PRIME SPECTRA OF QUANTIZED COORDINATE RINGS

K. R. Goodearl

This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common properties, there must be a common basis on which to treat them. The first part of the paper is expository. We survey a number of classes of quantized coordinate rings, as well as some related algebras that share common properties, and we record some of the basic properties known to occur for many of these algebras, culminating in stratifications of the prime spectra by the actions of tori of automorphisms. As our main interest is in the generic case, we assume various parameters are not roots of unity whenever convenient. In the second part of the paper, which is based on [20], we offer some support for the conjecture above, in the form of an axiomatic basis for the observed stratifications and their properties. At present, the existence of a suitable supply of normal elements is taken as one of the axioms; the search for better axioms that yield such normal elements is left as an open problem.

I. Quantized Coordinate Rings and Related Algebras

This part of the paper is an expository account of the prime ideal structure of algebras on the “quantized coordinate ring” side of the theory of quantum groups – quantizations of the coordinate rings of affine spaces, matrices, semisimple groups, symplectic and Euclidean spaces, as well as a few related algebras – quantized enveloping algebras of Borel and nilpotent subalgebras of semisimple Lie algebras, and quantized Weyl algebras. These algebras occur widely throughout the quantum groups literature, and different papers often investigate different versions. Thus, we begin by giving definitions of the most general versions of which we are aware; in quoting results from the literature, we will specify which version is under consideration. The

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

reader should bear in mind that many authors, when studying one version of a quantized coordinate algebra, use results proved for slightly different versions, on the understanding that the proofs carry over. This is especially prevalent with regard to quantized coordinate rings of semisimple groups; a detailed development covering the most general case would be a welcome addition to the literature.

Fix a base field k throughout. It need not be algebraically closed, and may have arbitrary characteristic.

1. Descriptions

This section is designed to be a reference source for descriptions of the main classes of the standard quantized coordinate rings currently studied, as well as some related algebras with similar properties. (The reader lacking a strong stomach for generators and relations should just skim this section and refer back to it as necessary.) For the sake of uniformity, and to emphasize that these algebras are deformations of classical coordinate rings, we label all quantized coordinate rings using notations of the form O_{•}(C), where • records one or more parameters and C records the name of the classical object. Thus, O_{q}(k^{n}) refers to a one-parameter quantization of the coordinate ring of affine n-space over a field k, while O_{λ,p}(GL_{n}(k)) refers to a multiparameter quantization of the coordinate ring of GL_{n}(k), and so on. Many different labels are used throughout the literature for these algebras, and we do not attempt to list the alternates here.

1.1. Quantum affine spaces. These are meant to be viewed as deformations of the coordinate rings O(k^{n}). Recall that O(k^{n}) is a (commutative) polynomial ring k[x_{1}, …, x_{n}], where x_{i} is the i-th coordinate function on k^{n}. In the present case, one deforms O(k^{n}) in a rather straightforward manner, by altering the commutativity relations x_{i}x_{j} = x_{j}x_{i} to “commutativity up to scalars”, that is, x_{i}x_{j} = q_{ij}x_{j}x_{i}. In order to prevent degeneracy, one needs some assumptions on the q_{ij}. In particular, they should be nonzero, and q_{ji} should equal q_{ij}^{−1} in order to prevent the pair of relations x_{i}x_{j} = q_{ij}x_{j}x_{i} and x_{j}x_{i} = q_{ji}x_{i}x_{j} from implying x_{i}x_{j} = 0. Thus, one assumes that q = (q_{ij}) is a multiplicatively antisymmetric n×n matrix over k, that is, q_{ii} = 1 and q_{ji} = q_{ij}^{−1} for all i, j. Given q, the corresponding multiparameter coordinate ring of quantum affine n-space over k is the k-algebra O_{q}(k^{n}) generated by elements x_{1}, …, x_{n} subject only to the relations x_{i}x_{j} = q_{ij}x_{j}x_{i} for i, j = 1, …, n. The standard single parameter version occurs when the q_{ij} for i < j are all equal to a fixed nonzero scalar q; in this case, we denote the algebra O_{q}(k^{n}).

Quantum affine spaces occur already in the work of Manin [37, Section 1, §2 and Section 4, §5]; a superalgebra version is discussed in [38, §1.4].

1.2. Quantum matrices. The set M_{n}(k) of n×n matrices over k, viewed as an algebraic variety, is just affine n^{2}-space, and its coordinate ring O(M_{n}(k)) is a polynomial ring in n^{2} indeterminates. Hence, one might expect to deform this algebra exactly as in (1.1). However, there is more structure in this case, and one seeks to preserve it as far as possible. Namely, O(M_{n}(k)) is a bialgebra with a comultiplication Δ : O(M_{n}(k)) → O(M_{n}(k)) ⊗ O(M_{n}(k)) which is effectively the transpose of the map M_{n}(k) × M_{n}(k) → M_{n}(k) given by matrix multiplication: If X_{ij} denotes the i, j-th coordinate function on M_{n}(k), then Δ(X_{ij}) = ∑_{l=1}^{n} X_{il} ⊗ X_{lj}. In particular, one would like deformations of O(M_{n}(k)), with appropriate sets of generators X_{ij}, which are also bialgebras in which the comultiplication of the X_{ij} is given by the equation above.

In addition, multiplication of matrices with row or column vectors from k^{n} induces on O(k^{n}) structures of left and right comodule over O(M_{n}(k)), and one would like some quantum affine spaces O_{q}(k^{n}) to be left and right comodules over the deformation of O(M_{n}(k)), with comodule structure maps behaving as in the classical case. In the single parameter case, there is a deformation O_{q}(M_{n}(k)) over which O_{q}(k^{n}) becomes both a left and a right comodule in the desired manner (see [38] and [52]). However, if one tries to make a multiparameter quantum affine space O_{q}(k^{n}) into a left \underline{and} right comodule over the desired type of deformation of O(M_{n}(k)), the latter deformation will usually be degenerate. To obtain nondegenerate deformations, one must use different multiparameter quantum affine spaces as left and right comodules, as discovered in [2] and [57], to which we refer the reader for a more complete discussion. The resulting algebras can be described as follows.

Let p = (p_{ij}) be a multiplicatively antisymmetric n × n matrix over k, and let λ be a nonzero element of k not equal to −1. The corresponding multiparameter coordinate ring of quantum n × n matrices over k is the kalgebra O_{λ,p}(M_{n}(k)) generated by elements X_{ij} (for i, j = 1, …, n) subject only to the following relations:

X_{ℓm}X_{ij} = \{ \begin{matrix}p_{ℓi}p_{jm}X_{ij}X_{ℓm} + (λ − 1)p_{ℓi}X_{im}X_{ℓj} & (ℓ > i, m > j) \\ λp_{ℓi}p_{jm}X_{ij}X_{ℓm} & (ℓ > i, m ≤ j) \\ p_{jm}X_{ij}X_{ℓm} & (ℓ = i, m > j). \end{matrix}

(We assume λ ≠ 0 to avoid obvious degeneracies in the second relation above. The assumption λ ≠ −1 is needed to ensure that O_{λ,p}(M_{n}(k)) has the same Hilbert series as a commutative polynomial algebra in n^{2} indeterminates [2, Theorem 1].) The standard single parameter version, denoted O_{q}(M_{n}(k)), occurs when the p_{ij} for i > j are all equal to a fixed nonzero scalar q, and λ = q^{−2}. In some references, such as [49] and [56], the roles of q and q^{−1} are interchanged, and/or q is squared.

Quantized coordinate rings for rectangular matrices are defined as subalgebras of those for square matrices. For instance, if m < n then O_{λ,p}(M_{m,n}(k)) is defined to be the k-subalgebra of O_{λ,p}(M_{n}(k)) generated by those X_{ij} with i ≤ m. There is a k-algebra retraction of O_{λ,p}(M_{n}(k)) onto O_{λ,p}(M_{m,n}(k)) whose kernel is generated by the X_{ij} with i > m; thus O_{λ,p}(M_{m,n}(k)) ≅ O_{λ,p}(M_{n}(k))/⟨X_{ij} | i > m⟩. Hence, results for O_{λ,p}(M_{n}(k)) easily extend to the rectangular case; we shall so extend results from the literature without explicit mention.

1.3. Quantum general linear groups. Within the algebraic variety M_{n}(k), the set GL_{n}(k) of invertible matrices forms an open subvariety, the complement of the variety defined by the vanishing of the determinant function D. The coordinate ring O(GL_{n}(k)) thus has the form O(M_{n}(k))[D^{−1}]. To construct a quantum analog, one inverts a “quantum determinant”, call it D_{λ,p}, in O_{λ,p}(M_{n}(k)). In order for the inversion process to work smoothly and avoid degeneracies, the powers of D_{λ,p} should form an Ore set. This occurs because the chosen D_{λ,p} turns out to be a normal element. (Recall that a normal element in a ring R is an element c such that cR = Rc.)

As in the classical case, the quantum determinant D_{λ,p} can be defined as a linear combination of products X_{1,π(1)}X_{2,π(2)} ⋯ X_{n,π(n)} as π runs through the symmetric group S_{n}. However, the coefficients are taken to be certain products of elements −p_{ij} rather than powers of −1. Specifically,

D_{λ,p} = ∑_{π∈S_{n}}( ∏_{\begin{matrix}1≤i<j≤n \\ π(i)>π(j) \end{matrix}}(−p_{π(i),π(j)}))X_{1,π(1)}X_{2,π(2)} ⋯ X_{n,π(n)}.

The motivation for this choice of D_{λ,p} comes from the construction of a quantized version of the exterior algebra on k^{n}; the n-th “quantum exterior power” of k^{n} is 1-dimensional, spanned by D_{λ,p} (see [2] for details). In the single parameter case, the quantum determinant D_{q} in O_{q}(M_{n}(k)) can be expressed as

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

where ℓ(π) denotes the length of the permutation π, that is, the minimum length of an expression for π as a product of adjacent transpositions (i, i+1). (Cf. [49, Lemma 4.1.1] but interchange q and q^{−1}.)

It has been computed that D_{λ,p} is a normal element of O_{λ,p}(M_{n}(k)) [2, Theorem 3]; in fact,

D_{λ,p}X_{ij} = λ^{j−i}(∏_{l=1}^{n} p_{jl}p_{li})X_{ij}D_{λ,p}

for all i, j. In the single parameter case, D_{q} is central (e.g., [49, Theorem 4.6.1]). It follows that the set of nonnegative powers of D_{λ,p} is a right and left Ore set in O_{λ,p}(M_{n}(k)). The multiparameter coordinate ring of quantum GL_{n}(k) is now defined to be the localization O_{λ,p}(GL_{n}(k)) = O_{λ,p}(M_{n}(k))[D_{λ,p}^{−1}]; in the single parameter case, O_{q}(M_{n}(k))[D_{q}^{−1}] is denoted O_{q}(GL_{n}(k)).

