Prime and Primitive Ideals of a Class of Iterated Skew Polynomial Rings

Journal of Algebra 244, 186–216 (2001) doi:10.1006/jabr.2001.8897, available online at http://www.idealibrary.com on

J. Gomez-Torrecillas´

Departamento de Algebra, Facultad de Ciencias, Universidad de Granada, E18071 Granada, Spain

and

L. EL Kaoutit

Département de Mathématiques, Université Abdelmalek Essaadi, Faculté des Sciences de Tétouan, B.P. 2121, Tétouan, Morocco

Communicated by Michel Brou´e

Received December 5, 2000

INTRODUCTION

Hodges and Levasseur described the primitive spectra of quantized coordinate rings of SL3 [14], and then of SLn [15]. These results established a close connection between primitive ideals, torus action, and Poisson geometry. 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 .

The foregoing description of the -strata applies to iterated Ore extensions 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 generated 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 algebra 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 ≤ ji 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

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. ideals of skew polynomial rings 191

q This class of algebras includes the quantum Weyl algebras An kC = n q 1 1 1 with q =cii=1, the coordinate rings of quantum symplectic 2×n 2 2 spaces qk C =q q 1q0, λji = q, and the coordinate 2×n −2 rings of quantum Euclidean spaces qk C =11q q0, −1 λji = q . Following [23, p. 39], we have

Lemma 1.2. Set zi = dxiyi − yixi for i = 1n and z0 = du. Then −1 zjyi = ciyizjzjxi = ci xizj i ≤ j −1 zjyi = d yizjzjxi = dxizj i>j

zjzi = zizj all i j −1 xiyi = ciyixi + λzi−1 i = 2nx1y1 = c1y1x1 + d z0

zi =cid − 1 yixi + dλzi−1 i = 2nz1 =c1d − 1 y1x1 + z0

Observe that δiyi=λzi−1 for all i>1. C The quantum space attached to Rn k is k Y X Y X , where Qn 1 1 n n Qn is the matrix deﬁned by (5)

Y1 X1 Y2 X2 ··· ··· Yn Xn  −1 −1 −1 −1 −1  Y1 1 c1 λ21 λ21c1 ··· λn1 λn1c1  −1 −1 −1 −1  X1  c1 1 λ21d λ21 c1d ··· λn1d λn1 c1d  Y  −1 −1 −1 −1  2  λ21 λ21 d 1 c2 ··· λn2 λn2c2  X  −1 −1 −1 −1 −1  2  λ21 c1 λ21c1 d c2 1 ··· λn2d λn2 c2d         −1 −1 −1  Yn λn1 λn1 dλn2 λn2 d ··· 1 cn X −1 −1 −1 −1 −1 −1 n λn1 c1 λn1c1 d λn2 c2 λn2c2 d ··· cn 1 −1 Notice that Y1u = duY1 and X1u = d uX1.

Remark 1.3. We have δiβi = cidβiδi for all i ≥ 1. So if cid is not a root of unity for every i = 1n, then [10, Theorem 2.3] each prime ideal of C Rn k is completely prime. C In order to classify the prime and the primitive ideals of Rn k we will suppose that for each i = 1n the scalar cid is not a root of unity. Denote by ℘n the following subset of R:   z1y1x1znynxn if z0 = 0 (6) ℘ = n  z1z2y2x2znynxn if not 192g omez-torrecillas´ and el kaoutit

Deﬁnition 1.4 [22]. A subset T of ℘n is said to be admissible if it sat- isﬁes the conditions:

(1) yi or xi ∈ T ⇔ zi and zi−1 ∈ T , for all i ≥ 2

(2) xi or yi ∈ T ⇔ z1 ∈ T ,ifz0 = 0.

For an admissible set T , let us denote by indT =i ∈1nzi ∈ T . An index i ∈ indT is said to be removable if T contains xi and yi;the set of removable indices is denoted by RemvT . If we denote T =i ∈ 1nyi ∈ T and T =j ∈1nxj ∈ T , then RemvT = if and only if T ∩ T =. We say that T is connected if for any i j ∈ indT suchthat i

T . Put ik = minindTkjk = maxindTk; we will always suppose that jk−1

Proposition 1.5. For k ≥ 2, let Q be an αkβk-stable prime ideal of Rk−1 such that zk−1 ∈/ Q. Then the ideals P = QRk + zkRk and QRk are prime extensions of Q to Rk. Proof. By [11, Theorem 10.3(ii)], applied to Q in the iterations

Rk/2 and Rk,wehaveQRk is a prime extension of Q. We apply [11, Theorem 10.3(iv)] to get the proposition. Consider fracRk/2/QRk/2= Ak/2, the Goldie quotient ring of Rk/2/QRk/2. We need to show that the extension of δk, denoted also by δk,toAk/2 is an inner βk-derivation, where βk is the extension of βk to Ak/2. It is sufﬁcient ideals of skew polynomial rings 193 to show this for the extension of δk to the skew Laurent polynomial ring ±1 −1 −1 Fk/2 =Rk−1/QYk αk. Put uk =1 − ckd λz¯k−1yk ∈ Fk/2, where z¯k−1 is the image of zk−1 in Fk/2. Then δkyk=ukyk − ckykuk and −1 −1 −1 −1 −1 −1 δkyk =−βkyk δkykyk = ukyk − ck yk uk. Therefore δk is an inner βk-derivation of Fk/2, hence δka=uka − βkauk, for all a ∈ Ak/2. So, by [9, Lemma 1.4], we have the following isomorphisms of k-algebras R k −1 ∼ ∼ Ck/2 = Ak/2xkβkδk = Ak/2xk − ukβk QRk where Ck/2 =Rk/2/QRk/2\0. Now, by [11, Theorem 10.3(iv)], the inverse image of the prime ideal xk − uk in Rk is a prime extension of −1 −1 QRk/2. Check that xk − uk =−ck1 − ckd z¯kyk , where z¯k is the image −1 of zk in Rk/QRk. So the inverse image of ¯zkyk in Rk is exactly the ideal P = QRk + zkRk.

