On 4n -dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras
aa r X i v : . [ m a t h . R A ] S e p On n -dimensional neither pointed nor semisimple Hopfalgebras and the associated weak Hopf algebras Jialei Chen a , Shilin Yang a † , Dingguo Wang b , Yongjun Xu b a College of Applied Sciences, Beijing University of Technology
Beijing 100124, P. R. China b School of Mathematical Sciences, Qufu Normal University
Qufu 273165, P. R. China
Abstract.
For a class of neither pointed nor semisimple Hopf algebras H n of dimension4 n , it is shown that they are quasi-triangular, which universal R -matrices are described.The corresponding weak Hopf algebras w H n and their representations are constructed.Finally, their duality and their Green rings are established by generators and relationsexplicitly. It turns out that the Green rings of the associated weak Hopf algebras are notcommutative even if the Green rings of H n are commutative. Keywords : quasi-triangularity, Green ring, weak Hopf algebra, representation.
Mathematics Subject Classification:
Introduction
Let k be an algebraically closed field of characteristic zero. In this paper, we will study repre-sentations of the class of neither pointed nor semisimple Hopf algebra H n of dimension 4 n (seeDefinition 1.1) and the associated weak Hopf algebras.The class of Hopf algebras H n plays an important role in constructing new Nichols algebras,new Hopf algebras and classifying Hopf algebras. Note that if n = 2 , a = 0, it is justthe unique neither pointed nor-semisimple 8-dimensional Hopf algebra ( A ′′ C ) ∗ (see [29]), or the12-dimensional Hopf algebra A ∗ (see [19]) up to isomorphism respectively. In [9], the authorsdetermined all finite-dimensional Hopf algebras over k whose coradical generated a Hopf subalgebraisomorphic to H . They also obtained new Nichols algebras of dimension 8 and new Hopf algebrasof dimension 64. Based on this, [32] determined all finite-dimensional Nichols algebras over thesemisimple objects in H H Y D and obtained some new Nichols algebras of non-diagonal type andnew Hopf algebras without the dual Chevalley property. By the equivalence M D ( H ) ≃ H H Y D ,the authors ([11],[33]) obtained some new Nichols algebras which were not of diagonal type andsome families of new Hopf algebras of dimension 216.As is well known, the classification of finite dimensional Hopf algebras over k is an importantopen problem. Since Kaplansky’s conjectures posed in 1975, several results on them have beenobtained (see [36, 24, 3, 21, 22, 10, 7, 4]). In [3], the authors proved that there were exactly4( q −
1) isomorphism classes of non-semisimple pointed Hopf algebras of dimension pq , of which,those Radford’s Hopf algebras (see [23]) occupied 1 /
4. It is remarked that the dual of H n is justthe Radford’s Hopf algebra in [23]. † Corresponding author: [email protected]
Supported by National Natural Science Foundation of China (Grant Nos.11701019, 11671024, 11871301)and the Beijing Natural Science Foundation (Grant No.1162002)
J. Chen, S. Yang, D. Wang, Y. Xu
Given a Hopf algebra H , the decomposition problem of tensor products of indecomposable mod-ules has attracted numerous attentions. In [8], Cibils classified the indecomposable modules over k Z n ( q ) /I d , and gave the decomposition formulas of the tensor product of two indecomposable k Z n ( q ) /I d -modules. Yang determined the representation type of a class of pointed Hopf algebras,classified all indecomposable modules of simple pointed Hopf algebra R ( q, a ), and gave the decom-position formulas of the tensor product of two indecomposable R ( q, a )-modules (see [34]). It isnoted that some results of R ( q, a ) were recently extended to more general case of pointed Hopfalgebras of rank one by Wang et al. (see [31]). Li and Hu described the Green rings of the 2-rankTaft algebra(at q = −