1.4. Quantum special linear groups. Note that the set SL_{n}(k) of n × n matrices with determinant 1 forms a closed subvariety of M_{n}(k), defined by the single equation D = 1. The coordinate ring O(SL_{n}(k)) thus has the form O(M_{n}(k))/(D − 1). To construct a quantum analog, we would thus factor out from O_{λ,p}(M_{n}(k)) the ideal generated by D_{λ,p} −1. However, unless D_{λ,p} is central, such a factor would be degenerate. Hence, quantum special linear groups are only defined for special choices of the parameters.

From the normality relations for D_{λ,p}, we see that D_{λ,p} is central if and only if

λ^{i} ∏_{l=1}^{n} p_{il} = λ^{j} ∏_{l=1}^{n} p_{jl}

for all i, j. In this case, the multiparameter coordinate ring of quantum SL_{n}(k) is defined to be the factor algebra

O_{λ,p}(SL_{n}(k)) = O_{λ,p}(M_{n}(k))/⟨D_{λ,p} − 1⟩.

In the single parameter case, where D_{q} is automatically central, the factor O_{q}(M_{n}(k))/⟨D_{q} − 1⟩ is denoted O_{q}(SL_{n}(k)).

1.5. Quantum semisimple groups. A systematic development of quantized coordinate rings for simple algebraic groups corresponding to the Dynkin diagrams A_{n}, B_{n}, C_{n}, D_{n} was given by Reshetikhin, Takhtadzhyan, and Fadeev in [53]. These algebras were expressed in terms of the entries of certain “R-matrices” attached to quantizations of the corresponding simple Lie algebras. Calculations of generators and relations for these quantized coordinate rings can be found in, e.g., [58]. Quantized coordinate rings for the exceptional groups, however, were not computed, due to the lack of explicit R-matrices in those cases (cf. [53, Remark 14, p. 212]; the G_{2} case is considered in [55]). The now standard approach to these algebras involves a quantization of the classical duality between the coordinate ring of a semisimple algebraic group and the enveloping algebra of its Lie algebra. Thus, one constructs quantized enveloping algebras of semisimple Lie algebras and defines the quantized coordinate rings of the corresponding semisimple groups as certain Hopf algebra duals. We outline this procedure.

(a) Let 𝔤 be a complex semisimple Lie algebra, and let G be a connected complex semisimple algebraic group with Lie algebra 𝔤. Both 𝔤 and G play symbolic roles; they serve mainly as suggestive labels. The only data needed from these objects are: a (symmetrized) Cartan matrix (d_{i}a_{ij}) for 𝔤; root and weight lattices Q ⊆ P corresponding to choices of Cartan subalgebra 𝔥 and root system for 𝔤; a lattice L lying between Q and P corresponding to the character group of a maximal torus of G with Lie algebra 𝔥; choices of simple roots α_{1}, …, α_{n} and fundamental weights ω_{1}, …, ω_{n}; and the (unique) bilinear pairing (−, −) : P × Q → ℤ such that (ω_{i}, α_{j}) = δ_{ij}d_{i} for all i, j. There is a unique extension of (−, −) to a symmetric bilinear form P × P → ℚ, which we also denote by (−, −).

Different authors base quantized coordinate algebras of G on different quantizations of the enveloping algebra U(𝔤). Thus we first describe a general one-parameter quantization of U(𝔤), as in [11, §0.3]. Let M be a lattice lying between Q and P (independent of L). Then let q ∈ k be a nonzero scalar, set q_{i} = q^{d_{i}} for i = 1, …, n, and assume that these q_{i} ≠ ±1. (In much of the literature on quantized enveloping algebras, k is taken to be ℂ, or a rational function field ℂ(q) or ℚ(q), or the algebraic closure of one of these rational function fields.) Let U^{0} be a copy of the group algebra kM, written as the k-algebra with basis {k_{λ} | λ ∈ M} where k_{0} = 1 and k_{λ}k_{μ} = k_{λ+μ} for λ, μ ∈ M. The single parameter quantized enveloping algebra of 𝔤 associated with the above choices is the k-algebra U_{q}(𝔤, M) generated by U^{0} and elements e_{1}, …, e_{n}, f_{1}, …, f_{n} satisfying the following relations for λ ∈ M and i, j = 1, …, n:

k_{λ}e_{i}k_{λ}^{−1} = q^{(λ,α_{i})}e_{i}  and  k_{λ}f_{i}k_{λ}^{−1} = q^{−(λ,α_{i})}f_{i}

e_{i}f_{j} − f_{j}e_{i} = δ_{ij} \frac{k_{α_{i}} − k_{α_{i}}^{−1}}{q_{i} − q_{i}^{−1}}

∑_{l=1}^{1−a_{ij}}(−1)^{l}\binom{1 − a_{ij}}{l}_{q_{i}}e_{i}^{1−a_{ij}−l}e_{j}e_{i}^{l} = 0 (i ≠ j)

∑_{l=1}^{1−a_{ij}}(−1)^{l}\binom{1 − a_{ij}}{l}_{q_{i}}f_{i}^{1−a_{ij}−l}f_{j}f_{i}^{l} = 0 (i ≠ j),

where \binom{1−a_{ij}}{l}_{q_{i}} is a q_{i}-binomial coefficient. The most commonly studied case is U_{q}(𝔤, Q), denoted just U_{q}(𝔤). This algebra is generated by e_{i}, f_{i}, k_{i} for i = 1, …, n, where k_{i} = k_{α_{i}}. At the other extreme, U_{q}(𝔤, P) is often denoted Ǔ_{q}(𝔤).

The algebra U_{q}(𝔤, M) is in fact a Hopf algebra, with comultiplication Δ, counit ϵ, and antipode S such that

Δ(k_{λ}) = k_{λ} ⊗ k_{λ}  ϵ(k_{λ}) = 1  S(k_{λ}) = k_{λ}^{−1}
Δ(e_{i}) = e_{i} ⊗ 1 + k_{α_{i}} ⊗ e_{i}  ϵ(e_{i}) = 0  S(e_{i}) = −k_{α_{i}}^{−1}e_{i}
Δ(f_{i}) = f_{i} ⊗ k_{α_{i}}^{−1} + 1 ⊗ f_{i}  ϵ(f_{i}) = 0  S(f_{i}) = −f_{i}k_{α_{i}}.

Hence, one can define the finite dual U_{q}(𝔤, M)^{∘} as in, e.g., [41, Definition 1.2.3]; this is a k-linear subspace of U_{q}(𝔤, M)^{∗} which becomes a Hopf algebra using the transposes of the multiplication, comultiplication, and antipode of U_{q}(𝔤, M) (e.g., [41, Theorem 9.1.3]).

(b) Single-parameter quantizations of O(G) are defined as subalgebras of U_{q}(𝔤, M)^{∘} generated by the “coordinate functions” of certain “highest weight” U_{q}(𝔤, M)-modules. It turns out that the resulting algebras are independent of the choice of M. Hence, we first describe quantizations of O(G) as subalgebras of U_{q}(𝔤)^{∘}; this has the advantage that no roots of q are required. For descriptions in terms of U_{q}(𝔤, M)^{∘}, see part (c). To avoid problems with certain calculations, one assumes that char(k) ≠ 2, and also char(k) ≠ 3 in case 𝔤 has a component of type G_{2}.

Here we only give a quantization of O(G) for the case that q is not a root of unity. The root of unity case requires defining a suitable algebra over a Laurent polynomial ring k_{0}[t^{±1}] and then specializing t to q (see, e.g., [35, Sections 7,8] or [12, Section 4]).

For each λ ∈ P^{+}, there is a finite dimensional simple U_{q}(𝔤)-module V (λ) with highest weight λ, that is, V (λ) is generated by an element u_{λ} satisfying k_{μ}u_{λ} = q^{(μ,λ)}u_{λ} for all μ ∈ Q and e_{i}u_{λ} = 0 for i = 1, …, n (see, e.g., [26, Theorem 5.10]). For v ∈ V (λ) and f ∈ V (λ)^{∗}, let c_{f,v}^{V (λ)} ∈ U_{q}(𝔤)^{∘}

denote the coordinate function defined by the rule c_{f,v}^{V (λ)}(u) = f(uv). The single parameter quantized coordinate ring of G is the k-subalgebra O_{q}(G, L) of U_{q}(𝔤)^{∘} generated by the c_{f,v}^{V (λ)} for λ ∈ L^{+}, f ∈ V (λ)^{∗}, and v ∈ V (λ). There is some redundancy in the notation O_{q}(G, L), since L is determined by G, but we prefer to emphasize L in this way. (Thus, the pair (G, L) is used as a label for that connected semisimple group with Lie algebra 𝔤 and weight lattice L.) The most commonly studied case corresponds to a simply connected group. This is the case L = P, and in this case we write O_{q}(G) for O_{q}(G, P).

When G = SL_{n}(ℂ), one obtains the algebra O_{q}(M_{n}(ℂ))/⟨D_{q} − 1⟩ as in (1.4) (e.g., [22, Theorem 1.4.1]; replace q^{2} by q). Thus, there is no ambiguity in the notation O_{q}(SL_{n}(ℂ)).

(c) In order to exhibit O_{q}(G, L) as a subalgebra of U_{q}(𝔤, M)^{∘}, one needs sufficient roots of q in k so that q^{(M,L)} ⊂ k. For λ ∈ L^{+}, there is then a simple finite dimensional U_{q}(𝔤, M)-module V (M, λ) with highest weight λ, where now the highest weight vector u_{λ} must satisfy k_{μ}u_{λ} = q^{(μ,λ)}u_{λ} for all μ ∈ M. Let us write O_{q}(G, L, M) for the k-subalgebra of U_{q}(𝔤, M)^{∘} generated by the coordinate functions of the V (M, λ) for λ ∈ L^{+}. Each V (M, λ) is also simple as a U_{q}(𝔤)-module; thus V (M, λ) becomes V (λ) by restriction, and coordinate functions of V (M, λ) map to coordinate functions of V (λ) by restriction. Therefore the restriction map U_{q}(𝔤, M)^{∘} → U_{q}(𝔤)^{∘} induces a k-algebra homomorphism O_{q}(G, L, M) → O_{q}(G, L). This map is an isomorphism, because

O_{q}(G, L, M) = ⨁_{λ∈L^{+}} C^{V (M,λ)}  and  O_{q}(G, L) = ⨁_{λ∈L^{+}} C^{V (λ)}

where C^{V (M,λ)} and C^{V (λ)} denote the k-linear spans of the coordinate functions on V (M, λ) and V (λ), respectively (cf. [29, §2.2], [30, §1.4.13], [24, §3.3]). One also needs the fact that C^{V (M,λ)} and C^{V (λ)} have the same k-di-mension, since both are isomorphic to V (λ) ⊗ V (λ)^{∗}.

For the reasons sketched above, we can – and do – choose to work with O_{q}(G, L). In [24, §3.3], for instance, the corresponding algebra – there denoted ℂ_{q}[G] – is defined as O_{q}(G, L, L).

