On the quasitriangular structures of abelian extensions of Z 2
aa r X i v : . [ m a t h . QA ] J u l ON THE QUASITRIANGULAR STRUCTURES OF ABELIANEXTENSIONS OF Z KUN ZHOU AND GONGXIANG LIU
Abstract.
The aim of this paper is to study quasitriangular structures on a classof semisimple Hopf algebras k G σ,τ kZ constructed through abelian extensions of k Z by k G for an abelian group G. We prove that there are only two forms of them.Using such description together with some other techniques, we get a completelist of all universal R -matrices on Hopf algebras H n , A n ,t and K (8 n, σ, τ ) (seeSection 2 for the definition of these Hopf algebras). Then we find a simple criterionfor a K (8 n, σ, τ ) to be a minimal quasitriangular Hopf algebra. As a product, someminimal quisitriangular semisimple Hopf algebras are found. Introduction
Throughout the paper we work over an algebraically closed field k of characteristic0. Our original motivation comes from the following well-known fact: There is oneto one correspondence between the conjugacy classes irreducible depth 2 inclusions ofhyperfinite II factors with finite index and the isomorphism classes of all Kac alge-bras, that is, finite dimensional C ∗ -Hopf algebras. Due to the undoubted importanceof quasitriangular structure, a natural question is: if a Kac algebra is quasitriangu-lar, then what can say about the corresponding subfactor? Soon, we realized that itseems quite hard to determine when a Hopf algebra is quasitriangular. For example,even for the very simple case, say, the eight-dimensional Kac-Paljutkin algebra K ,all quasitriangular structure on it weren’t until 2010 that Wakui figured them out[14].In this paper and subsequent works, we try to determine possible quasitriangularstructures on a class of semisimple Hopf algebras arising from exact factorizations offinite groups. The well-known eight-dimensional Kac-Paljutkin algebra K is a veryspecial case of them. The idea of constructing these semisimple Hopf algebras canbe tracked back to G. Kac [4]: Suppose that L = G Γ is an exact factorization of thefinite group L , into its subgroups G and Γ, such that G ∩ Γ = 1 . Associated to thisexact factorization and appropriate cohomology data σ and τ , there is a semisimplebicrossed product Hopf algebra H = k G σ,τ k Γ (see Section 2 for the definition and[9, 11, 12] for details and generalizations). The question of existence of quasitriangular
Mathematics Subject Classification.
Key words and phrases.
Quasitriangular Hopf algebra, Abelian extension. † Supported by NSFC 11722016. structures on k G σ,τ k Γ has been considered before. In 2011, S. Natale [13] provedthat if L is almost simple, then the extension admits no quasitriangular structure.But for our purpose, we want to find more concrete quasitriangular structures ratherthan absence of quasitriangular structures. So comparing the Natale’s viewpoint, weconsider the other extreme case: the almost commutative case. That is, we assumethat both G and Γ are commutative groups. As the start point, we further assumethat Γ is just the Z in this paper.At first, we find that there is a dichotomy on the forms of the quasitriangular struc-tures of k G σ,τ kZ . For convenience, we call one form trivial and the other formnontrivial. In principle, the trivial form corresponds to the bicharacters and thus isnot very complicated. The difficult point of the this paper is to determine the nontriv-ial forms. To do that, we get some necessary conditions for the existence of nontrivialforms. To state the applications of such observations, we give three classes of Hopfalgebras which are denoted by H n , A n ,t and K (8 n, σ, τ ) respectively. We needpoint out that the first two classes of Hopf algebras were studied by some authors[3, 8] before. As the main conclusion of this paper, all universal R -matrices on Hopfalgebras H n , A n ,t and K (8 n, σ, τ ) are given explicitly. In addition, we have iden-tified which are the minimal Hopf algebras among K (8 n, σ, τ ) (we show that there isalmost no minimal quasitriangular structures on H n and A n ,t ). When K (8 n, σ, τ )is a minimal quasitriangular Hopf algebra, we explicitly write out all minimal qua-sitriangular structures on it. As an application of the above conclusions, we find aclass of minimal quasitriangular Hopf algebras which are denoted by K (8 n, ζ ) ( n ≥ k G σ,τ kZ and give some examples of them. In Section 3, we show that thereare only two possible forms of quasitriangular structures on k G σ,τ kZ and give somenecessary conditions for k G σ,τ kZ preserving non-trivial quasitriangular structures.Using our necessary conditions, we can easily get all universal R -matrices on Hopfalgebras H n , A n ,t . Section 4 is devoting to figure out all universal R -matrices onHopf algebras K (8 n, σ, τ ). This section occupies most of parts of this paper due tothe situation becoming complicated than before. As the results, we not only get acomplete list of all quasitriangular structures on K (8 n, σ, τ ) but also find a simplecriterion to determine which one is minimal. Moreover, the new class of minimalquasitriangular Hopf algebras are also given in this section by using this criterion.All Hopf algebras in this paper are finite dimensional. For the symbol δ in Section 2,we mean the classical Kronecker’s symbol.2. Abelian extensions of Z and examples In this section, we recall the definition of k G σ,τ kZ , and then we give some examplesof k G σ,τ kZ for guiding our further research.2.1. The definition of k G σ,τ kZ . N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z Definition 2.1.
A short exact sequence of Hopf algebras is a sequence of Hopf algebrasand Hopf algebra maps (2.1) K ι −→ H π −→ A such that (i) ι is injective, (ii) π is surjective, (iii) Ker( π ) = HK + , K + is the kernel of the counit of K . In this situation it is said that H is an extension of A by K [9, Definiton 1.4]. Anextension (2.1) above such that K is commutative and A is cocommutative is calledabelian. In this paper, we only study the following special abelian extensions k G ι −→ A π −→ kZ , where G is a finite abelian group. Abelian extensions were classified by Masuoka (see[9, Proposition 1.5]), and the above A can be expressed as k G σ,τ kZ which is definedas follows.Let Z = { , x } be the cyclic group of order 2 and let G be a finite group. To givethe description of k G σ,τ kZ , we need the following data(i) ⊳ : Z → Aut( G ) is an injective group homomorphism.(ii) σ : G → k × is a map such that σ ( g ⊳ x ) = σ ( g ) for g ∈ G and σ (1) = 1.(iii) τ : G × G → k × is a unital 2-cocycle and satisfies that σ ( gh ) σ ( g ) − σ ( h ) − = τ ( g, h ) τ ( g ⊳ x, h ⊳ x ) for g, h ∈ G .We need remark that here (i) is not necessary for the definition of k G σ,τ kZ andour aim for adding this additional requirement is just to avoid making a commutativealgebra (in such case all quasitriangular structures are given by bicharacters and thusis known). Definition 2.2. [1, Section 2.2]
As an algebra, the Hopf algebra k G σ,τ kZ is gen-erated by { e g , x } g ∈ G satisfying e g e h = δ g,h e g , xe g = e g⊳x x, x = X g ∈ G σ ( g ) e g , g, h ∈ G. The coproduct, counit and antipode are given by ∆( e g ) = X h,k ∈ G, hk = g e h ⊗ e k , ∆( x ) = [ X g,h ∈ G τ ( g, h ) e g ⊗ e h ]( x ⊗ x ) ,ǫ ( x ) = 1 , ǫ ( e g ) = δ g, , S ( x ) = X g ∈ G σ ( g ) − τ ( g, g − ) − e g⊳x x, S ( e g ) = e g − , g ∈ G. Examples.
The following are some examples of k G σ,τ kZ and we will discussthem in next sections. KUN ZHOU AND GONGXIANG LIU
Example 2.3.
Let n ∈ N and assume that w is a primitive n th root of 1 in k . Then the generalized Kac-Paljutkin algebra H n [3, Section 2.2] belongs to k G σ,τ kZ . Bydefinition, the data ( G, ⊳, σ, τ ) of H n is given by the following way(i) G = Z n × Z n = h a, b | a n = b n = 1 , ab = ba i and a ⊳ x = b, b ⊳ x = a .(ii) σ ( a i b j ) = w ij for 1 ≤ i, j ≤ n .(iii) τ ( a i b j , a k b l ) = ( w ) jk for 1 ≤ i, j, k, l ≤ n .Among of them, if we take n = 2 then the resulting Hopf algebra is just the well-known Kac-Paljutkin 8-dimensional algebra K . That’s the reason why we call H n the generalized Kac-Paljutkin algebra.We give another kind of generalization of K , which is defined as follows and wedenote it by K (8 n, σ, τ ). Example 2.4.
Let n be a natural number. A Hopf algebra H belonging to k G σ,τ kZ is denoted by K (8 n, σ, τ ) if the data ( G, ⊳, σ, τ ) of H satisfies(i) G = Z n × Z = h a, b | a n = b = 1 , ab = ba i ;(ii) a ⊳ x = ab, b ⊳ x = b . If we take n = 1 and let σ ( a i b j ) = ( − ( i − j ) j and τ ( a i b j , a k b l ) = ( − j ( k − l ) for 1 ≤ i, j, k, l ≤
2, then we can easily check that the resulting 8-dimensional Hopf algebrais just the Kac-Paljutkin 8-dimensional algebra K . Therefore, we give another kindof generalization of K . Among of these Hopf algebras K (8 n, σ, τ ), the following classof Hopf algebras are particularly interesting for us since at least they will provide usa number of minimal quasitriangular Hopf algebras. Example 2.5.
Let n ∈ N such that n ≥ ζ is a primitive 2 n th rootof 1. A Hopf algebra H belonging to k G σ,τ kZ is denoted by K (8 n, ζ ) if the data( G, ⊳, σ, τ ) of H satisfies the following conditions(i) G = Z n × Z = h a, b | a n = b = 1 , ab = ba i and a ⊳ x = ab, b ⊳ x = b .(ii) σ ( a i b j ) = ( − i ( i − ζ i for 1 ≤ i ≤ n and 1 ≤ j ≤ τ ( a i b j , a k b l ) = ( − jk for 1 ≤ i, k ≤ n and 1 ≤ j, l ≤ K (16 , ζ ) is the 16 dimensional Hopf algebra H c : σ in [5, Section 3.1] .At last, we recall another kind of semisimple Hopf algebras for our research. Example 2.6.
Take an odd number n and let t be a primitive n th root of 1 in k ,then the Hopf algebras A n ,t were defined in [8, Definition 1.2]. By definition, theybelong to k G σ,τ kZ and the data ( G, ⊳, σ, τ ) of A n ,t can be described as follows(see [1, Section 2.3.4])(i) G = Z n × Z n = h a, b | a n = b n = 1 , ab = ba i and a ⊳ x = a − , b ⊳ x = b .(ii) σ ( g ) = 1, g ∈ G .(iii) τ ( a i b j , a k b l ) = t jk for 1 ≤ i, j, k, l ≤ n . N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z Forms of universal R -matrices In this section, we will prove that for k G σ,τ kZ there are at most two forms ofuniversal R -matrices. Based on this observation, we determine all possible quasitri-angular structures on generalized Kac-Paljutkin algebras H n (see Example 2.3) andsemisimple Hopf algebras A n ,t (see Example 2.6).3.1. Forms of Universal R -matrices. Recall that a quasitriangular Hopf algebrais a pair (
H, R ) where H is a Hopf algebra and R = P R (1) ⊗ R (2) is an invertibleelement in H ⊗ H such that(∆ ⊗ Id)( R ) = R R , (Id ⊗ ∆)( R ) = R R , ∆ op ( h ) R = R ∆( h ) , for h ∈ H . Here by definition R = P R (1) ⊗ R (2) ⊗ , R = P R (1) ⊗ ⊗ R (2) and R = P ⊗ R (1) ⊗ R (2) . The element R is called a universal R -matrix of H or aquasitriangular structure on H .To find the possible forms of universal R -matrices, we need the following Lemmas3.1-3.3 which will help us to check the braiding conditions. The first lemma is well-known. Lemma 3.1. [15, Proposition 12.2.11]
Let H be a Hopf algebra and R ∈ H ⊗ H . For f ∈ H ∗ , if we denote l ( f ) := ( f ⊗ Id)( R ) and r ( f ) := (Id ⊗ f )( R ) , then the followingstatements are equivalent (i) (∆ ⊗ Id)( R ) = R R and (Id ⊗ ∆)( R ) = R R . (ii) l ( f ) l ( f ) = l ( f f ) and r ( f ) r ( f ) = r ( f f ) for f , f ∈ H ∗ . Lemma 3.2.
Denote the dual basis of { e g , e g x } g ∈ G by { E g , X g } g ∈ G , that is, E g ( e h ) = δ g,h , E g ( e h x ) = 0 , X g ( e h ) = 0 , X g ( e h x ) = δ g,h for g, h ∈ G . Then the followingequations hold in the dual Hopf algebra ( k G σ,τ kZ ) ∗ : E g E h = E gh , E g X h = X h E g = 0 , X g X h = τ ( g, h ) X gh , g, h ∈ G. Proof.
Direct computations show that E g E h ( e k ) = E gh ( e k ) = δ gh,k , E g E h ( e k x ) = E gh ( e k x ) = 0for g, h, k ∈ G . As a result, we have E g E h = E gh . Similarly, one can get the last twoequations. (cid:3) Let k G σ,τ kZ as before. We need following two notions which will be used freelythroughout this paper. Let S := { g | g ∈ G, g ⊳ x = g } , T := { g | g ∈ G, g ⊳ x = g } . A very basic observation is:
Lemma 3.3.
We have S ⊆ T T where
T T = { gh | g, h ∈ T } .Proof. Clearly, for s ∈ S, t ∈ T , we have ts ∈ T . From the Definition 2.2 we know thatthe action ⊳ is injective, therefore T = ∅ . Let t ∈ T and it is obvious that S = t ( t − S )and hence S ⊆ T T . (cid:3) KUN ZHOU AND GONGXIANG LIU
With the help of
S, T , we find that
Lemma 3.4.
Let w : G × G → k , w : G × G → k , w : G × G → k , w : G × G → k be four maps and define R as follows R : = X g,h ∈ G w ( g, h ) e g ⊗ e h + X g,h ∈ G w ( g, h ) e g x ⊗ e h + X g,h ∈ G w ( g, h ) e g ⊗ e h x + X g,h ∈ G w ( g, h ) e g x ⊗ e h x. If R satisfies ∆( e g ) R = R ∆( e g ) for g ∈ G , then (i) w ( t, g ) = 0 , t ∈ T, g ∈ G . (ii) w ( g, t ) = 0 , t ∈ T, g ∈ G . (iii) w ( s, t ) = w ( t, s ) = 0 , s ∈ S, t ∈ T .Proof. Since ∆( e g ) R = R ∆( e g ) for g ∈ G , we have the following equations∆( e g )[ X h,k ∈ G w ( h, k ) e h x ⊗ e k ] = [ X h,k ∈ G w ( h, k ) e h x ⊗ e k ]∆( e g ) , (3.1) ∆( e g )[ X h,k ∈ G w ( h, k ) e h ⊗ e k x ] = [ X h,k ∈ G w ( h, k ) e h ⊗ e k x ]∆( e g ) , (3.2) ∆( e g )[ X h,k ∈ G w ( h, k ) e h x ⊗ e k x ] = [ X h,k ∈ G w ( h, k ) e h x ⊗ e k x ]∆( e g ) . (3.3)Firstly, we analyze equation (3.1) as follows∆( e g )[ X h,k ∈ G w ( h, k ) e h x ⊗ e k ] = X h,k ∈ Ghk = g w ( h, k ) e h x ⊗ e k , (3.4) [ X h,k ∈ G w ( h, k ) e h x ⊗ e k ]∆( e g ) = X h,k ∈ Ghk = g w ( h ⊳ x, k ) e h⊳x x ⊗ e k . (3.5)Note that if h ∈ T, k ∈ G such that hk = g , then e h x ⊗ e k will appear in (3.4) whilenot in (3.5). As a result w ( h, k ) = 0 for h ∈ T, k ∈ G and thus (i) has been proved.Similarly, for equation (3.2), there are the following equations∆( e g )[ X h,k ∈ G w ( h, k ) e h ⊗ e k x ] = X h,k ∈ Ghk = g w ( h, k ) e h ⊗ e k x, (3.6) [ X h,k ∈ G w ( h, k ) e h ⊗ e k x ]∆( e g ) = X h,k ∈ Ghk = g w ( h, k ⊳ x ) e h ⊗ e k⊳x x. (3.7)Observe that if h ∈ G, k ∈ T such that hk = g , then e h ⊗ e k x will appear in (3.6)while not in (3.7). Therefore w ( h, k ) = 0 for h ∈ G, k ∈ T and so (ii) is proved. N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z For equation (3.3), we obtain the following equations∆( e g )[ X h,k ∈ G w ( h, k ) e h x ⊗ e k x ] = X h,k ∈ Ghk = g w ( h, k ) e h x ⊗ e k x, (3.8) [ X h,k ∈ G w ( h, k ) e h x ⊗ e k x ]∆( e g ) = X h,k ∈ Ghk = g w ( h ⊳ x, k ⊳ x ) e h⊳x ⊗ e k⊳x x. (3.9)Note that if h ∈ S, k ∈ T , then e h x ⊗ e k x and e k x ⊗ e h x will appear in (3.8) and notin (3.9). This implies that w ( h, k ) = 0 for h ∈ S, k ∈ T . Similarly, one can find that w ( h, k ) = 0 for h ∈ T, k ∈ S . Therefore (iii) has been proved. (cid:3) Lemma 3.5.
Let R be the element given in Lemma 3.4 and assume that (∆ ⊗ Id)( R ) = R R , (Id ⊗ ∆)( R ) = R R . Then the following equations hold (i) w ( s , s ) = w ( s , s ) = w ( s , s ) = 0 , s , s ∈ S . (ii) w ( g, t ) w ( t , t ) = 0 , g ∈ G, t , t ∈ T . (iii) w ( t , g ) w ( t , t ) = 0 , g ∈ G, t , t ∈ T .Proof. We have known l ( X g ) l ( X h ) = l ( X g X h ) for g, h ∈ G due to Lemma 3.1. Let s ∈ S and we can find t , t ∈ T such that t t = s because of Lemma 3.3 and hencethe following equation holds l ( X t X t ) = τ ( t , t ) l ( X t t ) = τ ( t , t )[ X g ∈ G w ( t t , g ) e g + X s ∈ S w ( t t , s ) e s x ] . At the same time, l ( X t ) l ( X t ) = ( X t ∈ T w ( t , t ) e t x )( X t ∈ T w ( t , t ) e t x )= X t ∈ T w ( t , t ) w ( t , t ⊳ x ) e t x = X t ∈ T w ( t , t ) w ( t , t ⊳ x ) σ ( t ) e t . Since l ( X t ) l ( X t ) = l ( X t X t ), we get that w ( s, s ′ ) = w ( s, s ′ ) = 0 for s ′ ∈ S andthus w ( s, s ′ ) = w ( s, s ′ ) = 0 for s, s ′ ∈ S . Similarly by r ( X t ) r ( X t ) = r ( X t X t )one can get that w ( s, s ′ ) = 0 for s, s ′ ∈ S. Therefore, (i) is proved.It remains to show (ii) and (iii). We have known l ( E g ) l ( X t ) = 0 due to Lemma 3.2.However a direct computation shows that l ( E g ) l ( X t ) = P t ∈ T w ( g, t ) w ( t , t ) e t x .Therefore w ( g, t ) w ( t , t ) = 0 for g ∈ G, t , t ∈ T . Similarly, by r ( E g ) r ( X t ) = 0 weget that w ( t, g ) w ( t, t ) = 0 for g ∈ G, t , t ∈ T . These are exactly (ii), (iii). (cid:3) The following proposition shows that universal R -matrices of k G σ,τ kZ has onlytwo possible forms. Proposition 3.6.
