The Prime ideal Stratification and The Automorphism Group of U + r,s ( B 2 )
aa r X i v : . [ m a t h . QA ] S e p THE PRIME IDEAL STRATIFICATION AND THEAUTOMORPHISM GROUP OF U + r,s ( B ) XIN TANG
Abstract.
Let g be a finite dimensional complex simple Lie alge-bra, and let r, s ∈ C ∗ be transcendental over Q such that r m s n = 1implies m = n = 0. We will obtain some basic properties of thetwo-parameter quantized enveloping algebra U + r,s ( g ). In particular,we will verify that the algebra U + r,s ( g ) satisfies many nice proper-ties such as having normal separation, catenarity and Dixmier-Moeglin equivalence. We shall study a concrete example, the al-gebra U + r,s ( B ) in detail. We will first determine the normal ele-ments, prime ideals and primitive ideals for the algebra U + r,s ( B ),and study their stratifications. Then we will prove that the alge-bra automorphism group of the algebra U + r,s ( B ) is isomorphic to( C ∗ ) . Introduction
The two-parameter quantized enveloping algebra U r,s ( sl n ) has beenstudied in [7, 8, 29]. The two-parameter quantized enveloping algebras U r,s ( g ) have been studied for the finite dimensional complex simple Liealgebras g of other types in the literatures [4, 19] and the referencestherein. It is easy to see that two-parameter quantized envelopingalgebras U r,s ( g ) are close analogues of the one-parameter quantized en-veloping algebras U q ( g ). On the one hand, they share many commonfeatures with their one-parameter analogues. For instance, the two-parameter quantized enveloping algebras U r,s ( g ) are also Hopf algebraswith natural Hopf algebra structures, and they admit triangular de-compositions. Furthermore, these Hopf algebras can be realized as theDrinfeld doubles of their Hopf sub-algebras. On the other hand, theydo have some different features in both the structure and representationtheory [4, 5, 7, 8]. Date : July 4, 2018.2000
Mathematics Subject Classification.
Primary 17B37,16B30,16B35.
Key words and phrases.
Two-parameter quantized enveloping algebras, Primeideals, Primitive ideals, Stratifications, Automorphisms.This research project is partially supported by the ISAS mini-grant at Fayet-teville State University.
In order to study the structure of U r,s ( g ), one is led to first studythe two-parameter quantized enveloping algebras U + r,s ( g ). In references[28, 30], these algebras have been investigated from the point of viewof two-parameter Ringel-Hall algebras. Indeed, the algebra U + r,s ( g ) canbe presented as an iterated skew polynomial ring, and a PBW basishas been constructed for U + r,s ( g ). Furthermore, all the prime ideals ofthe algebra U + r,s ( g ) are proved to be completely prime ideals based ona mild condition on the parameters r, s .Historically, there has been much interest in the study of prime idealsand automorphism group of quantum algebras including the quantizedenveloping algebra U q ( g ) and their subalgebras, for example [1, 2, 3, 25].Recently, the automorphism group problem has been settled for the al-gebra U + q ( g ) in some special cases. The automorphism group of U + q ( sl )was settled by Alev and Dumas in [2] and by Caldero independentlyin [11]. Furthermore, the automorphism group of the algebra U + q ( sl )was determined in [22]; and the automorphism group of the algebra U + q ( B ) was determined in [20]. In the latter two cases, the central ele-ments and the normal elements (or equivalently the prime and primitiveideals) have played an important role in the determination of the au-tomorphism group. Unfortunately, it still remains a difficult questionto determine the automorphism group for any general U + q ( g ), althoughit was conjectured in [3] that the automorphism group is isomorphicto the semi-direct product of a torus with the group of algebra auto-morphisms induced by the symmetries of the Dynkin diagram. To geta better picture on the two-parameter quantized enveloping algebra U + r,s ( g ), it is natural to study the prime ideals and the algebra auto-morphisms of the algebra U + r,s ( g ). Once again, the determination ofthe normal elements might contribute to a better understanding of theprime and primitive ideals of U + r,s ( g ), and thus the automorphism groupof U + r,s ( g ).In this paper, as one step forward, we will first derive some back-ground information on prime and primitive ideals of the algebra U + r,s ( g ).Since the defining relations of the algebra U + r,s ( g ) are homogeneous,there is a torus (denoted by H ) acting on the algebra U + r,s ( g ). There-fore, we can apply Goodearl-Letzter stratification theory [15, 17] tostudy the prime and primitive spectra of the algebra U + r,s ( g ). Indeed,we will prove that the number of H− invariant prime ideals of the al-gebra U + r,s ( g ) is equal to the cardinality of the Weyl group associatedto the Lie algebra g . We will use the fact that the algebra U + r,s ( g ) isindeed a cocycle twist of the one-parameter quantized enveloping al-gebra U + q ( g ), whose H− invariant prime ideals were proved by Gorelik RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 3 [18] to be parameterized by the elements of the Weyl group. We willalso verify that the algebra U + r,s ( g ) have nice properties such as normalseparation and being catenary. Additionally, we will show the Dixmier-Moeglin equivalence holds for the algebra U + r,s ( g ); thus, the primitiveideals of U + r,s ( g ) can be recognized as the maximal elements in eachprime stratum.We will study a particular case, the algebra U + r,s ( B ) in terms ofits normal elements, prime ideals, primitive ideals, and automorphismgroup. We will first determine all the normal elements of U + r,s ( B ). Asa result, we determine all the prime and primitive ideals of U + r,s ( B )and describe their stratifications. In particular, we will prove that theautomorphism group of the algebra U + r,s ( B ) is a torus of rank two.As in the one-parameter case, one may conjecture that the automor-phism group of U + r,s ( g ) is isomorphic to the semi-direct product of atorus with the group of algebra automorphisms induced by the sym-metries of the corresponding Dynkin diagram. Both the result in [9]and the result in the current paper serve as positive evidences towardthe truth of this conjecture. In a paper [33] which is currently underpreparation, we will address the automorphisms and derivations of thetwo-parameter quantized enveloping algebra U + r,s ( sl ). In particular,we shall manage to prove that such a conjecture is true for the algebra U + r,s ( sl ).The paper is organized as follows. In