(d) Multiparameter versions of O_{q}(G, L) are obtained by twisting the multiplication. Let p : L × L → k^{×} be an alternating bicharacter, that is,

p(λ, λ) = 1
p(μ, λ) = p(λ, μ)^{−1}
p(λ, μ + μ′) = p(λ, μ)p(λ, μ′)

for λ, μ, μ′ ∈ L. There is a natural (L × L)-bigrading on O_{q}(G, L) (cf. [24, §3.3]), and the multiplication on O_{q}(G, L) can be twisted via p as in [24, §2.1]; the new multiplication, call it ·, is determined by the rule

a · b = p(λ, λ′)p(μ, μ′)^{−1}ab for a ∈ O_{q}(G, L)_{λ,μ} and b ∈ O_{q}(G, L)_{λ′,μ′}.

The vector space O_{q}(G, L), equipped with this new multiplication, is called a multiparameter quantized coordinate ring of G, denoted O_{q,p}(G, L). In the case L = P, we write simply O_{q,p}(G).

1.6. Quantized enveloping algebras of Borel and nilpotent subalgebras. Each of the quantized enveloping algebras U_{q}(𝔤, M) contains subalgebras which are viewed as quantizations of the enveloping algebras of Borel or nilpotent subalgebras of 𝔤. Unlike U_{q}(𝔤, M), these algebras behave much like O_{q}(G), and they play prominent roles in the study of O_{q}(G). Hence, we include them here under the rubric of “algebras similar to quantized coordinate rings”, along with quantized Weyl algebras (see the following subsection).

Carry over the objects and notation from (1.5a), in particular the Lie algebra 𝔤, lattices Q ⊆ M ⊆ P, the powers q_{i} = q^{d_{i}} ≠ ±1, and the group algebra U^{0} ⊆ U_{q}(𝔤, M). Let 𝔟^{+} and 𝔫^{+} denote the positive Borel and nilpotent subalgebras of 𝔤 corresponding to the choices above. The single parameter quantized enveloping algebra of 𝔟^{+} associated with the above choices is the k-algebra U_{q}(𝔟^{+}, M) generated by U^{0} and e_{1}, …, e_{n}. The corresponding single parameter quantized enveloping algebra of 𝔫^{+} is the k-subalgebra U_{q}(𝔫^{+}) of U_{q}(𝔟^{+}, M) generated by e_{1}, …, e_{n}; this algebra is often denoted U_{q}(𝔤)^{+} (it is, of course, independent of the choice of M).

There is an (M × M)-bigrading on the algebra U_{q}(𝔟^{+}, M) such that k_{λ} ∈ U_{q}(𝔟^{+}, M)_{−λ,λ} for λ ∈ M and e_{i} ∈ U_{q}(𝔟^{+}, M)_{−α_{i},0} for i = 1, …, n (see [24, Corollary 3.3]). Obviously, U_{q}(𝔫^{+}) is a bigraded subalgebra. Given an alternating bicharacter p : M ×M → k^{×}, we can twist the multiplications in U_{q}(𝔟^{+}, M) and U_{q}(𝔫^{+}) via p just as for O_{q}(G, L) above. The resulting algebras are the multiparameter quantized enveloping algebras of 𝔟^{+} and 𝔫^{+}, denoted U_{q,p}(𝔟^{+}, M) and U_{q,p}(𝔫^{+}). As in (1.5a), we omit M from the notation in the case M = Q.

There are two cases in which a quantized Borel is known to be a homomorphic image of a quantum semisimple group. Namely, when q is transcendental over ℚ, the algebra U_{q}(𝔟^{+}, P) is isomorphic to a factor algebra of O_{q}(G) [30, Corollary 9.2.12], and for q ∈ ℂ not a root of unity, U_{q,p}(𝔟^{+}, M) is a homomorphic image of O_{q,p^{−1}}(G, M)^{op} [24, Proposition 4.6].

1.7. Quantized Weyl algebras. Recall that the Weyl algebra A_{n}(k) can (at least in characteristic zero) be viewed as the algebra of polynomial differential operators on affine n-space, i.e., the algebra of all linear partial differential operators with polynomial coefficients on the coordinate ring O(k^{n}). Quantized versions of A_{n}(k) arose in Maltsiniotis’ development of “quantum differential calculus” [36], and can be viewed as algebras of “partial q-difference operators” on quantum affine spaces.

Let Q = (q_{1}, …, q_{n}) be a vector in (k^{×})^{n}, and let Γ = (γ_{ij}) be a multiplicatively antisymmetric n×n matrix over k. The multiparameter quantized Weyl algebra of degree n over k is the k-algebra A_{n}^{Q,Γ}(k) generated by elements x_{1}, y_{1}, …, x_{n}, y_{n} subject only to the following relations:

y_{i}y_{j} = γ_{ij}y_{j}y_{i}  (all i, j)
x_{i}x_{j} = q_{i}γ_{ij}x_{j}x_{i}  (i < j)
x_{i}y_{j} = γ_{ji}y_{j}x_{i}  (i < j)
x_{i}y_{j} = q_{j}γ_{ji}y_{j}x_{i}  (i > j)
x_{j}y_{j} = 1 + q_{j}y_{j}x_{j} + ∑_{l<j}(q_{l} − 1)y_{l}x_{l}  (all j).

See [28, §2.9] for a discussion of how to view elements of A_{n}^{Q,Γ}(k) as q-dif-ference operators on O_{Γ}(k^{n}).

1.8. Quantum symplectic spaces. The classical linear, symplectic, and orthogonal groups all act on the same affine space, k^{n}. We have already mentioned above that the relations in quantum matrices are partly chosen so that quantum affine spaces become comodules over quantum matrices, in a “standard” manner. It follows that quantum affine spaces are also comodules over quantum special linear groups. In order to develop an analogous situation for quantum symplectic and orthogonal groups, one needs, as it turns out, different deformations of O(k^{n}) than quantum affine spaces. These new algebras are called quantum symplectic and Euclidean spaces; they have (to our knowledge) only been defined in single parameter versions to date.

Let q be a nonzero scalar in k and n a positive integer. Set

(1, 2, …, n, n′, (n − 1)′, …, 1′) = (1, 2, …, n, n + 1, …, 2n)
(ρ(1), ρ(2), …, ρ(2n)) = (n, n − 1, …, 1, −1, −2, …, −n)
(ϵ_{1}, …, ϵ_{n}, ϵ_{n+1}, …, ϵ_{2n}) = (1, …, 1, −1, …, −1).

The one-parameter coordinate ring of quantum symplectic 2n-space over k (cf. [53, Definition 14] or [43, §1.1]) is the k-algebra O_{q}(𝔰𝔭 k^{2n}) generated by elements x_{1}, …, x_{2n} satisfying the following relations:

x_{i}x_{j} = qx_{j}x_{i}  (i < j; j ≠ i′)
x_{i}x_{i′} = x_{i′}x_{i} + (1 − q^{2}) ∑_{l=1}^{i′−1} q^{ρ(i′)−ρ(l)}ϵ_{i′}ϵ_{l}x_{l}x_{l′}  (i ≤ n)

(cf. [53, p. 210]). A simpler set of relations was found by Musson [43, §1.1] (cf. [45, §1.1]):

x_{i}x_{j} = qx_{j}x_{i}  (i < j; j ≠ i′)
x_{i}x_{i′} = q^{2}x_{i′}x_{i} + (q^{2} − 1) ∑_{l=1}^{i−1} q^{l−i}x_{l}x_{l′}  (i ≤ n).

1.9. Quantum Euclidean spaces. Let n ≥ 2 be an integer and q a nonzero scalar in k, such that q has a square root in k in case n is odd. Set m = ⌊n/2⌋, the integer part of n/2, and set i′ = n + 1 − i for 1 ≤ i ≤ n. Further, set

(ρ(1), …, ρ(n))

= \{ \begin{matrix}(m − \frac{1}{2}, m − \frac{3}{2}, …, \frac{1}{2}, 0, −\frac{1}{2}, −\frac{3}{2}, …, −m + \frac{1}{2}) & (n odd) \\ (m − 1, m − 2, …, 1, 0, 0, −1, −2, …, −m + 1) & (n even). \end{matrix}

The one-parameter coordinate ring of quantum Euclidean n-space over k (cf. [53, Definition 12] or [43, §2.1]) is the k-algebra O_{q}(𝔬 k^{n}) generated by elements x_{1}, …, x_{n} satisfying the following relations:

x_{i}x_{j} = qx_{j}x_{i}  (i < j; j ≠ i′)
x_{i}x_{i′} = x_{i′}x_{i} + \frac{1 − q^{2}}{1 + q^{n−2}} (q^{n−2} ∑_{l=1}^{i′−1} q^{ρ(i′)−ρ(l)}x_{l}x_{l′} − ∑_{l=i′}^{n} q^{ρ(i′)−ρ(l)}x_{l}x_{l′})
(i ≤ m).

A simpler set of relations was given in [43, §§2.1, 2.2]:

x_{i}x_{j} = qx_{j}x_{i}  (i < j; j ≠ i′)
x_{i}x_{i′} = x_{i′}x_{i} + (1 − q^{2}) ∑_{l=i+1}^{m} q^{l−i−2}x_{l}x_{l′}  (i ≤ m; n even)

x_{i}x_{i′} = x_{i′}x_{i} + (1 − q)q^{m−i−(1/2)}x_{m+1}^{2} + (1 − q^{2}) ∑_{l=i+1}^{m} q^{l−i−2}x_{l}x_{l′}

(i ≤ m; n odd).

For an alternative presentation of O_{q}(𝔬 k^{n}), see [46, Example 5; 47, Definition 3.1].

2. Some common properties

We record some fundamental properties common to the algebras discussed in Section 1. The cases we list are those for which we have located references in which a result appears, or from which it follows readily. Many cases which we conjecture should be included must be omitted at present, because they have not (to our knowledge) been addressed in the literature. We leave it to the reader to identify appropriate “missing” cases, and to formulate the corresponding conjectures. Perhaps more important than filling in specific cases is the problem of developing general theorems which could verify properties for all these algebras simultaneously.

2.1. Noetherian domains.

Theorem. All of the following algebras are noetherian domains: O_{q}(k^{n}); O_{λ,p}(M_{m,n}(k)); O_{λ,p}(GL_{n}(k)); O_{λ,p}(SL_{n}(k)); O_{q,p}(G, L) for q transcendental over ℚ or q ∈ ℂ not a root of unity; U_{q,p}(𝔟^{+}); U_{q,p}(𝔫^{+}); A_{n}^{Q,Γ}(k); O_{q}(𝔰𝔭 k^{2n}); O_{q}(𝔬 k^{n}).