Let R be the element given in Lemma 3.4 and assume that it is auniversal R -matrix of k G σ,τ kZ . Then R must belong to one of the following twocases: KUN ZHOU AND GONGXIANG LIU
Case 1: R = P g,h ∈ G w ( g, h ) e g ⊗ e h ; Case 2: R = P s ,s ∈ S w ( s , s ) e s ⊗ e s + P s ∈ S,t ∈ T w ( s, t ) e s x ⊗ e t + P t ∈ T,s ∈ S w ( t, s ) e t ⊗ e s x + P t ,t ∈ T w ( t , t ) e t x ⊗ e t x .Proof. Owing to Lemmas 3.4 and 3.5, we can assume that R has the following form: R = X g,h ∈ G w ( g, h ) e g ⊗ e h + X s ∈ S,t ∈ T w ( s, t ) e s x ⊗ e t + X t ∈ T,s ∈ S w ( t, s ) e t ⊗ e s x + X t ,t ∈ T w ( t , t ) e t x ⊗ e t x. If w ( t , t ) = 0 for all t , t ∈ T , then l ( X t ) = l ( X t ) = 0. Using Lemma 3.2 weknow that l ( X t ) l ( X t ) = l ( X t X t ) and as a result l ( X t X t ) = 0 for all t , t ∈ T .For s ∈ S , we can take t , t ∈ T such that s = t t . Hence we have that l ( X t X t ) = τ ( t , t )( P t ∈ T w ( s, t ) e t ) = 0 which implies that w ( s, t ) = 0 for s ∈ S, t ∈ T .Similarly, by r ( X t ) = r ( X t ) = 0 and r ( X t X t ) = P t ∈ T τ ( t , t ) w ( t, s ) e t , we have w ( t, s ) = 0 for s ∈ S, t ∈ T . Since w ( s, t ) = w ( t, s ) = 0 for s ∈ S, t ∈ T , we knowthat R = P g,h ∈ G w ( g, h ) e g ⊗ e h and therefore we get the first case.If there are t , t ′ ∈ T such that w ( t , t ′ ) = 0, then we will show that w ( t, g ) = w ( g, t ) = 0 for all g ∈ G, t ∈ T . For any g ∈ G , we have w ( g, t ′ ) w ( t , t ′ ) =0 by (ii) of Lemma 3.5 and as a result w ( g, t ′ ) = 0. Since R is invertible and( e t ⊗ e t ′ ) R = w ( t, t ′ ) e t x ⊗ e t ′ x , we know that w ( t, t ′ ) = 0 for t ∈ T . Next,we use (ii) and (iii) of Lemma 3.5 repeatedly. We have w ( t, g ) w ( t, t ′ ) = 0 dueto (iii) of Lemma 3.5. Thus w ( t, g ) = 0 for t ∈ T, g ∈ G . Since R is invertibleand ( e t ⊗ e t ) R = w ( t , t ) e t x ⊗ e t x for t , t ∈ T , we get that w ( t , t ) = 0for t , t ∈ T . Because w ( g, t ) w ( t , t ) = 0 by (ii) of Lemma 3.5, we know that w ( g, t ) = 0 for g ∈ G, t ∈ T and hence we get the second case. (cid:3) Remark 3.7.
For simple, we will call a universal R -matrix R in Case 1 (resp. Case2) of Proposition 3.6 by a trivial (resp. non-trivial ) quasitriagular structure.Recall that a minimal quasitriangular structure (see [15, Definition 12.2.14]) can beequivalently defined by: Assume that R is a universal R -matrix on Hopf algebra H ,and if we let H l := { l ( f ) | f ∈ H ∗ } and H r := { r ( f ) | f ∈ H ∗ } , then ( H, R ) isminimal if H = H l H r (see [15, Proposition 12.2.13]). In this case, we will say that R is a minimal quasitriangular structure on H and H is a minimal quasitriangular Hopfalgebra. Above proposition clearly implies the following corollary. Corollary 3.8.
Every minimal quasitriangular structure on k G σ,τ kZ is non-trivial. Universal R -matrices of H n , A n ,t . To determine all universal R -matricesof H n , A n ,t , we give necessary conditions for k G σ,τ kZ preserving a non-trivialquasitriangular structure firstly. For any finite set X , we use | X | to denote the numberof elements in X . N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z Proposition 3.9.
If there is a non-trivial quasitrianglar structure on k G σ,τ kZ ,then (i) | S | = | T | ; (ii) there is b ∈ S such that b = 1 and t ⊳ x = tb for t ∈ T ; (iii) | G | = 4 m for some m ∈ N + .Proof. Assume that R is a non-trivial quasitriangular structure on k G σ,τ kZ , thenwe have l ( E t ) l ( E t ) = P s ∈ S w ( t , s ) w ( t , s ) σ ( s ) e s for t , t ∈ T . In this situation,we claim that T T = S . In fact, suppose that there are t , t ∈ T satisfying t t ∈ T .Then it is easy to see that l ( E t t ) = P s ∈ S w ( t t , s ) e s x which contradicts to thefact l ( E t ) l ( E t ) = l ( E t t ) (Lemma 3.1). Thus we have T T = S . Take a t ∈ T . Weget that tT ⊆ S and thus | T | ≤ | S | . Since tS ⊆ T , | T | ≥ | S | . As a result we have | T | = | S | and thus (i) has been proved. Next we will show (ii). Take a t ∈ T, thenwe have T = t S . Let t ⊳ x = t and denote b = t − t , then we have b ∈ S by T T = S . Since t S ⊆ T and ( t s ) ⊳ x = ( t s ) b , we have t ⊳ x = tb for t ∈ T . It is easyto know that b = 1 since ⊳x is a group automorphism with order 2 and thus (ii) hasbeen proved. Now let’s show (iii). By definition, S is a subgroup of G and b = 1.Therefore we know that 2 | | S | . Since | G | = | S | + | T | and | S | = | T | , we can see that | G | = 4 m for some m ∈ N + . This completes the proof of (iii). (cid:3) Corollary 3.10. If | G | is an odd number or there are t , t ∈ T such that t − ( t ⊳x ) = t − ( t ⊳ x ) , then k G σ,τ kZ has no non-trivial quasitriangular structures. The following proposition determine all possible trivial quasitriangular structures.
Proposition 3.11. If R is a trivial quasitriangular structure on k G σ,τ kZ , then R must be given by the following way (i) R = P g,h ∈ G w ( g, h ) e g ⊗ e h for some bicharacter w on G ; (ii) w ( g ⊳ x, h ⊳ x ) = w ( g, h ) η ( g, h ) where η ( g, h ) = τ ( g, h ) τ ( h, g ) − for g, h ∈ G .Proof. We can assume that R = P g,h ∈ G w ( g, h ) e g ⊗ e h is a trivial quasitriangularstructure on it. Owing to (∆ ⊗ Id)( R ) = R R and (Id ⊗ ∆)( R ) = R R , we know(i). Expanding ∆ op ( x ) R = R ∆( x ), one can get (ii). (cid:3) The following examples are applications of above results and we get all universal R -matrices of H n , A n ,t respectively. Example 3.12.
Let H n as before in Example 2.3. Then by definition we find that a − ( a ⊳ x ) = a − b and b − ( b ⊳ x ) = b − a . We divide our consideration into two cases.1) If n = 2, then H is the 8-dimensional Kac-Paljutkin algebra K . All possiblequasitrigular structures on K were given in [14] (see see [14, Lemma 5.4]).2) If n >
2, then a − ( a ⊳ x ) = b − ( b ⊳ x ). Therefore H n has no non-trivial quasi-triangular structure by Corollary 3.10. Assume that R = P g,h ∈ G w ( g, h ) e g ⊗ e h is a trivial quasitriangular structure on H n , then w is a bicharacter on G and it satisfies the following equations by Proposition 3.11 w ( a, a ) n = 1 , w ( a, b ) n = 1 , (3.10) w ( b, a ) = w ( a, b ) , w ( b, b ) = w ( a, a ) . Using the above series of equations (3.10), we can get all universal R -matricesof H n ( n ≥
3) easily: let R be a universal R -matrix of H n , then R = X ≤ i,j,k,l ≤ n α ik + jl β il + jk e a i b j ⊗ e a k b l for some α, β ∈ k satisfying α n = β n = 1 . Example 3.13.
Let A n ,t as before in Example 2.6. Since | G | = n is odd, we knowthat A n ,t only has trivial quasitriangular structures on it by Corollary 3.10. Assumethat R = P g,h ∈ G w ( g, h ) e g ⊗ e h is a trivial quasitriangular structure on it, then weget that w is a bicharacter on G . It is easy to see that the condition (ii) of Proposition3.11 is equivalent to the following equations w ( a, a ) n = 1 , w ( a, b ) = t m , m ∈ N , n | (2 m − , (3.11) w ( b, a ) = w ( a, b ) − , w ( b, b ) n = 1 . Using the above series of equations (3.11), it is easy to see that all universal R -matrices of A n ,t are given by the following way: Let R be a universal R -matrix of A n ,t , then R = X ≤ i,j,k,l ≤ n α ik β jl t m ( il − jk ) e a i b j ⊗ e a k b l for some α, β ∈ k , m ∈ N satisfying α n = β n = 1 , n | (2 m − Quasitriangular structures on K (8 n, σ, τ )As another continuation of our Proposition 3.6, we want to determine all universal R -matrices of K (8 n, σ, τ ) in this section. As a consequence, we get a class of minimalquaistriangular semisimple Hopf algebras.4.1. Analysis of η . For a Hopf algebra K (8 n, σ, τ ), if we let η ( g, h ) = τ ( g, h ) τ ( h, g ) − for g, h ∈ G , then η is a bicharacter on G by τ is a 2-cocycle on the abelian group G .Becuase b = 1 and η is a bicharacter, we know that η ( a, b ) = 1 and η ( a, b ) = η ( b, a ).As a result, we have two cases for the value of η ( a, b ), that is, η ( a, b ) = 1 or η ( a, b ) = −
1. Both of them can occur. For example, for the dihedral group algebra k D (aspecial case of K (8 , σ, τ )) we have η ( a, b ) = 1. For the 8-dimensional Kac-Paljutkinalgebra K , we have η ( a, b ) = −
1. We found that the quasitriangular structures inthe two cases have similar expressions. To present our results conveniently, we assumethat η ( a, b ) = − R -matricesof K (8 n, σ, τ ) for the case η ( a, b ) = 1 in subsection 4.4.4.2. Trivial form.
The trivial quasitriangular structures on K (8 n, σ, τ ) can be de-termined easily. N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z Proposition 4.1.
Assume that R is a trivial quasitriangular structure on K (8 n, σ, τ ) ,then R = X ≤ i,k ≤ n, ≤ j,l ≤ α ik β il − jk ( − j ( k + l ) e a i b j ⊗ e a k b l for some α, β ∈ k satisfying α n = β = 1 . Proof.
Assume that R = P g,h ∈ G w ( g, h ) e g ⊗ e h is a trivial quasitriangular structureon K (8 n, σ, τ ), then w is a bicharacter on G and satisfies the following equations byProposition 3.11 w ( a, a ) n = 1 , w ( a, b ) = 1 . (4.1) w ( ab, ab ) = w ( a, a ) , w ( ab, b ) = − w ( a, b ) . If we let w ( a, a ) := α , w ( a, b ) := β , then it is not hard to see that the above seriesof equations (4.1) hold if and only if w ( a i b j , a k b l ) = α ik β il − jk ( − j ( k + l ) . This impliesour desired result. (cid:3) Nontrivial form.
In this subsection, we will give all nontrivial universal R -matrices of K (8 n, σ, τ ). By Corollary 3.8 and Proposition 3.9, if there is a non-trivialquasitriangular structure R on k G σ,τ kZ , then we have | S | = | T | and R has thefollowing form R = X s ,s ∈ S w ( s , s ) e s ⊗ e s + X s ∈ S,t ∈ T w ( s, t ) e s x ⊗ e t + X t ∈ T,s ∈ S w ( t, s ) e t ⊗ (4.2) e s x + X t ,t ∈ T w ( t , t ) e t x ⊗ e t x. Observe that if we let S = { s , · · · , s m } and T = { t , · · · , t m } , then the func-tions w i (1 ≤ i ≤
4) can be viewed as 4 matrices, which are ( w ( s i , s j )) ≤ i,j ≤ m ,( w ( t i , s j )) ≤ i,j ≤ m , ( w ( s i , t j )) ≤ i,j ≤ m , ( w ( t i , t j )) ≤ i,j ≤ m . So when we say w i (1 ≤ i ≤
4) we mean that they are matrices in the following content.To construct all non-trivial quasitriangular structures on K (8 n, σ, τ ), we introducethe following notations to simplify the calculation. Notation.
Let K (8 n, σ, τ ) as before, then we introduce symbols P i , λ i,j , λ i +1 ,j andthe function h as follows(i) P i := (cid:26) Q i − k =1 τ ( a, a k ) i ≥ , i ∈ N i = 0 or i = 1 ;(ii) λ i,j := P − i σ i ( a j ) , λ i +1 ,j := P − i +1 σ i ( a j ) , i, j ∈ N ;(iii) h ( t , t ) := τ ( t ,t ) τ ( t ⊳x,t ⊳x ) for t , t ∈ T ;We use these notations to construct universal R -matrices of K (8 n, σ, τ ) as follows.Let α, β ∈ k such that ( αβ ) n λ n, = 1 , β α = τ ( b, b ) τ ( b, a ) , and S j, := λ j +1 , α j +1 β j , S j, := h ( a, a j +1 b ) λ j +1 , α j β j +1 for j ∈ N . Now we canconstruct R α,β in the form of (4.2) through letting(i) w be given by (cid:26) w ( a i , a j ) = w ( a i b, a j ) = ( λ j, ) i ( αβ ) ij [ σ ( a j )] i w ( a i , a j b ) = − w ( a i b, a j b ) = ( λ j, ) i ( αβ ) ij [ σ ( a j )] i ,(ii) w be given by ( w ( a i , a j +1 ) = w ( a i , a j +1 b ) = λ i, j +1 [ S j, S j, ] i w ( a i b, a j +1 ) = − w ( a i b, a j +1 b ) = τ ( b,a ) βτ ( b,a i ) α λ i, j +1 [ S j, S j, ] i ,(iii) w be given by ( w ( a i +1 , a j ) = w ( a i +1 b, a j ) = λ j, i +1 [ S i, S i, ] j w ( a i +1 , a j b ) = − w ( a i +1 b, a j b ) = − τ ( b,a ) βτ ( b,a j ) α λ j, i +1 [ S i, S i, ] j ,(iv) w be given by w ( a i +1 , a j +1 ) = λ i +1 , j +1 S i +1 j, S ij, w ( a i +1 , a j +1 b ) = λ i +1 , j +1 S ij, S i +1 j, w ( a i +1 b, a j +1 ) = h ( a i +1 b, a j +1 ) λ i +1 , j +1 S ij, S i +1 j, w ( a i +1 b, a j +1 b ) = h ( a i +1 b, a j +1 b ) λ i +1 , j +1 S i +1 j, S ij, ,for 0 ≤ i, j ≤ ( n − Theorem 4.2.
The set of elements { R α,β | α, β ∈ k , ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) } gives all non-trivial quasitriangular structures on K (8 n, σ, τ ) . Remark 4.3. (1) Note the set S ′ := { ( α, β ) | α, β ∈ k , ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) } is not empty. Actually, we can show that | S ′ | = 4 n. This means that we have4 n -number of non-trivial quasitriangular structures on a K (8 n, σ, τ ).(2) The general idea of the proof of Theorem 4.2 is:Part 1: we need to show that R α,β is a universal R -matrix of K (8 n, σ, τ ) . This isgiven in Proposition 4.14. To show Proposition 4.14, we need Lemma 4.5 which givesan equivalent description of a universal R -matrix. Then we use Lemmas 4.6,4.7,4.11and 4.13 to verify this equivalent description. The following diagram illustrates the N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z relations between these lemmas and Proposition 4.14:Proposition 4 . ⇐ Lemma 4 . ⇐ Lemma 4 . . . ⇐ ( Lemma 4 . ⇐ Lemma 4 . . . we show that if R is a universal R -matrix of K (8 n, σ, τ ) , then R = R α,β forsome α, β ∈ k satisfying ( αβ ) n λ n, = 1 , β α = τ ( b,b ) τ ( b,a ) . This is the Proposition 4.19.The basic observation to prove Proposition 4.19 is Lemma 4.16 which states that R is essentially determined by the fourth matrix w . And we use Lemma 4.17 and Lemma4.18 to compute the w of R .The following content of this subsection is designed to prove Theorem 4.2 and westart with the properties of the notations we introduced. Lemma 4.4.
If we use the notation P i , λ i,j , λ i +1 ,j , h as above, then the followingequations hold: P i j P j i = σ ( a j ) i σ ( a i ) j , i, j ≥ , (4.3) P j i +1 σ ( a ) j σ ( a j ) i = P i j σ ( a i +1 ) j [ τ ( b, a ) τ ( b, a i +1 ) ] j , i, j ≥ , (4.4) P j i +1 σ ( a j +1 ) i σ ( a ) i [ τ ( b, a ) τ ( b, a j +1 ) ] i = P i j +1 σ ( a i +1 ) j σ ( a ) j [ τ ( b, a ) τ ( b, a i +1 ) ] j , i, j ≥ , (4.5) h ( t , t ) h ( t ⊳ x, t ⊳ x ) = 1 , h ( t , t ) = h ( t , t ) , t , t ∈ T, (4.6) h ( a, a i +1 b ) = τ ( b, a ) τ ( b, a i +1 ) , i ≥ . (4.7) Moreover if there are α, β ∈ k such that ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) , then wehave [ τ ( b,a ) βτ ( b,a i ) α ] = σ ( a i ) σ ( a i b ) , i ≥ .Proof. We use induction to prove (4.3). If i, j ∈ { , } , then P i j P j i = σ ( a j ) i σ ( a i ) j = 1.Assume that (4.3) hold for ( k, l ) ≤ ( i, j ) (we mean that k ≤ i and l ≤ j ). We considerthat case ( i, j + 1) at first. By P i j +2 P j +22 i = P i j [ τ ( a, a j ) τ ( a, a j +1 )] i P j +22 i = P i j P j i [ τ ( a , a j ) τ ( a, a )] i P i ( τ is a 2-cocycle)= P i j P j i P i P i τ ( a , a j ) i and σ ( a j +2 ) i σ ( a i ) j +1 = [ σ ( a j ) σ ( a ) τ ( a , a j ) ] i σ ( a i ) j σ ( a i )= P i j P j i σ ( a ) i σ ( a i ) τ ( a , a j ) i ( P i j P j i = σ ( a j ) i σ ( a i ) j by induction)= P i j P j i P i P i τ ( a , a j ) i , ( P i P i = σ ( a ) i σ ( a i ) by induction)we have P i j +2 P j +22 i = σ ( a j +2 ) i σ ( a i ) j +1 . Since the equation (4.3) is symmetric for i, j , we know thatit also holds for the case ( i + 1 , j ).We turn to the equation (4.4). By definition, P j i +1 σ ( a ) j σ ( a j ) i = [ P j i τ ( a, a i ) j ] σ ( a ) j σ ( a j ) i . Since P i j σ ( a i +1 ) j [ τ ( b, a ) τ ( b, a i +1 ) ] j = P j i σ ( a j ) i σ ( a i ) j σ ( a i +1 ) j [ τ ( b, a ) τ ( b, a i +1 ) ] j = P j i σ ( a j ) i [ σ ( a i +1 ) σ ( a i ) ] j [ τ ( b, a ) τ ( b, a i +1 ) ] j = P j i σ ( a j ) i [ σ ( a ) τ ( a, a i ) τ ( ab, a i )] j [ τ ( b, a ) τ ( b, a i +1 ) ] j = [ P j i τ ( a, a i ) j ] σ ( a ) j σ ( a j ) i [ τ ( ba, a i ) τ ( b, a ) τ ( b, a i +1 ) τ ( a, a i ) ] j = [ P j i τ ( a, a i ) j ] σ ( a ) j σ ( a j ) i , the equation (4.4) holds.For the equation (4.5), direct computations show that P j i +1 σ ( a j +1 ) i σ ( a ) i [ τ ( b, a ) τ ( b, a j +1 ) ] i = [ τ ( a, a i ) j P j i ] σ ( a j +1 ) i σ ( a ) i [ τ ( b, a ) τ ( b, a j +1 ) ] i = τ ( a, a i ) j σ ( a ) i P j i σ ( a j +1 ) i [ τ ( b, a ) τ ( b, a j +1 ) ] i = τ ( a, a i ) j σ ( a ) i [ P j +1 σ ( a ) i σ ( a i ) j ] (by (4.4))= τ ( a, a i ) j P i j +1 σ ( a i ) j = τ ( a, a i ) j [ P i j τ ( a, a j ) i ] σ ( a i ) j = [ τ ( a, a i ) j τ ( a, a j ) i ][ P i j σ ( a i ) j ]and P i j +1 σ ( a i +1 ) j σ ( a ) j [ τ ( b, a ) τ ( b, a i +1 ) ] j = [ τ ( a, a j ) i τ ( a, a i ) j ][ P j i σ ( a j ) i ] N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z = [ τ ( a, a i ) j τ ( a, a j ) i ][ P j i σ ( a j ) i ]= [ τ ( a, a i ) j τ ( a, a j ) i ][ P i j σ ( a i ) j ] (by (4.3)) . Therefore the equation (4.5) holds.To show the equation (4.6), we do the following calculations. By h ( t , t ) h ( t ⊳ x, t ⊳ x ) = τ ( t , t ) τ ( t ⊳ x, t ⊳ x ) h ( t ⊳ x, t ⊳ x )= τ ( t , t ) τ ( t ⊳ x, t ⊳ x ) τ ( t ⊳ x, t ⊳ x ) τ ( t , t )= τ ( t , t ) τ ( t ⊳ x, t ⊳ x ) τ ( t , t ) τ ( t ⊳ x, t ⊳ x )= σ ( t t ) σ ( t ) − σ ( t ) − σ ( t t ) σ ( t ) − σ ( t ) − = 1 , we have h ( t , t ) h ( t ⊳ x, t ⊳ x ) = 1. Since h ( t ⊳ x, t ⊳ x ) = τ ( t ⊳ x, t ⊳ x ) τ ( t , t )= [ τ ( t , t ) τ ( t ⊳ x, t ⊳ x ) ] − = h ( t , t ) − and h ( t , t ) h ( t ⊳ x, t ⊳ x ) = 1, we have h ( t , t ) = h ( t , t ). Therefore the equation(4.6) holds.For the last equation (4.7), we have h ( a, a i +1 b ) = τ ( a, a i +1 b ) τ ( a i +1 , ab ) = τ ( a, a i +1 b ) τ ( a i +1 , ba ) = τ ( a, a i +1 b ) τ ( b, a ) τ ( a i +1 , ba ) τ ( b, a )= τ ( a, a i +1 b ) τ ( b, a ) τ ( a i +1 b, a ) τ ( a i +1 , b ) = η ( a, a i +1 b ) τ ( b, a ) τ ( a i +1 , b )= − τ ( b, a ) τ ( a i +1 , b ) (by η ( a, a i +1 b ) = − τ ( b, a ) τ ( b, a i +1 ) (by η ( b, a i +1 ) = − . This means that we get the equation (4.7).Furthermore, if there are α, β ∈ k such that ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) , thenwe will show that τ ( b,a ) βτ ( b,a i ) α = σ ( a i ) σ ( a i b ) , i ≥
0. Firstly, we claim that σ ( b ) − = τ ( b, b ).In fact, by σ ( ab )( σ ( a ) σ ( b )) − = τ ( a, b ) τ ( ab, b ) and σ ( a ) = σ ( a ⊳ x ) = σ ( ab ), wehave σ ( b ) − = τ ( a, b ) τ ( ab, b ). Thanks to τ is a 2-cocycle, we have τ ( a, b ) τ ( ab, b ) = τ ( b, b ) τ ( a,
1) = τ ( b, b ) and thus σ ( b ) − = τ ( b, b ). Secondly, since[ τ ( b, a ) βτ ( b, a i ) α ] = [ τ ( b, a ) τ ( b, a i ) ] τ ( b, b ) τ ( b, a ) = τ ( b, b ) τ ( b, a i ) and σ ( a i ) σ ( a i b ) = σ ( a i ) σ ( a i ) σ ( b ) τ ( b, a i ) = 1 σ ( b ) τ ( b, a i ) together with σ ( b ) − = τ ( b, b ), we know that [ τ ( b,a ) βτ ( b,a i ) α ] = σ ( a i ) σ ( a i b ) . (cid:3) The following Lemma is used to prove Proposition 4.14.
