THE QUANTUM COORDINATE RING OF THE SPECIAL LINEAR GROUP

T. Levasseur and J. T. Stafford

Département de Mathématiques, Université de Poitiers, 86022 Poitiers Cedex, France. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA.

ABSTRACT. We prove that, even under the multiparameter definition of [AST], the quantum coordinate ring O_{q}(SL_{n}(k)) of the special linear group SL_{n}(k) satisfies most of the standard ring-theoretic properties of the classical coordinate ring O(SL_{n}(k)).

Key words and phrases: Quantum group, integrality, noncommutative regular ring, Gorenstein conditions.

AMS Mathematics Subject Classification(1991): 17B37, 81R50, 16U10, 16E10.

Typeset by AMS-TEX 2 T. LEVASSEUR AND J. T. STAFFORD

The research of the second author was supported in part by an NSF grant

THE RESULTS.

Fix a field k. Let O_{q} = O_{q}(SL_{n}(k)) be the (multiparameter) quantum coordinate ring of the special linear group SL_{n}(k) and let M_{q} = O_{q}(M_{n}(k)) be the corresponding quantum coordinate ring of all n × n matrices, as defined in [AST]. (The definition of these and other concepts used in this introduction are given in the next section). By definition, O_{q} = M_{q}/(Δ_{q} − 1), where Δ_{q} is a central element in M_{q} called the “quantum determinate”. One would like to assert that the standard properties of the classical coordinate ring O(SL_{n}(k)), for example integrality and finite global homological dimension, also hold for O_{q}. This is particularly true since it is easy to show that these properties do hold for M_{q} (this follows from the fact that, as is proved in [AST, pp.890-1], M_{q} is an iterated Ore extension of k in the sense of [Co, Section 12.2]). However, it is typically hard (and in the abstract impossible) to show that such properties pass to factor rings. The main aim of this note is to make the following observation, giving a different method for obtaining O_{q} from M_{q}:

PROPOSITION. Set O_{q}(GL_{n}(k)) = O_{q}(M_{n}(k))[Δ_{q}^{−1}] and let z be a central indeterminate. Then

O_{q}(SL_{n}(k)) ⊗_{k} k[z, z^{−1}] ≅ O_{q}(GL_{n}(k)).

One should interpret this result as a “quantum” analogue of the well known fact that SL_{n}(k) × k^{∗} ≅ GL_{n}(k). Once stated, this Proposition is almost trivial to prove. Its significance, however, is that many desirable properties pass from a ring to a central localization. Thus, for example, the Proposition allows one to prove:

COROLLARY. (i) O_{q} is a domain and a maximal order in its division ring of fractions. (ii) GKdim(O_{q}) = gldim(O_{q}) = n^{2} − 1. (iii) O_{q} is Auslander regular and CM. (iv) K_{0}(O_{q}) = ℤ.

At least for the standard 1-parameter version of O_{q}, the fact that O_{q} is a domain can be proved in several other ways, but all of them seem to require considerably more knowledge about the structure of O_{q}. For example, for most values of the quantum parameter and for any classical group G, a proof that O_{q}(G) is a domain can be obtained by combining [En] and [Jo, Lemme 9.11 ], while for G = SL_{n} it also follows from the appendix to [APW]. Also, D. Jordan and the authors have shown independently that M_{q}/(Δ_{q}) is a domain, from which it follows that O_{q} is a domain.

THE QUANTUM COORDINATE RING OF THE SPECIAL LINEAR GROUP 3

THE PROOFS.

Given a ring C, write C^{∗} for the set of units of C. Let q = {λ, q_{ij} : 1 ≤ i, j ≤ n} ⊂ k^{∗} be fixed, non-zero scalars that satisfy λ ≠ −1 and q_{ij} = q_{ji}^{−1} and q_{ii} = 1, for all 1 ≤ i, j ≤ n. Define O_{q}(k^{(n)}) to be the k-algebra with generators {x_{i} : 1 ≤ i ≤ n} and relations x_{j}x_{i} = q_{ji}x_{i}x_{j}, for all 1 ≤ i, j ≤ n. Define p = {λ, p_{ij}} by p_{ij} = λ^{−1}q_{ij}, for all i > j, and, as before, p_{ji} = p_{ij}^{−1} and p_{ii} = 1. Then M_{q} = O_{q}(M_{n}(k)) is defined to be the universal bi-algebra having O_{q}(k^{(n)}) as a left comodule algebra and O_{p}(k^{(n)}) as a right comodule algebra, in the sense of [Ma, Section 5]. Thus, M_{q} is the k-algebra with generators {x_{ij} : 1 ≤ i, j ≤ n} and relations defined by [AST, Equ. (8)]. The precise definition of these relations is not important, here, except that they are of the following form:

x_{ij}x_{ℓm} = \{ \begin{matrix}α_{ijℓm}x_{ℓm}x_{ij} + (λ − 1)λ^{−1}q_{im}x_{ℓj}x_{im} & if i > ℓ and j > m \\ α_{ijℓm}x_{ℓm}x_{ij} & otherwise \end{matrix}  (1)

for some α_{ijℓm} ∈ k^{∗}. We remark that the restrictions on the scalars q and p given above are precisely what is required for M_{q} to have the same Hilbert series as a polynomial ring in n^{2} variables (see [AST, Theorem 1]). The universal argument of [Ma, Section 8] shows that there exists a quantum determinant

Δ_{q} = ∑_{π∈S_{n}} α_{π}x_{1,π(1)}x_{2,π(2)} ⋯ x_{n,π(n)},  (2)

where the α_{π} ∈ k^{∗} are certain scalars and are defined in [AST, Equ. (15)]. By [AST, Theorem 3],

Δ_{q} is central ⟺ λ^{j} ∏_{m=1}^{n} q_{jm} = λ^{k} ∏_{m=1}^{n} q_{km} for all j, k.  (3)

Thus, O_{q}(SL_{n}(k)) = M_{q}/(Δ_{q} − 1) is defined precisely when (3) holds. For k sufficiently large, this gives a \binom{n−1}{2} + 1 parameter family of deformations of O(SL_{n}(k)) (see [ GGS, Section 14]). If S = {Δ_{q}^{r} : r ≥ 1}, we define O_{q}(GL_{n}(k)) = (M_{q})_{S} = M_{q}[Δ_{q}^{−1}].

Finally, if C is a commutative k-algebra, we define O_{q}(C) = O_{q}(SL_{n}(k)) ⊗_{k} C and M_{q}(C) = O_{q}(M_{n}(k)) ⊗_{k} C. The results of this paper actually hold if k is taken to be any Noetherian, commutative domain (in which case it is unnecessary to define O_{q}(C)) but, in order to prove this, one first needs to prove the corresponding generalization of [AST].

The standard, one parameter quantum coordinate ring O_{q}(SL_{n}(k)) of SL_{n}(k), as for example defined in [Ma] or [PW], is obtained by taking λ = q^{2} and q_{ij} = q for all i > j.

4 T. LEVASSEUR AND J. T. STAFFORD

PROOF OF THE PROPOSITION. Let C be a commutative k-algebra and μ ∈ C^{∗}. Then, as an C-algebra, M_{q}(C) = O_{q}(M_{n}(k)) ⊗_{k} C is still defined by the relations given in (1). The important point to note about these relations is that they are homogeneous in the set of variables {x_{1j} : 1 ≤ j ≤ n}. In other words, if a given relation from (1) has r occurrences of elements from the set {x_{1j}} occurring in one monomial, then every monomial in that relation has r occurrences of elements from {x_{1j}}. Thus, there is an C-algebra automorphism σ_{μ} of M_{q}(C) defined by

σ_{μ}(x_{1j}) = μ^{−1}x_{1j} and σ_{μ}(x_{ij}) = x_{ij}, for all 1 ≤ j ≤ n and 2 ≤ i ≤ n.

By the description of Δ_{q} in (2), one sees that σ_{μ}(Δ_{q}) = μ^{−1}Δ_{q}. Now assume that C = k[z, z^{−1}], for an indeterminate z. Then:

O_{q}(SL_{n}(k))[z, z^{−1}] ≅ O_{q}(SL_{n}(k)) ⊗_{k} k[z, z^{−1}] ≅ M_{q} ⊗_{k} k[z, z^{−1}]/(Δ_{q} − 1)
σ_{z} ; ≅ M_{q} ⊗_{k} k[z, z^{−1}]/(Δ_{q} − z)  ≅ M_{q}[Δ_{q}^{−1}].