Proof. In the cases O_{q}(k^{n}); O_{λ,p}(M_{m,n}(k)); A_{n}^{Q,Γ}(k); O_{q}(𝔰𝔭 k^{2n}); O_{q}(𝔬 k^{n}), one has only to observe that the algebra is an iterated skew polynomial ring over k. This is clear for O_{q}(k^{n}); for the other cases, see [2, pp. 890-891], [28, §§2.1, 2.8], [43, §§1.2, 2.3]. It follows that the localization O_{λ,p}(GL_{n}(k)) is a noetherian domain, and that the factor algebra O_{λ,p}(SL_{n}(k)) is noetherian. That the latter is a domain is proved in [34, Corollary].

For O_{q}(G) in the stated cases, see [29, Lemma 3.1, Proposition 4.1] and [30, Lemma 9.1.9, Proposition 9.2.2]. (An alternate proof of noetherianity is given in [4, Corollary 5.6].) The desired properties of O_{q}(G, L) are proved by the same arguments, and they carry over to O_{q,p}(G, L) by graded ring methods [24, Remark, p. 80] or by twisting results [60, Propositions 5.1, 5.2].

By [10, Corollary 1.8], U_{q}(𝔤) is a domain, and so U_{q}(𝔟^{+}) and U_{q}(𝔫^{+}) are domains. There are several ways to see that U_{q}(𝔟^{+}) and U_{q}(𝔫^{+}) are noetherian. For instance, it follows from the existence of a PBW basis for U_{q}(𝔤) (e.g., [26, Theorem 4.21]) that U_{q}(𝔤) is free as a right U_{q}(𝔟^{+})-module and as a right U_{q}(𝔫^{+})-module. Hence, the poset of left ideals of either subalgebra embeds in the poset of left ideals of U_{q}(𝔤), via I ↦ U_{q}(𝔤)I. This shows that U_{q}(𝔟^{+}) and U_{q}(𝔫^{+}) are left noetherian. There exists an antiautomorphism τ on U_{q}(𝔤) such that τ(e_{i}) = e_{i} for i = 1, …, n and τ(k_{λ}) = k_{λ}^{−1} for λ ∈ Q (e.g., [26, Lemma 4.6]). Since τ obviously stabilizes U_{q}(𝔟^{+}) and U_{q}(𝔫^{+}), these subalgebras must be right as well as left noetherian.

On the other hand, in view of [10, Proposition 1.7], given either A_{0} = U_{q}(𝔟^{+}) or A_{0} = U_{q}(𝔫^{+}), there is a sequence of algebras A_{0}, A_{1}, …, A_{N} such that each A_{i+1} is the associated graded ring of A_{i} with respect to some ℤ^{+}- filtration, and such that A_{N} is an iterated skew polynomial ring over either U^{0} or k. Since A_{N} is noetherian, it follows from standard filtered/graded techniques (e.g., [39, Theorem 1.6.9]) that A_{0} is noetherian.

Finally, the twisting results mentioned above ([60, Propositions 5.1, 5.2]) imply that U_{q,p}(𝔟^{+}) and U_{q,p}(𝔫^{+}) are noetherian domains. □

2.2. Complete primeness of prime factors. For any set {α_{i}} of nonzero scalars in k, let ⟨α_{i}⟩ denote the multiplicative subgroup of k^{×} generated by the α_{i}.

Theorem. Let A be one of the following algebras: O_{q}(k^{n}) with ⟨q_{ij}⟩ torsionfree; O_{λ,p}(M_{m,n}(k)) or O_{λ,p}(GL_{n}(k)) or O_{λ,p}(SL_{n}(k)) with ⟨λ, p_{ij}⟩ torsionfree; O_{q}(G) or U_{q}(𝔟^{+}) with q ∈ ℂ not a root of unity; U_{q}(𝔫^{+}) with q transcendental over ℚ; A_{n}^{Q,Γ}(k) with ⟨q_{i}, γ_{ij}⟩ torsionfree; O_{q}(𝔰𝔭 k^{2n}) or O_{q}(𝔬 k^{n}) with q not a root of unity. Then all prime ideals of A are completely prime, i.e., all prime factor rings of A are domains.

Proof. The cases O_{q}(k^{n}); O_{λ,p}(M_{m,n}(k)); A_{n}^{Q,Γ}(k); O_{q}(𝔰𝔭 k^{2n}); O_{q}(𝔬 k^{n}) follow from a general result about prime ideals of certain iterated skew polynomial rings: [18, Theorem 2.3]. See [18, Theorem 2.1], [16, Theorem 5.1], [8, Proposition II.1.2], [43, Corollary 1.2, Theorem 2.3] for further details. The cases O_{λ,p}(GL_{n}(k)) and O_{λ,p}(SL_{n}(k)) follow immediately. In addition, U_{q}(𝔫^{+}) is an iterated q-skew polynomial ring over k (see [54, Section 5] for the case k = ℚ(q) and extend scalars), and so in this case also the result follows from [18, Theorem 2.3] – see [54, Corollary to Theorem 3]. Finally, the cases O_{q}(G) and U_{q}(𝔟^{+}) are proved in [29, Theorems 11.4, 11.5]. □

2.3. Division rings of fractions.

Theorem. Let A be one of the algebras O_{λ,p}(M_{m,n}(k)); U_{q}(𝔰𝔩_{n}(k))^{+} with q not a root of unity; O_{q}(G) or U_{q}(𝔟^{+}, P) or U_{q}(𝔫^{+}) with q ∈ ℂ not a root of unity; A_{n}^{Q,Γ}(k); O_{q}(𝔰𝔭 k^{2n}); O_{q}(𝔬 k^{n}). Then Fract A ≅ Fract O_{q}(K^{t}) for some q, some field K ⊇ k, and some t.

Now let A be either O_{q}(M_{n}(k)) with q not a root of unity, or A_{n}^{Q,Γ}(k) with ⟨q_{i}, γ_{ij}⟩ torsionfree. If P is any prime ideal of A, then Fract(A/P) ≅ Fract O_{q}(K^{t}) for some q, some field K ⊇ k, and some t.

Proof. For the first set of cases, see [42, Theorem 1.24], [1, Théorème 2.15], [25, Theorem 3.5], [7, Theorems 3.3, 3.1], [28, §3.1], [43, Theorems 1.3, 2.4]. The case of Fract O_{q}(M_{n}(k)) = Fract O_{q}(GL_{n}(k)) was treated earlier in [9, Proposition 5] and [48, Theorem 3.8]. The case of U_{q}(𝔫^{+}) for q transcendental over ℂ was also obtained in [31]. For the cases of prime factors of O_{q}(M_{n}(k)) and A_{n}^{Q,Γ}(k), see [8, Théorèmes II.2.1, III.3.2.1]. □

2.4. Catenarity. Recall that (the prime spectrum of) a ring A is catenary if for any comparable prime ideals P ⊃ Q in A, all saturated chains of prime ideals from P to Q have the same length. Our discussion of this property will involve the following concept. The ring A, or its prime spectrum spec A, is said to have normal separation provided the following condition holds: For any proper inclusion P ⊃ Q of prime ideals of A, the factor P/Q contains a nonzero element which is normal in A/Q. In other words, there must exist an element c ∈ P \Q such that c+Q is normal in A/Q; we refer to the latter condition by saying that c is normal modulo Q.

Theorem. The following algebras are catenary: O_{q}(k^{n}); O_{q}(GL_{n}(ℂ)) and O_{q}(SL_{n}(ℂ)) with q ∈ ℂ not a root of unity; U_{q}(𝔫^{+}) with q transcendental over ℚ; A_{n}^{Q,Γ}(k) with no q_{i} a root of unity; O_{q}(𝔰𝔭 k^{2n}) and O_{q}(𝔬 k^{n}) with q not a root of unity.

Proof. These cases all follow from a general theorem based on Gabber’s methods: If A is an affine, noetherian, Auslander-Gorenstein, Cohen-Macaul-ay algebra with finite Gelfand-Kirillov dimension, and if spec A is normally separated, then A is catenary [17, Theorem 1.6]. See [17, Theorems 2.6, 3.13, 4.5, 4.8] and [46, Corollaries 12, 13] for the individual cases listed. □

2.5. Normal separation. As indicated in (2.4), normal separation plays a key role in proving catenarity. Normal separation in a ring A also implies Jategaonkar’s strong second layer condition (cf. [27, Theorem 3.3.16, Proposition 8.1.7, discussion p. 225]), from which it follows, in particular, that every module has a finite filtration such that associated primes in adjacent layers are linked in spec A (cf. [27, Theorem 9.1.2]). Other influences of the second layer condition on the representation theory of A are discussed in [5] and [27]. Finally, we mention that normal separation, in conjunction with several other properties typical of quantum coordinate rings, leads to a description of links and cliques in terms of lattices of automorphisms [3, Section 3].

Theorem. The following algebras have normal separation: O_{q}(k^{n}); O_{q}(G) with q transcendental over ℚ; O_{q,p}(G, L) with q ∈ ℂ not a root of unity; U_{q}(𝔟^{+}, P) and U_{q}(𝔫^{+}) over a rational function field ℂ(q); U_{q}(𝔟^{+}, P) with q transcendental over ℚ; U_{q,p}(𝔟^{+}, M) with q ∈ ℂ not a root of unity; A_{n}^{Q,Γ}(k) with no q_{i} a root of unity; O_{q}(𝔰𝔭 k^{2n}) and O_{q}(𝔬 k^{n}) with q not a root of unity.

Proof. For O_{q}(k^{n}), A_{n}^{Q,Γ}(k), O_{q}(𝔰𝔭 k^{2n}) and O_{q}(𝔬 k^{n}), see [17, Corollary 2.4, Theorem 3.12] and [46, Theorem 10]. The algebras U_{q}(𝔫^{+}) and U_{q}(𝔟^{+}, P) over ℂ(q) are actually polynormal, i.e., every ideal has a normalizing sequence of generators [6, Corollaires 3.2, 3.3].

Normal separation for A = O_{q}(G) or A = O_{q,p}(G, L) is a consequence of results in [30] and [24], as follows (cf. [3, Theorem 5.8]). In these cases, there is a partition

spec A = ⨆_{w∈W ×W} spec_{w} A,

where W is the Weyl group of G, described in [30, Corollary 9.3.9] and [24, Corollary 4.5]. For each w, there are an ideal I_{w} of A and an Ore set E_{w} of nonzero normal elements in A/I_{w} such that

spec_{w} A = {P ∈ spec A | P ⊇ I_{w} and (P/I_{w}) ∩ E_{w} = ∅}

[30, Proposition 10.3.2], [24, Theorem 4.4]. In particular, normal separation follows for primes P ⊃ Q such that Q ∈ spec_{w} A but P ∉ spec_{w} A. Further, the localization (A/I_{w})[E_{w}^{−1}] has central separation: for any primes P′ ⊃ Q′, there exists a central element in P′ \ Q′. In one of our cases, this follows from a sequence of results in [30], as shown in [3, §5.7]; in the other case, it is known that all ideals of (A/I_{w})[E_{w}^{−1}] are centrally generated [24, Theorem 4.15]. Central separation in (A/I_{w})[E_{w}^{−1}] together with normality of the elements of E_{w} yields normal separation for primes P ⊃ Q in the same spec_{w} A [3, Proposition 1.7]. An alternate proof of normal separation for O_{q}(G), based on Hopf algebra technology, is given in [33, Proposition 2.4].