section 1, we recall the defini-tion and some basic facts about the two-parameter quantized envelop-ing algebra U + r,s ( g ). In Section 2, we will study the normal elements,prime ideals, and primitive ideals for U + r,s ( g ). In Section 3, we treat aparticular case, the algebra U + r,s ( B ).1. The two-parameter quantized enveloping algebra U + r,s ( g )In this section, we recall the definition of the two-parameter quan-tized enveloping algebra U + r,s ( g ) and its Ringel-Hall algebra realization.In particular, we will recall how the algebra U + r,s ( g ) is presented as aniterated skew polynomial ring and the construction of its PBW basis.1.1. The definition of the algebra U + r,s ( g ) . Let g be a finite dimen-sional complex simple Lie algebra. The two-parameter quantum group U r,s ( g ) associated to the Lie algebra g has been studied by many au-thors in the literatures [7, 8, 4] therein. Moreover, the subalgebras U ≥ r,s ( g ) and U + r,s ( g ) of U r,s ( g ) have been further studied in [28, 30] from X. TANG the point of view of two-parameter Ringel-Hall algebras. In this sub-section, we will recall the definition of U + r,s ( g ) and some of its basicproperties.Let n be the rank of the Lie algebra g and let I = { , , · · · , n } . Letus denote by C = ( a ij ) i,j ∈ I the Cartan matrix associated to the Liealgebra g . Let { d i | i ∈ I } be a set of relatively prime positive integerssuch that d i a ij = d j a ji for i, j ∈ I . We will choose two complex numbers r, s ∈ C ∗ which are transcendental over Q in such a way that r m s n = 1implies that m = n = 0. And we will always set r i = r d i , s i = s d i .Let us denote by h− , −i the Ringel form (or Euler form) defined onthe root lattice Q ∼ = Z n where n is the rank of the Lie algebra g . Recallthat this bilinear form is defined as follows h i, j i : = h α i , α j i = d i a ij , if i < j,d i , if i = j, , if i > j. Note that there is a uniform definition proposed for the two-parameterquantum group U r,s ( g ) in [19], and we will recall the following defini-tion of the two-parameter quantized enveloping algebras U + r,s ( g ) usingthe notation in [19]. Definition 1.1. (See also [4, 7]) The two-parameter quantized envelop-ing algebra U + r,s ( g ) is defined to be the C − algebras generated by thegenerators e i subject to the following relations: − a ij X k =0 ( − k (cid:18) − a ij k (cid:19) r i s − i c ( k ) ij e − a ij − ki e j e ki = 0 , ( i = j )where c ( k ) ij = ( r i s − i ) k ( k − r k h j,i i s − k h i,j i for i = j , and for a symbol v , weset up the following notation:( n ) v = v n − v − , ( n ) v ! = (1) v (2) v · · · ( n ) v , (cid:18) nk (cid:19) v = ( n ) v !( k ) v !( n − k ) v ! , for n ≥ k ≥ , and (0) v ! = 1.In the case of the Lie algebra g = sl n +1 , the two-parameter quantizedenveloping algebra U + r,s ( sl n +1 ) is indeed generated by the generators e , e , · · · , e n subject to the following relations: e i e i +1 − ( r + s ) e i e i +1 e i + rse i +1 e i = 0 ,e i e i +1 − ( r + s ) e i +1 e i e i +1 + rse i +1 e i = 0 RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 5 for i = 1 , , · · · , n −
1. It is obvious that the two-parameter quantizedenveloping algebra U + r,s ( g ) is actually a subalgebra of the two-parameterquantum group U r,s ( g ).For later on purpose, we need to introduce a gradation on the algebra U + r,s ( g ). Let us denote by Z n the free abelian group of rank n , witha basis denoted by z , z , · · · , z n . Given any element a ∈ Z n , say a = P a i z i , we shall set | a | = P a i . By assigning to the generator e i the degree z i , the algebra U + r,s ( g ) shall become a Z n − graded algebra.Given any a ∈ Z n , we shall denote by U ± r,s ( g ) a the set of homogeneouselements of degree a in U ± r,s ( g ). It is easy to see that we have thefollowing decomposition of the algebra U + r,s ( g ): U + r,s ( g ) = M a U + r,s ( g ) a . Two-parameter Ringel-Hall algebra H r,s (Λ) . Let k be a fi-nite field. It is well-known that there is a finite dimensional Hereditary k − algebra Λ associated to the Lie algebra g . We denote by P the setof all isomorphism classes of finite dimensional Λ − modules. Thanksto the existence of Hall polynomials, a two-parameter twisted genericRingel-Hall algebra H r,s (Λ) has been constructed in [28, 30]. The al-gebra H r,s (Λ) has been successfully used to study the algebra U + r,s ( g ).More general work along the multiparameter quantized enveloping al-gebras and their realizations can be found in [26]. First of all, we willrecall the construction of H r,s (Λ), and some of its basic properties here.Let us denote by H r,s (Λ) the C − linear space spanned by the set { u α | α ∈ P} , and let us define a multiplication on the C − linear space H r,s (Λ) as follows: u α u β = X λ ∈P s −h α,β i F u λ u α u β ( rs − ) u λ , for any α, β ∈ P . Then it is easy to see that H r,s (Λ) is an associative C − algebra underthe above multiplication. In addition, H r,s (Λ) is a graded algebra whoseelements are graded by the dimension vectors. According to [30], wecan further choose elements X i in H r,s (Λ) for i = 1 , , · · · , m , where m is the number of positive roots, such that the following result is true. Theorem 1.1 (Theorem 2.3.2 in [30]) . The monomials X α (1)1 · · · X α ( m ) m with α (1) , · · · , α ( m ) ∈ N form a C − basis of the algebra H r,s (Λ) ; and X. TANG for i < j , we have X j X i = r h dim X i , dim X j i s −h dim X j , dim X i i X i X j + X I ( i,j ) c ( a i +1 , · · · , a j − ) X a i +1 i +1 · · · X a j − j − with coefficients c ( a i +1 , · · · , a j − ) in Q ( r, s ) . Here the index set I ( i, j ) is the set of sequences ( a i +1 , · · · a j − ) of natural numbers such that P j − t = i +1 a t a t = a i + a j . ✷ Note that by a prime ideal of A , we mean a proper two-sided ideal I ⊂ A such that aAb ⊂ I implies a ∈ I or b ∈ I . A prime ideal P is called completely prime if A/P is a domain, or equivalently ab ∈ I implies a ∈ P or b ∈ P . The previous theorem implies that H r,s (Λ)can be presented as an iterated skew polynomial ring, and thus it hasa PBW basis. Thanks to [17], we know that all prime ideals of H r,s (Λ)are completely prime under the condition that the multiplicative groupgenerated by r, s is torsion-free. Thanks to the results in [28, 30], onefurther knows that the two-parameter quantized enveloping algebra U + r,s ( g ) is indeed isomorphic to the two-parameter Ringel-Hall algebra H r,s (Λ). We should also mention that this isomorphism is indeed agraded isomorphism. Therefore, we shall have the following result. Theorem 1.2.