Thus, O_{q}(SL_{n}(k))[z, z^{−1}] ≅ O_{q}(GL_{n}(k)). □

Another way of viewing this result is as follows: Under the isomorphism, O_{q}[z, z^{−1}] ≅ O_{q}(GL_{n}(k)), the element z maps to Δ_{q}, and so, by inverting C = k[z]^{∗}, respectively D = k[Δ_{q}]^{∗}, we obtain O_{q}(SL_{n}(k(z))) ≅ (M_{q})_{D}.

Let M be a finitely generated module over a Noetherian k-algebra A. Then the Gelfand-Kirillov and homological dimensions of M will be denoted by GKdim(M), respectively hd(M). The global homological dimension of A will be written gldim(A). If the injective dimensions of _{A}A and A_{A} are finite, then they are equal, by [Za, Lemma A], and this integer will be denoted by injdim(A). If injdim(A) < ∞, then A is called Auslander-Gorenstein if A satisfies the following condition: For any integers 0 ≤ i < j and finitely generated (right) A-module M, one has Ext_{A}^{i}(N, A) = 0 for all (left) A-submodules N of Ext_{A}^{j}(M, A). If A is an Auslander-Gorenstein ring of finite global dimension, then A is called Auslander-regular. Set j(M) = min{j : Ext^{j}(M, A) ≠ 0}. The ring A is CM if j(M) + GKdim(M) = Gkdim(A) holds for all finitely generated A-modules M.

Before proving the Corollary, we need the following result that provides some more-or-less well known facts about these conditions.

Lemma. Suppose that R is a Noetherian ring that is Auslander-regular and CM. Let S = R[x; σ, δ] be an Ore extension, in the sense of [Co, Section 12.2]. Then:

(i) S is Auslander-regular.

THE QUANTUM COORDINATE RING OF THE SPECIAL LINEAR GROUP 5

(ii) Assume that R = ⨁_{i≥0} R_{i} is a connected graded k-algebra (thus R_{0} = k) such that

σ(R_{i}) ⊆ R_{i} for each i ≥ 0. Then S is CM. (iii) Let f be a central, regular element of R. Then R/fR is Auslander-Gorenstein and

CM. Proof: (i) This follows from [Ek, Theorem 4.2]. (ii) Filter S by degree in x and note that the corresponding graded ring gr(S) is iso-

morphic to R[y; σ]. The hypotheses on R ensure that gr(S) has the structure of a connected graded ring, defined by gr(S)_{n} = ⨁_{i+j=n} R_{i}y^{j}. Moreover, y is a normal, homogeneous element in gr(S) and gr(S)/y gr(S) ≅ R. Hence, by [Lv, Theorem 3.6], gr(S) is (graded) CM and, by part (i) gr(S) is Auslander-regular. If M is a finitely generated S-module, give M a good filtration and consider the associated graded gr(S)-module gr(M). Then, by [Bj, Theorem 4.3], j_{S}(M) = j_{gr(S)}(gr(M)) while, by [MS, Theorem 1.3], GKdim_{S}(M) = GKdim_{gr(S)}(gr(M)). Thus,

j_{S}(M) = GKdim(gr(S)) − GKdim(gr(M)) = GKdim(S) − GKdim(M)

and S is CM.

(iii) To avoid triviality, assume that f is not a unit. Set \overline{R} = R/fR. Since j_{R}(\overline{R}) = 1, the CM condition implies that GKdim(\overline{R}) = GKdim(R)−1. Let M be a finitely generated \overline{R}-module. By the Rees Lemma [Ro, Theorem 9.37], Ext_{R}^{j}(M, R) = Extj−1 ; \overline{R}(M, \overline{R}), for each j ≥ 1. It follows that \overline{R} is Auslander-Gorenstein and CM. □

The extra conditions in part (ii) of the Lemma are necessary since, in general, GKdim(M) ≠ GKdim(gr(M)). For example, suppose that R = ℂ[z, z^{−1}, y], where z and y are central indeterminates, and σ is the ℂ-automorphism of R defined by σ(z) = z but σ(y) = zy. Then, let S = R[x; σ, 0] and set M = S/(x − 1)S and N = S/xS (thus, N = gr(M) in the notation of the proof of part (ii) of the Lemma). It follows from [MS, Proposition 3.4] that j(M) = j(N) = 1 but GKdim(M) = 3 > 2 = GKdim(N).