Finally, the second U_{q}(𝔟^{+}, P) case and the U_{q,p}(𝔟^{+}, M) case follow from the cases of O_{q}(G) and O_{q,p}(G, L) (see the last paragraph of (1.6)). □

2.6. The Dixmier-Moeglin equivalence. Recall that a prime ideal P of a noetherian k-algebra A is rational provided the center of Fract(A/P) is algebraic over k. Recall also that P is locally closed in spec A provided the singleton {P } is closed in some Zariski-neighborhood, i.e., the intersection of the primes properly containing P is larger than P. One says that the algebra A satisfies the Dixmier-Moeglin equivalence if the sets of rational prime ideals, locally closed prime ideals, and primitive ideals all coincide (cf. [51]). The advantage of this equivalence is that when it holds, the primitive ideals of A can be identified without constructing any irreducible representations.

Theorem. The Dixmier-Moeglin equivalence holds in the following algebras: O_{q}(k^{n}); O_{λ,p}(M_{m,n}(k)); O_{λ,p}(GL_{n}(k)); O_{λ,p}(SL_{n}(k)); O_{q}(G) with q transcendental over ℚ and k algebraically closed; O_{q,p}(G, L) with q ∈ ℂ not a root of unity; U_{q}(𝔟^{+}, P) with q transcendental over ℚ; U_{q,p}(𝔟^{+}, M) with q ∈ ℂ not a root of unity; A_{n}^{Q,Γ}(k) with the q_{i} not roots of unity; O_{q}(𝔰𝔭 k^{2n}) and O_{q}(𝔬 k^{n}) with q not a root of unity.

Proof. For O_{q}(k^{n}), O_{λ,p}(M_{m,n}(k)), A_{n}^{Q,Γ}(k), O_{q}(𝔰𝔭 k^{2n}) and O_{q}(𝔬 k^{n}), see [19, Corollary 2.5, Theorem 3.2] and [20, Theorems 5.3, 5.5, 5.8, 5.11]. The cases O_{λ,p}(GL_{n}(k)) and O_{λ,p}(SL_{n}(k)) follow directly. The case O_{q,p}(G, L) follows from the results of [24] exactly as in the case O_{q}(SL_{n}(ℂ)), which was given explicitly in [23, Theorem 4.2]. The case O_{q}(G) follows from results in [30], as noted in [20, §2.4]. Finally, the cases U_{q}(𝔟^{+}, P) and U_{q,p}(𝔟^{+}, M) follow from the previous cases. □

3. Stratified spectra

In this section, we give a more extensive discussion of another feature common to many quantized coordinate algebras – a stratification of the prime spectrum in which each stratum is a classical scheme, in fact the prime spectrum of a (commutative) Laurent polynomial algebra. Such a stratification was first discovered in spec O_{q}(SL_{3}(ℂ)) by Hodges and Levasseur [22]; they soon extended it to O_{q}(SL_{n}(ℂ)) [23]. This was generalized by Joseph to O_{q}(G) [29, 30], and finally extended to O_{q,p}(G, L) by Hodges, Levasseur, and Toro [24]. In all these cases, the result followed from a long sequence of involved calculations specific to the algebra at hand. Later, it was noticed that some features of these stratifications could be tied to the action of a torus of automorphisms (cf. [3, Section 5 and Proposition 1.9]), and similar stratifications were observed in other quantized algebras (e.g., [3, Section 4], [19, §2.1, Theorem 2.3]). A general development of this type of stratification was begun in [20].

We begin with an outline of the main features of the stratification of spec O_{q,p}(G, L), followed by a discussion of the axiomatic framework into which this stratification nicely fits. Then we exhibit appropriate tori of automorphisms for the algebras from Section 1, and indicate the cases in which the details of the stratification have been worked out.

3.1. Let A be either the algebra O_{q}(G) with q transcendental over ℚ, or O_{q,p}(G, L) with q ∈ ℂ not a root of unity. As noted in (2.5), there is a partition

spec A = ⨆_{w∈W ×W} spec_{w} A,

where W is the Weyl group of G, described in [30, Corollary 9.3.9] and [24, Corollary 4.5]. For each w, there are an ideal I_{w} of A and an Ore set E_{w} of nonzero normal elements in A/I_{w} such that

spec_{w} A = {P ∈ spec A | P ⊇ I_{w} and (P/I_{w}) ∩ E_{w} = ∅}.

In fact, E_{w} is finitely generated, and so if e_{w} is the product (in some order) of a finite set of generators for E_{w}, we can write

spec_{w} A = {P ∈ spec A | P ⊇ I_{w}} ∩ {P ∈ spec A | e_{w} ∉ P },

an intersection of a closed and an open set. Thus spec_{w} A is a locally closed subset of spec A. Let A_{w} = (A/I_{w})[E_{w}^{−1}]. In the case of O_{q,p}(G, L) over ℂ, the center Z(A_{w}) is a Laurent polynomial ring over k [24, Theorem 4.14], the ideals of A_{w} are centrally generated [24, Theorem 4.15], and it follows that spec_{w} A is homeomorphic to spec Z(A_{w}). That such a homeomorphism also exists in the case of O_{q}(G) over k ⊇ ℚ(q) is not proved in [30] (a Laurent polynomial subalgebra Z_{w} ⊆ Z(A_{w}) is studied instead [30, §10.3.3]), but this can be deduced from the results there together with what we shall prove in Part II.

If prim A denotes the primitive spectrum of A (the set of primitive ideals, equipped with the Zariski topology), we obtain a corresponding partition

prim A = ⨆_{w∈W ×W} prim_{w} A,

where each prim_{w} A = prim A ∩ spec_{w} A is a locally closed subset of prim A. Further, prim_{w} A is precisely the set of maximal elements of spec_{w} A (this is given explicitly in [30, Theorem 10.3.7] for the first case, and is implicit in [24, Section 4] for the second; see also [3, Corollary 1.5]). Over an algebraically closed base field, a maximal torus H of G acts naturally as automorphisms of A [30, §10.3.8], [24, §3.3], and the sets prim_{w} A are precisely the H-orbits in prim A [30, Theorem 10.3.8], [24, Theorem 4.16].

3.2. The picture of spec A outlined in (3.1) can be conveniently organized in terms of the action of the group H. There is a unique minimal element J_{w} in spec_{w} A for each w ([30, Proposition 10.3.5]; implicit in [24, Section 4]), and it is easily checked that J_{w} is H-stable (cf. [3, §5.4]). Over an algebraically closed base field, it then follows that the J_{w} are precisely the H-stable prime ideals of A, and that each spec_{w} A consists precisely of those prime ideals P such that the intersection of the H-orbit of P equals J_{w} [3, Proposition 1.9].

These observations provide a framework that can be set up with respect to any group of automorphisms of any ring, as follows. As we shall see later, the special properties of this framework in the cases discussed in (3.1) appear in many other algebras as well.

3.3. H-stratifications. Let H be a group acting as automorphisms on a ring A. Recall that an H-prime ideal of A is any proper H-stable ideal J of A such that a product of H-stable ideals is contained in J only when one of the factors is contained in J. We shall write H-spec A for the set of H-prime ideals of A. For any ideal I of A, let (I : H) denote the intersection of the H-orbit of I, that is,

(I : H) = ⋂_{h∈H} h(I).

Alternatively, (I : H) is the largest H-stable ideal of A contained in I.

Observe that if P is a prime ideal of A, then (P : H) is an H-prime ideal. On the other hand, if A is noetherian and I is an H-prime ideal, it is easily checked that I is semiprime and that the prime ideals minimal over I form a single H-orbit (cf. [15, Remarks 4*,5*, p. 338]). In this case, I = (P : H) for any prime P minimal over I.

The H-stratum of any H-prime ideal J is the set

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

One defines H-strata in any subset of spec A by intersection with these strata. In particular, the H-stratum of J in prim A is the set

prim_{J} A = prim A ∩ spec_{J} A.

Since (P : H) ∈ H-spec A for each P ∈ spec A, the H-strata partition spec A, that is,

spec A = ⨆_{J∈H-spec A} spec_{J} A.

We refer to this partition as the H-stratification of spec A. Similarly, the partition of prim A into H-strata prim_{J} A is called the H-stratification of prim A.

3.4. Finite H-stratifications enjoy the minimal topological properties required of stratifications in algebraic and differential geometry, as follows. A stratification of an algebraic variety X is defined in [32, p. 56] to be a partition of X into a disjoint union of finitely many locally closed subsets (called strata) such that the closure of each stratum is a union of strata.

Lemma. Let H be a group acting as automorphisms on a ring A, and assume that H-spec A is finite.

(a) Each H-stratum in spec A is locally closed.

(b) The closure of each H-stratum in spec A is a union of H-strata.

(c) For each nonnegative integer d, the union of the H-strata corresponding to the elements of height at most d in H-spec A is open in spec A.

Proof. (a) Let J ∈ H-spec A, and let J′ be the intersection of the H-primes properly containing J (take J′ = A if J is maximal in H-spec A). Since J is H-prime and H-spec A is finite, J′ ⊋ J. Thus

spec_{J} A = {P ∈ spec A | P ⊇ J} ∩ {P ∈ spec A | P ⊉ J′}.

(b) If J ∈ H-spec A, the closure of spec_{J} A is just the set V (J) of those prime ideals of A containing J, and V (J) equals the union of the H-strata spec_{K} A as K runs over those H-primes containing J.

(c) Let Y be the set of elements of H-spec A of height greater than d, that is, those H-primes K from which there descends a chain K = K_{0} ⊋ K_{1} ⊋ ⋯ ⊋ K_{d+1} in H-spec A. The union of the H-strata spec_{J} A for J of height at most d in H-spec A is the set

{P ∈ spec A | P ⊉ ∩Y },

which is open in spec A. □

3.5. Tori of automorphisms. Whether or not our base field k is algebraically closed, we shall refer to any finite direct product of copies of the multiplicative group k^{×} as an (algebraic) torus. For each of the algebras A discussed in Section 1, there is a naturally occurring torus H acting as k-algebra automorphisms on A, as follows. The action of an element h ∈ H on an element a ∈ A will be denoted h.a; we leave it to the reader to verify in each case that there exist k-algebra automorphisms h.(−) as described, and that the rule h ↦ h.(−) is a group homomorphism from H to Aut A.