The algebra U + r,s ( g ) is an iterated skew polynomial ring.In particular, all prime ideals of U + r,s ( g ) are completely prime. ✷ In addition, one should note that the two-parameter quantized en-veloping algebra U + r,s ( g ) is indeed a cocycle twist of the one-parameterquantized enveloping algebra U + q ( g ). Let us explain in a little of de-tail. Note that the one-parameter quantized enveloping algebra U + q ( g )is isomorphic to the one-parameter Ringel-Hall algebra H v,v − (Λ) with v = q ; and the two-parameter quantized enveloping algebra U + r,s ( g )is isomorphic to the two-parameter Ringel-Hall algebra H r,s ( λ ). Bothisomorphisms are indeed graded isomorphisms. In addition, it is obvi-ous to see that both the algebra H v,v − (Λ) and the algebra H r,s (Λ) aregraded by the same group, and the algebra H r,s (Λ) is a cocycle twist ofthe H v,v − (Λ). Indeed, one can define the cocycle twist of the algebra H v,v − (Λ) as follows x ∗ y = ( v − s − ) h µ,ν i xy RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 7 for any x ∈ U + r,s ( g ) µ , y ∈ U + r,s ( g ) ν . Then it is easy to see that thiscocycle twist (the algebra with the new multiplication ∗ ) is isomorphicto the algebra H r,s (Λ) under the condition that rs − = v .1.3. The normal elements of the algebra U + r,s ( g ) . Recall the cen-tral elements and normal elements of the one-parameter quantized en-veloping algebra U + q ( g ) were described by Caldero in [10, 11]. Andthe central elements and normal elements have played an importantrole in the study of prime ideals and automorphisms of the algebra U + q ( g ). It is expected that the central elements and normal elementsof U + r,s ( g ) shall play an equally important role in the determination ofprime ideals and automorphisms of the algebra U + r,s ( g ).In this subsection, we shall derive some preliminary information onthe normal elements of the algebra U + r,s ( g ). In particular, we will provethat the normal elements of U + r,s ( g ) can be described as r − s − centralelements of U + r,s ( g ). We need to adopt an approach used by Caldero[11] for the one-parameter quantized enveloping algebra U + q ( g ). Thefollowing definition mimics the one stated in [11]. Definition 1.2.
Let A be a C − algebra and r, s ∈ C ∗ be transcendentalover Q such that r m s n = 1 implies m = n = 0. Two elements u, v ∈ A are called ( r, s ) − commuting if we have uv = r m s n vu for some integers m, n . An element in A is called ( r, s ) − central if it is ( r, s ) − commutingwith the natural generators of A .First of all, we have the following lemma. Lemma 1.1.
The normal elements of the algebra U + r,s ( g ) are exactlythe ( r, s ) − central elements. Proof: (The proof is an adoption of the one used by Caldero in[11]). First of all, it is obvious that the r − s − central elements of U + r,s ( g ) are normal elements of U + r,s ( g ). Now let a be a non-zero normalelement of U + r,s ( g ). Suppose that a = P β ∈Q + a β is a decomposition ofthe element a in the algebra U + r,s ( g ) = L β ∈Q + U + r,s ( g ) β where Q + is thepositive part of the root lattice. Since a is a normal element of U + r,s ( g ),we have aX i = X ′ i a , where X i are root vectors corresponding to thesimple roots of the Lie algebra g . By considering the weights, we shallhave a β X i = X ′ i a β for some X ′ i . Therefore, we shall have X ′ i = λ i X i for some λ i ∈ C . In particular, we shall have λ i = r m i s n i . Therefore,we have finished the proof of the statement. ✷ Quantum unique factorization domains.
The notion of anoncommutative unique factorization domain was introduced by Chat-ters [12] and were further studied in [13]. Recall that an element p of a X. TANG noetherian domain R is said to be prime if pR = Rp and pR is a heightone prime ideal of R and R/pR is an integral domain. A noetheriandomain R is called a unique factorization domain (noetherian UFD)provided that R has at least one height-one prime ideal, and everyheight-one prime ideal is generated by a prime element.Many examples including universal enveloping algebras of finite di-mensional solvable Lie algebras and their one-parameter quantizationsare noetherian UFDs. As a matter of fact, it has been proved thatmany iterated skew polynomial rings are UFDs under some mild con-ditions. In particular, the result ( Theorem 3.7 in [23]) states that if A is a torsion-free CGL-extension (Cauchon-Gooderal-Letzter extension)[17, 14], then A is a UFD. To proceed, we recall the following definitionof the CGL extensions from reference [23]. Definition 1.3 (Definition 3.1 in [23]) . An iterated skew polynomialextension A = k [ x ][ x ; σ , δ ] · · · [ x n ; σ n , δ n ] is said to be a CGL exten-sion (after Cauchon, Goodearl and Letzter) provided that the followinglist of conditions are satisfied: • With A j := k [ x ][ x ; σ , δ ] · · · [ x j ; σ j , δ j ] for each 1 ≤ j ≤ n ,each σ j is a k − automorphism of A j , each δ j is a locally nilpo-tent k − linear σ j − derivation of A j , and there exist nonrootsof unity q j ∈ k ∗ with σ j δ j = q j δ j σ j ; • For each i < j there exists a λ ji such that σ j ( x i ) = λ ji x i ; • There is a torus H = ( k ∗ ) r acting rationally on A by k − algebraautomorphisms; • The x i for 1 ≤ i ≤ n are H− eigenvectors; • There exist elements h , · · · , h n ∈ H such that h j ( x i ) = σ j ( x i )for j > i and such that the h j − eigenvalue of x j is not a root ofunity.Now it is easy to verify that we have the following result about thealgebra U + r,s ( g ). Theorem 1.3.
The algebra U + r,s ( g ) is a torsion-free CGL − extension.In particular, the algebra U + r,s ( g ) is a noetherian UFD. ✷ Prime and Primitive ideals of the algebra U + r,s ( g )2.1. The H -stratification of Spec ( U + r,s ( g )) and P rim ( U + r,s ( g )) . Inthis subsection, we will apply the stratification theory to the primespectrum and primitive spectrum of the algebra U + r,s ( g ). We denoteby Spec ( U + r,s ( g )) the set of all prime ideals of the algebra U + r,s ( g ). And RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 9 we denote by P rim ( U + r,s ( g )) the set of all primitive ideals of the al-gebra U + r,s ( g ). We first recall some results on the stratification theorydeveloped by Goodearl and Letzter [17, 15]. As a matter of fact, thestratification theory produces partitions of the prime and primitivespectra by using the action of a torus on the algebra U + r,s ( g ).Let H = ( C ∗ ) n , we define an H− action on the algebra U + r,s ( g ).Since the defining relations of the algebra U + r,s ( g ) are homogeneous,we can define a torus H− action on the algebra U + r,s ( g ). Indeed, let h = ( λ , λ , · · · , λ n ) ∈ H , we can define the action of h on the genera-tors e i of U + r,s ( g ) as follows: he i = λ i e i for i = 1 , , · · · , n . A non-zero element x in the algebra U + r,s ( g ) is calledan H− eigenvector of U + r,s ( g ) if we have hx ∈ C x for all h ∈ H . An ideal I of the algebra U + r,s ( g ) is called H− invariant if we have hI = I for all h ∈ H . We denote by H −
Spec ( U + r,s ( g )) the set of all H− invariantprime ideals of U + r,s ( g ). First of all, by a theorem of Goodearl andLetzter about iterated skew polynomial rings, we know that this set isfinite.In the one-parameter case, it was also proved by Gorelik in [18]that the set Spec H ( U + q ( g )) is parameterized by the elements of thecorresponding Weyl group. Thanks to the fact that the algebra U + r,s ( g )is a cocycle twist of the algebra U + q ( g ) and the cocycle twist establishesa one-to-one correspondence between the set of H− invariant primeideals of U + q ( g ) and the set of H− invariant prime ideals of U + r,s ( g ), wehave the following proposition. Proposition 2.1.
The number of H− invariant prime ideals of thealgebra U + r,s ( g ) is finite. In particular, the H− invariant prime idealsof the algebra U + r,s ( g ) are indexed by the elements of the Weyl groupassociated to g . ✷ Using Goodearl-Letzter stratification theory, the action of the torus H on the algebra U + r,s ( g ) allows us to construct a partition of the primespectrum Spec ( U + r,s ( g )). Suppose that J is an H− invariant prime idealof U + r,s ( g ), we denote by Spec J ( U + r,s ( g )) the H− stratum of the primespectrum Spec ( U + r,s ( g )) associated to the ideal J . Recall that we have Spec J ( U + r,s ( g )) : = { P ∈ Spec ( U + r,s ( g )) | ∩ h ∈H hP = J } . And the H− strata Spec J ( U + r,s ( g )) where J ∈ H − Spec ( U + r,s ( g )) form a partitionof the prime spectrum Spec ( U + r,s ( g )) as follows: Spec ( U + r,s ( g )) = ∪ J ∈H− Spec ( U + r,s ( g )) Spec J ( U + r,s ( g )) . Naturally, this partition induces a partition of the set
P rim ( U + r,s ( g ))of all (left) primitive ideals of U + r,s ( g ). Namely, for any J ∈ H − Spec ( U + r,s ( g )), we shall denote primitive stratum corresponding to J by P rim J ( U + r,s ( g )) : = Spec J ( U + r,s ( g )) ∩ P rim ( U + r,s ( g )) . Then it is obvious to see that the H− strata P rim J ( U + r,s ( g )) where J ∈ H − Spec ( U + r,s ( g )) form a partition of P rim ( U + r,s ( g )) as follows: P rim ( U + r,s ( g )) = ∪ J ∈H− Spec ( U + r,s ( g )) P rim J ( U + r,s ( g )) . Diximer-Moeglin equivalence, normal separation and cate-narity.
In this subsection, we establish the fact that the algebra U + r,s ( g )satisfies the Diximer-Moeglin equivalence; and it has normal separationand is catenary.Let k be any field and A be a k − algebra. Recall that a prime ideal P of a noetherian k − algebra A is called rational provided the center ofGoldie quotient F ract ( A/P ) is algebraic over the base field k . Recallthat a prime ideal P is called a locally closed point in the prime spec-trum spec ( A ) under the Zariski topology, provided the singleton { P } is a closed subset in some Zariski-neighborhood of the point P , i.e.,the intersection of these primes properly containing P is strictly largerthan P . We say that the algebra A satisfies the Dixmier-Moeglinequivalence if the sets of rational prime ideals, locally closed primeideals, and primitive ideals all coincide. A useful criterion for Dixmier-Moeglin equivalence has been established in [17] and we recall one ofits versions, which works for many iterated skew polynomial rings witha torus action.Let A = k [ y ][ y ; τ ; δ ] · · · [ y n ; τ n ; δ n ]be an iterated skew polynomial ring over the field k . Let H denotea group, whose elements act as k − algebra automorphisms on A suchthat variables y ; · · · ; y n are all H− eigenvectors. For 1 ≤ i ≤ n , let usset A i = k [ y ][ y ; τ ; δ ] · · · [ y i ; τ i ; δ i ]. The Theorem 4.7 in [17] statesthat the algebra A satisfies the Dixmier-Moeglin equivalence providedthat the algebra A satisfies the following three conditions:(1) There are infinitely many distinct eigenvalues for the action of H on y .(2) Each τ i is a k − algebra automorphism of A i − and each δ i is a k − linear τ i − derivation of A i − .(3) For 2 ≤ i ≤ n , there exists h i ∈ H such that the restriction of h i to A i − coincides with τ i and the h i − eigenvalue of y i is nota root of unity. RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 11 It is obvious that the element h acting on the algebra U + r,s ( g ) asan algebra automorphism, which satisfies the conditions as outlinedabove. Thus via Theorem 4.4 & Theorem 4.7 in [17], we can havethe following result.
Theorem 2.1.
The algebra U + r,s ( g ) satisfies the Dixmier-Moeglin equiv-alence. In particular, the primitive ideals of A are precisely the primeideals maximal within their H− strata. ✷ Recall that the prime spectrum of an algebra A is called catenaryif for any two prime ideals P ( Q ⊂ A , all saturated chains of primeideals from P to Q have the same length. In addition, the prime spec-trum of an algebra A is said to have normal separation if the followingcondition holds: For any proper inclusion P ( Q of prime ideals of A , the factor Q/P contains a nonzero element which is normal in thefactor algebra
A/P .Let A be a group-graded algebra. According to [15], in order to showthat the algebra A has normal separation, it suffices to show that thealgebra A has graded-normal separation. Recall that a graded-primeideal P of A is any proper homogeneous ideal P such that whenever I, J are homogeneous ideals of A with IJ ⊆ P , then either I ⊆ P or J ⊆ P . We say that a graded algebra A has graded-normal separationprovided that for any proper inclusion P ( Q of graded-prime idealsof A , there exists a homogeneous element c ∈ Q − P which is normalmodulo P . Once again, we can prove the following result using thefact that U + q ( g ) has normal separation and U + r,s ( g ) is a cocycle twist of U + q ( g ). Proposition 2.2.
The algebra U + r,s ( g ) satisfies the graded normal sep-aration, and therefore the normal separation. In particular, every non-zero prime ideal contains a non-zero normal element. ✷ We say that the Tauvel’s height formula holds in an algebra A pro-vided the following height ( P ) + GKdim ( A/P ) =
GKdim ( A )is true for any prime ideal P ( A .To proceed, we need to recall a result from [16]. Theorem 2.2 ( Theorem 1.6. in [16]) . Let R be an affine, noetherian,Auslander-Gorenstein, Cohen-Macaulay algebra over a field, with finiteGelfand-Kirillov dimension. If specR is normally separated, then R is catenary. If, in addition, R is a prime ring, then Tauvel’s heightformula holds. ✷ Let A be Noetherian k − algebra and let M be a finitely generatedmodule over A . Let us denote the Gelfand-Kirillov and homological di-mensions of M by GKdim ( M ), and respectively hd ( M ). Let us denotethe global homological dimension of A by gldim ( A ). Let us further de-note by injdim ( A ) the injective dimension of A . If injdim ( A ) < ∞ ,then A is called Auslander-Gorenstein provided that A satisfies thecondition: for any integer 0 ≤ i ≤ j and finitely generated (right) A − module M , we have Ext i ( M, A ) = 0 for all left A − submodules N of Ext j ( M, A ). If A is an Auslander-Gorenstein ring of finite globaldimension, then A is called Auslander-regular. Let us set j ( M ) = min { j : Ext j ( M, A ) = 0 } . The ring A is called Cohen-Macaulay (CM)if j ( M ) + GKdim ( M ) = GKdim ( A ) is true for all finitely generated A − modules M .Thanks to the iterated skew polynomial presentation of the algebra U + r,s ( g ), we can easily have the following result by using the Lemma established in [24] repeatedly.
Theorem 2.3.
The two-parameter quantized enveloping algebra U + r,s ( g ) is an affine noetherian C − algebra with a finite GK − dimension. More-over, the algebra U + r,s ( g ) is an Auslander-regular, Cohen-Macaulay do-main. ✷ To tie the loose ends up, we have the following result.
Theorem 2.4.
The two-parameter quantized enveloping algebra U + r,s ( g ) is catenary. In particular, Tauvel’s height formula holds for the algebra U + r,s ( g ) . ✷ Prime ideals and Automorphism group of the algebra U + r,s ( B )3.1. The algebras U + r,s ( sl ) and U + r,s ( B ) . Recall that r, s are chosenfrom C ∗ such that r, s are transcendental over Q and r m s n = 1 impliesthat m = n = 0. From reference [7], we shall recall the followingconstruction of the algebra U + r,s ( sl ). RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 13 Definition 3.1.
The two-parameter quantized enveloping algebra U + r,s ( sl )is generated by the generators e , e subject to the following relations: e e − ( r + s ) e e e + rse e = 0 ,e e − ( r + s ) e e e + rse e = 0 . Let us set the following new variables: X = e , X = e ,X = X X − sX X ,X ′ = e e − r e e . Then we shall have the following identities: X X = rX X , X X = r − X X . Let us further define some automorphisms τ , τ , τ and derivations δ , δ , δ as follows: τ ( X ) = rX , δ ( X ) = 0 τ ( X ) = s − X , τ ( X ) = r − X δ ( X ) = s − X , δ ( X ) = 0 . For the reader’s convenience, we will state the following well-knownresult.
Proposition 3.1.
The algebra U + r,s ( sl ) can be presented as an iteratedskew polynomial ring in that U + r,s ( sl ) ∼ = C [ X ][ X , τ , δ ][ X , τ , δ ] . In particular, the set { X i X j X k | i, j, k ≥ } forms a PBW-basis of U + r,s ( sl ) . The algebra U + r,s ( sl ) has a GK − dimension of , and everyprime ideal of U + r,s ( sl ) is completely prime. ✷ It is easy to see that the algebra U + r,s ( sl ) is a special example ofthe downup algebras as defined in [6]. The primitives ideals of U + r,s ( sl )were classified as a special case of the results in [27]. The automorphismgroup of U + r,s ( sl ) was determined to be isomorphic to ( C ∗ ) as a specialcase of the results recently established in [9]. In the reference [31], usingthe deleting derivation algorithm [14], we have embedded the algebra U + r,s ( sl ) into a quantum torus, which enables us to determine all thederivations of U + r,s ( sl ) and compute the first Hochschild cohomologygroup for U + r,s ( sl ). Though most of the results on primitive ideals andprime ideals of for the algebra U + r,s ( sl ) are well-known, we will recallsome of them in detail for the purpose of a smooth transition to the case of the algebra U + r,s ( B ). To state the results for U + r,s ( sl ), we willfollow the method used in reference [25].3.2. Normal and central elements of the algebra U + r,s ( sl ) . Firstof all, we will recall some detailed information on the normal and cen-tral elements of the algebra U + r,s ( sl ). We shall verify that the normalelements of U + r,s ( sl ) are exactly scalar multiples of monomials in thevariables X = e e − re e and X ′ = e e − se e .Let α , α be the simple roots of A . We denote by Q the root latticegenerated by α i and denote by Q + = { β = P i =1 β i α i ∈ Q | β i ≥ } .For any β ∈ Q + , we define U + r,s ( sl ) β = C − span { X l X m X n } with β = lα + m ( α + α ) + nα . Proposition 3.2. U + r,s ( sl ) = L β ∈Q U + r,s ( sl ) β . In particular, eachweight space U + r,s ( sl ) β is finite dimensional. ✷ Note that the U + r,s ( sl ) also admits a filtration such that in the asso-ciated graded algebra, we have the following identities. grX grX = sgrX grX ,grX grX = rgrX grX ,grX grX = r − grX grX . Let X = X i X j X k , then we have grX grX = r j s k grXgrX ,grX grX = r − j s − i grXgrX . First of all, we have the following obvious lemma.
Lemma 3.1. If x is an ( r, s ) − central element, then x is ( r, s ) − commutingwith X , X ′ . ✷ Now we have the following theorem on the normal elements andcentral elements of U + r,s ( sl ). Theorem 3.1. N = { c ( X ) i ( X ′ ) j | c ∈ C , i, j ≥ } is the set of allnormal elements of U + r,s ( sl ) . The center of U + r,s ( sl ) is C . ✷ RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 15 Prime and primitive spectra of the algebra U + r,s ( sl ) andtheir stratifications. The prime ideals of the quantum Heisenbergalgebra (isomorphic to U + q ( sl )) were completely described in [25]. Themethod used there can be carried over to our algebra U + r,s ( sl ) with aminor modification. The difference is that the algebra U + r,s ( sl ) hasa trivial center. Using the method in [25], we will give a completedetermination of all (completely) prime ideals of U + r,s ( sl ), and we willskip most of the details if no confusion arises.Note that the subalgebra of U + r,s ( sl ) generated by X , X or by X , X is a quantum plane. By abuse of notation, we will still denotethis subalgebra by C r,s [ X , X ]. For the prime ideals of a quantumplane, we have the following results from [25]. Lemma 3.2 (Lemma 2.1.2 in [25]) . Let P be a prime ideal of C r,s [ X , X ] such that P ∩ C [ X ] = 0 . Then P is a principle ideal generated by thenormal element X . ✷ Proposition 3.3 (Proposition 2.1.3 in [25]) . All prime ideals of thequantum plane C r,s [ X , X ] are given as follows: (0) , ( X ) , ( X );( X , X );( X , X − µ ) , ( X , X − ν ) . for µν = 0 . ✷ Now we can derive some results on prime ideals of the algebra U + r,s ( sl )similar to those in [25]. Lemma 3.3.