1) (see [14]). Chen, Van Oystaeyen and Zhang gave the Green rings of theTaft algebra H n ( q ) (see [5]). Li and Zhang extended the results of [5], computed the Green rings ofthe Generalized Taft Hopf algebras H n,d by generators and generating relations, and determinedall nilpotent elements in r ( H n,d ) (see [15]). Su and Yang (see [25]) characterized the representationring of small quantum group ¯ U q ( sl ) by generators and relations. It turns out that the represen-tation ring of ¯ U q ( sl ) is generated by infinitely many generators subject to a family of generatingrelations.The concept of weak Hopf algebra in the sense of Li was introduced by [13] in 1998 as a gen-eralization of Hopf algebra. Since then, many weak Hopf algebras or weak quantum groups wereconstructed, for example, Aizawa and Isaac ([1]) constructed weak Hopf algebras correspondingto U q ( sl n ) and Yang ([35]) constructed weak Hopf algebras w dq ( g ) corresponding to quantized en-veloping algebras U q ( g ) of a finite dimensional semisimple Lie algebra g . In [26], Su and Yang con-structed the weak Hopf algebra f H corresponding to the non-commutative and non-cocommutativesemisimple Hopf algebra H of dimension 8. They described the representation ring of f H andstudied the automorphism group of r ( f H ). In [27], Su and Yang studied the Green ring of the weakGeneralized Taft Hopf algebra r ( w s ( H n,d )), showing that the Green ring of the weak GeneralizedTaft Hopf algebra was much more complicated than its Grothendick ring.In the present paper, it is shown that H n is quasi-triangular, which universal R -matrices aredescribed. the weak Hopf algebras w H n and w H ∗ n corresponding to the Hopf algebra H n andits dual H ∗ n are constructed. Then their representations and Green rings are explicitly described.It turns out that the Green rings of the associated weak Hopf algebras are not commutative evenif the Green rings of H n are commutative.The paper is organized as follows. In Section 1, the definition of H n by generators and relationsis described first, then we prove that H n is quasitriangular and describe all universal R -matrices R explicitly. In Section 2, we compute the Green ring r ( H n ). In Section 3, we construct the weakHopf algebra w H n associated to H n . In Section 4, we study the representation ring r ( w H n ) of w H n by generators and relations explicitly. In Section 5, we consider the dual Hopf algebra H ∗ n and its weak Hopf algebra w H ∗ n , we also describe the representation rings r ( H ∗ n ) and r ( w H ∗ n ).Throughout this paper, we work over an algebraically closed field k of characteristic zero. Forthe theory of Hopf algebras and quantum groups, we refer to [18, 28, 12, 17].1. The non-semisimple non-pointed Hopf algebra H n First of all, let us give the defintion of the Hopf algebra H n . reen ring of the nonsemisimple (weak) Hopf Algebras Definition 1.1.
Let n ≥ and q be a primitive n -th root of unity. The Hopf algebra H n isdefined as follows. As an algebra it generated by z, x with relations z n = 1 , zx = qxz, x = 0 for any a ∈ k .The coalgebra structure is ∆( z ) = z ⊗ z + a (1 − q − ) z n +1 x ⊗ zx, ∆( x ) = x ⊗ z n ⊗ x ; ǫ ( z ) = 1 , ǫ ( x ) = 0 ,S ( z ) = z − , S ( x ) = − z n x. It is noted that if n = 1, H is just the 4-dimension Sweedler ′ s Hopf algebra.In general, we have∆( z i ) = z i ⊗ z i + a (1 − q − )(1 + q − + · · · + q − i − ) z n + i x ⊗ z i x. Let C i be the k -space spanned by z i , z n + i x, z i x, z n + i (1 ≤ i ≤ n −
1) and T = k ⊕ kz n ⊕ kz n x ⊕ kx. Lemma 1.2. C i is a simple subcoalgebra and as coalgebras H n = n − M i =1 C i ⊕ T and T ∼ = H as Hopf algebras.Proof. It is straightforward. (cid:3)
It follows that if a = 0, the Hopf algebra H n ( n ≥
2) is not pointed.
Example 1.3. If q is -th primitive root of unity and a = 2 , then H is just the unique neitherpointed nor semisimple -dimensional Hopf algebra ( A ′′ C ) ∗ (see [29]) up to isomorphism. Example 1.4. If q is -th primitive root of unity and a = 0 , then H is just the unique neitherpointed nor semisimple -dimensional Hopf algebra A ∗ (see [19]).By [34, Lemma 2.2, Theorem 2.1], H n is a Nakayama algebra with 2 n cyclic orientation andcyclic relations of length 2. In particular, it is of finite representation type.For every integer j , we set E j = 12 n n − X i =0 q − ij z i . It is easy to see that E , . . . , E n − list the distinct E ′ i s . Moreover, for 0 ≤ j, k < n , we have(1.1) E j z k = 1 n n − X i =0 q − ij z i + k = q jk n n − X i =0 q − ( i + k ) j z i + k ! = q jk E j . and xE i = E i +1 x. Lemma 1.5. { E , · · · , E n − } is a complete set of orthogonal idempotents of H n . J. Chen, S. Yang, D. Wang, Y. Xu
Proof.