Many of these tori arise as groups of ‘winding automorphisms’ of a bialgebra or Hopf algebra, as follows. Suppose that A is a bialgebra over k, and let A^{∧} be the set of linear characters on A, that is, k-algebra homomorphisms A → k. This set forms a monoid under convolution, the counit of A being the identity element of A^{∧}. For χ ∈ A^{∧}, there are k-algebra endomorphisms θ_{χ}^{l} and θ_{χ}^{r} on A given by the rules

θ_{χ}^{l}(a) = ∑_{(a)} a_{1}χ(a_{2})  and  θ_{χ}^{r}(a) = ∑_{(a)} χ(a_{1})a_{2}.

When χ is invertible in A^{∧}, say with inverse χ′, the endomorphisms θ_{χ}^{l} and θ_{χ}^{r} are automorphisms of A, with inverses θ_{χ′}^{l} and θ_{χ′}^{r} respectively, and we refer to them as the left and right winding automorphisms of A associated with χ. The maps χ ↦ θ_{χ}^{l} and χ ↦ θ_{χ}^{r} are a homomorphism and an antihomomorphism, respectively, from the group of units of A^{∧} to Aut A. The facts outlined above are well known when A is a Hopf algebra, in which case A^{∧} is a group under convolution (see, e.g., [30, §1.3.4]).

If C is a right comodule algebra over the bialgebra A (e.g., see [41, Definition 4.1.2]), with structure map C → C ⊗ A written as c ↦ ∑_{(c)} c_{0} ⊗ c_{1}, then each (invertible) character χ ∈ A^{∧} induces a k-algebra endomorphism (automorphism) of C according to the rule c ↦ ∑_{(c)} c_{0}χ(c_{1}). The analogous statements hold for a left comodule algebra over A.

1. Quantum affine spaces. For A = O_{q}(k^{n}), take H = (k^{×})^{n} and

(α_{1}, …, α_{n}).x_{i} = α_{i}x_{i}

for i = 1, …, n. These automorphisms can be viewed as winding automorphisms arising when A is made into a right comodule algebra over the bialgebra O_{λ,q}(M_{n}(k)), or into a left comodule algebra over O_{λ,λ^{−1}q}(M_{n}(k)).

2. Quantum matrices. For the algebra A = O_{λ,p}(M_{m,n}(k)), take H = (k^{×})^{m} × (k^{×})^{n} and

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

for i = 1, …, m and j = 1, …, n. In the case m = n, the automorphisms above are compositions of left and right winding automorphisms. Namely, for γ = (γ_{1}, …, γ_{n}) ∈ (k^{×})^{n} there is a character χ_{γ} ∈ A^{∧} such that χ_{γ}(X_{ij}) = δ_{ij}γ_{i}X_{ij} for all i, j, and the automorphism displayed above can be expressed as θ_{χ_{β}}^{l}θ_{χ_{α}}^{r} where β = (β_{1}, …, β_{n}) and α = (α_{1}, …, α_{n}). The map γ ↦ χ_{γ} embeds the torus (k^{×})^{n} in the group of units of A^{∧}; it is an isomorphism when λ and p are sufficiently generic.

3. Quantum general linear groups. For A = O_{λ,p}(GL_{n}(k)), we can take the torus H = (k^{×})^{n} × (k^{×})^{n} acting on O_{λ,p}(M_{n}(k)) as above, and extend the given automorphisms from O_{λ,p}(M_{n}(k)) to A, because D_{λ,p} is an H-eigenvector.

4. Quantum special linear groups. For A = O_{λ,p}(SL_{n}(k)) (assuming a suitable choice of λ, p so that D_{λ,p} is central), we must take a subgroup of the torus used for O_{λ,p}(M_{n}(k)), namely the stabilizer of D_{λ,p}:

H = {(α_{1}, …, α_{n}, β_{1}, …, β_{n}) ∈ (k^{×})^{n} × (k^{×})^{n} | α_{1}α_{2} ⋯ α_{n}β_{1}β_{2} ⋯ β_{n} = 1}.

The automorphisms of O_{λ,p}(M_{n}(k)) corresponding to elements of H all fix D_{λ,p} − 1 and therefore induce automorphisms of A.

5. Quantum semisimple groups. Let A = O_{q,p}(G, L) as described in (1.5). In the case k = ℂ, one works with a maximal torus of G with Lie algebra 𝔥 and character group L [24, §3.1]. In the general case, it is simplest to take the set H of group homomorphisms L → k^{×}; this is a group under pointwise multiplication, and it is a torus because L is a lattice. An action of H on A derives from an embedding of H into the character group A^{∧}. Namely, there is a monic group homomorphism ι : H → A^{∧} such that

ι(h)(c_{f,v}^{V (λ)}) = h(μ)f(v)

for λ ∈ P^{+}, μ ∈ P, v ∈ V (λ)_{μ}, f ∈ V (λ)^{∗} (see [24, §3.3]).

Therefore there is a group homomorphism H → Aut A such that h ↦ θ_{ι(h)}^{l}, which provides an action of H on A. This is the action used in [30] and [24]. For some purposes, however, an action of H × H is needed, given by (h_{1}, h_{2}) ↦ θ_{ι(h_{1})}^{l}θ_{ι(h_{2})}^{r} (see [3, §5.2]).

6. Quantized enveloping algebras of Borel and nilpotent subalgebras. For A = U_{q,p}(𝔫^{+}), take H = (k^{×})^{n} and

(α_{1}, …, α_{n}).e_{i} = α_{i}e_{i}

for i = 1, …, n. For A = U_{q,p}(𝔟^{+}, M), let H_{M} be the group of group homomorphisms M → k^{×} under pointwise multiplication and take H = H_{M} × (k^{×})^{n}, with

(h, α_{1}, …, α_{n}).k_{μ} = h(μ)k_{μ}
(h, α_{1}, …, α_{n}).e_{i} = α_{i}e_{i}

for μ ∈ M and i = 1, …, n.

7. Quantized Weyl algebras. For A = A_{n}^{Q,Γ}(k), take H = (k^{×})^{n} and

(α_{1}, …, α_{n}).x_{i} = α_{i}x_{i}
(α_{1}, …, α_{n}).y_{i} = α_{i}^{−1}y_{i}

for i = 1, …, n.

8. Quantum symplectic spaces. For A = O_{q}(𝔰𝔭 k^{2n}), take H = (k^{×})^{n} and

(α_{1}, …, α_{n}).x_{i} = α_{i}x_{i}
(α_{1}, …, α_{n}).x_{i′} = α_{i}^{−1}x_{i′}

for i = 1, …, n.

9. Quantum Euclidean spaces. For A = O_{q}(𝔬 k^{n}), set m = ⌊n/2⌋ and take H = (k^{×})^{m}, with

(α_{1}, …, α_{m}).x_{i} = α_{i}x_{i}  (i = 1, …, m)
(α_{1}, …, α_{m}).x_{i′} = α_{i}^{−1}x_{i′}  (i = 1, …, m)
(α_{1}, …, α_{m}).x_{m+1} = x_{m+1}  (if n is odd)

3.6. We close this part of the paper by recording cases in the literature where the prime spectrum of an algebra A from Section 1, stratified by a torus H from (3.5), fits into the pattern discussed in (3.1) and (3.2). Although the results quoted from [20] do not provide Ore sets of \underline{normal} elements, they can be modified to do so in all but one of the cases of interest here, as we shall show in Part II.

Theorem. Let A be one of the following algebras: O_{q}(k^{n}); O_{λ,p}(M_{m,n}(k)) or O_{λ,p}(GL_{n}(k)) or O_{λ,p}(SL_{n}(k)) with λ not a root of unity; O_{q}(G) with q transcendental over ℚ and k algebraically closed; O_{q,p}(G, L) with q ∈ ℂ not a root of unity; A_{n}^{Q,Γ}(k) with the q_{i} not roots of unity; O_{q}(𝔰𝔭 k^{2n}) or O_{q}(𝔬 k^{n}) with q not a root of unity. Let H be the corresponding torus acting on A as in (3.5). Then H-spec A is finite, and for each J ∈ H-spec A the following hold:

(a) There exists an Ore set E_{J} of regular elements in A/J such that spec_{J} A is homeomorphic to spec Z((A/J)[E_{J}^{−1}]) via localization and contraction.

(b) Z((A/J)[E_{J}^{−1}]) is a commutative Laurent polynomial algebra over an extension field of k.

(c) prim_{J} A equals the set of maximal elements of spec_{J} A.

(d) If k is algebraically closed, then prim_{J} A consists of a single H-orbit.

Proof. For O_{q}(k^{n}), see [19, Theorem 2.3, Proposition 2.11, Corollary 1.5]. For O_{λ,p}(M_{m,n}(k)), A_{n}^{Q,Γ}(k), O_{q}(𝔰𝔭 k^{2n}) and O_{q}(𝔬 k^{n}), see [20, Theorems 5.3, 5.5, 5.8, 5.11, 6.6, 6.8]. The cases O_{λ,p}(GL_{n}(k)) and O_{λ,p}(SL_{n}(k)) follow directly. For O_{q}(G) and O_{q,p}(G, L), see (3.1) and (3.2). □

II. An Axiomatic Development of Stratification Properties

This part of the paper is a variation on a theme developed in joint work with E. S. Letzter [20]. The goal is to show that, in the case of a torus H acting rationally on a noetherian k-algebra A, the H-stratification of spec A enjoys analogs of all the properties of the stratification presented in Theorem 3.6, under relatively mild hypotheses on A and H. Both developments (here and in [20]) include – at present – one hypothesis which requires substantial work to verify in the examples of interest. In [20, Section 6], that hypothesis is the complete primeness of the H-prime ideals of A. Here, we rely on a type of normal separation in H-spec A. One disadvantage is the “loss” of one example, namely O_{λ,p}(M_{m,n}(k)), for which normal separation has not been verified. (It is conjectured to hold, and is easily checked for the case m = n = 2.) However, the development here provides a better fit with the work of Hodges-Levasseur, Joseph, and Hodges-Levasseur-Toro on O_{q}(G) and O_{q,p}(G, L) in which the stratified picture of spec A first arose. For example, the localizations (A/J)[E_{J}^{−1}] in the present development coincide with the ones obtained in [22, 23, 29, 30, 24] in the appropriate cases, whereas the localizations used in [20, Section 6] are larger. Here we also obtain a few additional properties, namely that (A/J)[E_{J}^{−1}] is an affine k-al-gebra in appropriate circumstances, and that normal separation in H-spec A by H-eigenvectors implies normal separation in spec A.

The development of our theme is based on the equivalence between rational actions of (k^{×})^{r} and ℤ^{r}-gradings (see (5.1) for details). We begin by working out the basic properties of a stratification relative to gradedprime ideals for ℤ^{r}-graded rings satisfying an appropriate normal separation condition. This may have some independent interest in the context of groupgraded rings. Since it is at this stage where our variation differs in certain aspects from that in [20, Section 6], we provide full details. This is the content of Section 4. In Section 5, we translate these results into the context of (k^{×})^{r}-actions. That process is essentially the same as in [20, Section 6].