Corollary 1.6. The ideal zi is prime for all i ≥ 2.Ifz0 = 0, then z1 is a prime ideal too.

n Proof. It is clear that zi∩Ri = ziRi = Rizi is i+1-stable ideal for all i ≥ 1. So it sufﬁces to show that Rizi is a prime ideal of Ri. But this follows from Proposition 1.5 with Q = 0inRi−1 and i ≥ 2. For the ±1 case z0 = 0 and i = 1 consider F0 = ky1 and the element v1 =1 − −1 −1 c1d uy1 ∈ F0. We denote by δ1β1 the extensions of δ1β1 to F0, −1 −1 −1 −2 respectively. So δ1y1=v1y1 − c1y1v1 and δ1y1 =−c1 d uy1 ; thus −1 −1 −1 −1 δ1y =v1y − c y v1. Therefore δ1 is an inner β1-derivation of F0, 1 1 1 ∼1 and so F0x1β1δ1 = F0x1 − v1β1. By the equality x1 − v1 =−1 − −1 −1 −1 c1d c1z1y1 , the inverse image of the prime ideal z1y1 in R1 is the prime ideal z1R1. n Proposition 1.7. Let I be a j+1-stable prime ideal of Rj for jj+ 1. Then J = IRn +T is a prime ideal of Rn. Proof. Let k denote maxindT ; we use induction on k.Ifk = j + 2, then i = k, and T =zk.Aszj+1 ∈/ IRj+1, by Proposition 1.5, we have n that IRj+2 + zj+2Rj+2 is a prime j+3-stable ideal of Rj+2. Then J = IRj+2 + zj+2Rj+2Rn is a prime extension of IRj+2 + zj+2Rj+2. Now suppose that the proposition is true for any connected admissible set T , with maxindT

Put Jk−1 = IRk−1 + T Rk−1. So by induction hypothesis Jk−1Rn is a prime ideal of Rn. Recall that in each iteration every prime ideal is completely n prime; hence Jk−1 is a k-stable prime ideal of Rk−1. We claim that J is a prime extension to Rn of the prime ideal Jk−1. To show this we will consider the three cases listed below (1) If T = T ∪ykzk then, by the isomorphism Rk ∼ Rk−1 = ykαkxkβk Jk−1Rk Jk−1 the ideal Jk = Jk−1Rk + ykRk is a prime ideal of Rk. Check that Jk = n IRk + TRk, which is k+1-stable. Thus J = JkRn is a prime extension of Jk−1 to Rn. (2) The case T = T ∪xkzk is similar to the ﬁrst one by taking Jk = Jk−1Rk + xkRk. (3) If T = T ∪ykxkzk, then consider Jk = Jk−1Rk + ykRk + xkRk and Jk/2 = Jk−1Rk/2 + ykRk/2. It is clear that Jk = Jk/2Rk + xkRk (observe that zk−1 ∈ Jk−1) and Jk/2 is a prime extension of Jk−1 to Rk/2, which is βk-stable. Therefore Jk/2Rk is a prime ideal of Rk and we have the following algebra isomorphism: R Rk ∼ k/2 = xkβk Jk/2Rk Jk/2 n Hence Jk is a prime ideal of RK and Jk = IRk + TRk. This is a k+1- stable ideal, so JkRn = J is a prime extension of Jk to Rn. Thus J is a prime extension of Jk−1 to Rn.

A polynormal sequence a1as in a ring R is a sequence of elements of R suchthat a1 is normal in R, and each ak is normal modulo the ideal a1ak−1 for all k ≥ 2. An ideal I of R generated by a polynormal sequence is called a polynormal ideal. C Theorem 1.8. Let T be an admissible set of Rn k. Then T is a polynormal prime ideal. Proof. Clearly, T is a polynormal sequence. We prove that T is prime by induction on the number of connected components of T .IfT is a connected admissible set withmin indT > 1, then T is prime by Proposition 1.7.

Otherwise it is easy to see that T is a prime extension of T ∩ R1R1. Let now T = T1 ∪ T2 ···∪Tr , r>1, ik = minindTk, jk = maxindTk, k = 1r. Consider I =T ∪···∪T R as an ideal of R ;by 1 r−1 jr−1 jr−1 induction hypothesis IRn is a prime ideal of Rn. As in eachOre iteration RkRk/2, k = 1n, every prime ideal is completely prime, we have I is a prime ideal of R . Check that T =IR + T R ; applying Proposition 1.7 jr−1 n r n we have T is a prime ideal of Rn. ideals of skew polynomial rings 195

Now we need to compute the Gelfand–Kirillov dimension of the factor C C algebra of Rn k by an ideal generated by an admissible set. As Rn k isaPBWk-algebra with respect to the graded lexicographical order ≤deglex, we can use [2, Sect. 3; 3, Sect. 4]. C Lemma 1.9. Let T be an admissible set of Rn k. Then T is a two- sided Grobner¨ basis.

Proof. In this proof we use the notation of [2, 3]. Let T = T1 ∪···∪Tr be a connected decomposition of an admissible set T . It is clear that T l S u v=0 for any u ∈ Ts and v ∈ Tt with s = t ∈1r. Fix k ∈1r and consider Tk, i = minindTkj = maxindTk.By Lemma 1.2, zs is a semi-commuting element for every s = 1n.So T l k S zsvt =0 for every s ∈ij and t ∈i + 1j, where vt ∈ T l yt xt . By (3), we have also k S vsvt =0 for all t = s ∈i + 1j.It T l remains to show that k S ysxs=0 for all s ∈i + 1j∩RemvTk. l −1 −1 But S ysxs=cs xsys − ysxs = λcs zs−1 ∈ Tk. So from [3, Theorem 3.2], T T is a left Grobner¨ basis of RT . On the other hand, if t ∈ T then yt xt = T −1 −1 T T ct xt yt − λct zt−1 = 0, and if t ∈ T then xt yt = ct yt xt + λzt−1 = 0. So by [3, Remark 3.13] T is a two-sided Grobner¨ basis of T . Using the notations of [2, Sect. 3], the following lemma is easy to prove: Lemma 1.10. (1) Let E be a monoideal of p + generated by a min- imal set 1 k such that Supp i∩Supp j=for i = j. Then dimE=p − k. p r (2) Let E = B + , with B = k=1Bk be a disjoint union of subsets p p of . Consider Ek = Bk + for all k = 1r, and suppose that E has a set of generators of disjoint support (as in (1)). Then r dimE=p1 − r+ dimEi k=1 Let 5i = expyi and 5i = expxi for every i = 1n(see [2, Sect. 1]). C Proposition 1.11. Let T be an admissible set of Rn k. Then RC k GK dim n = 2n − lengthT T Proof. Using Lemma 1.9 and [2, Theorem 3.7; 3, Theorem 4.10] we have C GK dimRn k/T = dimexpT So it sufﬁces to compute dimexpT .LetT be a connected admissible set with i = minindT j = maxindT . By Lemma 1.9, expT is 196 gomez-torrecillas´ and el kaoutit