Since q − j is also an 2 n -th root of unity different from 1 if j = 0, we get n − X i =0 E i = 12 n n − X i =0 2 n − X j =0 q − ij z j = 12 n n − X j =0 n − X i =0 (cid:0) q − j (cid:1) i ! z j = 1 , Also, using (1.1), for 0 ≤ l, j < n : E j E l = 12 n n − X k =0 q − lk E j z k = 12 n n − X k =0 q − lk + jk E j = 12 n n − X k =0 (cid:0) q j − l (cid:1) k E j = ( E j if l = j l = j Hence, { E , · · · , E n − } is a complete set of orthogonal idempotents of H n . (cid:3) Quasi-triangular Hopf algebras play an important in the theory of Hopf algebras and quantumgroups, since they provide solutions to quantum Yang-Baxter equations. People try to constructquasi-triangular Hopf algebras and get a lot of results(see [29, 20, 6, 30, 16]). In this section, weshall show that H n is quasitriangular and give all universal R -matrices explicitly. First, we recallthe definition of quasi-triangular Hopf algebra.Let H be a finite dimensional Hopf algebra and R ∈ H ⊗ H an invertible element. The pair( H, R ) is said to be a quasi-triangular Hopf algebra and R is said to be a universal R -matrix of H ,if the following three conditions are satisfied.(i) ∆ ′ ( h ) = R ∆( h ) R − , for all h ∈ H ;(ii) (∆ ⊗ id )( R ) = R R ;(iii) ( id ⊗ ∆)( R ) = R R ;Here ∆ ′ = T ◦ ∆ , T : H ⊗ H → H ⊗ H, T ( a ⊗ b ) = b ⊗ a , and R ij ∈ H ⊗ H ⊗ H is given by R = R ⊗ R = 1 ⊗ R , R = ( T ⊗ id )( R ). Theorem 1.6. H n is a quasi-triangular Hopf algebra with universal R -matrix R = n − X i,j =0 ( − ij E i ⊗ E j + 2 a n − X i,j =0 ( − i ( j +1) E i x ⊗ E j x. Proof.
Let R ∈ H n ⊗ H n be a universal R -matrix, and T = k h z | z n = 1 i . First of all, we claimthat R ∈ T ⊗ T + ( T ⊗ T )( x ⊗ x ) . Indeed, we assume that R = X h ∈ T h ⊗ X h + X h ∈ T hx ⊗ Y h , X h , Y h ∈ H n . Note that ∆( z n ) = z n ⊗ z n and ∆ cop ( z n ) R = R ∆( z n ) , The relations zx = qxz implies that xz n = − z n x. From this relation, it follows that X h ∈ T and Y h ∈ T x.