4. Graded-normal separation

Here we build a stratification in the context of rings graded by a free abelian group, with a normal separation hypothesis. This hypothesis will allow us to construct certain localizations which are graded-simple (i.e., have no nontrivial homogeneous ideals). Hence, we begin by analyzing the prime spectra of some graded-simple rings.

Lemma 4.1. Let R be a graded-simple ring graded by an abelian group G.

(a) The center Z(R) is a homogeneous subring of R, strongly graded by the subgroup G_{Z} = {x ∈ G | Z(R)_{x} ≠ 0} of G.

(b) Every nonzero homogeneous element of Z(R) is invertible.

(c) As Z(R)-modules, Z(R) is a direct summand of R.

(d) Suppose that G_{Z} is a free abelian group of finite rank. Choose a basis {g_{1}, …, g_{n}} for G_{Z}, and choose a nonzero element z_{j} ∈ Z(R)_{g_{j}} for each j. Then Z(R) = Z(R)_{1}[z_{1}^{±1}, …, z_{n}^{±1}], a Laurent polynomial ring over the field Z(R)_{1}.

Proof. It is obvious that Z(R) is a homogeneous subring of R, graded by the set G_{Z}. Part (b) follows from the graded-simplicity of R, and then it is clear that G_{Z} is a subgroup of G and that Z(R) is strongly G_{Z}-graded. This proves part (a). Part (d) is routine (see the proof of [20, Lemma 6.3(c)]).

Note that S = ⨁_{x∈G_{Z}} R_{x} is a homogeneous subring of R containing Z(R). Further, S is a left S-module direct summand of R, with complement M = ⨁_{x∈G\G_{Z}} R_{x}. On the other hand, as in [20, Lemma 6.3(c)], S is a free Z(R)-module with a basis including 1, whence Z(R) is a Z(R)-module direct summand of S. Part (c) follows. □

Proposition 4.2. Let R be a graded-simple ring graded by an abelian group G. Then there exist bijections between the sets of ideals of R and Z(R), given by contraction and extension, that is,

I ↦ I ∩ Z(R)  and  J ↦ JR.

Proof. It is clear from Lemma 4.1(c) that JR ∩ Z(R) = J for every ideal J of Z(R). It remains to show that (I ∩ Z(R))R = I for any ideal I of R. Set J = I ∩ Z(R), and suppose that I ≠ JR. Pick an element x ∈ I \ JR of minimal length, say length n. Then x = x_{1} + ⋯ + x_{n} for some nonzero elements x_{i} ∈ R_{g_{i}}, where g_{1}, …, g_{n} are distinct elements of G.

Now Rx_{1}R is a nonzero homogeneous ideal of R, so Rx_{1}R = R by gradedsimplicity and ∑_{j} a_{j}x_{1}b_{j} = 1 for some a_{j}, b_{j} ∈ R. Express each a_{j}, b_{j} as a sum of homogeneous elements, and substitute these expressions in the terms a_{j}x_{1}b_{j}. This expands the sum ∑_{j} a_{j}x_{1}b_{j} in the form ∑_{t} c_{t}x_{1}d_{t} with all c_{t}, d_{t} homogeneous. Hence, after relabelling, we may assume that the a_{j} and b_{j} are all homogeneous, say of degrees e_{j} and f_{j}, respectively.

Comparing identity components in the equation ∑_{j} a_{j}x_{1}b_{j} = 1, we obtain

∑_{e_{j}g_{1}f_{j}=1} a_{j}x_{1}b_{j} = 1.

Thus, after deleting all other a_{j}x_{1}b_{j} terms, we may assume that e_{j}g_{1}f_{j} = 1 for all j. Since G is abelian, it follows that e_{j}f_{j} = g_{1}^{−1} for all j.

Set x′ = ∑_{j} a_{j}xb_{j}, and note that x′ ∈ I. Further,

x′ = ∑_{i=1}^{n} ∑_{j} a_{j}x_{i}b_{j}

where ∑_{j} a_{j}x_{i}b_{j} ∈ R_{g_{1}^{−1}g_{i}} for each i. As a result, x′ is an element whose support is contained in {1, g_{1}^{−1}g_{2}, …, g_{1}^{−1}g_{n}} and whose identity component is 1. Hence, x_{1}x′ is an element of I whose support is contained in {g_{1}, …, g_{n}} and whose g_{1}-component is x_{1}. Comparing this element with x, we see that x−x_{1}x′ is an element of I whose support is contained in {g_{2}, …, g_{n}}. By the minimality of n, we must have x − x_{1}x′ ∈ JR, and so x′ ∉ JR. Therefore, after replacing x by x′, there is no loss of generality in assuming that g_{1} = 1 and x_{1} = 1.

\underline{Claim}: There do not exist g ∈ G and a nonzero element y ∈ I whose support is properly contained in the set {gg_{1}, …, gg_{n}}.

Suppose there do exist such g and y. Say the gg_{s}-component of y is nonzero. As above, there exists an element of the form y′ = ∑_{j} a_{j}yb_{j} whose support is properly contained in {g_{s}^{−1}g_{1}, …, g_{s}^{−1}g_{n}} and whose identity component is 1. Then y′ ∈ I, and x − x_{s}y′ is an element of I whose support is properly contained in {g_{1}, …, g_{n}}. Since y′ and x − x_{s}y′ are elements of I of length less than n, they must lie in JR, by the minimality of n. But then x ∈ JR, contradicting our assumptions. Therefore the claim is proved.

Finally, consider any homogeneous element r ∈ R, say r ∈ R_{g}. Then rx − xr is an element of I with support contained in {gg_{2}, …, gg_{n}}, and so rx − xr = 0 by the claim. It follows that x ∈ Z(R) and so x ∈ J, contradicting our assumption that x ∉ JR.

Therefore I = JR. □

Corollary 4.3. If R is a graded-simple ring graded by an abelian group, then contraction and extension provide mutually inverse homeomorphisms between spec R and spec Z(R). □

4.4. Graded-normal separation. Let R be a group-graded ring. Recall that a graded-prime ideal of R is any proper homogeneous ideal P such that whenever I, J are homogeneous ideals of R with IJ ⊆ P, then either I ⊆ P or J ⊆ P. We say that R has graded-normal separation provided that for any proper inclusion P ⊃ Q of graded-prime ideals of R, there exists a homogeneous element c ∈ P \ Q which is normal modulo Q.

Theorem 4.5. Let R be a right noetherian ring graded by an abelian group G, and assume that R has graded-normal separation.

Let J be a graded-prime ideal of R, set

S_{J} = {P ∈ spec R | J is the largest homogeneous ideal contained in P },

and let E_{J} be the multiplicative set of all the nonzero homogeneous normal elements in the ring R/J.

(a) E_{J} is a right and left denominator set of regular elements in R/J.

(b) The localization map R → R/J → R_{J} = (R/J)[E_{J}^{−1}] induces a homeomorphism of S_{J} onto spec R_{J}.

(c) Contraction and extension induce mutually inverse homeomorphisms between spec R_{J} and spec Z(R_{J}).

(d) If G is free abelian of rank r < ∞, then Z(R_{J}) is a commutative Laurent polynomial ring over the field Z(R_{J})_{1} (the identity component of Z(R_{J}) in the induced G-grading), in r or fewer indeterminates.

Proof. Without loss of generality, J = 0. Consequently, R is now a gradedprime ring, i.e., 0 is a graded-prime ideal of R.

(a) For x ∈ E_{J}, observe that r.ann_{R}(x) is a homogeneous ideal such that (xR) r.ann_{R}(x) = 0. Since R is graded-prime and x ≠ 0, we must have r.ann_{R}(x) = 0. Similarly, l.ann_{R}(x)=0; thus all elements of E_{J} are regular. The Ore condition follows directly from normality.

(b) If P ∈ spec R and P_{0} is the largest homogeneous ideal contained in P, then P_{0} is a graded-prime ideal. Because of graded-normal separation, P_{0} is nonzero precisely when P_{0} ∩ E_{J} is nonempty. Hence, S_{J} consists precisely of those prime ideals of R disjoint from E_{J}. Standard localization theory (e.g., [21, Theorem 9.22]) now yields part (b).

(c) Since the elements of E_{J} are homogeneous, the G-grading on R extends (uniquely) to a G-grading on R_{J}. Because of the noetherian hypothesis on R, two-sided ideals I in R induce two-sided ideals IR_{J} in R_{J} (e.g., [21, Theorem 9.20(a)]). The standard arguments concerning contractions of prime ideals (e.g., [21, Theorem 9.20(c)]) now show that any graded-prime ideal of R_{J} must contract to a graded-prime ideal of R. Since every nonzero gradedprime ideal of R meets E_{J}, we thus see that R_{J} has no nonzero graded-prime ideals. On the other hand, any maximal proper homogeneous ideal of R_{J} is graded-prime. Therefore R_{J} must be graded-simple. Part (c) now follows from Corollary 4.3.

(d) This follows from Lemma 4.1(d). □

Corollary 4.6. Let R be a right noetherian ring graded by an abelian group G. If R has graded-normal separation, then R also has normal separation.

Proof. Let P ⊃ Q be a proper inclusion of prime ideals of R. Let P_{0} and Q_{0} be the largest homogeneous ideals contained in P and Q, respectively, and note that P_{0} and Q_{0} are graded-prime ideals of R. Obviously P_{0} ⊇ Q_{0}.

If P_{0} ≠ Q_{0}, then by assumption there exists a homogeneous element c ∈ P_{0} \ Q_{0} which is normal modulo Q_{0}. In particular, c ∈ P and c is normal modulo Q. Since RcR is a homogeneous ideal of R, not contained in Q_{0}, we see that RcR ⊈ Q. Thus in this case we are done.

Now assume that P_{0} = Q_{0}. It is harmless to pass to R/Q_{0}, and hence there is no loss of generality in assuming that P_{0} = Q_{0} = 0. Consequently, 0 is now a graded-prime ideal of R.

Let S denote the localization R_{0} = R[E_{0}^{−1}], in the notation of Theorem 4.5. By part (b) of the theorem, P S ⊃ QS is a proper inclusion of prime ideals of S. Part (c) then implies that there exists a central element z ∈ P S \ QS. Write z = cx^{−1} for some c ∈ P and x ∈ E_{0}, and note that c ∉ Q. Since xR = Rx, we have zxR = zRx in S. But z commutes with R, and so zxR = Rzx, that is, cR = Rc. Therefore c is a normal element of R, and normal separation is proved. □

5. Stratification under rational torus actions

We now turn to rational actions by tori as automorphisms of noetherian algebras. Under suitable normal separation hypotheses, the results of the previous section apply, yielding a picture of the corresponding stratifications that incorporates the features of the examples discussed in Section 3.

Throughout this section, we assume that our base field k is infinite. In the examples of interest, this is automatic due to the presence of non-roots of unity.