generated by the elements 5i + 5i5k if k ∈ T , and 5l if l ∈ T , where k l = i + 1j. Apply Lemma 1.10(1) to get dimexpT = 2n − lengthT .IfT is not connected, consider T = T1 ∪···∪Tr the connected decomposition of T . Then, by Lemma 1.9, r expT = expTk k=1 and expT has a set of generators of disjoint support. So by Lemma 1.10(2) we have r dimexpT = 2n1 − r+ dimexpT k=1 r = 2n1 − r+ 2n − lengthTk k=1 = 2n − lengthT

2. THE -ACTION AND THE -PRIME IDEALS

Let k be an infinite field. We will define a rational action of an alge- C braic k-torus on Rn k, which will be shown to satisfy [13, 4.1]. Thus C [13, Theorem 6.6, Corollary 6.9] apply to the algebra Rn k. As in [8], we will show that the -prime ideals are precisely the ideals generated by admissible sets. We need to establisha k-algebra isomorphism between a C localization of a factor algebra Rn k/T , for a fixed admissible set T , and a localization of a quantum space attached to T . C Remark 2.1. Let G denote a group acting on Rn k as k-algebra automorphisms. Assume that yixi, i = 1n are G-eigenvectors. If h ∈ G then, by (3), the action of h has one of the following forms  −1  hyi = ηiyi and hxi = ηi xi if u = 0 (7) or  −1 hyi = ηiyi and hxi = ηi θxi if u = 0; where ηi is the h-eigenvalue of yi, i = 1n, and θ is the common h- eigenvalue of z1zn. In conclusion, the group G can be replaced by a subgroup of the algebraic torus k×n or of the torus k×n × k×. In what follows will denote the torus k×n or k×n+1, depending on the value of u. Definition 2.2. We define the following rational action of on C Rn k: ideals of skew polynomial rings 197

× n (1) If u = 0, then =k and for any h =h1 hn∈ we take

hyi = hiyi −1 hxi = hi xi × n+1 (2) If u = 0, then =k and for any h =h1hnhn+1∈ we take

hyi = hiyi −1 hxi = hi hn+1xi Remark 2.3. The actions given in Deﬁnition 2.2 satisfy the hypothesis of [13, 4.1(c)]. Let us show this claim; recall that each cid is not a root of unity. In the case u = 0 we have d = 1, so for any j ∈1n the restriction to Rj−1 and Rj/2 of the action of the following elements of

j h =λj1λjj−1cj 11 j −1 −1 g =λj1 c1λjj−1cj−1cj 11 gives the automorphisms αjβj, respectively. If u = 0, then for any j ∈ 1n we take

j h =λj1λjj−1cjd 11d j −1 −1 −1 g =λj1 c1λjj−1cj−1cj 1d

j j It is clear that αjβj are the restriction of h g , respectively. Hence both C actions on Rn k satisfy [13, 4.1]. Notice that if d = 1 and u = 0, then we can use the torus k×n instead of k×n+1 withtheobvious action. But if d = 1, we have to enlarge the size of the acting torus in order to “place” the parameter d. The action deﬁned in [13, 5.4] for the quantum Euclidean 2×n space qk does not satisfy [13, 4.1(c)]; see the following example for n = 2. C 2×2 Example 2.4. Let R2 k=qk with q not a root of unity. If we suppose that =k×2 and the -action satisﬁes [13, 4.1(c)], then there exists h =h1h2∈ such that the restriction of h to ky1x1 coincides withthe k-algebra automorphism α2. Thus −1 α2y1=q y1 = hy1 = h1y1 −1 −1 α2x1=q x1 = hx1 = h1 x1 this means that q2 = 1, in spite of the assumption that q is not a root of unity. 198 gomez-torrecillas´ and el kaoutit

C In the rest of this section the algebra Rn k will be denoted by R.By Remark 2.3 and [13, Proposition 4.2], each -prime ideal of R is completely prime and there exist at most 22n. Recall that R satisﬁes the Nullstellensatz, so, by [13, Corollary 6.9], R satisﬁes the Dixmier–Moeglin Equivalence. The primitive ideals of R are precisely those maximal within their -strata. Our next goal is to describe the -prime ideals in terms of admissible sets. We need to control the Gelfand–Kirillov dimension of certain local- izations of R. Proposition 2.5. Let W be any subset of 1n. Consider the multiplicative subset of R generated by ykk∈ W . Then is a right Ore set and the Gelfand–Kirillov dimension of R −1 equals 2n. Proof. Compare with[8, Proposition 2.1]. By [9, Lemma 1.4; 18, Lemma 4.1], is a right Ore set of R and so GKdimR≤GKdimR −1. So we will prove the converse inequality. Consider the k-algebra S generated by y1x1ynxn satisfying the relations (3) and new variables ;kk∈ W , withthefollowing additional relations −1 −1 ;i;j = λji ;j;ixi;j = λji d;jxi j>i ; x = λ−1c x ; y; = λ ; y j>i (8) i j ji i j i i j ji j i ;iyj = λjiyj;i j>i ;iyj = yi;i = 1

k−1 −1 k−1−l 2 k−3 k−1 2 ;kxk = ckxk;k + λd dλ cld − 1y1x1;k + d λ z0;k l=1

−1 There is a surjective homomorphism of algebras S → R sending yi to −1 −1 yixi to xi, and ;k to yk . Then GKdimR ≤GkdimS. We claim that GKdimS=2n. Order the variables ; < ···<;

2n+m By [7, Proposition 3.2], S can be endowed withan ≤w-filtration such that the 2n+m-graded algebra GS is semi-commutative, namely, it is generated by finitely many homogeneous elements ; ; y x y x k1 km 1 1 n n ideals of skew polynomial rings 199 in addition yk;k = 0 for every k ∈ W . Therefore GS is a factor of the coordinate algebra of an 2n + m-dimensional quantum affine space by the ideal generated by the elements yk;kk ∈ W . By [2, Theorem 4.4.7; 3, Theorem 4.10], it is clear that GKdimGS =2n + m − m, and by [7, Corollary 2.12] we have GKdimS = GKdimGS =2n.

Fix an admissible set T = T1 ∪ T2 ∪···∪Tr with ik = minindTk, jk = maxindTk. For a prime ideal P of R we denote by htP the height of P. Let us now compute the height of an ideal generated by the admissible set T . Proposition 2.6. Let T be an admissible set of R. Then R htT = lengthT =2n − GK dim T Proof. Let us ﬁrst prove the proposition for a connected admissible set T . Consider j = maxindT and i = minindT ; we use induction on j.Ifj = i = 1 then the possible admissible sets are z1 if z0 = 0or z1x1 z1y1 and z1y1x1 if z0 = 0. Using Corollary 1.6 and the deﬁnition of thelengthin conjunction with[21, Theorem4.1.11] we get the result in this case. Suppose that the proposition holds for all connected admissible T suchthatmax indT

0T T vj=T vj ∈yjxj or 0T T yjT yjxj=T By Theorem 1.8 we know that T is generated by a polynormal sequence. So using [21, Theorem 4.1.11] the chains above are maximal. Therefore htT = htT + 5, where 5 ∈1 2. Now, the result is clear by induction hypothesis. Let T be an admissible set and T1 ∪···∪Tr its connected decomposition. We show the proposition in this case by induction on r. For r = 1 the proposition has been already proved. Suppose that r>1, so T = T ∪ Tr where T = T1 ∪···∪Tr−1. We denote ir = minindTr . So, by Proposition 1.5, we have a chain of prime ideals

0⊆T T z ···T T =T ir r 200 gomez-torrecillas´ and el kaoutit

This chain is maximal because T is a polynormal ideal. The number of prime ideals between T zi and T is exactly the number of the variables r ylxll ∈ indTr .SowehavehtT = htT + lengthTr ; hence by induction hypothesis and the deﬁnition of the length we have htT = lengthT . The second equality follows from Proposition 1.11.

Let us denote by T the multiplicative set of R generated by yjj∈ / T ; by Proposition 2.5 this is a right Ore set of R.LetQT be a submatrix of the matrix Qn deﬁned by deleting the rows and the columns corresponding to the variables x y ∈ T and x k = 1r.Ifz = 0 and x ∈/ T we i i ik 0 1 will not delete the row and the column corresponding to the variable x1. Consider AT the quantum space associated to the matrix QT , considered as a subalgebra of the quantum space A = k Y X Y X attached Qn 1 1 n n to R. The multiplicative set T of AT generated by all the Yk’s is a right −1 Ore set, so consider BT = AT T . We will denote by a¯ the image of a ∈ R in R . Consider the k-algebra homomorphism = ! R "−1 → B given by T T T T T

=T ¯yk=Yk (for all k)

=T ¯xk=Xk (if k − 1 ∈ indT k≥ 2) −1 −1 =T ¯xk=Xk +1 − ckd λZk−1Yk (if 2 ≤ k and k k − 1 ∈/ indT ) −1 −1 =T ¯x1=X1 +1 − c1d Z0Y1 −1 −1 =T ¯xk=1 − ckd λZk−1Yk (if k ∈ indT and k − 1 ∈/ indT ), −1 where Z0 = d z0 = u Zk =ckd − 1YkXk. It is clear that T ∩T = −1 −1 ∼ "−1 so, by [6, Proposition 3.6.15], R T /T T = R/T T . Composing =T with this last isomorphism we get a new map which we also denote by =T . A similar algebra homomorphism was given in [27, Sect. 3.2] in the case of quantized Weyl algebras. Proposition 2.7. The mapping R −1 = ! T → B T −1 T T T is a k-algebra isomorphism.

Proof. It is clear that =T is surjective, and T ⊆ker=T .So −1 −1 −1 GK dimBT ≤GK dimR T /T T . We claim that GK dimR T / −1 −1 T T ≤GK dimBT . From [18, Lemma 3.16], we have GK dimR T / −1 −1 T T ≤GK dimR T −htT . Using Proposition 2.5 and −1 −1 Proposition 2.6 we have GK dimR T /T T ≤2n − lengthT . We know that GK dimAT =2n − lengthT =GK dimBT .So −1 −1 GK dimR T /T T ≤GK dimBT . Since T is a completely prime ideal, Theorem 1.8, it follows from [18, Proposition 3.15] that =T is a k-algebra isomorphism. ideals of skew polynomial rings 201

Consider the algebraic torus and its action R as in Deﬁnition 2.2. For any subset X ⊆1n= n, we denote by X the torus h h ∈ k× if =k×n+1 = i i∈X∪n+1 i X × × n hii∈X hi ∈ k if =k .

Let T be an admissible set of R; we denote by InT the set of indices of the variables that appear in AT InT ⊆ n. Deﬁne the following action of the torus = on A .If =k×n+1, then for any h ∈ InT T T i i∈InT ∪n+1 T , h Y = h Y i i∈InT ∪n+1 l l l h X = h−1h X i i∈InT ∪n+1 k k n+1 k If =k×n, then for any h ∈ , i i∈InT T h Y = h Y i i∈InT l l l h X = h−1X i i∈InT k k k

Consider the canonically extended action of T to the localization BT = −1 AT . For each h ∈ T , we have the following composite map

R = h =−1 R "−1 −→T B −→ B −→T "−1 T T T T T T where h denotes the extension of h to BT . Definition 2.8. We define the action of the torus on R "−1 as T T T follows. Given h ∈ T , define −1 hx ==T h=T x for every x ∈ R "−1. T T The following lemma is clear.

Lemma 2.9. Consider T as a factor group of the torus . The action of T induced on R/T by that of coincides with the restriction of the action deﬁned in Deﬁnition 2.8. Proposition 2.10. There is a bijection ζ between − SpecR and

nR, the set of all the admissible sets of R, deﬁned by

ζ ! − SpecR−→ nR

J −→ J ∩ ℘n With the inverse map −1 ζ ! nR−→ − SpecR T −→ T 202 gomez-torrecillas´ and el kaoutit

Proof. To show that ζ and ζ−1 are well deﬁned, we use Theorem 1.8, Remark 2.3, and [13, Proposition 4.2]. Clearly ζζ−1 = id; let us show that −1 ζ ζ = id. Consider J ∈ − SpecR suchthat J ∩ ℘n = T and suppose " that T J.LetT = n\T =j1jr BT is an iterated Ore extension of the form ±1 ±1 B = k"Y Y X β ···X β T Q j1 jr i1 i1 it it ±1 ±1 where k"Y Y is the McConnell–Pettit algebra associated to a Q j1 jr suitable matrix Q". To complete the proof we use the same arguments as in the proof of [8, Proposition 2.5] with the following element h ∈ T . Fix l ∈1t− 1;if =k×n+1, then we take  −1  λi j cj i= jk i  jkil k l −1 h = λi i ci i= ik i jkil k l h = −1 i  λi i ci i= ik

Corollary 2.11. The number of -prime ideals is 1 √ √ 2 + 2n +2 − 2n u = 0 2 or 1 √ √ √ 2 + 2n+1 −2 − 2n+1 u = 0 2 2 Proof. Similar to that [8, Corollary 2.6; 27, Proposition 3.1.16].

3. THE -STRATIFICATION

C In this section, we work out the -stratiﬁcation (1) of SpecRn k. First, we give a simpler description of each -stratum. Let T be an admis- C sible set of Rn k and let us denote C C SpecT Rn k = P ∈ SpecRn kP ∩ ℘n = T ideals of skew polynomial rings 203

C Lemma 3.1. Let J be an -prime ideal of Rn k and let T be the admissible set such that J =T . Then

C C SpecT Rn k = SpecJ Rn k Proof. Analogous to [8, Lemma 3.1], using Proposition 2.10. C Proposition 3.2. The -stratiﬁcation of SpecRn k is given by

C C (9) SpecRn k = SpecT Rn k T admissible Proof. This is a consequence of Proposition 2.10 and Lemma 3.1.

The k-algebra obtained by localizing BT at all Xk that appear in AT is C the McConnell–Pettit k-algebra PQT . We will denote Rn k by R.As in [8] consider T the composite map −1 R = R → T →T B A→ PQ −1 T T T T

Remark 3.3 Let J be an -prime ideal of R and J ∩ ℘n = T .LetT −1 −1 denote the inverse image in R T /T T of the multiplicative set of BT generated by all the Xk’s. This is a right Ore set and the corresponding ∼ localization RT satisﬁes that RT = PQT . Clearly RT ⊆ RJ , where J =T −1 and RJ =R/JJ J is the set of all non-zero homogeneous elements withrespect to a certain n-grading (see [13, Theorem 6.6]). In the general case one cannot expect RT = RJ . The following is a counterexample: take C 2×2 n = 2R2 k=qk T =z2; then the homogeneous element 3 y¯1 +¯y2 of degree 1 0 0∈ is not invertible in RT .

Theorem 3.4. Let T be an admissible set. Then T induces a homeomorphism between SpecPQT and SpecT R deﬁned by −1 T ! SpecPQT → SpecT R −1 → T −1 Proof. Notice that T is prime because every prime ideal in R or −1 PQT is completely prime. So it sufﬁces to show that T ∈SpecT R −1 for all ∈ SpecPQT . Put T ∩℘n = T ; it is clear that T ⊆ T .We show the other inclusion by contradiction. So suppose that there exists k ∈ indT and k/∈ indT . Hence, modulo T ,wehavez¯k = 0. So =T ¯zk= ckd − 1YkXk ∈ , which is a contradiction with ∈ SpecPQT . Now, if indT =indT then there exists xk ∈ T suchthat =T ¯xk=Xk ∈ , also a contradiction. We need to show the injectivity. So let and be −1 −1 two elements of SpecPQT suchthat T =T . It is clear that 204 gomez-torrecillas´ and el kaoutit

−1 −1 −1 −1 −1 T ∩ T = T ∩ T =. Apply =T to T T /T T = −1 −1 −1 T T /T T to get = . For the surjectivity we take P ∈ −1 −1 SpecT R, and we put = =T P T /T T ;so ∩ T =. Sup- pose that there exists an indeterminate Xk of PQT suchthat Xk ∈ . Therefore if =T ¯xk=Xk then xk ∈ P with k/∈ T ; this is a contradic- −1 −1 tion with P ∈ SpecT R.If=T ¯xk=Xk +1 − ckd λZk−1Yk , then −1 −1 −1 =T Xk=−1 − ckd y¯k z¯k.Sok/∈ indT and zk ∈ P; this is also a contradiction with P ∈ SpecT R. We have shown that the extension of −1 to PQT is a prime ideal, which is the inverse image of P by T .

Corollary 3.5. Let T be an admissible set. Then SpecT R is homeomorphic to SpecZPQT , where ZPQT is the center of PQT .

Proof. By [12, Corollary 1.5(b)], the contraction → ∩ ZPQT gives a homeomorphism between SpecPQT and SpecZPQT .The corollary is consequence of Theorem 3.4.

e Let be a prime ideal of ZPQT ; we denote by its extension to PQT .BymaxZPQT we will denote the set of maximal ideals of ZPQT . Now we give the analogue of [8, Theorem 3.10]. Theorem 3.6. Let

=T T is an admissible set, ∈ SpecZPQT and

=T T is an admissible set, ∈ maxZPQT

If the parameters cid i = 1n are not roots of unity, then the map −1 e T → T deﬁnes a bijection between and the prime spectrum C SpecRn k whose restriction to is a bijection onto the primitive C spectrum PrimRn k.