Lemma 4.5.
Denote the dual of K (8 n, σ, τ ) by H ∗ , then R is a universal R -matrixof K (8 n, σ, τ ) if and only if the following equations hold τ ( s , s ) = τ ( s , s ) , s , s ∈ S, (4.8) w ( s, t ⊳ x ) = w ( s, t ) η ( s, t ) , s ∈ S, t ∈ T, (4.9) w ( t ⊳ x, s ) = w ( t, s ) η ( t, s ) , s ∈ S, t ∈ T, (4.10) τ ( t , t ) w ( t ⊳ x, t ⊳ x ) = τ ( t ⊳ x, t ⊳ x ) w ( t , t ) , t , t ∈ T, (4.11) l ( f ) l ( f ) = l ( f f ) , r ( f ) r ( f ) = r ( f f ) , f , f ∈ H ∗ . (4.12) Proof.
On the one hand, we have the following equation∆ op ( x ) R = [ X g,h ∈ G τ ( h, g ) e g ⊗ e h ]( x ⊗ x ) R = [ X s ,s ∈ S τ ( s , s ) w ( s , s ) e s ⊗ e s + X s ∈ S,t ∈ T τ ( t, s ) w ( s, t ⊳ x ) e s x ⊗ e t + X t ∈ T,s ∈ S τ ( s, t ) w ( t ⊳ x, s ) e t ⊗ e s x + X t ,t ∈ T τ ( t , t ) w ( t ⊳ x, t ⊳ x ) e t x ⊗ e t x ]( x ⊗ x ) , On the other hand, the following equation hold R ∆( x ) = R [ X g,h ∈ G τ ( g, h ) e g ⊗ e h ]( x ⊗ x )= [ X s ,s ∈ S τ ( s , s ) w ( s , s ) e s ⊗ e s + X s ∈ S,t ∈ T τ ( s, t ) w ( s, t ) e s x ⊗ e t + X t ∈ T,s ∈ S τ ( t, s ) w ( t, s ) e t ⊗ e s x + X t ,t ∈ T τ ( t ⊳ x, t ⊳ x ) w ( t , t ) e t x ⊗ e t x ]( x ⊗ x ) . Therefore, ∆ op ( x ) R = R ∆( x ) holds if and only if equations (4.8)-(4.11) hold. Thelast equation is just Lemma 3.1. (cid:3) N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z We use the following Lemmas 4.6 and 4.7 to check that R α,β satisfies the first fourequalities of Lemma 4.5. Lemma 4.6.
Let K (8 n, σ, τ ) as above, then τ ( s , s ) = τ ( s , s ) for all s , s ∈ S .Proof. Recall the η was defined by η ( g, h ) = τ ( g, h ) τ ( h, g ) − and η is a bicharacter on G , thus η ( a, b ) = η ( a,
1) = 1 and η ( b, a ) = η (1 , a ) = 1. Since S = { a i b j | i, j ≥ } and η ( a i b j , a k b l ) = η ( a, b ) il η ( b, a ) jk = 1, we have τ ( s , s ) = τ ( s , s ) for all s , s ∈ S . (cid:3) Lemma 4.7.
The w i (1 ≤ i ≤ of R α,β satisfy the following equations w ( s, t ⊳ x ) = w ( s, t ) η ( s, t ) , s ∈ S, t ∈ T, (4.13) w ( t ⊳ x, s ) = w ( t, s ) η ( t, s ) , s ∈ S, t ∈ T, (4.14) w ( t , t ) = h ( t , t ) w ( t ⊳ x, t ⊳ x ) , t , t ∈ T. (4.15) Proof.
To simplify the calculation, we analyze η furthermore. Recall that η is bichar-acter on G and we have assumed η ( a, b ) = η ( b, a ) = − η ( s, t ) η ( s, t ⊳ x ) = τ ( s, t ) τ ( t, s ) η ( s, t ⊳ x ) = τ ( s, t ) τ ( t, s ) τ ( s, t ⊳ x ) τ ( t ⊳ x, s )= τ ( s, t ) τ ( s ⊳ x, t ⊳ x ) τ ( t, s ) τ ( t ⊳ x, s ⊳ x ) = σ ( st ) σ ( s ) − σ ( t ) − σ ( ts ) σ ( t ) − σ ( s ) − = 1 , we know that η ( s, t ⊳ x ) = η ( s, t ) − . This observation can help us to simply the proof.Indeed, we find that if w ( s, t ⊳ x ) = w ( s, t ) η ( s, t ) then we have w ( s, t ) = w ( s, t ⊳x ) η ( s, t ⊳ x ) automatically. By S = { a i b j | i, j ≥ } and T = { a i +1 b j | i, j ≥ } ,this discussion tells us that to show the equation (4.13) we only need to show thefollowing two special cases: w ( a i , a j +1 ⊳ x ) = w ( a i , a j +1 ) η ( a i , a j +1 ) , i, j ≥ , (4.16) w ( a i b, a j +1 ⊳ x ) = w ( a i b, a j +1 ) η ( a i b, a j +1 ) , i, j ≥ . (4.17)Using the same arguments (we have η ( t⊳x, s ) = η ( t, s ) − similarly and h ( t ⊳x, t ⊳x ) = h ( t , t ) − by (4.6)), to show the equations (4.14) and (4.15) we only need to showthe following equations w ( a i +1 ⊳ x, a j ) = w ( a i +1 , a j ) η ( a i +1 , a j ) , i, j ≥ , (4.18) w ( a i +1 ⊳ x, a j b ) = w ( a i +1 , a j b ) η ( a i +1 , a j b ) , i, j ≥ , (4.19) w ( a i +1 b, a j +1 ) = h ( a i +1 b, a j +1 ) w (( a i +1 b ) ⊳ x, a j +1 ⊳ x ) , i, j ≥ , (4.20) w ( a i +1 b, a j +1 b ) = h ( a i +1 b, a j +1 b ) w (( a i +1 b ) ⊳ x, ( a j +1 b ) ⊳ x ) , i, j ≥ . (4.21)We will check them one by one. Since w ( a i , a j +1 ⊳ x ) = w ( a i , a j +1 b ) = λ i, j +1 [ S j, S j, ] i w ( a i , a j +1 ) η ( a i , a j +1 ) = w ( a i , a j +1 ) = λ i, j +1 [ S j, S j, ] i , we have w ( a i , a j +1 ⊳ x ) = w ( a i , a j +1 ) η ( a i , a j +1 ) and therefore the equation(4.16) holds. Since w ( a i b, a j +1 ⊳ x ) = w ( a i b, a j +1 b )= τ ( b, a ) βτ ( b, a i ) α λ i, j +1 [ S j, S j, ] i and w ( a i b, a j +1 ) η ( a i b, a j +1 ) = w ( a i b, a j +1 )( − − τ ( b, a ) βτ ( b, a i ) α λ i, j +1 [ S j, S j, ] i ( − τ ( b, a ) βτ ( b, a i ) α λ i, j +1 [ S j, S j, ] i , we have w ( a i b, a j +1 ⊳ x ) = w ( a i b, a j +1 ) η ( a i b, a j +1 ) and therefore the equation(4.17) holds. Since w ( a i +1 ⊳ x, a j ) = w ( a i +1 b, a j )= λ j, i +1 [ S i, S i, ] j and w ( a i +1 , a j ) η ( a i +1 , a j ) = w ( a i +1 , a j )= λ j, i +1 [ S i, S i, ] j , we have w ( a i +1 ⊳ x, a j ) = w ( a i +1 , a j ) η ( a i +1 , a j ) and therefore the equation(4.18) holds. Since w ( a i +1 ⊳ x, a j b ) = w ( a i +1 b, a j b )= τ ( b, a ) βτ ( b, a j ) α λ j, i +1 [ S i, S i, ] j and w ( a i +1 , a j b ) η ( a i +1 , a j b ) = w ( a i +1 , a j b )( − − τ ( b, a ) βτ ( b, a j ) α λ j, i +1 [ S i, S i, ] j ( − τ ( b, a ) βτ ( b, a j ) α λ j, i +1 [ S i, S i, ] j , we have w ( a i +1 ⊳ x, a j b ) = w ( a i +1 , a j b ) η ( a i +1 , a j b ) and therefore the equation(4.19) holds. Since w ( a i +1 b, a j +1 ) = h ( a i +1 b, a j +1 ) λ i +1 , j +1 S ij, S i +1 j, = h ( a i +1 b, a j +1 ) w ( a i +1 , a j +1 b )and w ( a i +1 b, a j +1 b ) = h ( a i +1 b, a j +1 b ) λ i +1 , j +1 S i +1 j, S ij, = h ( a i +1 b, a j +1 b ) w ( a i +1 , a j +1 ) , N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z the equation (4.20) and the equation (4.21) hold. (cid:3) The following Lemma 4.8 is completely prepared to prove Lemma 4.9.
Lemma 4.8.
Let R α,β as above, then [ w ( a, t ) w ( a, t ⊳ x ) σ ( t )] n = P n , w ( b, t ) = τ ( b, b ) , (4.22) w ( a, t ) w ( b, t ⊳ x ) = τ ( a, b ) w ( ab, t ) , w ( a, t ) w ( b, t ) = τ ( b, a ) w ( ab, t ) , (4.23) [ w ( a, t ) w ( a, t ⊳ x ) σ ( t )] i = P i w ( a i , t ) , (4.24) w ( a i , t ) w ( b, t ) = τ ( a i , b ) w ( a i b, t ) , (4.25) w ( a, t ) i +1 w ( a, t ⊳ x ) i σ ( t ) i = P i +1 w ( a i +1 , t ) , (4.26) w ( a i +1 , t ) w ( b, t ⊳ x ) = τ ( a i +1 , b ) w ( a i +1 b, t ) , (4.27) w ( a, s ) n σ ( s ) n = 1 , w ( b, s ) = 1 , (4.28) w ( b, s ) w ( a, s ) = w ( ab, s ) , (4.29) w ( a, s ) i σ ( s ) i = w ( a i , s ) , w ( a i , s ) w ( b, s ) = w ( a i b, s )(4.30) w ( a, s ) i +1 σ ( s ) i = w ( a i +1 , s ) , w ( a i +1 , s ) w ( b, s ) = w ( a i +1 b, s ) . (4.31) Proof.
We will show (4.22) at first. Since T = { a i +1 , a i +1 b | i ≥ } , we need to showthe following equations:[ w ( a, a j +1 ) w ( a, a j +1 b ) σ ( a j +1 )] n = P n , [ w ( a, a j +1 b ) w ( a, a j +1 ) σ ( a j +1 b )] n = P n . Since σ ( a j +1 ) = σ ( a j +1 ⊳ x ) by the definition of σ and σ ( a j +1 ⊳ x ) = σ ( a j +1 b ), wehave σ ( a j +1 ) = σ ( a j +1 b ). This implies that we only need to show the first equation.Since [ w ( a, a j +1 ) w ( a, a j +1 b ) σ ( a j +1 )] n = [ S j, S j, σ ( a j +1 )] n , [ S j, S j, σ ( a j +1 )] n = [( λ j +1 , α j +1 β j ) S j, σ ( a j +1 )] n = [( λ j +1 , α j +1 β j )( h ( a, a j +1 b ) λ j +1 , α j β j +1 ) σ ( a j +1 )] n = h ( a, a j +1 b ) n σ ( a j +1 ) n λ n j +1 , ( αβ ) (2 j +1) n and λ n j +1 , ( αβ ) (2 j +1) n = λ n j +1 , [( αβ ) n ] j +1 = λ n j +1 , [ λ − n, ] j +1 = [ P − j +1 σ ( a ) j ] n [ λ n, ] − j − = [ P − n j +1 σ ( a ) jn ][ λ n, ] − j − = [ P − n j +1 σ ( a ) jn ][ P − n σ ( a ) n ] − j − = P − n j +1 P j +12 n σ ( a ) − n , we have[ w ( a, a j +1 ) w ( a, a j +1 b ) σ ( a j +1 )] n = h ( a, a j +1 b ) n σ ( a j +1 ) n P − n j +1 P j +12 n σ ( a ) − n . By the equation (4.4), we have P l k +1 σ ( a ) l σ ( a l ) k = P k l σ ( a k +1 ) l [ τ ( b, a ) τ ( b, a k +1 ) ] l . (4.32)Let k = j and l = n , then the equation (4.32) above becomes (by noting that a n = 1) P n j +1 σ ( a ) n = P j n σ ( a j +1 ) n [ τ ( b, a ) τ ( b, a j +1 ) ] n . (4.33)Therefore we have[ w ( a, a j +1 ) w ( a, a j +1 b ) σ ( a j +1 )] n = h ( a, a j +1 b ) n P − n j +1 P j n σ ( a j +1 ) n σ ( a ) − n P n = h ( a, a j +1 b ) n [ τ ( b, a j +1 ) τ ( b, a ) ] n P n = P n where for the last equality we use the equation (4.7). Since[ w ( b, a j +1 )] = [ − τ ( b, a ) βα ] = τ ( b, a ) [ βα ] = τ ( b, a ) τ ( b, b ) τ ( b, a ) = τ ( b, b ) , [ w ( b, a j +1 b )] = [ τ ( b, a ) βα ] = τ ( b, a ) [ βα ] = τ ( b, a ) τ ( b, b ) τ ( b, a ) = τ ( b, b )and T = { a i +1 , a i +1 b | i ≥ } , we have w ( b, t ) = τ ( b, b ) by noting that T = { a i +1 , a i +1 b | i ≥ } . Therefore, we get the equations (4.22).Now let us show the equations (4.23). Since τ ( a, b ) w ( ab, a j +1 ) = τ ( a, b )[ h ( ab, a j +1 ) S j, ]= τ ( a, b ) h ( ab, a j +1 )[ h ( a, a j +1 b ) λ j +1 , α j β j +1 ]= τ ( a, b ) h ( ab, a j +1 )[ h ( a, a j +1 b ) βα S j, ]= τ ( a, b )[ h ( ab, a j +1 ) h ( a, a j +1 b )] βα S j, = τ ( a, b )[ h ( ab, a j +1 ) h (( ab ) ⊳ x, a j +1 ⊳ x )] βα S j, = τ ( a, b ) βα S j, (by (4.6))= − τ ( b, a ) βα S j, (by η ( a, b ) = − S j, )( − τ ( b, a ) βα )= w ( a, a j +1 ) w ( b, a j +1 b ) , N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z we have w ( a, a j +1 ) w ( b, a j +1 b ) = τ ( a, b ) w ( ab, a j +1 ) . (4.34)Since w ( a, a j +1 b ) w ( b, a j +1 ) = S j, w ( b, a j +1 ) = S j, [ τ ( b, a ) βα ]= [ h ( a, a j +1 b ) λ j +1 , α j β j +1 ][ τ ( b, a ) βα ]= [ h ( a, a j +1 b ) βα S j, ][ τ ( b, a ) βα ]= τ ( b, a ) h ( a, a j +1 b )( βα ) S j, = τ ( b, a ) τ ( b, a ) τ ( b, a j +1 ) ( βα ) S j, = τ ( b, a ) τ ( b, a ) τ ( b, a j +1 ) τ ( b, b ) τ ( b, a ) S j, = τ ( b, b ) τ ( b, a j +1 ) S j, and τ ( a, b ) w ( ab, a j +1 b ) = τ ( a, b )[ h ( ab, a j +1 b ) S j, ]= τ ( a, b ) τ ( ab, a j +1 b ) τ ( a j +1 , a ) S j, = τ ( a, b ) τ ( ab, a j +1 b ) τ ( a j +1 , a ) S j, = τ ( b, a j +1 b ) τ ( a, a j +1 ) τ ( a j +1 , a ) S j, = τ ( b, a j +1 b ) S j, = τ ( b, ba j +1 ) S j, = τ ( b, ba j +1 ) τ ( b, a j +1 ) τ ( b, a j +1 ) S j, = τ ( b, b ) τ ( b, a j +1 ) S j, , we have w ( a, a j +1 b ) w ( b, a j +1 ) = τ ( a, b ) w ( ab, a j +1 b ) . (4.35)By equations (4.34) and (4.35), we have w ( a, t ) w ( b, t ⊳ x ) = τ ( a, b ) w ( ab, t ) for t ∈ T = { a i +1 , a i +1 b | i ≥ } . Since w ( b, a j +1 ) w ( a, a j +1 ) = ( τ ( b, a ) βα ) w ( a, a j +1 )= ( − w ( b, a j +1 b )) w ( a, a j +1 ) = − w ( a, a j +1 ) w ( b, a j +1 b )= − τ ( a, b ) w ( ab, a j +1 ) (by (4.34))= τ ( b, a ) w ( ab, a j +1 )and w ( b, a j +1 b ) w ( a, a j +1 b ) = ( − τ ( b, a ) βα ) w ( a, a j +1 b )= ( − w ( b, a j +1 )) w ( a, a j +1 b )= − w ( a, a j +1 b ) w ( b, a j +1 )= − τ ( a, b ) w ( ab, a j +1 b ) (by (4.35))= τ ( b, a ) w ( ab, a j +1 b ) , we have w ( b, t ) w ( a, t ) = τ ( b, a ) w ( ab, t ) and thus the equations (4.23) hold.Next, we will show (4.24). Since[ w ( a, a j +1 ) w ( a, a j +1 b ) σ ( a j +1 )] i = [ S j, S j, σ ( a j +1 )] i and P i w ( a i , a j +1 ) = P i [ λ i, j +1 ( S j, S j, ) i ]= P i λ i, j +1 ( S j, S j, ) i = P i ( P − i σ ( a j +1 ) i )( S j, S j, ) i = σ ( a j +1 ) i ( S j, S j, ) i = [ S j, S j, σ ( a j +1 )] i , we have [ w ( a, a j +1 ) w ( a, a j +1 b ) σ ( a j +1 )] i = P i w ( a i , a j +1 ) . (4.36)Similarly, by[ w ( a, a j +1 b ) w ( a, a j +1 ) σ ( a j +1 b )] i = [ w ( a, a j +1 b ) w ( a, a j +1 ) σ ( a j +1 )] i = P i w ( a i , a j +1 ) (by (4.36))= P i w ( a i , a j +1 b ) , we have [ w ( a, a j +1 b ) w ( a, a j +1 ) σ ( a j +1 b )] i = P i w ( a i , a j +1 b ). This means thatwe get the equation (4.24).We turn to the proof of the equation (4.25). Since w ( a i , a j +1 ) w ( b, a j +1 ) = [ λ i, j +1 ( S j, S j, ) i ] w ( b, a j +1 )= [ λ i, j +1 ( S j, S j, ) i ][ τ ( b, a ) βα ]= τ ( b, a ) βα λ i, j +1 ( S j, S j, ) i N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z and τ ( a i , b ) w ( a i b, a j +1 ) = τ ( a i , b )[ τ ( b, a ) τ ( b, a i ) βα λ i, j +1 ( S j, S j, ) i ]= τ ( a i , b ) τ ( b, a i ) [ τ ( b, a ) βα λ i, j +1 ( S j, S j, ) i ]= η ( a i , b )[ τ ( b, a ) βα λ i, j +1 ( S j, S j, ) i ]= τ ( b, a ) βα λ i, j +1 ( S j, S j, ) i , we have w ( a i , a j +1 ) w ( b, a j +1 ) = τ ( a i , b ) w ( a i b, a j +1 ) . (4.37)Since w ( a i , a j +1 b ) w ( b, a j +1 b ) = w ( a i , a j +1 ) w ( b, a j +1 b )= w ( a i , a j +1 )[ − w ( b, a j +1 )]= − w ( a i , a j +1 ) w ( b, a j +1 )= − τ ( a i , b ) w ( a i b, a j +1 ) (by (4.37))= τ ( a i , b )[ − w ( a i b, a j +1 )]= τ ( a i , b ) w ( a i b, a j +1 b ) , we have w ( a i , a j +1 b ) w ( b, a j +1 b ) = τ ( a i , b ) w ( a i b, a j +1 b ). Therefore the equa-tion (4.25) is proved.We will show the equation (4.26). Since w ( a, a j +1 ) i +1 w ( a, a j +1 b ) i σ ( a j +1 ) i = S i +1 j, S ij, σ ( a j +1 ) i = σ ( a j +1 ) i S i +1 j, S ij, and P i +1 w ( a i +1 , a j +1 ) = P i +1 [ λ i +1 , j +1 S i +1 j, S ij, ]= P i +1 [ P − i +1 σ ( a j +1 ) i ] S i +1 j, S ij, = σ ( a j +1 ) i S i +1 j, S ij, , we have w ( a, a j +1 ) i +1 w ( a, a j +1 b ) i σ ( a j +1 ) i = P i +1 w ( a i +1 , a j +1 ) . (4.38)Since w ( a, a j +1 b ) i +1 w ( a, a j +1 ) i σ ( a j +1 b ) i = S i +1 j, S ij, σ ( a j +1 b ) i = σ ( a j +1 ) i S i +1 j, S ij, and P i +1 w ( a i +1 , a j +1 b ) = P i +1 [ λ i +1 , j +1 S ij, S i +1 j, ]= P i +1 [ P − i +1 σ ( a j +1 ) i ] S ij, S i +1 j,
14 KUN ZHOU AND GONGXIANG LIU = σ ( a j +1 ) i S i +1 j, S ij, , we have w ( a, a j +1 b ) i +1 w ( a, a j +1 ) i σ ( a j +1 b ) i = P i +1 w ( a i +1 , a j +1 b ) . (4.39)Combining the equation (4.38) together with the equation (4.39), we get the equation(4.26).Now let us go to the proof of the equation (4.27). Directly we have w ( a i +1 , a j +1 ) w ( b, a j +1 b ) = [ λ i +1 , j +1 S i +1 j, S ij, ] w ( b, a j +1 b )= [ λ i +1 , j +1 S i +1 j, S ij, ][ − τ ( b, a ) βα ]= [ λ i +1 , j +1 S i +1 j, S ij, ][ τ ( a, b ) βα ]= τ ( a, b ) βα [ λ i +1 , j +1 S i +1 j, S ij, ]and τ ( a i +1 , b ) w ( a i +1 b, a j +1 ) = τ ( a i +1 , b )[ h ( a i +1 b, a j +1 ) λ i +1 , j +1 S ij, S i +1 j, ]= τ ( a i +1 , b )[ h ( a i +1 b, a j +1 ) λ i +1 , j +1 S i +1 j, S ij, S j, S j, ]= [ τ ( a i +1 , b ) h ( a i +1 b, a j +1 ) S j, S j, ][ λ i +1 , j +1 S i +1 j, S ij, ] . To prove w ( a, a j +1 ) w ( b, a j +1 b ) = τ ( a i +1 , b ) w ( a i +1 b, a j +1 ), we need only toshow that τ ( a, b ) βα = τ ( a i +1 , b ) h ( a i +1 b, a j +1 ) S j, S j, . In fact, by S j, S j, = h ( a, a j +1 b ) λ j +1 , α j β j +1 λ j +1 , α j +1 β j = h ( a, a j +1 b ) βα , we have τ ( a i +1 , b ) h ( a i +1 b, a j +1 ) S j, S j, = τ ( a i +1 , b ) h ( a i +1 b, a j +1 ) h ( a, a j +1 b ) βα = τ ( a i +1 , b ) τ ( a i +1 b, a j +1 ) τ ( a j +1 b, a i +1 ) h ( a, a j +1 b ) βα = τ ( a i +1 , b ) τ ( a i +1 b, a j +1 ) τ ( a j +1 b, a i +1 ) h ( a, a j +1 b ) βα = τ ( a i +1 , ba j +1 )) τ ( b, a j +1 ) τ ( a j +1 b, a i +1 ) h ( a, a j +1 b ) βα = η ( a i +1 , ba j +1 ) τ ( b, a j +1 ) h ( a, a j +1 b ) βα = − τ ( b, a j +1 ) h ( a, a j +1 b ) βα N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z = − τ ( b, a j +1 ) τ ( a, a j +1 b ) τ ( a j +1 , ab ) βα = − τ ( b, a j +1 ) τ ( a, a j +1 b ) τ ( a j +1 , ab ) βα = − τ ( ab, a j +1 ) τ ( a, b ) τ ( a j +1 , ab ) βα = − η ( ab, a j +1 ) τ ( a, b ) βα = τ ( a, b ) βα . Therefore, we get w ( a i +1 , a j +1 ) w ( b, a j +1 b ) = τ ( a i +1 , b ) w ( a i +1 b, a j +1 ) . (4.40)Now, w ( a i +1 , a j +1 b ) w ( b, a j +1 ) = [ h ( a i +1 , a j +1 b ) λ i +1 , j +1 S i +1 j, S ij, ] w ( b, a j +1 b )= h ( a i +1 , a j +1 b ) S j, S j, w ( a i +1 , a j +1 ) w ( b, a j +1 b )= h ( a i +1 , a j +1 b ) S j, S j, w ( a i +1 , a j +1 b )[ − τ ( b, a ) βα ]= h ( a i +1 , a j +1 b ) S j, S j, w ( a i +1 , a j +1 b )[ − w ( b, a j +1 )]= − h ( a i +1 , a j +1 b ) S j, S j, w ( a i +1 , a j +1 b ) w ( b, a j +1 )= − h ( a i +1 , a j +1 b ) S j, S j, τ ( a j +1 , b ) w ( a i +1 b, a j +1 )= τ ( a i +1 , b )[ − h ( a i +1 , a j +1 b ) S j, S j, w ( a i +1 b, a j +1 )]and τ ( a i +1 , b ) w ( a i +1 b, a j +1 b ) = τ ( a i +1 , b )[ h ( a i +1 b, a j +1 b ) λ i +1 , j +1 S i +1 j, S ij, ]= τ ( a i +1 , b )[ h ( a i +1 b, a j +1 b ) w ( a i +1 , a j +1 )] . Therefore similarly to get our desired equation we need only to prove the following − h ( a i +1 , a j +1 b ) S j, S j, w ( a i +1 b, a j +1 ) = h ( a i +1 b, a j +1 b ) w ( a i +1 , a j +1 ) . (4.41)As a matter of fact, by h ( a i +1 , a j +1 b ) w ( a i +1 b, a j +1 ) = h ( a i +1 , a j +1 b ) h ( a i +1 b, a j +1 ) λ i +1 , j +1 S ij, S i +1 j, = λ i +1 , j +1 S ij, S i +1 j, (by (4.6))= S j, S j, w ( a i +1 b, a j +1 ) , we have − h ( a i +1 , a j +1 b ) S j, S j, w ( a i +1 b, a j +1 ) = − S j, S j, [ h ( a i +1 , a j +1 b ) w ( a i +1 b, a j +1 )]= − S j, S j, S j, S j, w ( a i +1 b, a j +1 )= − [ S j, S j, ] w ( a i +1 b, a j +1 )= − [ h ( a, a j +1 b ) βα ] w ( a i +1 b, a j +1 )= − h ( a, a j +1 b ) τ ( b, b ) τ ( b, a ) w ( a i +1 b, a j +1 )= − [ τ ( b, a ) τ ( b, a j +1 ) ] τ ( b, b ) τ ( b, a ) w ( a i +1 b, a j +1 )= − τ ( b, b ) τ ( b, a j +1 ) w ( a i +1 b, a j +1 ) . Moreover, since h ( a i +1 b, a j +1 b ) w ( a i +1 , a j +1 ) = τ ( a i +1 b, a j +1 b ) τ ( a j +1 , a i +1 ) w ( a i +1 , a j +1 )= τ ( a i +1 b, ba j +1 ) τ ( a j +1 , a i +1 ) w ( a i +1 , a j +1 )= τ ( a i +1 b, ba j +1 ) τ ( b, a j +1 ) τ ( a j +1 , a i +1 ) τ ( b, a j +1 ) w ( a i +1 , a j +1 )= τ ( a i +1 , a j +1 ) τ ( a i +1 b, b ) τ ( a j +1 , a i +1 ) τ ( b, a j +1 ) w ( a i +1 , a j +1 )= η ( a i +1 , a j +1 ) τ ( a i +1 b, b ) τ ( b, a j +1 ) w ( a i +1 , a j +1 )= τ ( a i +1 b, b ) τ ( b, a j +1 ) w ( a i +1 , a j +1 )= τ ( a i +1 b, b ) τ ( a j +1 , b ) τ ( b, a j +1 ) τ ( a j +1 , b ) w ( a i +1 , a j +1 )= τ ( b, b ) τ ( b, a j +1 ) τ ( a j +1 , b ) w ( a i +1 , a j +1 )= τ ( b, b ) τ ( b, a j +1 )( − τ ( b, a j +1 )) w ( a i +1 , a j +1 )= − τ ( b, b ) τ ( b, a j +1 ) w ( a i +1 , a j +1 ) , we have − h ( a i +1 , a j +1 b ) S j, S j, w ( a i +1 b, a j +1 ) = h ( a i +1 b, a j +1 b ) w ( a i +1 , a j +1 ) . N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z Therefore we have proved the following equation w ( a i +1 , a j +1 b ) w ( b, a j +1 ) = τ ( a i +1 , b ) w ( a i +1 b, a j +1 b ) . (4.42)Combining the equation (4.40) together with the equation (4.42), the equation (4.27)is proved.We will show the equation (4.28). Since[ w ( a, a i )] n = [ λ i, ( S , S , ) i ] n = [ λ i, ( αβ ) i ] n = λ n i, ( αβ ) in = λ n i, [( αβ ) n ] i = λ n i, [ λ − n, ] i = λ n i, [ P n σ ( a ) − n ] i = λ n i, P i n σ ( a ) − in = ( P − i σ ( a ) i ) n P i n σ ( a ) − in = P − n i σ ( a ) in P i n σ ( a ) − in = P − n i P i n = σ ( a n ) i σ ( a i ) n (by (4.3))= 1 σ ( a i ) n , we have [ w ( a, a i )] n = σ ( a i ) n and therefore[ w ( a, a i )] n σ ( a i ) n = 1 . (4.43)Since [ w ( a, a i b )] n = [ − τ ( b, a ) βτ ( b, a i ) α λ i, ( S , S , ) i ] n = [ − τ ( b, a ) βτ ( b, a i ) α ] n [ λ i, ( S , S , ) i ] n = [ − τ ( b, a ) τ ( b, a i ) ] n [ τ ( b, b ) τ ( b, a ) ] n [ λ i, ( S , S , ) i ] n = [ τ ( b, b ) τ ( b, a i ) ] n [ λ i, ( S , S , ) i ] n = [ τ ( b, b ) τ ( b, a i ) ] n w ( a, a i ) n , we have [ w ( a, a i b )] n σ ( a i b ) n = [ τ ( b, b ) τ ( b, a i ) ] n w ( a, a i ) n σ ( a i b ) n = [ τ ( b, b ) τ ( b, a i ) ] n σ ( a i ) − n σ ( a i b ) n (by (4.43))= τ ( b, b ) n [ σ ( a i b ) σ ( a i ) τ ( b, a i ) ] n = τ ( b, b ) n σ ( b ) n = [ τ ( b, b ) σ ( b )] n . To show that [ w ( a, a i b )] n σ ( a i b ) n = 1, we need only to prove that τ ( b, b ) σ ( b ) = 1.Since σ ( ab ) σ ( a ) − σ ( b ) − = τ ( a, b ) τ ( ab, b ) and σ ( a ) = σ ( a ⊳ x ) = σ ( ab ), we have σ ( b ) − = τ ( a, b ) τ ( ab, b ). Thanks to τ being a 2-cocycle, we have τ ( a, b ) τ ( ab, b ) = τ ( b, b ) τ ( a,
1) = τ ( b, b ) and thus σ ( b ) − = τ ( b, b ) . (4.44)Therefore [ w ( a, a i b )] n σ ( a i b ) n = 1 . (4.45)Combining the equation (4.43) and the equation (4.45) we get the first part of (4.28)by nothing that S = { a i , a i b | i ≥ } . Since by definition w ( b, a i ) = 1 and w ( b, a i b ) = −
1, we get the rest of (4.28).The proof the equation (4.29) is easy. Since by definition w ( b, a i ) = 1, w ( b, a i b ) = −
1, we have w ( a, a i ) w ( b, a i ) = w ( a, a i ) w ( a, a i ) w ( b, a i ) = w ( a, a i ) = w ( ab, a i ) ,w ( a, a i b ) w ( b, a i b ) = − w ( a, a i )( −
1) = w ( ab, a i b ) . Therefore, we get the equation (4.29) by S = { a i , a i b | i ≥ } .Now let’s prove the equation (4.30). Since w ( a, a j ) i σ ( a j ) i = [ λ j, ( S , S , ) j ] i σ ( a j ) i = [ λ j, ( αβ ) j ] i σ ( a j ) i = λ i j, ( αβ ) ij σ ( a j ) i = w ( a i , a j ) , we have w ( a, a j ) i σ ( a j ) i = w ( a i , a j ) . (4.46)Since w ( a, a j b ) i σ ( a j b ) i = [ − τ ( b, a ) βτ ( b, a j ) α λ j, ( S , S , ) j ] i σ ( a j b ) i = [( − τ ( b, a ) βτ ( b, a j ) α ) w ( a, a j )] i σ ( a j b ) i = [ − τ ( b, a ) βτ ( b, a j ) α ] i w ( a, a j ) i σ ( a j b ) i = [ − τ ( b, a ) βτ ( b, a j ) α ] i w ( a, a j ) i σ ( a j ) i σ ( a j b ) i σ ( a j ) i = [ − τ ( b, a ) βτ ( b, a j ) α ] i w ( a i , a j ) σ ( a j b ) i σ ( a j ) i = [ − τ ( b, a ) βτ ( b, a j ) α ] i σ ( a j b ) i σ ( a j ) i w ( a i , a j )= [ − τ ( b, a ) βτ ( b, a j ) α ] i σ ( a j b ) i σ ( a j ) i w ( a i , a j b ) , N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z we have w ( a, a j b ) i σ ( a j b ) i = [ − τ ( b,a ) βτ ( b,a j ) α ] i σ ( a j b ) i σ ( a j ) i w ( a i , a j b ), and thus we needonly to prove [ − τ ( b,a ) βτ ( b,a j ) α ] i σ ( a j b ) i σ ( a j ) i = 1 if we want to show that w ( a, a j b ) i σ ( a j b ) i = w ( a i , a j b ). In fact, we have[ τ ( b, a ) βτ ( b, a j ) α ] σ ( a j b ) σ ( a j ) = τ ( b, a ) τ ( b, a j ) τ ( b, b ) τ ( b, a ) σ ( a j b ) σ ( a j )= τ ( b, b ) τ ( b, a j ) σ ( a j b ) σ ( a j )= τ ( b, b ) σ ( a j b ) σ ( a j ) τ ( b, a j ) = τ ( b, b ) σ ( b )= 1 , (by (4.44)) . Therefore w ( a, a j b ) i σ ( a j b ) i = w ( a i , a j b ) . (4.47)Combining the equation (4.46) and the equation (4.47), we get that w ( a, s ) i σ ( s ) i = w ( a i , s )(4.48)for s ∈ S. By w ( a i , a j ) w ( b, a j ) = w ( a i , a j ) = w ( a i , a j ) = w ( a i b, a j ) ,w ( a i , a j b ) w ( b, a j b ) = − w ( a i , a j b ) = w ( a i b, a j b ) , we know that w ( a i , s ) w ( b, s ) = w ( a i b, s ) . (4.49)Since the equation (4.48) and the equation (4.49) hold, the proof of (4.30) is done.Now we turn to the proof of the last equation (4.31) hold. Direct computations showthat w ( a, a j ) i +1 σ ( a j ) i = [ λ j, ( S , S , ) j ] i +1 σ ( a j ) i = [ P − j σ ( a ) j ( S , S , ) j ] i +1 σ ( a j ) i = [ P − j σ ( a ) j ( αβ ) j ] i +1 σ ( a j ) i = P − (2 i +1)2 j σ ( a ) j (2 i +1) ( αβ ) j (2 i +1) σ ( a j ) i = [ P − (2 i +1)2 j σ ( a ) j σ ( a j ) i ][ σ ( a ) ij ( αβ ) j (2 i +1) ] . Since w ( a i +1 , a j ) = λ j, i +1 [ S i, S i, ] j = P − j σ ( a i +1 ) j [ S i, S i, ] j and S i, S i, = ( λ i +1 , α i +1 β i ) S i,
10 KUN ZHOU AND GONGXIANG LIU = ( λ i +1 , α i +1 β i )( h ( a, a i +1 b ) λ i +1 , α i β i +1 )= h ( a, a i +1 b ) λ i +1 , α i +1 β i +1 = h ( a, a i +1 b )[ P − i +1 σ ( a ) i ] α i +1 β i +1 = h ( a, a i +1 b ) P − i +1 σ ( a ) i α i +1 β i +1 , we have w ( a i +1 , a j ) = P − j σ ( a i +1 ) j h ( a, a i +1 b ) j P − j i +1 [ σ ( a ) ij ( αβ ) j (2 i +1) ] . Since P j i +1 σ ( a ) j σ ( a j ) i = P i j σ ( a i +1 ) j [ τ ( b,a ) τ ( b,a i +1 ) ] j and h ( a, a i +1 b ) = τ ( b,a ) τ ( b,a i +1 ) byLemma 4.4, we have P − (2 i +1)2 j σ ( a ) j σ ( a j ) i = P − j σ ( a i +1 ) j h ( a, a i +1 b ) j P − j i +1 and there-fore w ( a, a j ) i +1 σ ( a j ) i = w ( a i +1 , a j ) . (4.50)Since w ( a, a j b ) = − τ ( b, a ) βτ ( b, a j ) α λ j, ( S , S , ) j = − τ ( b, a ) βτ ( b, a j ) α w ( a, a j )and σ ( a j b ) = σ ( a j ) σ ( a j b ) σ ( a j ) , we have w ( a, a j b ) i +1 σ ( a j b ) i = [ − τ ( b, a ) βτ ( b, a j ) α w ( a, a j )] i +1 [ σ ( a j ) σ ( a j b ) σ ( a j ) ] i = ( − τ ( b, a ) βτ ( b, a j ) α ) i +1 [ w ( a, a j ) i +1 σ ( a j ) i ][ σ ( a j b ) σ ( a j ) ] i = ( − τ ( b, a ) βτ ( b, a j ) α ) i +1 w ( a i +1 , a j )[ σ ( a j b ) σ ( a j ) ] i = ( − τ ( b, a ) βτ ( b, a j ) α ) i [ − τ ( b, a ) βτ ( b, a j ) α w ( a i +1 , a j )][ σ ( a j b ) σ ( a j ) ] i = ( − τ ( b, a ) βτ ( b, a j ) α ) i w ( a i +1 , a j b )[ σ ( a j b ) σ ( a j ) ] i = ( − τ ( b, a ) βτ ( b, a j ) α ) i [ σ ( a j b ) σ ( a j ) ] i w ( a i +1 , a j b )= w ( a i +1 , a j b ) (by [ τ ( b, a ) βτ ( b, a j ) α ] = σ ( a j ) σ ( a j b ) in Lemma 4 . . Therefore we have w ( a, a j b ) i +1 σ ( a j b ) i = w ( a i +1 , a j b ) . (4.51)By (4.50), (4.51) and S = { a i , a i b | i ≥ } , we have w ( a, s ) i +1 σ ( s ) i = w ( a i +1 , s ) . (4.52) N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z Since w ( a i +1 , a j ) w ( b, a j ) = w ( a i +1 , a j ) = w ( a i +1 b, a j ) ,w ( a i +1 , a j b ) w ( b, a j b ) = − w ( a i +1 , a j b ) = w ( a i +1 b, a j b )and S = { a i , a i b | i ≥ } , we have w ( a i +1 , s ) w ( b, s ) = w ( a i +1 b, s ) . (4.53)Combining the equation (4.52) together with the equation (4.53), we prove the lastequation (4.31). (cid:3) The following Lemma 4.9 and Lemma 4.10 are prepared to prove Lemma 4.11.
Lemma 4.9.