5.1. Rational torus actions. Let A be a k-algebra, and let H be a torus over k, say H = (k^{×})^{r}. An action of H on A by k-algebra automorphisms is said to be rational provided A is a directed union of finite dimensional H-stable subspaces V_{i} such that the induced group homomorphisms H → GL(V_{i}) are morphisms of algebraic varieties. Let Ĥ denote the set of rational characters of H, that is, algebraic group morphisms H → k^{×}. Then Ĥ is an abelian group under pointwise multiplication. Since k is infinite, Ĥ is a lattice of rank r, with a basis given by the component projections (k^{×})^{r} → k^{×}.

The rationality of an H-action implies that A is spanned by H-eigenvec-tors [44, Chapter 5, Corollary to Theorem 36], and that the eigenvalues of these eigenvectors are rational. This yields a k-vector space decomposition A = ⨁_{x∈Ĥ} A_{x}, where A_{x} denotes the x-eigenspace of A. (Up to this point, everything holds for a rational action by k-linear transformations.) Since H acts by automorphisms, A_{x}A_{y} ⊆ A_{xy} for all x, y ∈ Ĥ. Thus we have a grading of A by the free abelian group Ĥ, such that the homogeneous elements are the H-eigenvectors in A.

Conversely, any grading of A by ℤ^{r} arises in this fashion, as noted in [50, p. 784]. To see this, identify ℤ^{r} with Ĥ, and let A = ⨁_{x∈Ĥ} A_{x} be an Ĥ-grading. Let each h ∈ H act on A as the k-linear transformation with eigenvalue x(h) on A_{x} for all x, that is, h.a = x(h)a for a ∈ A_{x}. This yields a rational action of H on A by k-algebra automorphisms, such that each A_{x} is the x-eigenspace. The reader may take this as the definition of a rational H-action if so desired.

In case k is algebraically closed and A is noetherian, a remarkable theorem of Moeglin-Rentschler and Vonessen [40, Théorème 2.12(ii)], [59, Theorem 2.2] states that H acts transitively on the H-strata of rational ideals in A, that is, each H-stratum of rational ideals is a single H-orbit. This holds, in fact, for any algebraic group H, not just tori. If, further, the Dixmier-Moeglin equivalence holds, then each H-stratum in prim A is a single H-orbit. For the very special case where H is a torus and all H-prime ideals of A are completely prime, a direct proof of this transitivity result is given in [20, Theorem 6.8]. Here we shall derive such a result using normal separation in place of complete primeness – see Theorem 5.5(c).

5.2. Normal separation in H-spec. Assume that we have a torus H acting rationally on a k-algebra A by k-algebra automorphisms. In this situation, we shall assume that A has been equipped with the natural Ĥgrading as in (5.1). With respect to this grading, the homogeneous ideals of A are precisely the H-stable ideals, and so the graded-prime ideals coincide with the H-prime ideals. We say that H-spec A has normal H-separation provided that for any proper inclusion P ⊃ Q of H-prime ideals, there exists an H-eigenvector c ∈ P \ Q which is normal modulo Q. This condition is just graded-normal separation with respect to the Ĥ-grading as defined in (4.4).

The term ‘H-normal separation’ we would reserve for a slightly stronger condition involving ‘H-normal’ elements. If Q is an H-stable ideal of A, an element c ∈ A is said to be H-normal modulo Q in case there exists h ∈ H such that ca − h(a)c ∈ Q for all a ∈ A. It is easily checked that each of the homogeneous components of such an element c is also H-normal modulo Q. Now H-spec A has H-normal separation provided that for any proper inclusion P ⊃ Q of H-prime ideals of A, there exists an element c ∈ P \ Q which is H-normal modulo Q. This strengthening of normal H-separation is not needed in our proofs, but it does hold in almost all the examples discussed in Sections 2 and 3, and is conjectured to hold in the remaining one, namely O_{λ,p}(M_{n}(k)).

We note also that since the homogeneous ideals of A coincide with the Hstable ideals, the largest homogeneous ideal of A contained in a given ideal I is just (I : H). Hence, the strata of spec A with respect to graded-prime ideals, as in Theorem 4.5, coincide with the H-strata. Thus Theorem 4.5 and Corollary 4.6 yield the following results.

Theorem 5.3. Let A be a right noetherian k-algebra, and let H be a torus of rank r, acting rationally on A by k-algebra automorphisms. Assume that H-spec A has normal H-separation. Then spec A has normal separation.

Now let J be an H-prime ideal of A, let E_{J} be the multiplicative set of all nonzero normal H-eigenvectors in A/J, and set A_{J} = (A/J)[E_{J}^{−1}].

(a) The localization map A → A/J → A_{J} induces a homeomorphism of spec_{J} A onto spec A_{J}.

(b) Contraction and extension induce mutually inverse homeomorphisms between spec A_{J} and spec Z(A_{J}).

(c) The ring Z(A_{J}) is a commutative Laurent polynomial ring over the field Z(Fract A/J)^{H} (the fixed subfield of Z(Fract A/J) under the induced action of H), in r or fewer indeterminates.

Proof. These statements follow immediately from Theorem 4.5 and Corollary 4.6 except for the description of the coefficient field in part (c). According to Theorem 4.5(d), this field equals Z(A_{J})_{1}. By definition of the Ĥ-grading, Z(A_{J})_{1} = Z(A_{J})^{H}, which is clearly contained in Z(Fract A/J)^{H}. Thus, it only remains to prove the reverse inequality.

As in the proof of Theorem 4.5(c), A_{J} is graded-simple with respect to the Ĥ-grading, and so it is H-simple. Given any u ∈ Z(Fract A/J)^{H}, observe that the set I = {a ∈ A_{J} | au ∈ A_{J}} is a nonzero H-stable ideal of A_{J}. By H-simplicity, I = A_{J}, whence u ∈ A_{J}. Therefore u ∈ Z(A_{J})^{H}, as desired. □

5.4. To compare Theorem 5.3 with [20, Theorem 6.6], note that the latter theorem only applies to completely prime H-prime ideals of A. That restriction is due to the lack of any known graded Goldie theorem for graded-prime rings (cf. [20, §6.1]), whereas one can easily prove a graded Ore theorem [20, Lemma 6.2]. We have finessed the graded Goldie problem here by assuming a suitable supply of normal elements. Other advantages of this assumption include the following result.

Proposition. Let A, H, J, E_{J}, A_{J} be as in Theorem 5.3. If H-spec A is finite and A is an affine k-algebra, then A_{J} is an affine k-algebra.

Proof. After passing to A/J, we may assume that J = 0. If E_{J} consists of units, then A_{J} = A and we are done. Otherwise, A contains some proper nonzero H-stable ideals (generated by nonzero normal H-eigenvectors which are not units). Hence, the maximal proper H-stable ideals of A are nonzero, and these are, in particular, H-prime. In other words, 0 is not the only H-prime ideal of A.

Let J_{1}, …, J_{n} be the nonzero H-prime ideals of A. By our normal H-sep-aration assumption, each J_{i} contains a nonzero normal H-eigenvector c_{i}. Set c = c_{1}c_{2} ⋯ c_{n}, a normal H-eigenvector contained in all nonzero H-primes of A. Since J = 0 is H-prime, c ≠ 0.

The action of H on A extends naturally to an action by k-algebra automorphisms on the localization B = A[c^{−1}]. Since any H-prime of A[c^{−1}] contracts to an H-prime of A, we see that 0 is the only H-prime of B. Thus B is H-simple.

If e ∈ E_{J}, then since eA is an ideal of A and A is right noetherian, eB is an ideal of B. But eB is nonzero and H-stable, whence eB = B. Thus e is right invertible in B, and hence invertible. Therefore A_{J} = B, which is clearly affine. □

5.5. Recall that a noetherian k-algebra A satisfies the Nullstellensatz (over k) [39, Chapter 9] provided that A is a Jacobson ring and the endomorphism rings of all simple A-modules are algebraic over k. In that case, it follows from the Jacobson condition that every locally closed prime of A is primitive, and from [14, Lemma 4.1.6] that every primitive ideal of A is rational. As in [20, Section 6], the presence of the Nullstellensatz yields the following refinements to Theorem 5.3.

Theorem. Let A be a noetherian k-algebra satisfying the Nullstellensatz, and let H be a torus acting rationally on A by k-algebra automorphisms. Assume that H-spec A is finite and satisfies normal H-separation.

(a) A satisfies the Dixmier-Moeglin equivalence, and the primitive ideals of A are precisely the maximal elements of the H-strata in spec A.

(b) The fields Z(Fract A/J)^{H} occurring in Theorem 5.3(c) are all algebraic over k.

(c) If k is algebraically closed, each H-stratum in prim A consists of a single H-orbit.

Proof. Except for minor modifications, the proofs are the same as in [20, Theorem 6.8, Corollary 6.9].

Since H-spec A is finite, maximal elements of H-strata are locally closed in spec A [20, §2.2(ii)]. Hence, to prove part (a) it remains to show that rational ideals are maximal within their H-strata in spec A.

Let J ∈ H-spec A, and define E_{J} and A_{J} as in Theorem 5.3. Then Z(A_{J}) is a Laurent polynomial ring of the form k_{J}[z_{1}^{±1}, …, z_{n}^{±1}], where k_{J} = Z(Fract A/J)^{H}. By the previous paragraph, any maximal element of spec_{J} A is locally closed, and such primes are rational because of the Nullstellensatz. Hence, there exist rational ideals in spec_{J} A.

Now let P be any rational ideal in spec_{J} A, and set Q = P A_{J} ∩ Z(A_{J}). Then Z(A_{J})/Q embeds in Z(Fract A/P), which is algebraic over k by assumption. Hence, k_{J} is algebraic over k (thus establishing part (b)), and Q is a maximal ideal of Z(A_{J}). It follows that P is maximal in spec_{J} A, which completes part (a).

If k is algebraically closed, one checks that the induced action of H on Z(A_{J}) incorporates all automorphisms such that z_{1} ↦ β_{1}z_{1}, …, z_{n} ↦ β_{n}z_{n} for arbitrary β_{i} ∈ k^{×}. Hence, H acts transitively on max Z(A_{J}), and part (c) follows. □

5.6. Theorems 5.3 and 5.5 can be used to derive all cases of Theorem 3.6 except that of O_{λ,p}(M_{m,n}(k)). All the necessary hypotheses except for normal separation can be verified fairly readily from the basic descriptions of the algebras. Further, these hypotheses are also known to hold in O_{λ,p}(M_{m,n}(k)), where normal separation is conjectured. Thus, we conclude by focusing on normal separation as a key problem:

Problem. Find general hypotheses on a noetherian k-algebra A equipped with a rational action of a torus H by k-algebra automorphisms which

(a) imply that H-spec A is finite and satisfies normal H-separation;

(b) are readily proved for the algebras described in Section 1.

Acknowledgement

We thank Ken Brown, Iain Gordon, Tim Hodges, David Jordan and Thierry Levasseur for helpful correspondence and references.

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

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