Let P be a prime ideal of U + r,s ( sl ) such that P ∩ C r,s [ X , X ] =0 , then P is the one of the following: ( X ) , ( X , X ) , ( X , X ) , ( X , X − µ, X ) , ( X , X − λ, X ) , ( X , X , X ) . with λ, µ ∈ C ∗ . Proof: (See the proof of
Lemma 2.2.1 in [25]) Let P ⊂ U + r,s ( sl )be a prime ideal such that P ∩ C r,s [ X , X ] = 0. If X ∈ P , then X ∈ P . Suppose X ∈ P , then the ideal P/ ( X ) is a prime ideal of U + r,s ( sl ) / ( X ) which is isomorphic to a quantum plane C [ X , X ].Thus we have that P is one of the following prime ideals of U + r,s ( sl ):( X ) , ( X , X ) , ( X , X ) , ( X , X − λ, X ) , ( X , X , X − µ ) , ( X , X , X )with λµ = 0.If X − λ ∈ P for some λ = 0, then we have ( X − λ ) X − sX ( X − λ ) ∈ P . Thus we have X + ( s − λX ∈ P . Since (1 − s ) λ = 0, wehave that X ∈ P .To finish the proof, we need to show that the situation ( X − λ, X ) ⊂ P can not happen for any λ = 0. Suppose we have that ( X − µ, X ) ⊂ P for some µ = 0. Then we have X ∈ P , which implies that we shallhave µ = 0, a contradiction. Thus we have finished the proof. ✷ Because of the symmetry between the variables X , X ′ , we can alsohave the following lemma. Lemma 3.4.
Let P be a prime ideal of U + r,s ( sl ) such that P ∩ C r,s [ X , X ′ ] =0 and X ′ ∈ P , then P is given by replacing X by X ′ in the previouslemma. ✷ So far, we have described all prime ideals of the algebra U + r,s ( sl )which have non-trivial intersections with the subalgebra C r,s [ X , X ]or C r,s [ X , X ′ ]. Thus the problem is reduced to further describe thoseprime ideals of U + r,s ( sl ) which have trivial intersections with the sub-algebras C r,s [ X , X ] and C r,s [ X , X ′ ]. We will show that such a primeideal is indeed the zero ideal. The crucial point is to show that everynon-zero two-sided ideal of U + r,s ( sl ) contains at least one nonzero nor-mal element, and thus any non-zero prime ideal contains either X or X ′ . However, this is true due to the fact that U + r,s ( sl ) satisfies normalseparation. Thus, we have the following proposition. Proposition 3.4.
Let P ⊂ U + r,s ( sl ) be a nonzero prime ideal of U + r,s ( sl ) ,then P contains a non-zero normal element of U + r,s ( sl ) . ✷ Now we are ready to state the main result about prime ideals of thealgebra U + r,s ( sl ). Theorem 3.2.
All prime ideals of the algebra U + r,s ( sl ) are given asfollows: (0) , ( X ) , ( X ′ ) , ( X , X ) , ( X , X ) , ( X , X , X − µ ) , ( X , X , X ) , ( X , X − λ, X ) . RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 17 Proof:
Let P ⊂ U + r,s ( sl ) be a non-zero prime ideal of U + r,s ( sl ). Then P contains a non-zero normal element. Since P is completely prime,then P contains either X or X . Thus the theorem follows. ✷ To summarize everything, we state the following result.
Theorem 3.3.
The algebra U + r,s ( sl ) is catenary and satisfies the nor-mal separation. In addition, Tauvel’s height formula holds for the al-gebra U + r,s ( sl ) . ✷ By Spec ( U + r,s ( sl )), we denote the set of all prime ideals of U + r,s ( sl ).By P rim ( U + r,s ( sl )), we denote the set of all primitive ideals of U + r,s ( sl ).Note that the torus H = ( C ∗ ) is acting on the algebra U + r,s ( sl ) viaalgebra automorphisms. In order to describe the H− stratification ofthe prime ideal spectrum, we need to single out all the H− invariantprime ideals, which are indeed homogeneous ideals. It is easy to seethat we have the following proposition. Proposition 3.5.
The H− invariant prime ideals of the algebra U + r,s ( sl ) are given as follows: (0) , ( X ) , ( X ′ ) , ( X , X ) , ( X , X ) , ( X , X , X ) . ✷ Now we describe the H− stratification of the prime ideal spectrumfor the algebra U + r,s ( sl ). Namely, we have the following theorem. Theorem 3.4.
The H− stratification of all the prime ideals of the al-gebra U + r,s ( sl ) is given as follows: Spec (0) ( U + r,s ( sl )) = { (0) } ; Spec ( X ) ( U + r,s ( sl )) = { ( X ) } ; Spec ( X ′ ) ( U + r,s ( sl )) = { ( X ′ ) } ; Spec ( X ,X ) ( U + r,s ( sl )) = { ( X , X ) } ∪ { ( X , X − β, X ) | β ∈ C ∗ } ; Spec ( X ,X ) ( U + r,s ( sl )) = { ( X , X ) } ∪ { ( X − α, X , X ) | α ∈ C ∗ } ; Spec ( X ,X ,X ) ( U + r,s ( sl )) = { ( X , X , X ) } . ✷ From [9], we have G = Aut ( U + r,s ( sl )) ∼ = ( C ∗ ) . Note that the group G is acting on the set of prime ideals. Therefore, we shall further havethe following result. Proposition 3.6.
The prime spectrum of the algebra U + r,s ( sl ) has H− invariant strata which are exactly G − orbits in the set Spec ( U + r,s ( sl )) . ✷ The primitive spectrum of U + r,s ( sl ) and its H− stratification. Recall the primitive ideals of the downup algebras have been investi-gated by Praton in [27], where the primitive ideals are constructed interms of the annihilators of irreducible representations of the downupalgebras. In this section, we give a determination using the Dixmier-Moeglin equivalence.
Proposition 3.7. (See also [27] ) The primitive ideals of the algebra U + r,s ( sl ) are given as follows: (0) , ( X ) , ( X ′ ) , ( X , X , X − β ) , ( X , X − α, X ) , ( X , X , X ) . for α, β ∈ C ∗ . ✷ In addition, one can easily obtain the following result.
Theorem 3.5.
The H− stratification of the primitive ideals is given asfollows: P rim (0) ( U + r,s ( sl )) = { (0) } ; P rim ( X ) ( U + r,s ( sl )) = { ( X ) } ; P rim ( X ′ ) ( U + r,s ( sl )) = { ( X ′ ) } ; P rim ( X ,X ) ( U + r,s ( sl )) = { ( X , X − β, X ) | β ∈ C ∗ } ; P rim ( X ,X ) ( U + r,s ( sl )) = { ( X − α, X , X ) | α ∈ C ∗ } ; P rim ( X ,X ,X ) ( U + r,s ( sl )) = { ( X , X , X ) } . ✷ In particular, we shall further have the following result.
Corollary 3.1.
The primitive strata of U + r,s ( sl ) are exactly G − orbitsin the set of all primitive ideals. Furthermore, the ideal (0) is a primi-tive ideal of U + r,s ( sl ) , and thus the algebra U + r,s ( sl ) is a primitive ring. ✷ Prime ideals and primitive ideals of the algebra U + r,s ( B ) . First of all, we need to recall the following definition from reference [4].
Definition 3.2.
The two-parameter quantized enveloping algebra U + r,s ( B )is defined to be the C − algebra generated by the generators e , e sub-ject to the following relations: e e − ( r + s ) e e e + r s e e = 0 ,e e − ( r + rs + s ) e e e + rs ( r + rs + s ) e e e − r s e e = 0 . RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 19 Now we further recall some basic properties of the algebra U + r,s ( B )as established in [32]. First of all, we need to fix more notation bysetting the following new variables: X = e , X = e = e e − r e e ,X = e e − s − e e , X = e . Now we recall the following result from reference [32], whose proofis a straightforward calculation.
Lemma 3.5.
The following identities hold. (1) X X = s X X ; (2) X X = r s X X ; (3) X X = rsX X ; (4) X X = r X X + X ; (5) X X = s X X − s X ; (6) X X = r − s − X X . ✷ In addition, let us define some algebra automorphisms τ , τ , and τ ,and some derivations δ , δ , and δ as follows: τ ( X ) = s − X , δ ( X ) = 0 ,τ ( X ) = r − s − X , τ ( X ) = r − s − X ,δ ( X ) = 0 , δ ( X ) = 0 ,τ ( X ) = r − X , τ ( X ) = S − X , τ ( X ) = r − s − X ,δ ( X ) = − r − X , δ ( X ) = X , δ ( X ) = 0 . ✷ Then we have the following well-known result on a basis of the alge-bra U + r,s ( B ). Theorem 3.6.
The algebra U + r,s ( B ) can be presented as an iteratedskew polynomial ring. In particular, we have the following result U + r,s ( sl ) ∼ = C [ X ][ X , τ , δ ][ X , τ , δ ][ X , τ , δ ] . The set { X a X b X c X d | a, b, c, d ∈ Z ≥ } forms a PBW-basis of the algebra U + r,s ( B ) . In particular, the algebra U + r,s ( B ) has a GK − dimension of . ✷ Furthermore, using the graded algebra gr ( U + r,s ( B )) of U + r,s ( B ) asso-ciated to its obvious filtration, it is easy to verify directly the followingresult. Corollary 3.2.
The center of the algebra U + r,s ( B ) is reduced to thebase field C . ✷ Following the convention in the case of U + q ( B ), we may also denotethe element X by Z . In addition, we shall further set the followingvariable: Z ′ = ( X ( X +( s − − r − s − ) X X ) − s ( X +( s − − r − s − ) X X ) X ) . For convenience, let us further set a new variable W = X + ( s − − r − s − ) X X . Then we shall have the following lemma, which can beverified by brutal force. Lemma 3.6.
The following identities can be verified to hold in U + r,s ( B ) . (1) X W = r s W X + (1 − r − s ) X ; (2) X W = s W X ; (3) X W = W X ; (4) X W = s − W X ; (5) X Z ′ = r s Z ′ X ; (6) X Z ′ = Z ′ X ; (7) X Z ′ = r − s − Z ′ X ; (8) X Z ′ = r − s − Z ′ X . ✷ As a result of the previous identities, we know that both the elements Z and Z ′ are r − s − central elements of the algebra U + r,s ( B ). Thus both Z and Z ′ are normal elements of the algebra U + r,s ( B ) because normalelements of the algebra U + r,s ( B ) are just r − s − central elements of thealgebra U + r,s ( B ). In addition, we shall have the following lemma, whichdescribes all the normal elements of the algebra U + r,s ( B ). Lemma 3.7.
The normal elements of U + r,s ( B ) are of the form αZ m ( Z ′ ) n where α ∈ C and m, n ∈ Z ≥ . Proof:
Let u be a normal element of U + r,s ( B ). Then u is an r − s central element of U + r,s ( B ). Let us embed the algebra U + r,s ( B ) into the RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 21 subalgebra C r,s [ e ± , e ± , Z, Z ′ ] of the Goldie quotient ring of U + r,s ( B ).Therefore, the element u takes the following format u = X α a,b,m,n e a e b Z m ( Z ′ ) n . Since the element u is r − s − commuting with the elements e , e , and e = e e − r e e , the element u shall r − s − commute with the elements e and e . Via direct calculations, we can prove that u = αe a e b Z m ( Z ′ ) n . Furthermore, using the fact that the element u is r − s − commutingwith e , via direct calculations, we can prove that the element u indeedtakes the following format u = αZ m ( Z ′ ) n for some α ∈ C as desired. Therefore, we have finished the proof. ✷ Furthermore, using the general properties previously established for U + r,s ( g ), we shall have the following result. Proposition 3.8.
Every non-zero prime ideal of U + r,s ( B ) either con-tains Z or Z ′ . In particular, the algebra U + r,s ( B ) satisfies the normalseparation, is catenary; and the Tauvel’s Formula and the Dixmier-Moeglin equivalence hold. ✷ The next proposition relates the algebra U + r,s ( B ) to the algebra U + r,s ( sl ) as previously discussed. Proposition 3.9.
Let ( Z ) be the two-sided ideal of the algebra U + r,s ( B ) generated by the normal element Z , then we have the following U + r,s ( B ) / ( Z ) ∼ = U + r,s ( sl ) as an algebra. ✷ In addition, using the theory developed in [15, 17], we can easilyverify the following proposition.
Proposition 3.10.
The ideal ( Z ′ ) generated by the normal element Z ′ is a prime ideal of the algebra U + r,s ( B ) . Furthermore, the ideal ( Z ′ ) isa primitive ideal of the algebra U + r,s ( B ) . Proof:
Since the algebra U + r,s ( B ) can be embedded into a subalgebra C r,s [ e ± , e ± , Z, Z ′ ] of the Goldie quotient ring of U + r,s ( B ), the quotientalgebra U + r,s ( B ) / ( Z ′ ) can be embedded into the ring C r,s [ e ± , e ± , Z ]which is obviously a domain. Therefore, we have proved that ( Z ′ ) is a (completely) prime ideal of U + r,s ( B ). In addition, we can verify thatthe center of the localization of U + r,s ( B ) / ( Z ′ ) with respect to the Oreset associated to the prime ideal Z ′ (see [15] for the information onthis Ore set) is the base field C . Using Theorem 5.3. in [15], we canprove that ( Z ′ ) is the only prime ideal in the stratum associated tothe H− invariant ideal ( Z ′ ). As a result, the prime ideal ( Z ′ ) is also aprimitive ideal of U + r,s ( B ). ✷ In addition, due to the fact that the algebra U + r,s ( B ) satisfies the nor-mal separation and αZ m Z ′ n are the only normal elements of U + r,s ( B ),we have the following result. Proposition 3.11.
The ideal (0) is the only element in
Spec (0) ( U + r,s ( B )) .Therefore, we know that that the prime ideal (0) is a primitive ideal ofthe algebra U + r,s ( B ) . Therefore, the algebra U + r,s ( B ) is primitive. ✷ In addition, it is easy to see that we have the following.