PROOF OF THE COROLLARY. Order the generators x_{ij} of M_{q} lexicographically and consider the corresponding chain of rings

k⟨x_{11}⟩ ⊂ k⟨x_{11}⟩⟨x_{12}⟩ ⊂ ⋯ ⊂ M_{q}.  (†)

If R ⊂ S = R⟨x⟩ is a successive pair of rings from this chain, then [AST, pp.890-1] shows that there exists a k-algebra automorphism τ and a τ-derivation δ of R such that S is isomorphic to the Ore extension R[x; τ, δ]. Thus M_{q} is an iterated Ore extension.

6 T. LEVASSEUR AND J. T. STAFFORD

(i) By [MR, Theorem 1.2.9], M_{q} is a Noetherian domain. Since O_{q} = M_{q}/(Δ_{q} − 1), certainly O_{q} is Noetherian. By the Proposition, O_{q}[z, z^{−1}] ≅ (M_{q})_{S} is a domain, and hence so is its subring O_{q}. By [MaR, Proposition V.2.5], M_{q} is a maximal order in its division ring of fractions D; that is, if M_{q} ⊆ T ⊆ D, for some ring T such that aT b ⊆ M_{q} for some non-zero elements a, b ∈ M_{q}, then T = M_{q}. By [MaR, Proposition IV.2.1], (M_{q})_{S} is also a maximal order in D. It follows easily from the Proposition that O_{q} is a maximal order. This proves part (i) of the corollary.

(ii) and (iii) By [AST, Proposition 2, and its proof], GKdim(M_{q}) = gldim(M_{q}) = n^{2}. Thus, by the Proposition and [MR, Theorem 7.5.3(iv)],

gldim(O_{q}) = gldim(O_{q}[z, z^{−1}]) − 1 = gldim((M_{q})_{S}) − 1 ≤ gldim(M_{q}) − 1 = n^{2} − 1.

By (1), M_{q} has the structure of a connected graded ring, by giving each x_{ij} degree one. If R ⊂ S = R[x; σ, δ] are a pair of successive rings in the chain (†), then this induces, on R, the structure of connected graded ring R = ⨁_{i≥0} R_{i} and implies that σ(R_{j}) ⊆ R_{j}, for each j. Thus, by part (ii) of the Lemma and induction, M_{q} is Auslander-regular and CM. Now regard O_{q} as M_{q}/(Δ_{q} − 1). Then, part (iii) of the Lemma implies that O_{q} is Auslander-Gorenstein and CM, with GKdim(O_{q}) = GKdim(M_{q}) − 1 = n^{2} − 1. Since gldim(O_{q}) < ∞, this implies that O_{q} is Auslander regular.

It remains to show that gldim(O_{q}) ≥ n^{2} − 1. Consider the factor ring

A = O_{q}/(x_{ij} : i ≠ j, x_{ℓℓ} − 1 : ℓ ≠ 1).

The description of the relations of O_{q} in (1) and (2) imply that A ≅ k[x_{11}]/(x_{11} − γ), for some γ ∈ k^{∗}. Therefore, both A and O_{q} have a 1-dimensional module S. (This also follows from [AST, Theorem 3].) But, by the CM condition, j(S) = GKdim(O_{q})−GKdim(S) = n^{2} − 1. Thus, gldim(O_{q}) ≥ n^{2} − 1. This completes the proof of parts (ii) and (iii) of the Corollary.

(iv) By the proof of (ii), gldim(M_{q}) < ∞. Thus, by [MR, Corollary 12.3.6 and Theorem 12.6.13], K_{0}(M_{q}) = ℤ. Therefore, by the Proposition and [MR, Proposition 12.1.12], K_{0}(O_{q}[z, z^{−1}]) = K_{0}((M_{q})_{S}) = ℤ. By [MR, Corollary 12.3.6], this implies that K_{0}(O_{q}) = ℤ. □

THE QUANTUM COORDINATE RING OF THE SPECIAL LINEAR GROUP 7

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