambridge Univ. Press, Cambridge, 1980.

15. M. Vancliff, Primitive and Poisson spectra of twists of polynomial rings, Algebras and Representation Theory (to appear).

Department of Mathematics, University of California, Santa Barbara, CA 93106 E-mail address: goodearl@math.ucsb.edu

Department of Mathematics, Texas A&M University, College Station, TX 77843 E-mail address: letzter@math.tamu.edu