Hence R can be written as R = R ′ + ˆ R where R ′ ∈ T ⊗ T and ˆ R ∈ ( T ⊗ T )( x ⊗ x ) . Let R ′ = n − X i,j =0 a ij E i ⊗ E j ∈ T ⊗ T. Note that ( ǫ ⊗ id )( ˆ R ) = 0 and ( ǫ ⊗ id )( R ) = 1, therefore ( ǫ ⊗ id )( R ′ ) = 1. Thus we have a i = a j = 1 for all i, j = 0 , , · · · , n − . reen ring of the nonsemisimple (weak) Hopf Algebras Moreover, since ∆ cop ( x ) R = R ∆( x ), and ∆ cop ( x ) ˆ R = 0 = ˆ R ∆( x ), we see that ∆ cop ( x ) R ′ = R ′ ∆( x ), n − X i,j =0 a ij E i ⊗ xE j + n − X i,j =0 a ij xE i ⊗ z n E j = n − X i,j =0 a ij E i x ⊗ E j + n − X i,j =0 a ij E i z n ⊗ E j x Hence we get n − X i,j =0 a ij E i ⊗ E j +1 x + n − X i,j =0 ( − j a ij E i +1 x ⊗ E j = n − X i,j =0 ( − i a ij E i ⊗ E j x + n − X i,j =0 a ij E i x ⊗ E j . This implies that a i,j − = ( − i a ij , and a i − ,j = ( − j a ij and we have a ij = ( − ij . Then any universal R -matrix R of H n can be expressed by R = n − X i,j =0 ( − ij E i ⊗ E j + ˆ R, where ˆ R can be written as ˆ R = n − X i,j =0 b ij E i x ⊗ E j x, b ij ∈ k. It is noted that ∆( z ) = (1 ⊗ a ( q − z n x ⊗ x )( z ⊗ z ) . Compute both side of the equation ∆ cop ( z ) R = R ∆( z ) , then it is straightforward to see that the left hand side is n − X i,j =0 ( − ij q i + j E i ⊗ E j + n − X i,j =0 h a ( q − − ( i − j − j q i + j − + b ij q i + j i E i x ⊗ E j x, and the right hand side is n − X i,j =0 ( − ij q i + j E i ⊗ E j + n − X i,j =0 (cid:2) a ( q − − i + ij q i + j − + b ij q i + j − (cid:3) E i x ⊗ E j x. Comparing the two-hand side of the above equation, we have a ( q − − ( i − j − j q i + j − + b ij q i + j = a ( q − − i + ij q i + j − + b ij q i + j − , and b ij = 2 a ( − ij + i . Hence, if R is a universal R -matrix of H n , then R must be equal to R = n − X i,j =0 ( − ij E i ⊗ E j + 2 a n − X i,j =0 ( − i ( j +1) E i x ⊗ E j x. By direct computations we see that (∆ ⊗ id ) ( R ) = R R and ( id ⊗ ∆) ( R ) = R R . Hence R is a universal R -matrix of H n . (cid:3) J. Chen, S. Yang, D. Wang, Y. Xu Indecomposable representations of H n From this section, we always assume that a = 0 in Definition 1.1. The situation for a = 0 canbe considered similarly. Let H = H n and M i be the 2-dimensional cyclic H -module with bases { v i , v i } , where i ∈ Z n . The multiplication of x and z in H provides the actions on M i , that is x ( v i , v i ) = ( v i , v i ) ! ,z ( v i , v i ) = ( v i , v i ) q i q i +1 ! . For any i ∈ Z n , let S i be the 1-dimensional cyclic H -module with base { v i } , with the action x · v i =0 , z · v i = q i v i . Up to isomorphism, { M i | i ∈ Z n } provides the complete list of isomorphism classesof indecomposable H -modules with two dimension. Then we have the following decompositionformulas of the tensor product of two indecomposable H -modules. Theorem 2.1.
Let i, j ∈ Z n , then as H -modules, we have (1) S i ⊗ S j ∼ = S i + j (mod2 n ) . (2) S i ⊗ M j ∼ = M i + j (mod2 n ) . (3) M i ⊗ M j ∼ = M i + j (mod2 n ) ⊕ M i + j +1(mod2 n ) . Proof.
Recall that ∆( z ) = z ⊗ z + a (1 − q − ) z n +1 x ⊗ zx, and ∆( x ) = x ⊗ z n ⊗ x , for i ∈ Z n ,let σ ( i ) = ( − i , we have(1) x · ( v i ⊗ v j ) = 0, z · ( v i ⊗ v j ) = q i + j v i ⊗ v j , therefore S i ⊗ S j ∼ = S i + j (mod2 n ) . (2) For j, k ∈ { , } and i ∈ Z n , x · ( v i ⊗ v jk ) = ( σ ( i ) v i ⊗ v j , k = 1 , , k = 2 .z · ( v i ⊗ v jk ) = ( q i + j v i ⊗ v jk , k = 1 ,q i + j +1 v i ⊗ v jk , k = 2 . so we have S i ⊗ M j ∼ = M i + j (mod2 n ) . (3) For k, l ∈ { , } and i ∈ Z n , note that x · ( v i ⊗ v j ) = v i ⊗ v j + σ ( i ) v i ⊗ v j ,x · ( v i ⊗ v j ) = v i ⊗ v j ,x · ( v i ⊗ v j ) = σ ( i + 1) v i ⊗ v j ,x · ( v i ⊗ v j ) = 0 ,z · ( v ik ⊗ v jl ) = q i + j (cid:0) v i ⊗ v j + a ( q − σ ( i + 1) v i ⊗ v j (cid:1) , k + l = 2 q i + j +1 v ik ⊗ v