Theorem 3.7. (1)
The algebra U + r,s ( B ) has H− invariant primeideals, which are listed as follows: (0) , ( Z ) , ( Z ′ ) , ( e ) , ( e ) , ( e ) , ( e ) , ( e , e ) . (2) The stratification of the prime ideals of the algebra U + r,s ( B ) isgiven as follows: Spec (0) ( U + r,s ( B )) = { (0) } ; Spec ( Z ) ( U + r,s ( B )) = { ( Z ) } ; Spec ( Z ′ ) ( U + r,s ( B )) = { ( Z ′ ) } ; Spec ( e ) ( U + r,s ( B )) = { ( e ) } ; Spec ( e ) ( U + r,s ( B )) = { ( e ) } ; Spec ( e ) ( U + r,s ( B )) = { ( e ) } ∪ { ( e , e − µ ) | µ ∈ C ∗ } ; Spec ( e ) ( U + r,s ( B )) = { ( e ) } ∪ { ( e − λ, e ) | λ ∈ C ∗ } ; Spec ( e ,e ) ( U + r,s ( B )) = { ( e , e ) } . ✷ As a result, we have the following description of the primitive idealsof the algebra U + r,s ( B ). RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 23 Theorem 3.8.
The stratification of the primitive ideals of the algebra U + r,s ( B ) is given as follows: P rim (0) ( U + r,s ( B )) = { (0) } ; P rim ( Z ) ( U + r,s ( B )) = { ( Z ) } ; P rim ( Z ′ ) ( U + r,s ( B )) = { ( Z ′ ) } ; P rim ( e ) ( U + r,s ( B )) = { ( e ) } ; P rim ( e ) ( U + r,s ( B )) = { ( e ) } ; P rim ( e ) ( U + r,s ( B )) = { ( e , e − µ ) | µ ∈ C ∗ } ; P rim ( e ) ( U + r,s ( B )) = { ( e − λ, e ) | λ ∈ C ∗ } ; P rim ( e ,e ) ( U + r,s ( B )) = { ( e , e ) } . ✷ The automorphism group of the algebra U + r,s ( B ) . As thedefining relations of the algebra U + r,s ( B ) are homogeneous in the givengenerators e , e , there is an N − grading on the algebra U + r,s ( B ) ob-tained by assigning to e i the degree 1. Let U + r,s ( B ) = M i ∈ N U + r,s ( B ) i be the corresponding decomposition, with U + r,s ( B ) i being the subspaceof homogeneous elements of degree i. In particular, we have that U + r,s ( B ) = C and U + r,s ( B ) is the two-dimensional space spannedby the generators e , e . For any t ∈ N , we can set U + r,s ( B ) ≥ t = L i ≥ t U + r,s ( B ) i and define V ≤ t = L i ≤ t U + r,s ( B ) i . We say that thenonzero element u ∈ U + r,s ( B ) ≥ t − U + r,s ( B ) ≤ t − has degree t , and write deg ( u ) = t . Since the algebra U + r,s ( B ) is a domain, we have deg ( uv ) = deg ( u ) + deg ( v ) for u, v = 0.Now we recall the following definition from reference [22]. Definition 3.3.
Let A = L i ∈ N A i be an N − graded C − algebra with A = C , which is generated as an algebra by A = C x ⊕ · · · ⊕ C x n . Iffor each i ∈ { , ..., n } , there exists 0 = a ∈ A and a scalar q i such that x i a = q i ax i , then we say that A is an N − graded algebra with enough q − commutation relations.According to [22], it is easy to see that we have the following results. Proposition 3.12.
The algebra U + r,s ( B ) endowed with the gradation asjust defined, is a connected N − graded algebra with enough q − commutationrelations. ✷ Corollary 3.3.
Let σ ∈ Aut ( U + r,s ( B )) and x ∈ U + r,s ( B ) d − , then σ ( x ) = y d + y y>d where y d ∈ U + r,s ( B ) d − and y d ∈ U + r,s ( B ) ≥ d +1 . ✷ In order to proceed, we need to establish some basic identities. Firstof all, recall that we have denoted by e = e e − s e e . We have thefollowing lemma. Lemma 3.8.
Let us assign degree to both of the generators e , e .Then we have Z ′ = r s (1 − r − s ) Ze + (1 − r − s )(1 − r − s ) e e e + r − s (1 − r − s ) e = r (1 − r − s )( s Z + (1 − r − s ) e e ) e + (1 − r − s ) e = ( rs ) − (1 − r − s ) e e + rs (1 + r − s ) Ze = ( rs ) − (1 − r − s )( e ) + ue for some homogeneous element u ∈ U + r,s ( B ) of degree . Proof:
These identities can be verified via brutal force calculations,and we will not repeat them here. ✷ Lemma 3.9.
Let σ ∈ Aut ( U + r,s ( B )) be an algebra automorphism of U + r,s ( B ) , then we have the following σ ( Z ) = λZ, σ ( Z ′ ) = µZ ′ for some λ, µ ∈ C ∗ . Proof:
Since the normal elements of the algebra U + r,s ( B ) are just αZ m ( Z ′ ) n and the normal elements are sent to normal elements by thealgebra automorphism σ , we have that σ ( Z ) = λZ or σ ( Z ) = λZ ′ forsome λ ∈ C ∗ . Via the commuting relations between e , e and Z, Z ′ ,we can further prove that σ ( Z ) = λZ and σ ( Z ′ ) = µZ ′ as desired. ✷ Theorem 3.9.
Let σ ∈ Aut ( U + r,s ( B )) . Then we have the following σ ( e ) = α e , σ ( e ) = α e for some α , α ∈ C ∗ . In particular, we have the following Aut ( U + r,s ( B )) ∼ = ( C ∗ ) . Proof: (Our proof follows the argument used in [21] for the case of U + q ( B )) Let σ ∈ Aut ( U + r,s ( B )) be an algebra automorphism of thealgebra U + r,s ( B ). Based on the graded arguments, we shall have thefollowing σ ( X i ) = α i X i + u i RIME IDEALS AND AUTOMORPHIMS OF U + r,s ( B ) 25 for i = 1 , , , u i is of higher degree than the X i unless u i iszero.For i = 1 , d i = deg ( σ ( e i )) and d = σ ( s Z + (1 − r − s ) e e ). Let us further set d ′ = deg ( σ ( e )). We have that d , d ≥ d ≥ d , d ′ ≥
2. It suffices to show that d = d = 1.Since Z is fixed by σ up to a non-zero scalar, we have that deg ( σ ( Z )) =3. Suppose that we have d + d >
3. Since Z ′ is also fixed by σ up toa non-zero scalar, we have that d + d = d + d + d = 2 d >
4, whichimplies d + d = d >
2. Thus we have that d + d ′ >
4. So we havethat 3 + d = d + d ′ , which implies that d + d ′ = 3. This implies that d = 1 and d ′ = 2. If d + d = 3, then we have d = 1 , d = 2. Sowe always have d = 1 and either d = 2 or d ′ = 2. In either case, weshall have d = 1 as desired. Using the commuting relations between( e ) , e and Z , and the fact that σ fixes Z up to a non-zero scalar, wecan finish the proof of the theorem. ✷ It is obviously that the automorphism group
Aut ( U + r,s ( B )) acts onthe set of primitive ideals. In particular, we have the following result. Proposition 3.13.
The
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