Let R α,β as above, then l ( X a ) n = P n l ( X ) , l ( X b ) = τ ( b, b ) l ( X ) , (4.54) l ( X a ) l ( X b ) = τ ( a, b ) l ( X ab ) , l ( X b ) l ( X a ) = τ ( b, a ) l ( X ab ) , (4.55) l ( X a ) i = P i l ( X a i ) , l ( X a i ) l ( X b ) = τ ( a i , b ) l ( X a i b ) , (4.56) l ( X a ) i +1 = P i +1 l ( X a i +1 ) , l ( X a i +1 ) l ( X b ) = τ ( a i +1 , b ) l ( X a i +1 b ) , (4.57) l ( X a ) l ( X ) = l ( X a ) , l ( X b ) l ( X ) = l ( X b ) , (4.58) l ( E a ) n = l ( E ) , l ( E b ) = l ( E ) , (4.59) l ( E a ) l ( E b ) = l ( E ab ) , l ( E b ) l ( E a ) = l ( E ab ) , (4.60) l ( E a ) i = l ( E a i ) , l ( E a i ) l ( E b ) = l ( E a i b ) , (4.61) l ( E a ) i +1 = l ( E a i +1 ) , l ( E a i +1 ) l ( E b ) = l ( E a i +1 b ) , (4.62) l ( E a ) l ( E ) = l ( E a ) , l ( E b ) l ( E ) = l ( E b )(4.63) l ( X ) l ( E ) = 0 , l ( E ) l ( X ) = 0 . (4.64) Proof.
We will show the equation (4.54). Since l ( X a ) n = [ X t ∈ T w ( a, t ) e t x ] n = [( X t ∈ T w ( a, t ) e t x ) ] n = [ X t ∈ T w ( a, t ) w ( a, t ⊳ x ) e t x ] n = [ X t ∈ T w ( a, t ) w ( a, t ⊳ x ) σ ( t ) e t ] n = X t ∈ T [ w ( a, t ) w ( a, t ⊳ x ) σ ( t )] n e t = X t ∈ T P n e t (by (4.22))and P n l ( X ) = P n X t ∈ T w (1 , t ) e t = X t ∈ T P n e t , we have l ( X a ) n = P n l ( X ). Since l ( X b ) = [ X t ∈ T w ( b, t ) e t ] = X t ∈ T w ( b, t ) e t = X t ∈ T τ ( b, b ) e t (by (4.22))and τ ( b, b ) l ( X ) = τ ( b, b ) X t ∈ T w (1 , t ) e t = X t ∈ T τ ( b, b ) e t , we have l ( X b ) = τ ( b, b ) l ( X ). This means that the equation (4.54) hold.Next, we want to show the equation (4.55). Since l ( X a ) l ( X b ) = ( X t ∈ T w ( a, t ) e t x ) l ( X b ) = ( X t ∈ T w ( a, t ) e t x )( X t ∈ T w ( b, t ) e t )= X t ∈ T w ( a, t ) w ( b, t ⊳ x ) e t x = X t ∈ T τ ( a, b ) w ( ab, t ) e t x (by (4.23))and τ ( a, b ) l ( X ab ) = τ ( a, b ) X t ∈ T w ( ab, t ) e t x = X t ∈ T τ ( a, b ) w ( ab, t ) e t x, we have l ( X a ) l ( X b ) = τ ( a, b ) l ( X ab ). By l ( X b ) l ( X a ) = ( X t ∈ T w ( b, t ) e t ) l ( X a ) = ( X t ∈ T w ( b, t ) e t )( X t ∈ T w ( a, t ) e t x )= X t ∈ T w ( b, t ) w ( a, t ) e t x = X t ∈ T τ ( b, a ) w ( ab, t ) e t x (by (4.23))and τ ( a, b ) l ( X ab ) = τ ( a, b ) X t ∈ T w ( ab, t ) e t x = X t ∈ T τ ( b, a ) w ( ab, t ) e t x (by (4.23)) , we have l ( X b ) l ( X a ) = τ ( b, a ) l ( X ab ) and the equation (4.55) hold.We turn to the proof of the equation (4.56). Due to l ( X a ) i = [ X t ∈ T w ( a, t ) e t x ] i = [( X t ∈ T w ( a, t ) e t x ) ] i = [ X t ∈ T w ( a, t ) w ( a, t ⊳ x ) e t x ] i N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z = [ X t ∈ T w ( a, t ) w ( a, t ⊳ x ) e t σ ( t )] i = X t ∈ T [ w ( a, t ) w ( a, t ⊳ x ) σ ( t )] i e t = X t ∈ T P i w ( a i , t ) e t (by (4.24))and P i l ( X a i ) = P i X t ∈ T w ( a i , t ) e t = X t ∈ T P i w ( a i , t ) e t , we have l ( X a ) i = P i l ( X a i ). Since l ( X a i ) l ( X b ) = ( X t ∈ T w ( a i , t ) e t ) l ( X b ) = ( X t ∈ T w ( a i , t ) e t )( X t ∈ T w ( b, t ) e t )= X t ∈ T w ( a i , t ) w ( b, t ) e t = X t ∈ T τ ( a i , b ) w ( a i b, t ) e t (by (4.25))and τ ( a i , b ) l ( X a i b ) = τ ( a i , b ) X t ∈ T w ( a i b, t ) e t = X t ∈ T τ ( a i , b ) w ( a i b, t ) e t , we have l ( X a i ) l ( X b ) = τ ( a i , b ) l ( X a i b ) and thus (4.56) holds.For the equation (4.57), we find that l ( X a ) i +1 = [ X t ∈ T w ( a, t ) e t x ] i +1 = [( X t ∈ T w ( a, t ) e t x ) ] i [ X t ∈ T w ( a, t ) e t x ]= [( X t ∈ T w ( a, t ) w ( a, t ⊳ x ) e t x )] i [ X t ∈ T w ( a, t ) e t x ]= [ X t ∈ T w ( a, t ) w ( a, t ⊳ x ) σ ( t ) e t ] i [ X t ∈ T w ( a, t ) e t x ]= [ X t ∈ T w ( a, t ) i w ( a, t ⊳ x ) i σ ( t ) i e t ][ X t ∈ T w ( a, t ) e t x ]= X t ∈ T w ( a, t ) i +1 w ( a, t ⊳ x ) i σ ( t ) i e t x = X t ∈ T P i +1 w ( a i +1 , t ) e t x (by (4.26))and P i +1 l ( X a i +1 ) = P i +1 X t ∈ T w ( a i +1 , t ) e t x = X t ∈ T P i +1 w ( a i +1 , t ) e t x. Therefore, l ( X a ) i +1 = P i +1 l ( X a i +1 ). By l ( X a i +1 ) l ( X b ) = ( X t ∈ T w ( a i +1 , t ) e t x ) l ( X b ) = ( X t ∈ T w ( a i +1 , t ) e t x )( X t ∈ T w ( b, t ) e t )= X t ∈ T w ( a i +1 , t ) w ( b, t ⊳ x ) e t x = X t ∈ T τ ( a i +1 , b ) w ( a i +1 b, t ) e t (by (4.27))and τ ( a i +1 , b ) l ( X a i +1 b ) = τ ( a i +1 , b ) X t ∈ T w ( a i +1 b, t ) e t = X t ∈ T τ ( a i +1 , b ) w ( a i +1 b, t ) e t , we have l ( X a i +1 ) l ( X b ) = τ ( a i +1 , b ) l ( X a i +1 b ) and the equation (4.57) holds.Now we prove the equation (4.58). From l ( X a ) l ( X ) = ( X t ∈ T w ( a, t ) e t x ) l ( X ) = ( X t ∈ T w ( a, t ) e t x )( X t ∈ T w (1 , t ) e t )= ( X t ∈ T w ( a, t ) e t x )( X t ∈ T e t ) = X t ∈ T w ( a, t ) e t x = l ( X a ) , we have l ( X a ) l ( X ) = l ( X a ). Due to l ( X b ) l ( X ) = ( X t ∈ T w ( b, t ) e t ) l ( X ) = ( X t ∈ T w ( b, t ) e t )( X t ∈ T w (1 , t ) e t )= ( X t ∈ T w ( b, t ) e t ) = l ( X b ) , we have l ( X b ) l ( X ) = l ( X b ). Thus we have the equation (4.58).Now we consider the equation (4.59). Based on l ( E a ) n = ( X s ∈ S w ( a, s ) e s x ) n = [( X s ∈ S w ( a, s ) e s x ) ] n = [ X s ∈ S w ( a, s ) e s x ] n = [ X s ∈ S w ( a, s ) σ ( s ) e s ] n = X s ∈ S w ( a, s ) n σ ( s ) n e s = X s ∈ S e s (by (4.28))and l ( E ) = X s ∈ S w (1 , s ) e s = X s ∈ S e s , N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z we get that l ( E a ) n = l ( E ). Owning to l ( E b ) = [ X s ∈ S w ( b, s ) e s ] = X s ∈ S w ( b, s ) e s = X s ∈ S e s (by (4.28))and l ( E ) = X s ∈ S w (1 , s ) e s = X s ∈ S e s , we have l ( E b ) = l ( E ) and thus the equation (4.59) holds.For the equation (4.60), we find that l ( E a ) l ( E b ) = ( X s ∈ S w ( a, s ) e s x ) l ( E b ) = ( X s ∈ S w ( a, s ) e s x )( X s ∈ S w ( b, s ) e s )= X s ∈ S w ( a, s ) w ( b, s ) e s x = X s ∈ S w ( ab, s ) e s x (by (4.29))= l ( E ab ) , and l ( E b ) l ( E a ) = ( X s ∈ S w ( b, s ) e s ) l ( E a ) = ( X s ∈ S w ( b, s ) e s )( X s ∈ S w ( a, s ) e s x )= X s ∈ S w ( b, s ) w ( a, s ) e s x = X s ∈ S w ( ab, s ) e s x (by (4.29))= l ( E ab ) . We get the equation (4.60).For the equation (4.61), direct computations show that l ( E a ) i = ( X s ∈ S w ( a, s ) e s x ) i = [( X s ∈ S w ( a, s ) e s x ) ] i = [ X s ∈ S w ( a, s ) e s x ] i = [ X s ∈ S w ( a, s ) σ ( s ) e s ] i = X s ∈ S w ( a, s ) i σ ( s ) i e s = X s ∈ S w ( a i , s ) e s (by (4.30))= l ( E a i ) , and l ( E a i ) l ( E b ) = ( X s ∈ S w ( a i , s ) e s x ) l ( E b ) = ( X s ∈ S w ( a i , s ) e s )( X s ∈ S w ( b, s ) e s )= X s ∈ S w ( a i , s ) w ( b, s ) e s = X s ∈ S w ( a i b, s ) e s (by (4.30))= l ( E a i b ) . Therefore the equation (4.61) holds.Now we are going to prove the equation (4.62). By l ( E a ) i +1 = ( X s ∈ S w ( a, s ) e s x ) i +1 = [( X s ∈ S w ( a, s ) e s x ) ] i ( X s ∈ S w ( a, s ) e s x )= [ X s ∈ S w ( a, s ) e s x ] i ( X s ∈ S w ( a, s ) e s x )= [ X s ∈ S w ( a, s ) σ ( s ) e s ] i ( X s ∈ S w ( a, s ) e s x )= [ X s ∈ S w ( a, s ) i σ ( s ) i e s ]( X s ∈ S w ( a, s ) e s x )= X s ∈ S w ( a, s ) i +1 σ ( s ) i e s x = X s ∈ S w ( a i +1 , s ) e s x (by (4.31) of Lemma 4 . l ( E a i +1 ) , and l ( E a i +1 ) l ( E b ) = ( X s ∈ S w ( a i +1 , s ) e s x ) l ( E b )= ( X s ∈ S w ( a i +1 , s ) e s x )( X s ∈ S w ( b, s ) e s )= X s ∈ S w ( a i +1 , s ) w ( b, s ) e s x = X s ∈ S w ( a i +1 b, s ) e s x (by (4.31) of Lemma 4 . l ( E a i +1 b ) , we get the desired equation (4.62).For the equation (4.63), we have l ( E a ) l ( E ) = ( X s ∈ S w ( a, s ) e s x ) l ( E ) = ( X s ∈ S w ( a, s ) e s x )( X s ∈ S w (1 , s ) e s ) N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z = ( X s ∈ S w ( a, s ) e s x )( X s ∈ S e s ) = X s ∈ S w ( a, s ) e s x = l ( E a )and l ( E b ) l ( E ) = ( X s ∈ S w ( a, s ) e s x ) l ( E ) = ( X s ∈ S w ( b, s ) e s x )( X s ∈ S w (1 , s ) e s )= ( X s ∈ S w ( b, s ) e s x )( X s ∈ S e s ) = X s ∈ S w ( b, s ) e s x = l ( E b ) . This implies the equation (4.63).For the last equation (4.64), we find that l ( X ) l ( E ) = ( X s ∈ S w (1 , s ) e t ) l ( E )= ( X s ∈ S w (1 , s ) e t )( X s ∈ S w (1 , s ) e s )= 0and l ( E ) l ( X ) = ( X s ∈ S w (1 , s ) e s ) l ( X )= ( X s ∈ S w (1 , s ) e s )( X s ∈ S w (1 , s ) e t )= 0 . This implies that the equation (4.64) holds. (cid:3)
Let K (8 n, σ, τ ) as before, we associate a free object with it as follows. Recall thatthe data G of K (8 n, σ, τ ) is h a, b | a n = b = 1 , ab = ba i . We define A G as a free k algebra generated by set { x , x a , x b , e , e a , e b } , and let I G be the ideal generated by { x na − Π n − i =1 τ ( a, a i ) x , x b − τ ( b, b ) x , x b x a − η ( a, b ) x a x b , x a x − x a , x b x − x b , e na − e , e b − e , e b e a − e a e b , e a e − e a , e b e − e b , x e , e x } . Then we have the followinglemma. Lemma 4.10.
Denote the dual Hopf algebra of K (8 n, σ, τ ) by H ∗ , then H ∗ ∼ = A G /I G as an algebra.Proof. Since Lemma 3.2, we have X na = Π n − i =1 τ ( a, a i ) X , X b = τ ( b, b ) X ,X b X a = η ( a, b ) X a X b , X a X = X a , X b X = X b ,E na = E , E b = E , E b E a = E a E b ,E a E = E a , E b E = E b , X E = 0 , E X = 0 . then we can define an algebra map π : A G /I G → H ∗ by setting π ( x ) = X , π ( x a ) = X a , π ( x b ) = X b , π ( e ) = E , π ( e a ) = E a , π ( e b ) = E b , by using the definition of A G /I G , we will show that π is bijective.Firstly we show that π is surjective. Since definition of H ∗ , it is linear spanned by { X g , E g | g ∈ G } . By Lemma 3.2, { X g , E g | g ∈ G } are contained in the linear spacespanned by { X ia X jb , E ia E jb | i, j ∈ N } . Therefore H ∗ is generated by { X a , X b , E a , E b } as an algebra. Since π is an algebra map and { X a , X b , E a , E b } ⊆ Im π , we know π issurjective.Secondly we show that π is injective. Note that x g x = x x g = x g and e g e = e e g = e g for all g ∈ G , we have e g x h = ( e g e )( x x h ) = e g ( e x ) x h = 0 and x h e g =( x h x )( e e g ) = x h ( x e ) e g = 0. Then we can see that A G /I G is linear spanned by { X ia X jb , E ia E jb | ≤ i ≤ n, ≤ j ≤ } and as a result we know that dim( A G /I G ) ≤ n . Since dim( H ∗ ) = 8 n , we have dim( A G /I G ) ≤ dim( H ∗ ). But we have provedthat π is surjective and thus dim( A G /I G ) ≥ dim( H ∗ ). Then we have dim( A G /I G ) =dim( H ∗ ). Since π is surjective, we know that π is injective. (cid:3) We use the following Lemmas 4.11 and 4.13 to show that R α,β satisfies the lastequation of Lemma 4.5. Lemma 4.11.
Let R α,β as above, then l ( f ) l ( f ) = l ( f f ) for f , f ∈ H ∗ where H ∗ is the dual of K (8 n, σ, τ ) .Proof. Due to Lemmas 4.9 and 4.10, the following map is an algebra map: π : H ∗ → H, X l ( X ) , X a l ( X a ) , X b l ( X b ) ,E l ( E ) , E a l ( E a ) , E b l ( E b ) . To show that l ( f ) l ( f ) = l ( f f ) for f , f ∈ H ∗ , we only need to show that π ( f ) = l ( f ) for all f ∈ H ∗ . By Lemma 4.9, we know that l ( X a ) i = P i l ( X a i ) , l ( X a i ) l ( X b ) = τ ( a i , b ) l ( X a i b ) ,l ( X a ) i +1 = P i +1 l ( X a i +1 ) , l ( X a i +1 ) l ( X b ) = τ ( a i +1 , b ) l ( X a i +1 b ) ,l ( E a ) i = l ( E a i ) , l ( E a i ) l ( E b ) = l ( E a i b ) ,l ( E a ) i +1 = l ( E a i +1 ) , l ( E a i +1 ) l ( E b ) = l ( E a i +1 b ) . Due to π is an algebra map, we have π ( X a ) i = P i π ( X a i ) , π ( X a i ) π ( X b ) = τ ( a i , b ) π ( X a i b ) ,π ( X a ) i +1 = P i +1 π ( X a i +1 ) , π ( X a i +1 ) π ( X b ) = τ ( a i +1 , b ) π ( X a i +1 b ) ,π ( E a ) i = π ( E a i ) , π ( E a i ) π ( E b ) = π ( E a i b ) ,π ( E a ) i +1 = π ( E a i +1 ) , π ( E a i +1 ) π ( E b ) = π ( E a i +1 b )But we have the following equations by the definition of ππ ( x ) = l ( X ) , π ( x a ) = l ( X a ) , π ( x b ) = l ( X b ) ,π ( e ) = l ( E ) , π ( e a ) = l ( E a ) , π ( e b ) = l ( E b ) . N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z Therefore we have π ( f ) = l ( f ) for all f ∈ H ∗ and we have completed the proof. (cid:3) In order to get another side version of above lemma, we need the following observation.
Lemma 4.12.