jl , k + l = 3; q i + j +2 v i ⊗ v j , k + l = 4 . Let w = v i ⊗ v j , w = v i ⊗ v j , and w = v i ⊗ v j − aσ ( i + 1) v i ⊗ v j ,w = v i ⊗ v j + σ ( i ) v i ⊗ v j , reen ring of the nonsemisimple (weak) Hopf Algebras then we have x ( w , w ) = ( w , w ) ! ,z ( w , w ) = ( w , w ) q i + j +1 q i + j +2 ! , and x ( w , w ) = ( w , w ) ! ,z ( w , w ) = ( w , w ) q i + j q i + j +1 ! . Therefore, M i ⊗ M j ∼ = M i + j (mod2 n ) ⊕ M i + j +1(mod2 n ) . (cid:3) Let H be a finite dimensional Hopf algebra and M and N be two finite dimensional H -modules.Recall that the Green ring or the representation ring r ( H ) of H can be defined as follows. As agroup r ( H ) is the free Abelian group generated by the isomorphism classes of the finite dimensional H -modules M , modulo the relations [ M ⊕ N ] = [ M ] + [ N ]. The multiplication of r ( H ) is givenby the tensor product of H -modules, that is, [ M ][ N ] = [ M ⊗ N ]. Then r ( H ) is an associative ringwith identity given by [ k ε ], the trivial 1-dimensional H -module. Note that r ( H ) is a free abeliangroup with a Z -basis { [ M ] | M ∈ ind ( H ) } , where ind ( H ) denotes the set of finite dimensionalindecomposable H -modules.Denote [ S ] = b , [ M ] = c . Corollary 2.2.
The Green ring r ( H n ) is a commutative ring generated by b and c . The set { b k | ≤ k ≤ n − } ∪ { b i c | ≤ i ≤ n − } forms a Z -basis for r ( H n ) .Proof. Firstly, r ( H n ) is a commutative ring since H n is a quasitriangular Hopf algebra. ByTheorem 2.1, b n = 1 and there is a one to one correspondence between the set { b i | ≤ i ≤ n − } and the set of one-dimensional simple H n module { [ S i ] | ≤ i ≤ n − } . Besides, for all0 ≤ i ≤ n −
1, [ S i ] c = [ M i ], hence [ M i ] = b i c and all the two-dimensional simple H n modules { [ M i ] | ≤ i < j ≤ n − } are obtained. (cid:3) Theorem 2.3.
The Green ring r ( H n ) is isomorphic to the quotient ring of the ring Z [ x , x ] module the ideal I generated by the following elements x n − , x − x x − x Proof.
By Corollary 2.2, r ( H n ) is generated by b and c . Hence there is a unique ring epimorphismΦ : Z [ x , x ] → r ( H n )such that Φ( x ) = b = [ S ] , Φ( x ) = c = [ M ] . Since b n = 1 , c = bc + c, bc = cb, we have Φ( x n −
1) = 0 , Φ( x − x x − x ) = 0 , Φ( x x − x x ) = 0 . J. Chen, S. Yang, D. Wang, Y. Xu
It follows that Φ( I ) = 0 , and Φ induces a ring epimorphismΦ : Z [ x , x ] /I → r ( H n ) , such that Φ( v ) = Φ( v ) for all v ∈ Z [ x , x ], where v = π ( v ) (natural epimorphism π : Z [ x , x ] → Z [ x , x ] /I ). As r ( H n ) is a free Z -module of rank 4 n , with a Z -basis { b i | ≤ i ≤ n − } ∪ { b j c | ≤ j ≤ n − } , we can define a Z -module homomorphism:Ψ : r ( H n ) → Z [ x , x ] /I,b i c → x i x = x i x , b j → x j = x j , ≤ i, j ≤ n − . Observe that as a free Z -module, Z [ x , x ] /I is generated by elements x i x and x j , ≤ i, j ≤ n − , we have ΨΦ( x i x ) = ΨΦ( x i x ) = Ψ( b i c ) = x i x , ΨΦ( x j ) = ΨΦ( x j ) = Ψ( b j ) = x j , for all 0 ≤ i, j ≤ n − . Hence ΨΦ = id , and Φ is injective. Thus, Φ is a ring isomorphism. (cid:3) Weak Hopf algebras corresponding to H n Firstly, we recall the concept of weak Hopf algebra given by Li(see [13]). By definition, aweak Hopf algebra is k -bialgebra H with a map T ∈ hom( H, H ) such that T ∗ id ∗ T = T and id ∗ T ∗ id = id , where ∗ is the convolution map in hom( H, H ).Let w H n be the algebra generated by Z, X with relations Z n +1 = Z, ZX = qXZ, X = 0 . Theorem 3.1. w H n is a noncommutative and noncocommutative weak Hopf algebra with comul-tiplication, counit and the weak antipode T as follows ∆( Z ) = Z ⊗ Z + a (1 − q − ) Z n +1 X ⊗ ZX, ∆( X ) = X ⊗ Z n ⊗ X ; ǫ ( Z ) = 1 , ǫ ( X ) = 0 ,T ( Z ) = Z n − , T ( X ) = − Z n X. Proof.