C Proof. The bijection between and SpecRn k follows from Theorem 3.4, Corollary 3.5, and the stratiﬁcation (9). The bijection C between and PrimRn k follows from [12, Corollary 1.5(c)], taking C into account that Rn k and PQT are Jacobson algebras. Remark 3.7. From Theorem 3.6 we deduce that the determination of C the prime and primitive spectra of Rn k depends on the computation of the basis of the k-algebras ZPQT where T runs the set of all admissible sets. Some assumptions on the matrix Qn allow sucha determination by C using [12]. Let us see what happens in case Rn k is one of the following 2×n q 2×n k-algebras: qk An k, and qk . ideals of skew polynomial rings 205

C 2×n (1) Consider Rn k=qk where q is not a root of unity. Discussing the quantum linear system attached to an admissible set T (see ∼ k Deﬁnition 4.3), it was shown, in [8, Corollary 3.9], that ZPQT = k , where k = ocompT is the number of connected components of odd lengthin theconnected decomposition of T . An effective computation of the primitive ideals in the algebraically closed case was given in [8, Corollary 3.12]. C q (2) The second example is Rn k=An k, which was studied by Goodearl [9] with n = 1, after by Rigal [27, Proposition 3.2.3] and, inde- pendently, by Akhazavidegan and Jordan [1, Proposition 4.12] for any n.

In this case the McConnell–Pettit algebra PQT , where T ℘n, is simple. See [1, Proposition 4.9; 27, Proposition 2.3.8] for a characterization of this simplicity in terms of the matrix QT . Then ZPQT = k for all T ℘n q (see Proposition 4.1.) By Corollary 3.5, this means that SpecT An k = T for any T ℘n. In the following section we study the case when T = ℘n and we give the primitive ideals in the algebraically closed case. C 2×n (3) The third example is Rn k=qk with q not a root of unity. In Section 4 we will show, by solving the quantum linear system ∼ k attached to T , that ZPQT = k , where k is computed from T by easy combinatorial arguments (see Corollary 4.8). For the primitive ideals we give, in the algebraically closed case, an effective method to compute them (see Corollary 4.10).

q 2×n 4. APPLICATION TO An k AND qk

C q Let Rn k=An kC=q1qn 1 1 1,soz0 = 1. As in [27], × we denote by Bn the subgroup of k generated by λij and qi for i j∈ 2 1n and i

T = ℘n =z1y2x2z2ynxnzn We have = ¯y =Y and = ¯x =1 − q −1Y −1 which implies ℘n 1 1 ℘n 1 1 1 that ZPQ = kY ±1. Let us denote by the set of the ideals of ℘n 1 ℘n q An k containing strictly ℘ that are the inverse images by of the n ℘n ±1 non-zero prime ideals of kY1 . By Corollary 3.5, we have q (10) Spec An k=℘n ∪ ℘n ℘n 206 gomez-torrecillas´ and el kaoutit

Corollary 4.2. If B is free of rank 1 nn + 1, then n 2 q SpecAn k = T ∪ ℘n T admissible and q PrimAn k = T ∪ ℘n T =℘nT admissible

q In particular An k is a primitive k-algebra. If k is algebraically closed, then q SpecAn k = T ∪℘nx1 − α T admissible and q PrimAn k = T ∪℘nx1 − α

T =℘nT admissible where α ∈ k×. Proof. In the general case this is a consequence of Theorem 3.6 and Proposition 4.1 in conjunction with(9) and (10). If k is algebraically closed, let α ∈ k×.So

−1 −1 −1 Y1 − α=℘ x1 − α 1 − q1 ℘n n thus =℘ x − α−11 − q −1α ∈ k×. Therefore ℘n n 1 1

q −1 −1 Spec An k=℘n℘nx1 − α 1 − q1 ℘n

2×n C The k-algebra qk is obtained as Rn k with

−2 −1 C =11q q0λji = q 1 ≤ i

2×n From now on, qk will be denoted by R. The deﬁning relations for R are

−1 −1 yjyi = q yiyjyjxi = q xiyj j>i x x = qx x xy = qy x j>i (11) j i i j j i i j i−1 2 l−i xiyi = yixi +1 − q q ylxl l=1 ideals of skew polynomial rings 207

−2 The normal elements are zi = q xiyi − yixii= 1nz0 = 0, and

zjyi = yizjzjxi = xizj i ≤ j 2 −2 zjyi = q yizjzjxi = q xizj i>j zjzi = zizj all i j xiyi = yixi + qzi−1 i = 1n −2 −1 zi =q − 1yixi + q zi−1 i = 1nz0 = 0

Finally the matrix Qn is

Y1 X1 Y2 X2 ··· Yn Xn  −1 −1  Y1 11qq ··· qq X  11qq−1 ··· qq−1  1   Y  q−1 q−1 11··· qq−1  2   X  qq11··· qq−1  2        −1 −1 −1 −1  Yn q q q q ··· 11 Xn qqqq··· 11

Let T be an admissible set of Rn = R and T = T1 ∪···∪Tr ik = minindTkjk = maxindT k = 1r its connected decomposi- n tion. We denote by AT the quantum space attached to Rn/T withthe kij associated matrix QT =q 1≤ij≤t , where kij ∈0 1 − 1.If1≤ l ≤ t then the symbol Vl will denote a variable Xl for l ∈ik + 1jk\T ,a variable Yl for l ∈ik + 1jk\T k = 1r, and the absence of a n variable when l ∈ RemvT . Let us denote T =kij 1≤ij≤t ∈ Mt×t . 2×n Deﬁnition 4.3. Let T be an admissible set of qk and consider n the associated matrix T =kij 1≤ij≤t .Thequantum linear system associ- n ated to T is the linear system of equations over the integers T m = 0, where m ∈ t . n t n We denote by NullT the torsion free abelian group m ∈ T m = 0. Order the variables Y1