Let R α,β as above, then (i) l ( X t ) = r ( X t ) , t ∈ T ; (ii) l ( E t ) = r ( E t⊳x ) t ∈ T ; (iii) l ( E s ) = r ( E s ) s ∈ S .Proof. We will show (i) at first. By definition, l ( X t ) = P t ′ ∈ T w ( t, t ′ ) e t ′ x and r ( X t ) = P t ′ ∈ T w ( t ′ , t ) e t ′ x . To show (i), we need to prove that w ( t, t ′ ) = w ( t ′ , t ). Directcomputations show that w ( a i +1 , a j +1 ) = λ i +1 , j +1 S i +1 j, S ij, = λ i +1 , j +1 [ λ j +1 , α j +1 β j ] i +1 S ij, = λ i +1 , j +1 [ λ j +1 , α j +1 β j ] i +1 [ h ( a, a j +1 b ) λ j +1 , α j β j +1 ] i = λ i +1 , j +1 λ i +12 j +1 , h ( a, a j +1 b ) i α ( i +1)( j +1)+ ij β j ( i +1)+ i ( j +1) = P − i +1 σ ( a j +1 ) i λ i +12 j +1 , h ( a, a j +1 b ) i α ( i +1)( j +1)+ ij β j ( i +1)+ i ( j +1) = P − i +1 σ ( a j +1 ) i P − (2 i +1)2 j +1 σ ( a ) j ( i +1) h ( a, a j +1 b ) i α ij + i + j +1 β ij + i + j and similarly w ( a j +1 , a i +1 ) = P − j +1 σ ( a i +1 ) j P − (2 j +1)2 i +1 σ ( a ) i ( j +1) h ( a, a i +1 b ) j α ij + i + j +1 β ij + i + j . By (4.5) in Lemma 4.4, we find that w ( a i +1 , a j +1 ) = w ( a j +1 , a i +1 ) . (4.65)To show (i), we also need consider other cases. By w ( a i +1 , a j +1 b ) = λ i +1 , j +1 S ij, S i +1 j, = λ i +1 , j +1 S i +1 j, S ij, S j, S j, = w ( a i +1 , a j +1 ) S j, S j, = S j, S j, w ( a i +1 , a j +1 )and w ( a j +1 b, a i +1 ) = h ( a j +1 b, a i +1 ) λ j +1 , i +1 S ji, S j +1 i, = h ( a j +1 b, a i +1 )[ λ j +1 , i +1 S j +1 i, S ji, ] S i, S i, = h ( a j +1 b, a i +1 ) w ( a j +1 , a i +1 ) S i, S i, = h ( a j +1 b, a i +1 ) S i, S i, w ( a j +1 , a i +1 )and w ( a i +1 , a j +1 ) = w ( a j +1 , a i +1 ) , to show that w ( a i +1 , a j +1 b ) = w ( a j +1 b, a i +1 ) we just need to prove that S j, S j, = h ( a j +1 b, a i +1 ) S i, S i, . In fact, h ( a j +1 b, a i +1 ) S i, S i, = h ( a j +1 b, a i +1 ) h ( a, a i +1 b ) λ i +1 , α i β i +1 λ i +1 , α i +1 β i = h ( a j +1 b, a i +1 ) h ( a, a i +1 b ) βα = h ( a j +1 b, a i +1 ) τ ( b, a ) τ ( b, a i +1 ) βα (by Lemma 4 . τ ( a j +1 b, a i +1 ) τ ( a i +1 b, a j +1 ) τ ( b, a ) τ ( b, a i +1 ) βα = τ ( a j +1 b, a i +1 ) τ ( b, a ) τ ( a i +1 b, a j +1 )( − τ ( a i +1 , b )) βα = τ ( a j +1 b, a i +1 ) τ ( b, a ) τ ( a i +1 , ba j +1 )( − τ ( b, a j +1 )) βα = τ ( a j +1 b, a i +1 ) τ ( b, a ) τ ( a i +1 , ba j +1 )( − τ ( b, a j +1 )) βα = η ( a j +1 b, a i +1 ) τ ( b, a ) − τ ( b, a j +1 ) βα = − τ ( b, a ) − τ ( b, a j +1 ) βα = τ ( b, a ) τ ( b, a j +1 ) βα = h ( a, a j +1 b ) βα (by (4.7)) , and S j, S j, = h ( a, a j +1 b ) λ j +1 , α j β j +1 λ j +1 , α j +1 β j = h ( a, a j +1 b ) βα . Therefore we have w ( a i +1 , a j +1 b ) = w ( a j +1 b, a i +1 ) . (4.66)Since w ( a i +1 b, a j +1 b ) = h ( a i +1 b, a j +1 b ) λ i +1 , j +1 S i +1 j, S ij, = h ( a i +1 b, a j +1 b ) w ( a i +1 , a j +1 ) ,w ( a j +1 b, a i +1 b ) = h ( a j +1 b, a i +1 b ) λ j +1 , i +1 S j +1 i, S ji, = h ( a j +1 b, a i +1 b ) w ( a j +1 , a i +1 ) N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z = h ( a j +1 b, a i +1 b ) w ( a i +1 , a j +1 ) (by (4.65))and h ( a i +1 b, a j +1 b ) = h ( a j +1 b, a i +1 b ) (by (4.6)) , we have w ( a i +1 b, a j +1 b ) = w ( a j +1 b, a i +1 b ) . (4.67)Combining the equations (4.65),(4.66),(4.67) and T = { a i +1 , a i +1 b | i ≥ } , we knowthat w ( t, t ′ ) = w ( t ′ , t ) for all t, t ′ ∈ T . Thus (i) is proved.To show (ii). By l ( E t ) = P s ∈ S w ( t, s ) e s x and r ( E t⊳x ) = P s ∈ S w ( s, t ⊳ x ) e s x , weneed to show that w ( t, s ) = w ( s, t ⊳ x ) for s ∈ S, t ∈ T. By definition, w ( a i +1 , a j ) = λ j, i +1 [ S i, S i, ] j = w ( a j , a i +1 b )and w ( a i +1 b, a j ) = λ j, i +1 [ S i, S i, ] j = w ( a j , a i +1 )and w ( a i +1 , a j b ) = − τ ( b, a ) βτ ( b, a j ) α λ j, i +1 [ S i, S i, ] j = w ( a j b, a i +1 b )and w ( a i +1 b, a j b ) = τ ( b, a ) βτ ( b, a j ) α λ j, i +1 [ S i, S i, ] j = w ( a j b, a i +1 ) . By S = { a i , a i b | i ≥ } and T = { a i +1 , a i +1 b | i ≥ } , w ( t, s ) = w ( s, t ⊳ x ) and(ii) is proved.At last, let us show (iii). Similarly, by l ( E s ) = P s ′ ∈ S w ( s, s ′ ) e s ′ and r ( E s ) = P s ′ ∈ S w ( s ′ , s ) e s ′ , to show (iii) we need to show that w ( s, s ′ ) = w ( s ′ , s ) for s, s ′ ∈ S. Since w ( a i , a j ) = ( λ j, ) i ( αβ ) ij [ σ ( a j )] i = [ P − j σ ( a ) j ] i ( αβ ) ij [ σ ( a j )] i = P − i j σ ( a j ) i σ ( a ) ij ( αβ ) ij ,w ( a j , a i ) = P − j i σ ( a i ) j σ ( a ) ij ( αβ ) ij and P i j P j i = σ ( a j ) i σ ( a i ) j , (by Lemma 4 . we have w ( a i , a j ) = w ( a j , a i ) . (4.68)By the definition of w , we have w ( a i b, a j ) = w ( a i , a j ) and w ( a j , a i b ) = w ( a j , a i ). Because w ( a i , a j ) = w ( a j , a i ), we know that w ( a i b, a j ) = w ( a j , a i b ) . (4.69)Similarly, owing to the definition of w , we have w ( a i b, a j b ) = − w ( a i , a j ) and w ( a j b, a i b ) = − w ( a j , a i ). Using w ( a i , a j ) = w ( a j , a i ) again, we can getthat w ( a i b, a j b ) = w ( a j b, a i b ) . (4.70)Combining these equations (4.68),(4.69) and (4.70), we obtain that l ( E s ) = r ( E s ) andthus (iii) has been proved. (cid:3) Based on this observation, we have
Lemma 4.13.
Let R α,β as above, then r ( f ) r ( f ) = r ( f f ) for f , f ∈ H ∗ where H ∗ is the dual of K (8 n, σ, τ ) .Proof. Since H ∗ = h X t , X s , E t , E s | s ∈ S, t ∈ T i as linear space, we have to show thefollowing equations: r ( X t ) r ( X t ) = r ( X t X t ) , r ( X t ) r ( X s ) = r ( X s X t ) , (4.71) r ( X s ) r ( X t ) = r ( X t X s ) , r ( X s ) r ( X s ) = r ( X s X s ) , (4.72) r ( E t ) r ( E t ) = r ( E t E t ) , r ( E t ) r ( E s ) = r ( E s E t ) , (4.73) r ( E s ) r ( E t ) = r ( E t E s ) , r ( E s ) r ( E s ) = r ( E s E s ) , (4.74) r ( X g ) r ( E h ) = 0 , r ( E h ) r ( X g ) = 0(4.75)where s, s , s ∈ S and t, t , t ∈ T and g, h ∈ G . To show that r ( X t ) r ( X t ) = r ( X t X t ), we need to prove the following equation: w ( t ′ , t ) w ( t ′ ⊳ x, t ) σ ( t ) = τ ( t , t ) w ( t ′ , t t ) . (4.76)Since Lemma 4.11, we have l ( X t ) l ( X t ) = τ ( t , t ) l ( X t t ). Because l ( X t ) l ( X t ) = τ ( t , t ) l ( X t t ), l ( X t ) l ( X t ) = ( X t ′ ∈ T w ( t , t ′ ) e t ′ x )( X t ′ ∈ T w ( t , t ′ ) e t ′ x )= X t ′ ∈ T w ( t , t ′ ) w ( t , t ′ ⊳ x ) e t ′ x = X t ′ ∈ T w ( t , t ′ ) w ( t , t ′ ⊳ x ) σ ( t ′ ) e t ′ = X t ′ ∈ T w ( t ′ , t ) w ( t ′ ⊳ x, t ) σ ( t ′ ) e t ′ (by (i) of Lemma 4 . X t ′ ∈ T w ( t ′ ⊳ x, t ) w ( t ′ , t ) σ ( t ′ ) e t ′ , N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z and τ ( t , t ) l ( X t t ) = τ ( t , t )( X t ′ ∈ T w ( t t , t ′ ) e t ′ )= τ ( t , t )( X t ′ ∈ T w ( t ′ ⊳ x, t t ) e t ′ )= X t ′ ∈ T τ ( t , t ) w ( t ′ ⊳ x, t t ) e t ′ , we have w ( t ′ ⊳ x, t ) w ( t ′ , t ) σ ( t ′ ) = τ ( t , t ) w ( t ′ ⊳ x, t t ) for all t ′ ∈ T . Since( t ′ ⊳ x ) ∈ T if t ′ ∈ T , we know that w ( t ′ , t ) w ( t ′ ⊳ x, t ) σ ( t ′ ⊳ x ) = τ ( t , t ) w ( t ′ , t t ),but σ ( t ′ ⊳ x ) = σ ( t ′ ) by definition of σ , so we have w ( t ′ , t ) w ( t ′ ⊳ x, t ) σ ( t ′ ) = τ ( t , t ) w ( t ′ , t t ) . Since r ( X t ) r ( X t ) = ( X t ′ ∈ T w ( t ′ , t ) e t ′ x )( X t ′ ∈ T w ( t ′ , t ) e t ′ x )= X t ′ ∈ T w ( t ′ , t ) w ( t ′ ⊳ x, t ) e t ′ x = X t ′ ∈ T w ( t ′ , t ) w ( t ′ ⊳ x, t ) σ ( t ′ ) e t ′ = X t ′ ∈ T τ ( t , t ) w ( t ′ , t t ) e t ′ (by (4.76))= τ ( t , t ) X t ′ ∈ T w ( t ′ , t t ) e t ′ = τ ( t , t ) r ( X t t )= r ( X t X t ) , we have r ( X t ) r ( X t ) = r ( X t X t ). Then we will show that r ( X t ) r ( X s ) = r ( X s X t ).Since r ( X t ) r ( X s ) = ( X t ′ ∈ T w ( t ′ , t ) e t ′ x )( X t ′ ∈ T w ( t ′ , s ) e t ′ ) = X t ′ ∈ T w ( t ′ , t ) w ( t ′ ⊳ x, s ) e t ′ x = X t ′ ∈ T w ( t, t ′ ) w ( t ′ ⊳ x, s ) e t ′ x = X t ′ ∈ T w ( t, t ′ ) w ( s, t ) e t ′ x = ( X t ′ ∈ T w ( s, t ) e t ′ )( X t ′ ∈ T w ( t, t ′ ) e t ′ x )= l ( X s ) l ( X t ) = τ ( s, t ) l ( X st ) (by Lemma 4 . τ ( s, t ) r ( X st ) (by Lemma 4 . r ( X s X t ) , we have r ( X t ) r ( X s ) = r ( X s X t ) and therefore the equation (4.71) holds. Then we will show that r ( X s ) r ( X t ) = r ( X t X s ) and r ( X s ) r ( X s ) = r ( X s X s ). Since r ( X s ) r ( X t ) = ( X t ′ ∈ T w ( t ′ , s ) e t ′ )( X t ′ ∈ T w ( t ′ , t ) e t ′ x )= X t ′ ∈ T w ( t ′ , s ) w ( t ′ , t ) e t ′ x = X t ′ ∈ T w ( s, t ′ ⊳ x ) w ( t ′ , t ) e t ′ x = X t ′ ∈ T w ( s, t ′ ⊳ x ) w ( t, t ′ ) e t ′ x = X t ′ ∈ T w ( t, t ′ ) w ( s, t ′ ⊳ x ) e t ′ x = ( X t ′ ∈ T w ( t, t ′ ) e t ′ x )( X t ′ ∈ T w ( s, t ′ ) e t ′ ) = l ( X t ) l ( X s )= τ ( t, s ) l ( X st ) (by Lemma 4 . τ ( t, s ) r ( X st ) (by Lemma 4 . r ( X t X s ) , we have r ( X s ) r ( X t ) = r ( X t X s ). Since l ( X s ) l ( X s ) = τ ( s , s ) l ( X s s ), l ( X s ) l ( X s ) = ( X t ′ ∈ T w ( s , t ′ ) e t ′ )( X t ′ ∈ T w ( s , t ′ ) e t ′ )= X t ′ ∈ T w ( s , t ′ ) w ( s , t ′ ) e t ′ , and τ ( s , s ) l ( X s s ) = τ ( s , s )( X t ′ ∈ T w ( s s , t ′ ) e t ′ )= X t ′ ∈ T τ ( s , s ) w ( s s , t ′ ) e t ′ , we have w ( s , t ′ ) w ( s , t ′ ) = τ ( s , s ) w ( s s , t ′ ) . (4.77)Since r ( X s ) r ( X s ) = ( X t ′ ∈ T w ( t ′ , s ) e t ′ )( X t ′ ∈ T w ( t ′ , s ) e t ′ ) = X t ′ ∈ T w ( t ′ , s ) w ( t ′ , s ) e t ′ = X t ′ ∈ T w ( s , t ′ ⊳ x ) w ( s , t ′ ⊳ x ) e t ′ (by (ii) of Lemma 4 . X t ′ ∈ T τ ( s , s ) w ( s s , t ′ ⊳ x ) e t ′ (by (4.77))= X t ′ ∈ T τ ( s , s ) w ( t ′ , s s , ) e t ′ (by (ii) of Lemma 4 . τ ( s , s ) r ( X s s ) = r ( X s X s ) , we have r ( X s ) r ( X s ) = r ( X s X s ) and therefore the equation (4.72) has been proved.Then we will show that r ( E t ) r ( E t ) = r ( E t E t ) and r ( E t ) r ( E s ) = r ( E s E t ). Since r ( E t ) r ( E t ) = l ( E t ⊳x ) r ( E t ) = l ( E t ⊳x ) l ( E t ⊳x ) (by Lemma 4 . N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z = l ( E t ⊳x E t ⊳x ) (by Lemma 4 . l ( E ( t t ) ⊳x ) = l ( E t t )= r ( E t t ) (by (iii) of Lemma 4 . r ( E t ) r ( E s ) = l ( E t⊳x ) r ( E s ) = l ( E t⊳x ) l ( E s ) (by Lemma 4 . l ( E t⊳x E s ) (by Lemma 4 . l ( E ( ts ) ⊳x ) = r ( E ts ) (by Lemma 4 . r ( E s E t ) , we have r ( E t ) r ( E t ) = r ( E t E t ) and r ( E t ) r ( E s ) = r ( E s E t ). Therefore the equation(4.73) holds.Then we will show that r ( E s ) r ( E t ) = r ( E t E s ) and r ( E s ) r ( E s ) = r ( E s E s ). Since r ( E s ) r ( E t ) = r ( E s ) l ( E t⊳x )= l ( E s ) l ( E t⊳x ) (by Lemma 4 . l ( E s E t⊳x ) (by Lemma 4 . l ( E ( st ) ⊳x ) = r ( E st ) (by Lemma 4 . r ( E t E s ) , and r ( E s ) r ( E s ) = l ( E s ) l ( E s ) (by Lemma 4 . l ( E s E s ) (by Lemma 4 . l ( E s s ) = r ( E s s ) = r ( E s E s ) , we have r ( E s ) r ( E t ) = r ( E t E s ) and r ( E s ) r ( E s ) = r ( E s E s ) and hence the equation(4.74) holds.Then we will show that r ( X g ) r ( E h ) = 0 and r ( E h ) r ( X g ) = 0 for g, h ∈ G . Since r ( X t ) = X t ′ ∈ T w ( t ′ , t ) e t ′ x, r ( X s ) = X t ′ ∈ T w ( t ′ , s ) e t ′ ,r ( E t ) = X s ′ ∈ S w ( s ′ , t ) e s ′ x, r ( E s ) = X s ′ ∈ S w ( s ′ , s ) e s ′ , we know that r ( X g ) r ( E h ) = 0 and r ( E h ) r ( X g ) = 0 for g, h ∈ G and therefore theequation (4.75) holds. (cid:3) With the above preparation, we can prove that
Proposition 4.14.
The element R α,β is a universal R -matrix of K (8 n, σ, τ ) .Proof. By Lemmas 4.6 and 4.7, we have τ ( s , s ) = τ ( s , s ) , s , s ∈ S,w ( s, t ⊳ x ) = w ( s, t ) η ( s, t ) , s ∈ S, t ∈ T, w ( t ⊳ x, s ) = w ( t, s ) η ( t, s ) , s ∈ S, t ∈ T,τ ( t , t ) w ( t ⊳ x, t ⊳ x ) = τ ( t ⊳ x, t ⊳ x ) w ( t , t ) , t , t ∈ T. By Lemmas 4.11 and 4.13, we have l ( f ) l ( f ) = l ( f f ) , r ( f ) r ( f ) = r ( f f ) , f , f ∈ H ∗ . Due to Lemma 4.5, R α,β is a universal R -matrix of K (8 n, σ, τ ). (cid:3) Now we turn to the proof of Proposition 4.19. Let R be a universal R -matrix of k G σ,τ kZ , and let H l = { l ( f ) | f ∈ H ∗ } and H r = { r ( f ) | f ∈ H ∗ } where H ∗ is thedual of k G σ,τ kZ . Note that both H l and H r are subalgebras of H ∗ . Lemma 4.15.
We have H l = h l ( X t ) , l ( E t ) | t ∈ T i and H r = h r ( X t ) , r ( E t ) | t ∈ T i as algebras.Proof. By Lemma 3.2 and S = T T , we know that H ∗ = h X t , E t | t ∈ T i as an algebra.Define two maps π : H ∗ → H l , f l ( f ) π ′ : H ∗ → H r , f r ( f ) . It is not hard to see that both of them are surjective algebra maps. (cid:3)
Now let R w and R v be two non-trivial universal R -matrices of k G σ,τ kZ which aredenoted by the following way: R w = P s ,s ∈ S w ( s , s ) e s ⊗ e s + P s ∈ S,t ∈ T w ( s, t ) e s x ⊗ e t + P t ∈ T,s ∈ S w ( t, s ) e t ⊗ e s x + P t ,t ∈ T w ( t , t ) e t x ⊗ e t x and R v = P s ,s ∈ S v ( s , s ) e s ⊗ e s + P s ∈ S,t ∈ T v ( s, t ) e s x ⊗ e t + P t ∈ T,s ∈ S v ( t, s ) e t ⊗ e s x + P t ,t ∈ T v ( t , t ) e t x ⊗ e t x .We find that Lemma 4.16.
Given R w , R v as above, then R w = R v if and only if w ( t , t ) = v ( t , t ) for t , t ∈ T .Proof. Let l w : H ∗ → H defined by l w ( f ) = ( f ⊗ Id)( R w ) and let l v : H ∗ → H definedby l v ( f ) = ( f ⊗ Id)( R v ), then R w = R v if and only if l w = l v . By Lemma 3.2 and S = T T , we get that H ∗ = h X t , E t | t ∈ T i as an algebra. This implies that l w = l v if and only if l w ( X t ) = l v ( X t ) and l w ( E t ) = l v ( E t ) for t ∈ T . Since l w ( X t ) = X t ∈ T w ( t, t ′ ) e t ′ x, l v ( X t ) = X t ∈ T v ( t, t ′ ) e t ′ x, t ∈ T,l w ( E t ) = X t ∈ T w ( t, s ) e s x, l v ( E t ) = X t ∈ T v ( t, s ) e s x, t ∈ T, N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z we know that l w = l v if and only if w ( t, t ′ ) = v ( t, t ′ ) and w ( t, s ) = v ( t, s ) for s ∈ S, t, t ′ ∈ T . To complete the proof, we will show that if w ( t, t ′ ) = v ( t, t ′ ) for all t, t ′ ∈ T then w ( t, s ) = v ( t, s ) for all s ∈ S, t ∈ T . Let r w : ( H ∗ ) op → H defined by r w ( f ) = (Id ⊗ f )( R w ) and let r v : ( H ∗ ) op → H defined by r v ( f ) = (Id ⊗ f )( R v ), then r w and r v are algebra maps by Lemma 3.1, and therefore we have r w ( X s ) r w ( X t ) = τ ( t, s ) r w ( X st ) , r v ( X s ) r v ( X t ) = τ ( t, s ) r v ( X st ) , s ∈ S, t ∈ T. Since r w ( X s ) r w ( X t ) = ( X t ′ ∈ T w ( t ′ , s ) e t ′ )( X t ′ ∈ T w ( t ′ , t ) e t ′ x )= X t ′ ∈ T w ( t ′ , s ) w ( t ′ , t ) e t ′ x,r v ( X s ) r v ( X t ) = ( X t ′ ∈ T v ( t ′ , s ) e t ′ )( X t ′ ∈ T v ( t ′ , t ) e t ′ x )= X t ′ ∈ T v ( t ′ , s ) v ( t ′ , t ) e t ′ x and τ ( t, s ) r w ( X st ) = τ ( t, s )( X t ′ ∈ T w ( t ′ , ts ) e t ′ x )= X t ′ ∈ T τ ( t, s ) w ( t ′ , st ) e t ′ x,τ ( t, s ) r v ( X st ) = τ ( t, s )( X t ′ ∈ T v ( t ′ , ts ) e t ′ x )= X t ′ ∈ T τ ( t, s ) v ( t ′ , st ) e t ′ x, we have w ( t ′ , s ) = τ ( t, s ) w ( t ′ , st ) w ( t ′ , t ) , s ∈ S, t, t ′ ∈ T,v ( t ′ , s ) = τ ( t, s ) v ( t ′ , st ) v ( t ′ , t ) , s ∈ S, t, t ′ ∈ T. As a result we know that if w ( t, t ′ ) = v ( t, t ′ ) for all t, t ′ ∈ T then w ( t, s ) = w ( t, s )for all s ∈ S, t ∈ T . (cid:3) The following Lemma 4.17 and Lemma 4.18 are used to compute the fourth matrix w of R . Lemma 4.17.
Assume that R is a universal R -matrix of K (8 n, σ, τ ) . If we let w ( a, a ) = α and w ( a, ab ) = β , then ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) . Proof.
Since l ( X a ) n = P n l ( X ) and l ( X a ) n = [ X t ∈ T w ( a, t ) e t x ] n = [( X t ∈ T w ( a, t ) e t x ) ] n = [ X t ∈ T w ( a, t ) w ( a, t ⊳ x ) e t x ] n = [ X t ∈ T w ( a, t ) w ( a, t ⊳ x ) σ ( t ) e t ] n = X t ∈ T w ( a, t ) n w ( a, t ⊳ x ) n σ ( t ) n e t ,P n l ( X ) = P n X t ∈ T w (1 , t ) e t = P n X t ∈ T e t = X t ∈ T P n e t , we have w ( a, t ) n w ( a, t ⊳ x ) n σ ( t ) n = P n . Take t = a , we get that w ( a, a ) n w ( a, ab ) n σ ( a ) n = P n . Since w ( a, a ) = α,w ( a, ab ) = β and λ n, = P − n σ ( a ) n , we have ( αβ ) n λ n, = 1.Since l ( X b ) l ( X a ) = τ ( b, a ) l ( X ab ), l ( X b ) l ( X a ) = ( X t ∈ T w ( b, t ) e t )( X t ∈ T w ( a, t ) e t x )= X t ∈ T w ( b, t ) w ( a, t ) e t x and τ ( b, a ) l ( X ab ) = τ ( b, a ) X t ∈ T w ( ab, t ) e t x = X t ∈ T τ ( b, a ) w ( ab, t ) e t x, we must have w ( b, t ) w ( a, t ) = τ ( b, a ) w ( ab, t ) for t ∈ T . Through letting t = a ,then the equality w ( b, t ) w ( a, t ) = τ ( b, a ) w ( ab, t ) becomes w ( b, a ) w ( a, a ) = τ ( b, a ) w ( ab, a ) . (4.78)We claim w ( ab, a ) = β . By the equation (4.11), we have w ( t , t ) = h ( t , t ) w ( t ⊳x, t ⊳ x ). Through letting t = a, t = ab , the equality w ( t , t ) = h ( t , t ) w ( t ⊳x, t ⊳ x ) becomes w ( ab, a ) = h ( ab, a ) w ( a, ab ). By h ( ab, a ) = τ ( ab,a ) τ ( ab,a ) = 1, we have w ( ab, a ) = w ( a, ab ) = β . Due to the equation w ( b, a ) w ( a, a ) = τ ( b, a ) w ( ab, a )and w ( ab, a ) = β , we know that w ( b, a ) = τ ( b, a ) βα . (4.79)Since l ( X b ) = τ ( b, b ) l ( X ) and l ( X b ) = ( X t ∈ T w ( b, t ) e t ) = X t ∈ T w ( b, t ) e t , N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z τ ( b, b ) l ( X ) = τ ( b, b ) X t ∈ T w (1 , t ) e t = X t ∈ T τ ( b, b ) e t , we have w ( b, t ) = τ ( b, b ). Taking t = a , w ( b, t ) = τ ( b, b ) becomes w ( b, a ) = τ ( b, b ). Since w ( b, a ) = τ ( b, b ) and w ( b, a ) = τ ( b, a ) βα , we have β α = τ ( b,b ) τ ( b,a ) . (cid:3) Lemma 4.18.
Assume that R is a universal R -matrix of K (8 n, σ, τ ) . If we let w ( a, a ) = α and w ( a, ab ) = β , then w of R must have the following expression: w ( a i +1 , a j +1 ) = λ i +1 , j +1 S i +1 j, S ij, w ( a i +1 , a j +1 b ) = λ i +1 , j +1 S ij, S i +1 j, w ( a i +1 b, a j +1 ) = h ( a i +1 b, a j +1 ) λ i +1 , j +1 S ij, S i +1 j, w ( a i +1 b, a j +1 b ) = h ( a i +1 b, a j +1 b ) λ i +1 , j +1 S i +1 j, S ij, for ≤ i, j ≤ ( n − .Proof. Since r ( X a ) i +1 = P i +1 r ( X a i +1 ) and r ( X a ) i +1 = [ X t ∈ T w ( t, a ) e t x ] i +1 = [ X t ∈ T w ( t, a ) e t x ] i [ X t ∈ T w ( t, a ) e t x ]= [( X t ∈ T w ( t, a ) e t x ) ] i [ X t ∈ T w ( t, a ) e t x ]= [ X t ∈ T w ( t, a ) w ( t ⊳ x, a ) e t x ] i [ X t ∈ T w ( t, a ) e t x ]= [ X t ∈ T w ( t, a ) w ( t ⊳ x, a ) σ ( t ) e t ] i [ X t ∈ T w ( t, a ) e t x ]= [ X t ∈ T w ( t, a ) i w ( t ⊳ x, a ) i σ ( t ) i e t ][ X t ∈ T w ( t, a ) e t x ]= X t ∈ T w ( t, a ) i +1 w ( t ⊳ x, a ) i σ ( t ) i e t x,P i +1 r ( X a i +1 ) = P i +1 X t ∈ T w ( t, a i +1 ) e t x = X t ∈ T P i +1 w ( t, a i +1 ) e t x, we have w ( t, a ) i +1 w ( t⊳x, a ) i σ ( t ) i = P i +1 w ( t, a i +1 ). Taking t = a in this equation,we get that α i +1 β i σ ( a ) i = P i +1 w ( a, a i +1 ). Since by definition λ i +1 , = P − i +1 σ ( a ) i ,we know w ( a, a i +1 ) = S i, . (4.80)Similarly, if we let t = ab , then w ( t, a ) i +1 w ( t ⊳ x, a ) i σ ( t ) i = P i +1 w ( t, a i +1 ) be-comes β i +1 α i σ ( a ) i = P i +1 w ( ab, a i +1 ) and so we have w ( ab, a i +1 ) = λ i +1 , α i β i +1 .Since w ( t , t ) = h ( t , t ) w ( t ⊳ x, t ⊳ x ) by Lemma 4.5, let t = a and t = a i +1 b we have w ( a, a i +1 b ) = h ( a, a i +1 b ) w ( ab, a i +1 ). Because w ( ab, a i +1 ) = λ i +1 , α i β i +1 , we find that w ( a, a i +1 b ) = S i, . (4.81)Since l ( X a ) i +1 = P i +1 l ( X a i +1 ), l ( X a ) i +1 = [ X t ∈ T w ( a, t ) e t x ] i +1 = [ X t ∈ T w ( a, t ) e t x ] i [ X t ∈ T w ( a, t ) e t x ]= [( X t ∈ T w ( a, t ) e t x ) ] i [ X t ∈ T w ( a, t ) e t x ]= [ X t ∈ T w ( a, t ) w ( a, t ⊳ x ) e t x ] i [ X t ∈ T w ( a, t ) e t x ]= [ X t ∈ T w ( a, t ) w ( a, t ⊳ x ) σ ( t ) e t ] i [ X t ∈ T w ( a, t ) e t x ]= [ X t ∈ T w ( a, t ) i w ( a, t ⊳ x ) i σ ( t ) i e t ][ X t ∈ T w ( a, t ) e t x ]= X t ∈ T w ( a, t ) i +1 w ( a, t ⊳ x ) i σ ( t ) i e t x and P i +1 l ( X a i +1 ) = P i +1 X t ∈ T w ( a i +1 , t ) e t x = X t ∈ T P i +1 w ( a i +1 , t ) e t x, we have w ( a, t ) i +1 w ( a, t ⊳ x ) i σ ( t ) i = P i +1 w ( a i +1 , t ). Now let t = a j +1 , then w ( a, t ) i +1 w ( a, t ⊳ x ) i σ ( t ) i = P i +1 w ( a i +1 , t ) becomes w ( a, a j +1 ) i +1 w ( a, a j +1 b ) i σ ( a j +1 ) i = P i +1 w ( a i +1 , a j +1 ) . Since w ( a, a j +1 ) = S j, by (4.80), w ( a, a j +1 b ) = S j, by (4.81) and λ i +1 , j +1 = P − i +1 σ ( a j +1 ) i , we know that w ( a i +1 , a j +1 ) = λ i +1 , j +1 S i +1 j, S ij, . (4.82)Similarly, if we let t = a j +1 b , then w ( a, t ) i +1 w ( a, t ⊳ x ) i σ ( t ) i = P i +1 w ( a i +1 , t )becomes w ( a, a j +1 b ) i +1 w ( a, a j +1 ) i σ ( a j +1 b ) i = P i +1 w ( a i +1 , a j +1 b ) . Since w ( a, a j +1 ) = S j, by equation (4.80), w ( a, a j +1 b ) = S j, by equation (4.81), λ i +1 , j +1 = P − i +1 σ ( a j +1 ) i and σ ( a j +1 b ) = σ ( a j +1 ), we know that w ( a i +1 , a j +1 b ) = λ i +1 , j +1 S ij, S i +1 j, . (4.83) N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z Due to w ( t , t ) = h ( t , t ) w ( t ⊳ x, t ⊳ x ) by (4.11), this implies that if we let t = a i +1 b and t = a j +1 then w ( a i +1 b, a j +1 ) = h ( a i +1 b, a j +1 ) w ( a i +1 , a j +1 b ).Therefore the equation (4.83) implies that w ( a i +1 b, a j +1 ) = h ( a i +1 b, a j +1 ) λ i +1 , j +1 S ij, S i +1 j, . (4.84)Moreover, if we let t = a i +1 b and t = a j +1 b , then w ( t , t ) = h ( t , t ) w ( t ⊳x, t ⊳ x ) gives w ( a i +1 b, a j +1 b ) = h ( a i +1 b, a j +1 b ) w ( a i +1 , a j +1 ) . So the equation (4.82) tells us that w ( a i +1 b, a j +1 b ) = h ( a i +1 b, a j +1 b ) λ i +1 , j +1 S i +1 j, S ij, . (4.85)Putting the equations (4.82),(4.83),(4.84),(4.85) together, we get what we want. (cid:3) Using Lemma 4.16-Lemma 4.18, we can prove that
Proposition 4.19. If R is a universal R -matrix of K (8 n, σ, τ ) , then R = R α,β forsome α, β ∈ k such that ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) .Proof. Assume that R = R w is a non-trivial universal R -matrix of K (8 n, σ, τ ). If wetake w ( a, a ) = α and w ( a, ab ) = β , then ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) by Lemma4.17. Therefore we can define a R α,β as before. By Lemma 4.18, we know that w of R w is the same with the fourth matrix of R α,β . Owing to Lemma 4.16, we get that R w = R α,β . (cid:3) The case η ( a, b ) = 1 . We assume that η ( a, b ) = 1 in this subsection, and wewill give all universal R -matrices of K (8 n, σ, τ ) in this subsection. Due to the proofis almost the same as the case η ( a, b ) = −
1, we only give the results here withoutproofs.
Proposition 4.20.
Assume that R is a trivial quasitriangular structure on K (8 n, σ, τ ) ,then R = X ≤ i,k ≤ n, ≤ j,l ≤ α ik β il − jk e a i b j ⊗ e a k b l for some α, β ∈ k satisfying α n = β = 1 . To give all non-trivial quasitriangular structures on K (8 n, σ, τ ), we just need tochange the R α,β of Proposition 4.14 a little. That is to say if we take α, β ∈ k such that ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) and let S j, = λ j +1 , α j +1 β j , S j, = h ( a, a j +1 b ) λ j +1 , α j β j +1 for j ∈ N , then we can construct R ′ α,β in the form of (4.2) through letting (i) w be given by (cid:26) w ( a i , a j ) = w ( a i b, a j ) = ( λ j, ) i ( αβ ) ij [ σ ( a j )] i w ( a i , a j b ) = w ( a i b, a j b ) = ( λ j, ) i ( αβ ) ij [ σ ( a j )] i ,(ii) w be given by ( w ( a i , a j +1 ) = w ( a i , a j +1 b ) = λ i, j +1 [ S j, S j, ] i w ( a i b, a j +1 ) = w ( a i b, a j +1 b ) = τ ( b,a ) βτ ( b,a i ) α λ i, j +1 [ S j, S j, ] i ,(iii) w be given by ( w ( a i +1 , a j ) = w ( a i +1 b, a j ) = λ j, i +1 [ S i, S i, ] j w ( a i +1 , a j b ) = w ( a i +1 b, a j b ) = τ ( b,a ) βτ ( b,a j ) α λ j, i +1 [ S i, S i, ] j ,(iv) w be given by w ( a i +1 , a j +1 ) = λ i +1 , j +1 S i +1 j, S ij, w ( a i +1 , a j +1 b ) = λ i +1 , j +1 S ij, S i +1 j, w ( a i +1 b, a j +1 ) = h ( a i +1 b, a j +1 ) λ i +1 , j +1 S ij, S i +1 j, w ( a i +1 b, a j +1 b ) = h ( a i +1 b, a j +1 b ) λ i +1 , j +1 S i +1 j, S ij, ,for 0 ≤ i, j ≤ ( n − Proposition 4.21.
The set of elements { R ′ α,β | α, β ∈ k , ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) } gives all non-trivial quasitriangular structures on K (8 n, σ, τ ) . A class of minimal quasitriangular Hopf algebras.
In this subsection, wefirstly identify all minimal quasitriangular Hopf algebras among K (8 n, σ, τ ) for thecase η ( a, b ) = − K (8 n, σ, τ ) to be minimalquasitriangular Hopf algebras. Secondly we study minimal quasitriangular structureson K (8 n, σ, τ ) for the case η ( a, b ) = 1 and we found that K (8 n, σ, τ ) has no minimalquasitriangular structure on it. Furthermore, if K (8 n, σ, τ ) is minimal for the case η ( a, b ) = −
1, then we will write down all its minimal quasitriangular structures onit. And using these results we can prove that K (8 n, ζ ) are minimal if n ≥ K (8 n, ζ ) ( n ≥ K isminimal and give all universal R -matrices on it.Since we also feel interested in minimal quasitriangular Hopf algebras, we give asufficient condition for k G σ,τ kZ to be a minimal quasitriangular Hopf algebra. Lemma 4.22.
Let k G σ,τ kZ as above and let R as above 4.2, and if R is a universal R -matrix on k G σ,τ kZ satisfying (i) the linear subspace spanned by { l ( X t ) | t ∈ T } equals to the linear subspacespanned by { r ( X t ) | t ∈ T } , N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z (ii) the linear subspace spanned by { l ( E t ) | t ∈ T } equals to the linear subspacespanned by { r ( E t ) | t ∈ T } ,then ( k G σ,τ kZ , R ) is minimal if and only if w i (1 ≤ i ≤ are four non-degeneratedmatrices.Proof. Firstly, we will show that H l = H r and H l H r = H l . By Lemma 4.15, H l = h l ( X t ) , l ( E t ) | t ∈ T i and H r = h r ( X t ) , r ( E t ) | t ∈ T i . By conditions (i) and (ii), wehave H l = H r and therefore H l H r = H l .Secondly, we will show that H l = H if and only if w i (1 ≤ i ≤
4) are non-degeneratedmatrices. If H l = H , then l : H ∗ → H is bijective where l ( f ) = ( f ⊗ Id)( R ).Since { X s , E s , X t , E t | s ∈ S, t ∈ T } is a base of H ∗ by definition, we get that { l ( X s ) , l ( E s ) , l ( X t ) , l ( E t ) | s ∈ S, t ∈ T } is a base of H . But by definition, l ( E s ) = X s ′ ∈ T w ( s, s ′ ) e s ′ , l ( X s ) = X t ∈ T w ( s, t ) e t ,l ( E t ) = X t ∈ T w ( t, s ) e s x, l ( X t ) = X t ∈ T w ( t, t ′ ) e t ′ x. Therefore w i (1 ≤ i ≤
4) are non-degenerated matrices. Now if w i (1 ≤ i ≤
4) arenon-degenerated matrices, then { l ( X s ) , l ( E s ) , l ( X t ) , l ( E t ) | s ∈ S, t ∈ T } is a base of H and thus l : H ∗ → H l is bijective. As a result we know that dim( H l ) = dim( H )which implies that H l = H . (cid:3) Using Lemma 4.22 and Lemma 4.12, we can get that
Corollary 4.23.
Let R α,β as above in Proposition 4.14, then R α,β is a minimalquasitriangular structure on K (8 n, σ, τ ) if and only if its w i (1 ≤ i ≤ are non-degenerated matrices.Proof. By Lemma 4.12, we have l ( X t ) = r ( X t ) and l ( E t ) = r ( E t⊳x ) for t ∈ T .Therefore R α,β satisfies that the condition of Lemma 4.22. Now using Lemma 4.22again and we get the result. (cid:3) We are not very satisfied with this conclusion due to the absence of an easy criteriafor the non-degeneracies of these matrices w i (1 ≤ i ≤ Lemma 4.24.