Firstly, it can be shown by direct calculations that the following relations hold:∆( Z ) n +1 = ∆( Z ) , ∆( Z )∆( X ) = q ∆( X )∆( Z ) , ∆( X ) = 0 ,ǫ ( Z ) n +1 = ǫ ( Z ) , ǫ ( Z ) ǫ ( X ) = qǫ ( X ) ǫ ( Z ) , ǫ ( X ) = 0 , Therefore, ∆ and ǫ can be extended to algebra morphism from w H n to w H n ⊗ w H n and from w H n to k respectively. We also have(∆ ⊗ id )∆( Y ) = ( id ⊗ ∆)∆( Y ) , ( ǫ ⊗ id ) ǫ ( Y ) = ( id ⊗ ǫ ) ǫ ( Y ) = Y for Y = X, Z . It follows that w H n is a bialgebra.Secondly, we prove that in the bialgebra w H n , the map T can define a weak antipode in thenatural way. To see this, note that the map T : w H n → w H n op keeps the defining relations:( T ( Z )) n +1 = (( Z ) n − ) n +1 = Z n − = T ( Z ) , ( T ( X )) = ( − Z n X ) = 0 . reen ring of the nonsemisimple (weak) Hopf Algebras T ( X ) T ( Z ) = ( − Z n X )( Z ) n − = q − n ( Z ) n − ( − Z n x ) = qT ( Z ) T ( X ) . It follows that the map T can be extended to an anti-algebra homomorphism T : w H n → w H n .Besides, it is easy to see that in w H n ,( id ∗ T ∗ id )( Z ) = ZT ( Z ) Z = Z n +1 = Z = id ( Z ) , ( T ∗ id ∗ T )( Z ) = T ( Z ) ZT ( Z ) = Z zn − = T ( Z ) . and ( id ∗ T ∗ id )( X ) = µ ( id ⊗ T ⊗ id )( X ⊗ ⊗ Z n ⊗ X ⊗ Z n ⊗ Z n ⊗ X )= X + z n T ( X ) + z n T ( z ) n X = X − z n X + z n X = id ( X ) , ( T ∗ id ∗ T )( X ) = µ ( T ⊗ id ⊗ T )( X ⊗ ⊗ Z n ⊗ X ⊗ Z n ⊗ Z n ⊗ X )= T ( X ) + T ( Z n ) X + T ( Z n ) Z n T ( X )= − Z n X + Z n X − Z n X = − Z n X = T ( X ) . On the other hand, we have id ∗ T ( X ) = X + Z n T ( X ) = X − Z n X = X (1 − Z n ) ,T ∗ id ( X ) = T ( X ) + T ( Z ) n X = − Z n X + Z n X = 0 . and id ∗ T ( Z ) = ZT ( Z ) + a (1 − q − ) Z n +1 XT ( ZX ) = Z n + a (1 − q − ) Z n +1 X ( − Z n X ) Z n − = Z n = T ( Z ) . These arguments show that for any h ∈ w H n we have id ∗ T ( h ) and T ∗ id ( h ) are in the centerof w sn,d . Now, if a, b ∈ w sn,d and T ∗ id ∗ T ( a ) = T ( a ) , T ∗ id ∗ T ( b ) = T ( b ) ,id ∗ T ∗ id ( a ) = a, id ∗ T ∗ id ( b ) = b, one can check that T ