Journal of Algebra 244, 186–216 (2001) doi:10.1006/jabr.2001.8897, available online at http://www.idealibrary.com on Prime and Primitive Ideals of a Class of Iterated Skew Polynomial Rings J. Gomez-Torrecillas´ Departamento de Algebra, Facultad de Ciencias, Universidad de Granada, E18071 Granada, Spain and L. EL Kaoutit D´epartement de Math´ematiques, Universit´e Abdelmalek Essaadi, Facult´e des Sciences de T´etouan, B.P. 2121, T´etouan, Morocco Communicated by Michel Brou´e Received December 5, 2000 INTRODUCTION Hodges and Levasseur described the primitive spectra of quantized coor- dinate rings of SL3 [14], and then of SLn [15]. These results established a close connection between primitive ideals, torus action, and Poisson geom- etry. The proofs relied on explicit computations involving generators and relations. Subsequently, Josephgeneralized theHodges–Levasseur program to semisimple algebraic quantum groups [17]. Hodges et al. then expanded Joseph’s work to include certain multiparameter deformations [16]. These papers rely less on concrete calculations and more on deeper, more conceptual, techniques. It is a natural and important question, then, as to how the preceding theory might apply to other algebras, particularly other algebras arising in the study of quantum groups. Goodearl and Letzter established parallel results for certain iterated skew polynomial rings [13], withapplication to quantum Weyl algebras and to some quantum coordinate rings. 186 0021-8693/01 $35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved. ideals of skew polynomial rings 187 There have also been several papers (including [22–26]) by S. Q. Oh and collaborators that further the above program for certain other quantum coordinate rings. The present paper fits into the above framework by, first, extending the work of Oh on primitive spectra [22, 26] to the prime spectra of the iterated Ore extensions introduced in [23], and second, providing a more detailed version of the Goodearl–Letzter study for these cases [13]. Goodearl and Letzter’s general framework is to consider some group acting as automorphisms on a ring R which give the set -Spec(R) consist- ing of all -prime ideals of R.The-stratification of the prime spectrum SpecR is then defined as (1) SpecR= SpecJ R j∈-SpecR where each stratum Spec R consists of those prime ideals P of R such J that h∈ hP=J. In the case that is a torus of rank r acting rationally on a noetherian algebra R over an infinite field k (see [13] for details), the strata SpecJ R corresponding to completely prime -invariant ideals J of R are described in [13, Theorem 6.6] as follows. (a) For eachcompletely prime -invariant ideal J of R, there exists an Ore set J in the algebra R/J such that the localization map R → R/J → −1 RJ =R/JJ induces a homeomorphism of SpecJ R onto SpecRJ . (b) Contraction and extension induce mutually inverse homeomor- phisms between SpecRJ and SpecZRJ , where ZRJ is the centre of RJ . (c) ZRJ is a commutative Laurent polynomial ring over an extension of k,inr or fewer indeterminates. The foregoing description of the -strata applies to iterated Ore exten- sions of k under suitable conditions [13, Sect. 4]. For some quantized coordinate rings, the aforementioned general stratification of the prime spectrum can be worked out in detail. This non-trivial research has been 2×n done for the coordinate algebras of quantum symplectic spaces qk C in [8]. These algebras belong to the class of algebras Rn k introduced 2×n in [23], which also includes the coordinate rings qk of quantum q euclidean spaces and the quantum Weyl algebras An k. 188 gomez-torrecillas´ and el kaoutit The aim of this note is to give a detailed description of the prime spectra of C the k-algebras Rn k (see Definition 1.1), where C =c1cndλu × n+2 is an element of k × k, suchthat d = 1ifu = 0 and =λij λii = 1is a multiplicatively anti-symmetric matrix withentries in k×. We first define a rational action of the torus =k×r , where r = n if C u = 0 and r = n + 1ifu = 0, on the k-algebra Rn k for an infinite base C field k, and we show that [13, Theorem 6.6] applies to Rn k for any C -prime ideal. The algebra Rn k is filtered witha finite-dimensional filtration withsemi-commutative associated graded algebra, (see, e.g., [4, C Section 3; 5]) which implies, by [19, Theorem 3.8], that Rn k satisfies the Nullstellensatz over an arbitrary field k, so [13, Corollary 6.9] applies C to Rn k. In a second step we give a more explicit description of the -stratification C of the spectra of Rn k in the following aspects. (1) We prove that the -prime ideals are just the ideals generated by the admissible sets in the sense of [22]. More explicitly, consider the C finite subset ℘n of Rn k as defined later in (6). The map J → J ∩ ℘n C gives a bijection between the -prime ideals of Rn k and the admissible subsets of ℘n (Proposition 2.10). (2) For each -prime ideal J, let T = J ∩ ℘n the corresponding admissible set. We give explicitly a McConnell–Pettit k-algebra PQT , which is strictly contained in RJ , such that the Jthstratum is described as C C SpecJ Rn k=P ∈ SpecRn k P ∩ ℘n = T and it is homeomorphic to the spectrum of PQT (Theorem 3.4). (3) By using [12], we obtain that each stratum is homeomorphic to the spectrum of the centre ZPQT of P QT for a suitable admissible set C 2×n T . In the particular case Rn k=qk , we give an explicit method to compute the number of indeterminates in the Laurent polynomial ring ZPQT over k, for any admissible set T (Corollary 4.8). The first obstacle is to prove the nice properties of the ideals gener- ated by the admissible sets as in [22]. Using well known results from [11], we prove that each admissible set T generates a polynormal prime ideal T . We compute the Gelfand–Kirillov dimension of the factor algebras C Rn k/T by using Grobner–Basis¨ techniques; see [3]. An explicit C homomorphism T connecting Rn k and the McConnell–Pettit alge- bra PQT is given. Sucha mapping was used by Rigal in thecase of quantum Weyl algebras [27] (see also [26] for a similar morphism in the quantum euclidean case). ideals of skew polynomial rings 189 2×2 × FIG. 1. The prime spectrum of qk (k is algebraically closed, α γ ∈ k ). Our methods allow us to give an effective description (modulo Commu- 2×n tative Algebra) of Specqk for eachgiven n (Corollary 4.9). This is 2×n possible because eachprime ideal in thestratum Spec T qk is rec- ognized as the inverse image under the algebra homomorphism T .Inthe algebraically closed case, we give an effective method to compute the primi- 2×n 2×2 tive ideals of qk . As an illustration, we compute Specqk and 2×2 Primqk (see Fig. 1). Using the epimorphism defined in [23, Exam- 2n+1 ple 5], we determine the prime spectrum of qk (q has a square root 3 in k), and we compute Specqk as an example (see Fig. 2). 3 × FIG. 2. The prime spectrum of qk k is algebraically closed, α γ ∈ k ). 190 gomez-torrecillas´ and el kaoutit 1. DEFINITION AND BASIC PROPERTIES Throughout this note we will consider different quantum spaces, so we will use some convenient notation. Let =λij be a p × p matrix −1 withentries in k suchthat λii = 1 and λji = λji . Consider the k-algebra kt1tp generated by t1tp subject to the relations titj = λij tjti. This is called the coordinate algebra of the p-dimensional quantum affine space associated to and it is the iterated Ore extension (2) kt1tp=kt1t2 σ2···tp σp where σitj=λij tj for every 1 ≤ j<i≤ p. This k-algebra is a noethe- rian domain, and its skew field of fractions is denoted by kt1tp. A useful intermediate algebra is the McConnell–Pettit algebra P= ± ± kt1 tp (see [20]). Definition 1.1. Let k be a field and let n be a strictly positive × n+2 integer. Let C =c1c2cndλu be an element of k × k with d = 1ifu = 0. Consider a multiplicatively anti-symmetric matrix × =λji1≤i<j≤n withentries in k suchthat λii = 1 for all i = C 1n. Define Rn k to be the finitely generated k-algebra with generators y1x1ynxn, satisfying the following relations −1 yjyi = λjiyiyjyjxi = λji dxiyj j>i x x = λ c−1d−1x x xy = λ−1c y x j>i (3) j i ji i i j j i ji i i j i−1 i−1−l i−1 xiyi = ciyixi + λ λd cld − 1ylxl +dλ u i ≥ 1 l=1 c This algebra was defined by Oh in [23]. By [23, p. 39], Rn k is an iterated Ore extension c R0 ⊆ R1 ⊆···⊆Rn k=Rn where R0 = k and Rk = Rk/2xkβkδkRk/2 = Rk−1ykαk for all k ≥ 1, and αiβi are algebra automorphisms defined by −1 αjyi=λjiyiαjxi=λji dxii<j (4) −1 −1 −1 βjyi=λji ciyiβjxi=λjici d xii<j βiyi=ciyi and each δi is a left βi-derivation defined by i−1 i−1−l i−1 δiyi=λ λd cld − lylxl +λd u i > 1 l=1 δiRi−1=0i≥ 1 and δ1y1=u n By k we denote the set αkβkδkαnβnδn for each k = 1n.