This CnT is the number of consecutive disjoint submatrices of the form 0 5 of the tridiagonal of n , where 5 ∈1 −1. For example, if n = 3 −5 0 T and T =x z ∪z then A3 = k Y Y X Y hence C T =2; 1 1 3 T QT 1 2 2 3 3 208 gomez-torrecillas´ and el kaoutit the couples are Y1Y2 X2Y3.Ifn = 4T =x1z1y2z2∪z4 then A4 = k Y X Y X Y and the couples are Y X X Y , thus T QT 1 2 3 3 4 1 2 3 4 C4T =2. Remark 4.4. (1) Let T be an admissible set suchthatRemv T =, and T = T1 ∪ T2 ∪···∪Tr the connected decomposition of T with ik = minindTkjk = maxindTkk= 1r. Suppose that jr = n − 1, so An = AmY V V V Y X m= i − 1. Then T T ir ir +1 n−2 n−1 n n r Cn−1T +1ifn − ir − 1isodd CnT = Cn−1T if n − ir − 1 is even. If we put T = Rn−2 ∩ T then CnT =Cn−2T +1. (2) Let T be an admissible set with j = maxindT ≤ m − 1m

(1) Suppose that jr 1 and put T = T1 ∪···∪Tr−1. Then CnT =CmT +n − ir +1 where m = ir − 1.

Proof. (1) If jr

Here we distinguish two cases. The ﬁrst case is that n − ir − 1 is even; by Remark 4.4(1) (with n − 1) applied to T ,wehaveCn−1T =Cn−2T +1. Clearly CnT =Cn−2T +2, whence CnT =Cn−1T +1. Check that ideals of skew polynomial rings 209

CnT =Cn−1T . Hence CnT =CnT +1. Applying Remark 4.4(1) to T ,wegetCnT =Cn−1T +1. The second case is that n − ir − 1is odd, so CnT =Cn−2T +1CnT =Cn−2T +1 and thus, CnT = CnT =Cn−1T +1 by Remark 4.4(1) applied to T .

(2) If jr = n, then An = AmY V V m= i − 1 T T ir ir +1 n r So if n − ir is even then CnT =CmT +p + 1 with n − ir = 2p.The same is true if n − ir = 2p + 1. t Lemma 4.6. Let A ∈ Mm×mv ∈ Mm×1v ∈ M1×m be the transpose of v and 5 5 ∈1 −1. Then   A5 v5v rank  −5 vt 0 5  = rank A + 2 −5vt −5 0 and   Av−vv  −vt 001 Av rank = rank + 2  vt 001 −vt 0 −vt −1 −10 Proof. Compute the ranks by using the minors and suitable row and column elementary operations.

n Proposition 4.7. Let T be an admissible set of R and T ∈ Mt×t the n associated matrix. Then rank T = 2 × CnT . Proof. We use induction on n. The cases n = 1 2 are easy. Suppose that the result holds for all admissible sets in Rmm

RemvT =and let T = T1 ∪ T2 ∪···∪Tr be a connected decomposition with ik = minindTkjk = maxindTkk = 1r. We will consider the different possible cases. The notation v stands for a column vector for all its entries equal to 1, and vt is its transpose.

Case 1. If jr

By Lemma 4.6 we have n−2 v (12) rank n = rank T + 2 T −vt 0 Put T ∪z if j

and consider T as an admissible set of Rn−1.Wehave n−2 v n−1 = T T −vt 0

n−1 By the induction hypothesis we have rank T = 2 × Cn−1T . Use n Lemma 4.5(1) and (12) to get rank T = 2 × CnT .

Case 2. If jr = n − 1 then  n−2  T 5v v −v  5vt 01−1  n =   T −vt −10 0 vt 100

where 5 ∈1 −1 and T = T ∩ Rn−2. So Lemma 4.6 implies that

n n−2 (13) rank T = rank T + 2

n−2 By the induction hypothesis we have rank T = 2 × Cn−2T . Then n rank T = 2 × CnT by Remark 4.4(1) and (13). Case 3. If jr = n put T = T1 ∪···∪Tr−1m= ir − 1. Then  m  T v5i +1v ··· 5n−1v5nv t r  −v 0 5i +1 ··· 5n−1 5n   t r   −5i +1v −5i +1 0 ··· 5n−1 5n  n  r r  T =      t  −5n−1v −5n−1 −5n−1 ··· 0 5n t −5nv −5n −5n ··· −5n 0 where 5k ∈1 −1k = ir + 1n. Apply Lemma 4.6 several times to get   m v  rank T +n − i if n − i is even rank n = −vt 0 r r T   m rank T +n − ir + 1 if n − ir is odd. ideals of skew polynomial rings 211

So we have   m−1   T v −vv  t   −v 001  rank  t  +n − ir if n − ir is even  v 001  t n −v −1 −10 rank T =     m−1 v −v  T   t   rank −v 00 +n − ir + 1 if n − ir is odd. vt 00 Apply Lemma 4.6 again to get m−1 v (14) rank n = rank T + 2p + 1 T −vt 0 where p =n − ir . Put   T ∪zm if jr−1

Considered as an admissible set of Rm, the matrix associated to T is of the form m−1 v m = T T −vt 0

m Hence by the induction hypothesis we have rank T = 2 × CmT .Ifwe apply Lemma 4.5(1) to T with n = m + 1, then Cm+1T =CmT +1; hence

m (15) rank T = 2Cm+1T −1

By Lemma 4.5(2) we have CnT =CmT +p + 1, and by Remark 4.4(2) we have Cm+1T =CmT +1, because jr−1

n Let T be an admissible set of R and T ∈ Mt×t and let

α α1 αt t U = U1 ···Ut =α1αt ∈ 212 gomez-torrecillas´ and el kaoutit

n be the k-basis of PQT , where the Ul’s denote the variables in AT .Let T T m1 mk n be the basis of NullT . By Corollary 4.8, we have k = NnT . Using [12, 1.3] we get

mT ±1 mT ±1 ZPQT = kU 1 U k

mT ±1 mT ±1 This is a Laurent polynomial ring in the variables U 1 U k ; N T thus it is canonically isomorphic to the group algebra k n .