The matrices w i (1 ≤ i ≤ of R α,β are non-degenerated if and onlyif ( σ ( a ) αβ ) σ ( a ) τ ( a, a ) − is a primitive n th root of 1.Proof. To calculate the matrices w i (1 ≤ i ≤ S, T in the following way s j := a j − , s n + j := a j − b, t j := a j − , t n + j := a j − b, where 1 ≤ i, j ≤ n . For convenience, let us denote four matrices A i (1 ≤ i ≤
4) asfollows A := ( w ( s i , s j )) ≤ i,j ≤ n A := ( w ( s i , t j )) ≤ i,j ≤ n A := ( w ( t i , s j )) ≤ i,j ≤ n A := ( w ( t i , t j )) ≤ i,j ≤ n . Firstly, we determine when ( w ( s i , s j )) ≤ i,j ≤ n is non-degenerate. For 1 ≤ i, j ≤ n ,we find that w ( s i , s n + j ) = w ( a i − , a j − b ) = ( λ j − , ) i − ( αβ ) i − j − [ σ ( a j − )] i − = w ( a i − , a j − ) = w ( s i , s j ) ,w ( s n + i , s j ) = w ( a i − b, a j − ) = ( λ j − , ) i − ( αβ ) i − j − [ σ ( a j − )] i − = w ( a i − , a j − ) = w ( s i , s j )and w ( s n + i , s n + j ) = w ( a i − b, a j − b ) = − ( λ j − , ) i − ( αβ ) i − j − [ σ ( a j − )] i − = − w ( a i − , a j − ) = w ( s i , s j ) . Therefore we have( w ( s i , s j )) ≤ i,j ≤ n = (cid:18) A A A − A (cid:19) ∼ (cid:18) A A (cid:19) . Here “ ∼ ” means that two matrices can be gotten each other through elementary oper-ations. Thus ( w ( s i , s j )) ≤ i,j ≤ n is non-degenerate if and only if A is non-degenerate.Since l ( E s ) l ( E s ) = l ( E s s ) and r ( E s ) r ( E s ) = r ( E s s ), we get that w is bichar-acter on S . Let w ( a , a ) = γ , then we have γ = ( λ , ) ( αβ ) σ ( a ) = ( P − σ ( a )) ( αβ ) σ ( a )= ( τ ( a, a ) − σ ( a )) ( αβ ) σ ( a ) = ( σ ( a ) αβ ) σ ( a ) τ ( a, a ) − and w ( s i , s j ) = w ( a i − , a j − )= w ( a , a ) ( i − j − = γ ( i − j − . Therefore we have A = ( γ ( i − j − ) ≤ i,j ≤ n . So A is non-degenerate if and only if γ m = 1 for 1 ≤ m ≤ ( n − w is bicharacter on S , we have w ( a , a ) n = w ( a n , a ) = 1 and thus γ n = 1. So we get that ( w ( s i , s j )) ≤ i,j ≤ n is non-degeneratedif and only if the following condition holds( σ ( a ) αβ ) σ ( a ) τ ( a, a ) − is a primitive nth root of 1.(4.86)Secondly, we determine when ( w ( s i , t j )) ≤ i,j ≤ n is non-degenerate. Assume that1 ≤ i, j ≤ n and let δ := τ ( b, a ) βα , then we have w ( s i , t n + j ) = w ( a i − , a j − b )= w ( a i − , a j − )= w ( s i , t j )and w ( s i + n , t j ) = w ( a i − b, a j − ) N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z = τ ( b, a ) βτ ( b, a i − ) α λ i − , j − [ S j − , S j − , ] i − = τ ( b, a i − ) − τ ( b, a ) βα λ i − , j − [ S j − , S j − , ] i − = τ ( b, a i − ) − δλ i − , j − [ S j − , S j − , ] i − = τ ( b, a i − ) − δw ( a i − , a j − )= τ ( b, a i − ) − δw ( s i , t j ) . Therefore w ( s i , t n + j ) = w ( s i , t j ) and w ( s i + n , t j ) = τ ( b, a i − ) − δw ( s i , t j ). Since w ( s i + n , t j + n ) = w ( a i − b, a j − b )= − w ( a i − b, a j − )= − w ( s i + n , t j )= − τ ( b, a i − ) − δw ( s i , t j ) , we have w ( s i + n , t j + n ) = − τ ( b, a i − ) − δw ( s i , t j ). Let B = ( b ij ) ≤ i,j ≤ n be a n × n matrix defined by b ij = τ ( b, a i − ) − δw ( s i , t j ), then we have( w ( s i , t j )) ≤ i,j ≤ n = (cid:18) A A B − B (cid:19) ∼ (cid:18) A A A − A (cid:19) ∼ (cid:18) A A (cid:19) . Therefore ( w ( s i , t j )) ≤ i,j ≤ n is non-degenerate if and only if A is non-degenerate.Let α j := w ( a , a j − ), then we have α j = P − σ ( a j − ) S j − , S j − , by definition.By w ( s i , t j ) = w ( a i − , a j − )= λ i − , j − [ S j − , S j − , ] i − = P − i − σ ( a j − ) i − [ S j − , S j − , ] i − = P − i − P i − P − ( i − σ ( a j − ) i − [ S j − , S j − , ] i − = P − i − P i − [ P − σ ( a j − ) S j − , S j − , ] i − = P − i − P i − α ( i − j , we have A = ( P − i − P i − α ( i − j ) ≤ i,j ≤ n ∼ ( α ( i − j ) ≤ i,j ≤ n . Therefore A is non-degenerated if and only if α j α i = 1 for 1 ≤ i < j ≤ n . Directcomputations show that α j α i = P − σ ( a j − ) S j − , S j − , P − σ ( a i − ) S i − , S i − , = σ ( a j − ) S j − , S j − , σ ( a i − ) S i − , S i − ,
16 KUN ZHOU AND GONGXIANG LIU and S j − , S j − , S i − , S i − , = ( λ j − , α j β j − ) S j − , S i − , S i − , = ( λ j − , α j β j − )( h ( a, a j − b ) λ j − , α j − β j ) S i − , S i − , = h ( a, a j − b )( λ j − , ) ( αβ ) j − h ( a, a i − b )( λ i − , ) ( αβ ) i − = ( λ j − , ) ( λ i − , ) h ( a, a j − b ) h ( a, a i − b ) ( αβ ) j − i ) = P − j − σ ( a ) j P − i − σ ( a ) i h ( a, a j − b ) h ( a, a i − b ) ( αβ ) j − i ) = P − j − P − i − h ( a, a j − b ) h ( a, a i − b ) [ σ ( a ) αβ ] j − i ) . This implies that α j α i = σ ( a j − ) σ ( a i − ) P − j − P − i − h ( a, a j − b ) h ( a, a i − b ) [ σ ( a ) αβ ] j − i ) . (4.87)By Lemma 4.4, P l k +1 σ ( a ) l σ ( a l ) k = P k l σ ( a k +1 ) l [ τ ( b, a ) τ ( b, a k +1 ) ] l , k, l ≥ , (4.88) h ( a, a k +1 b ) = τ ( b, a ) τ ( b, a k +1 )and thus P l k +1 σ ( a ) l σ ( a l ) k = P k l σ ( a k +1 ) l [ h ( a, a k +1 b )] l , k, l ≥ . (4.89)Taking k = ( j −
1) and l = 1, then the equation (4.89) becomes P j − σ ( a ) σ ( a ) j − = P j − σ ( a j − )[ h ( a, a k +1 b )] and thus σ ( a j − ) P − j − h ( a, a j − b ) = P − j − σ ( a ) σ ( a ) j − . (4.90)Due to the equations (4.87) and (4.90), we have α j α i = P − j − σ ( a ) σ ( a ) j − P − i − σ ( a ) σ ( a ) i − [ σ ( a ) αβ ] j − i ) = [( σ ( a ) αβ ) σ ( a ) τ ( a, a ) − ] j − i . Since we have showed that ( w ( s i , t j )) ≤ i,j ≤ n is non-degenerate if and only if A isnon-degenerate and A is non-degenerate if and only if α j α i = 1 for 1 ≤ i < j ≤ n , weknow that ( w ( s i , t j )) ≤ i,j ≤ n is non-degenerate if and only if the following conditionholds [( σ ( a ) αβ ) σ ( a ) τ ( a, a ) − ] m = 1 for all 1 ≤ m ≤ ( n − . (4.91) N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z Now we turn to the consideration of the non-degeneracy of ( w ( t i , s j )) ≤ i,j ≤ n . Thanksto Lemma 4.12, we have l ( E t ) = r ( E t⊳x ) for all t ∈ T . Since l ( E t ) = P s ∈ S w ( t, s ) e s x and r ( E t⊳x ) = P s ∈ S w ( s, t ⊳ x ) e s x , we have w ( t, s ) = w ( s, t ⊳ x ). From this obser-vation, we have w ( t i , s j ) = w ( s j , t i + n ) , w ( t i + n , s j ) = w ( s j , t i ) ,w ( t i , s j + n ) = w ( s j + n , t i + n ) , w ( t i + n , s j + n ) = w ( s j + n , t i ) , for 1 ≤ i, j ≤ n . Therefore, ( w ( t i , s j )) ≤ i,j ≤ n is non-degenerate if and only if( w ( s i , t j )) ≤ i,j ≤ n is non-degenerate and thus ( w ( t i , s j )) ≤ i,j ≤ n is non-degenerateif and only if the condition (4.91) holds.Lastly we will determine when ( w ( t i , t j )) ≤ i,j ≤ n is non-degenerate. By Lemma 4.11,we have l ( X b ) l ( X a i − ) = τ ( b, a i − ) l ( X a i − b ). Because l ( X b ) l ( X a i − ) = ( X t ∈ T w ( b, t ) e t )( X t ∈ T w ( a i − , t ) e t x )= X t ∈ T w ( b, t ) w ( a i − , t ) e t x and τ ( b, a i − ) l ( X a i − b ) = τ ( b, a i − ) X t ∈ T w ( a i − b, t ) e t x = X t ∈ T τ ( b, a i − ) w ( a i − b, t ) e t x, we have w ( b, t ) w ( a i − , t ) = τ ( b, a i − ) w ( a i − b, t ). Through taking t = a j − , weget that w ( b, a j − ) w ( a i − , a j − ) = τ ( b, a i − ) w ( a i − b, a j − ) and therefore w ( t i + n , t j ) = τ ( b, a i − ) − δw ( t i , t j ) , ≤ i, j ≤ n. (4.92)And if we let t = a j − b , then w ( b, t ) w ( a i − , t ) = τ ( b, a i − ) w ( a i − b, t ) becomes w ( b, a j − b ) w ( a i − , a j − b ) = τ ( b, a i − ) w ( a i − b, a j − b ), and hence we have w ( t i + n , t j + n ) = − τ ( b, a i − ) − δw ( t i , t j + n ) 1 ≤ i, j ≤ n. (4.93)Similarly, owing to Lemma 4.11, we have r ( X b ) r ( X a i − ) = τ ( a i − , b ) r ( X a i − b ). By r ( X b ) r ( X a i − ) = ( X t ∈ T w ( t, b ) e t )( X t ∈ T w ( t, a i − ) e t x )= X t ∈ T w ( t, b ) w ( t, a i − ) e t x,τ ( a i − , b ) r ( X a i − b ) = τ ( a i − , b ) X t ∈ T w ( t, a i − b ) e t x = X t ∈ T τ ( a i − , b ) w ( t, a i − b ) e t x, we have w ( t, b ) w ( t, a i − ) = τ ( a i − , b ) w ( t, a i − b ). Through taking t = a j − ,we get that w ( a j − , b ) w ( a j − , a i − ) = τ ( a i − , b ) w ( a j − , a i − b ). Note that we already have τ ( a j − , b ) = − τ ( b, a j − ) and w ( t i , b ) = − δ , so w ( t i , t j + n ) = τ ( b, a j − ) − δw ( t i , t j ) , ≤ i, j ≤ n. (4.94)From the equations (4.92), (4.93) and (4.94), we know that( w ( t i , t j )) ≤ i,j ≤ n = (cid:18) A CD E (cid:19) , where C, D, E are n × n matrices defined by C : = ( τ ( b, a j − ) − δw ( t i , t j )) ≤ i,j ≤ n ,D : = ( τ ( b, a i − ) − δw ( t i , t j )) ≤ i,j ≤ n ,E : = ( − τ ( b, a i − ) − δw ( t i , t j + n )) ≤ i,j ≤ n . Therefore we have( w ( t i , t j )) ≤ i,j ≤ n = (cid:18) A CD E (cid:19) ∼ (cid:18) A CA − C (cid:19) ∼ (cid:18) A C (cid:19) ∼ (cid:18) A A (cid:19) . As a result we get that ( w ( t i , t j )) ≤ i,j ≤ n is non-degenerate if and only if A is non-degenerate. Since w ( t i , t j ) = w ( a i − , a j − )= λ i − , j − S ij − , S i − j − , = P − i − σ ( a j − ) i − S ij − , S i − j − , , we have A = ( P − i − σ ( a j − ) i − S ij − , S i − j − , ) ≤ i,j ≤ n ∼ ( σ ( a j − ) i − S i − j − , S i − j − , ) ≤ i,j ≤ n . Therefore A is non-degenerate if and only if β j β i = 1 for all 1 ≤ i < j ≤ n where β j = σ ( a j − ) S j − , S j − , . Recall that α j = P − σ ( a j − ) S j − , S j − , . Thus β j β i = α j α i and we get that A is non-degenerate if and only if α j α i = 1 for all 1 ≤ i < j ≤ n . Butwe have proved α j α i = 1 for all 1 ≤ i < j ≤ n if and only if the condition (4.91) holds(see the proof for the non-degeneracy of w ). Since conditions 4.86, 4.91, we get whatwe want. (cid:3) Using the Lemma 4.24, we can give a very simple criterion for K (8 n, σ, τ ) to be aminimal quasitriangular Hopf algebra for the case η ( a, b ) = − Theorem 4.25. If K (8 n, σ, τ ) such that η ( a, b ) = − , then it is minimal if and onlyif there is a ω ∈ k such that ω n = P n and ω σ ( a ) τ ( a, a ) − is a primitive n th rootof 1. Moreover, if K (8 n, σ, τ ) is minimal, then all minimal quasitriangular structureson it can be given by { R α,β | α = − ω τ ( a,a ) σ ( a ) , β = ωσ ( a ) α , ω ∈ k such that ω n = P n and ω σ ( a ) τ ( a, a ) − is a primitive n th root of } . Proof.
Firstly, we will show that if there is a ω ∈ k such that ω n = P n and ω σ ( a ) τ ( a, a ) − is a primitive n th root of 1 , N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z then K (8 n, σ, τ ) is minimal. Since k is algebraically closed, we can find a α ∈ k suchthat α = w τ ( b,a ) σ ( a ) τ ( b,b ) . Define β := wσ ( a ) α . Since( αβ ) n λ n, = ( αβ ) n ( P − n σ ( a ) n ) = ( αβσ ( a )) n P − n = ω n P − n = 1and β α = ω σ ( a ) α = ω σ ( a ) τ ( b, b ) σ ( a ) ω τ ( b, a ) = τ ( b, b ) τ ( b, a ) , we can define R α,β by using this α and β . By Proposition 4.14, R α,β is a univer-sal R -matrix of K (8 n, σ, τ ). By ( σ ( a ) αβ ) σ ( a ) τ ( a, a ) − = ω σ ( a ) τ ( a, a ) − and ω σ ( a ) τ ( a, a ) − is a primitive n th root of 1, R α,β is a minimal quasitriangular struc-ture on K (8 n, σ, τ ) by above Lemma 4.24.Conversely, assume that K (8 n, σ, τ ) is minimal. Then we need to find a ω ∈ k suchthat ω n = P n and ω σ ( a ) τ ( a, a ) − is a primitive n th root of 1. Now assume that R α,β is a minimal quasitriangular structure on K (8 n, σ, τ ). Define ω := αβσ ( a ). Wewill show that this ω satisfies our requirements. Since ( αβ ) n λ n, = 1 and λ n, = P − n σ ( a ) n , we have ( αβσ ( a )) n = P n and therefore ω n = P n . Because R α,β isminimal quasitriangular structure, we know ( σ ( a ) αβ ) σ ( a ) τ ( a, a ) − is a primitive n th root of 1 which implies that ω σ ( a ) τ ( a, a ) − is primitive n th root of 1. Hencewe have showed ω is a needed one.At last, we need to show that if K (8 n, σ, τ ) is minimal, then all minimal quasitri-angular structures on it can be given by { R α,β | α = − ω τ ( a,a ) σ ( a ) , β = ωσ ( a ) α , ω ∈ k such that ω n = P n and ω σ ( a ) τ ( a, a ) − is a primitive n th root of 1 } . For conve-nience, we denote the set { R α,β | α = − ω τ ( a,a ) σ ( a ) , β = ωσ ( a ) α , ω ∈ k such that ω n = P n and ω σ ( a ) τ ( a, a ) − is a primitive n th root of 1 } by Q . If R α,β is a minimalquasitriangular structure on it, then we can define ω := αβσ ( a ). It is not hard tosee that α = − ω τ ( a,a ) σ ( a ) and β = ωσ ( a ) α . Therefore R α,β ∈ Q . If R α,β ∈ Q , then wehave ( αβ ) n λ n, = 1 and β α = τ ( b,b ) τ ( b,a ) due to the calculations above. Therefore, R α,β is a universal R -matrix of K (8 n, σ, τ ) by Proposition 4.14. In addition, Lemma 4.24implies that R α,β is a minimal quasitriangular structure. (cid:3) To use Theorem 4.25 more conveniently, we give the following corollary.
Corollary 4.26.
Let K (8 n, σ, τ ) as before in Theorem 4.25. If τ ( a, a i ) = 1 for i ∈ N ,then K (8 n, σ, τ ) is minimal if and only if there is a ω ∈ k such that ω n = 1 and ω σ ( a ) is a primitive n th root of 1. Moreover, if K (8 n, σ, τ ) is minimal, then allminimal quasitriangular structures on it can be given by { R α,β | α = − ω σ ( a ) , β = ωσ ( a ) α , ω ∈ k such that ω n = 1 and ω σ ( a ) is a primitive n th root of } . Using Corollary 4.26, we can give a class of minimal quasitriangular Hopf algebras asfollows
Corollary 4.27.
Let K (8 n, ζ ) be the Hopf algebras given in Example 2.5, then wehave the following conclusions: (i) if n is even and n ≥ , then K (8 n, ζ ) is minimal and all minimal quasi-triangular structures on it can be given by { R α,β | α = ω ζ , β = ωαζ , ω ∈ k such that ω n = 1 and − ( ωζ ) is primitive n th root of } . (ii) if n is odd or n = 2 , then K (8 n, ζ ) is not minimal.Proof. Firstly, we show (i). By the definition of K (8 n, ζ ), σ ( a ) = − ζ . If n iseven and bigger than 4, we can find a ω ∈ k such that ω n = 1 and ω = − ω σ ( a ) = ζ and thus ω σ ( a ) is a primitive n th root of 1. ByCorollary 4.26, we know that K (8 n, ζ ) is minimal and all minimal quasitriangularstructures on it can be given by { R α,β | α = ω ζ , β = ωαζ , ω ∈ k such that ω n =1 and − ( ωζ ) is primitive n th root of 1 } .Secondly, we show (ii). If n is odd, then we have ( ω σ ( a )) n = [ − ( ωζ ) ] n = − ω ∈ k such that ω n = 1. Hence ω σ ( a ) is not a primitive n th root of 1.As a result K (8 n, ζ ) is not minimal by Corollary 4.26. If n = 2, then σ ( a ) = 1. Let ω ∈ k such that ω = 1. Thus ω σ ( a ) = 1 which is not a primitive 2th root of 1.Therefore K (16 , ζ ) is not minimal by applying Corollary 4.26 again. (cid:3) As an application of Theorem 4.25, we use the following example to illustrate ourresults.
Example 4.28.
Let K as before in Example 2.3, and let ˜ a = a, ˜ b = ab , thenwe get that G = h ˜ a, ˜ b | ˜ a = ˜ b = 1 , ˜ a ˜ b = ˜ b ˜ a i and ˜ a ⊳ x = ˜ a ˜ b, ˜ b ⊳ x = ˜ b . Thisimplies that K belongs to K (8 n, σ, τ ) and such that η (˜ a, ˜ b ) = −
1. It can be seenthat σ (˜ a ) = 1 and n = 1. Since σ (˜ a ) = 1, K is minimal by Corollary 4.26.Moreover, all minimal quasitriangular structures on it can be given by { R α,β | α, β ∈ k such that α = − , αβ = 1 } . Combined with Proposition 4.1, we can get thatall universal R -matrices on K by { R α,β | α, β ∈ k such that α = − , αβ = 1 } ∪{ P ≤ i,j,k,l ≤ γ ik + jl δ il + jk ( − jk e a i b j ⊗ e a k b l | γ, δ ∈ k such that γ = δ = 1 } . Thisresult is the same as the result in [14, Lemma 5.4].For completeness, we give the following result for the case η ( a, b ) = 1. Proposition 4.29.
Let K (8 n, σ, τ ) as before. If η ( a, b ) = 1 , then K (8 n, σ, τ ) is notminimal.Proof. Let R ′ α,β as before in Proposition 4.21, then we will show that R ′ α,β is not aminimal quasitriangular structure on K (8 n, σ, τ ) and therefore we complete the proof.Similar to the proof of Lemma 4.12, we know that l ( X t ) = r ( X t ) and l ( E t ) = r ( E t⊳x )for t ∈ T in this case. And hence R ′ α,β such that the condition of Lemma 4.22. We N THE QUASITRIANGULAR STRUCTURES OF ABELIAN EXTENSIONS OF Z claim that the w of R ′ α,β is not a non-degenerated matrix. To show this, let us denotea matrix A as follows A := ( w ( s i , s j )) ≤ i,j ≤ n , where s i = a i − , s j = a j − . Assume 1 ≤ i, j ≤ n and let s n + i = a i − b , then wehave w ( s i , s n + j ) = w ( a i − , a j − b )= ( λ j − , ) i − ( αβ ) i − j − [ σ ( a j − )] i − = w ( a i − , a j − )= w ( s i , s j )and w ( s n + i , s j ) = w ( a i − b, a j − )= ( λ j − , ) i − ( αβ ) i − j − [ σ ( a j − )] i − = w ( a i − , a j − )= w ( s i , s j )and w ( s n + i , s n + j ) = w ( a i − b, a j − b )= ( λ j − , ) i − ( αβ ) i − j − [ σ ( a j − )] i − = w ( a i − , a j − )= w ( s i , s j ) . Therefore we have ( w ( s i , s j )) ≤ i,j ≤ n = (cid:18) A A A A (cid:19) ∼ (cid:18) A A (cid:19) . Thus w is non-degenerated and we know that R ′ α,β is not minimal quasitriangularstructure on K (8 n, σ, τ ) by Lemma 4.22. (cid:3) References [1] A. Abella, Some advances about the existence of compact involutions in semisimple Hopf alge-bras, S˜ao Paulo J. Math. Sci. 13 (2019), no. 2, 628-651.[2] D. E. Evans, M. Pugh, Braided subfactors, spectral measures, planar algebras, and Calabi-Yaualgebras associated to SU(3) modular invariants, Progress in operator algebras, noncommutativegeometry, and their applications, Theta Ser. Adv. Math., 15, Theta, Bucharest, 2012.[3] D. Pansera, A class of semisimple Hopf algebras acting on quantum polynomial algebras, Rings,modules and codes, 303-316, Contemp. Math., 727, Amer. Math. Soc., Providence, RI, 2019.[4] G. I. Kac, Group extensions which are ring groups, Mat. Sb. (N.S.) 76 (1968), 473-496.[5] Y. Kashina, Classification of semisimple Hopf algebras of dimension 16, J. Algebra 232 (2000),no. 2, 617-663.[6] Y. Kashina, On semisimple Hopf algebras of dimension 2 m , Algebr. Represent. Theory 19 (2016),no. 6, 1387-1422. [7] A. Masuoka, Semisimple Hopf algebras of dimension 6, 8, Israel J. Math. 92 (1995), no. 1-3,361-373.[8] A. Masuoka, Some further classification results on semisimple Hopf algebras, Comm. Algebra 24(1996), no. 1, 307-329.[9] A. Masuoka, Hopf algebra extensions and cohomology, New directions in Hopf algebras, 167-209,Math. Sci. Res.Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.[10] A. Masuoka, Extensions of Hopf algebras, Deformation of group schemes and applications tonumber theory (Japanese) (Kyoto, 1995). ˜Surikaisekikenky˜usho K˜oky˜uroku No. 942 (1996), 53-65.[11] A. Masuoka, Extensions of Hopf algebras and Lie bialgebras, Trans. Amer. Math. Soc. 352(2000), no. 8, 3837-3879.[12] A. Masuoka, Cohomology and coquasi-bialgebra extensions associated to a matched pair ofbialgebras, Adv. Math. 173 (2003), no. 2, 262-315.[13] S. Natale, On quasitriangular structures in Hopf algebras arising from exact group factorizations,Comm. Algebra 39 (2011), no. 12, 4763-4775.[14] M. Wakui, Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite di-mension, J. Pure Appl. Algebra 214 (2010), no. 6, 701-728.[15] D. E. Radford, Hopf Algebras, World Scientific, Series on Knots and Everything, 49. WorldScientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. xxii+559 pp.[16] D. E. Radford, Minimal quasitriangular Hopf algebras, J. Algebra 157 (1993), no. 2, 285-315.[17] S. Suzuki, A family of braided cosemisimple Hopf algebras of finite dimension, Tsukuba J. Math.22 (1998), no. 1, 1-29. Department of Mathematics, Nanjing University, Nanjing 210093, China
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