2×n Corollary 4.9. Consider qk where q is not a root of unity. Let

N T =T T is an admissible set ∈ Speck n and

N T =T T is an admissible set ∈ maxk n

−1 e Then the map T → T deﬁnes a bijection between and 2×n 2×n Specqk whose restriction to is a bijection onto Primqk . Proof. Apply Theorem 3.6 and Corollary 4.8. From now on, we suppose that k is algebraically closed. Let T be an admis- T T sible set and let m1 mk k = NnT , be a basis of NullT .The maximal ideals of ZPQT are of the form

mT mT =U 1 − λ1U k − λk

× k where =λ1λk∈k . By Corollary 4.9, the primitive ideals of 2×n −1 e qk are of the form T , when T ranges over the set of all admissible sets. We shall exhibit a procedure to compute them from the solutions of the quantum systems deﬁned in Deﬁnition 4.3. t For m =m1mt ∈ we denote 1 m+ = m +m m +m 2 1 1 t t and 1 m− = m −m m −m 2 1 1 t t where m is the absolute value of m ∈ . Then the inverse image of in

AT is

+ − + − mT −mT mT −mT (16) U 1 − λ1U 1 U k − λkU k ideals of skew polynomial rings 213

2×n For each s = 1k, let Y T λ denote an element of k suchthat ms s q

T + T − ms −ms = Y T λ +T = U − λ U T ms s s Then

−1 e =T Y T λ Y T λ T m1 1 mk k

2×n This gives a description of Primqk .

2×n Corollary 4.10. The primitive ideals of qk , when q is not a root 2×n of unity, are the maximal elements of each stratum SpecT qk , where T is an admissible set. If k is algebraically closed, then they are of the form

T Y T λ Y T λ m1 1 mk k

× k where k = NnT and =λ1λk∈k . Example 4.11. In this example we compute the prime and primitive 2×2 × spectra of qk , where q ∈ k is not a root of unity and k is algebraically closed. Eachof the14 strata can be explicitly described. As an illustration, we compute here two of them. First, consider the stratum corresponding to and z y x . We know that A = k Y X Y X ; solving 1 1 1 Q2 1 1 2 2 the attached quantum linear system, we get the basis of Null which −1 ±1 ±1 is −1 1 0 0 0 0 1 1. Thus ZPQ2 = kY1 X1 Y2X2 , hence the maximal ideals corresponding to the -stratum are z2 − γ x1 − × αy1, where α γ ∈ k . Let us denote by the set of the prime ideals of 2×2 qk that are the inverse images by of the non-zero prime but not −1 ±1 ±1 maximal ideals of kY1 X1 Y2X2 .So

2×2 Specqk =0∪II ∈ ∪z2 − γ x1 − αy1

Analogously, for T =z1y1x1,weget

2×2 SpecT qk =y1x1∪JJ ∈ ∪y1x1y2 − γ x2 − α

2×2 where is the set of the ideals of qk strictly containing y1x1 that are the inverse images by T of the non-zero prime but not maximal ±1 ±1 × ideals of ZPQT = ky2 x2 and α γ ∈ k . For any other T ,the algebra ZPQT is one-dimensional, and the computations are straight- 2×2 forward. The lattice of prime ideals of qk is drawn in Fig. 1. The primitive ideal generated by a set A is denoted by A, while prime but not primitive ideals are denoted by A. A line connecting two prime ideal means inclusion. When both ideals belong to the same stratum, we use a wavy line. 214 gomez-torrecillas´ and el kaoutit

Following [23, Example 5], the coordinate ring of quantum Euclidean 2×n+1 space qk , when q has a square root in k,isthek-algebra generated by 2n + 1 variables ω y1x1ynxn satisfying the following relations

−1 −1 yjyi =q yiyjyjxi =q xiyj j>i

xjxi =qxixjxjyi =qyixj j>i (17) −1 yiω=q ωyixiω=qωxi all i i−1 2 l−i 1−i −1/2 1/2 2 xiyi = yixi +1 − q q ylxl + q q − q ω l=1 This k-algebra is an iterated skew polynomial ring

2×n+1 qk =kωy1α1x1β1δ1···ynαnxnβnδn where αiβi are algebra automorphisms and δi are left βi-derivations, −2 deduced from the relations (17). Observe that βiδi = q δiβi for all C −2 −1 i ≥ 1. Consider Rn+1 k with C =11q q0λij = q for C 2×n+1 1 ≤ i

2×n+1 2×n+1 φ ! qk −→qk

1/2 −1 sending y1 → q 1 + q ω x1 → ω yi → yi−1xi → xi−1 i ≥ 2, and 1/2 −1 kerφ=y1 − q 1 + q x1. Denote by

2×n+1 2×n+1 Spec0qk =P ∈ Specqk kerφ⊆P

2×n+1 2×n+1 Clearly, Specqk is homeomorphic to Spec0qk . Example 4.12. Here we apply the foregoing homeomorphism to com- 3 pute the prime spectrum of qk when q has a square root in k and it × is not a root of unity. Let β ∈ k and denote by ηβ the automorphism of 2×2 qk sending y1 → βy1y2 → y2 and xi → xii = 1 2. Consider the epimorphism

2×2 3 φβ ! qk −→qk

1/2 −1 sending y2 → βy1x2 → x1y1 → q 1 + q ω, and x1 → ω. It is clear −1 −1/2 × that kerφβηβ=x1 − β q 1 + qy1. Now ﬁx α ∈ k and put β = −1 −1/2 α q 1 + q so kerφβηβ=x1 − αy1. Using Fig. 1 we get the lattice 3 of prime ideals of qk ; see Fig. 2. There, is the set of the prime ideals 3 × of qk which are the image under φβηβ of elements of αγ ∈ k , and −2 −2 −1/2 1/2 2 ˇz1 =q − 1βy1x1 + q q − q ω . ideals of skew polynomial rings 215

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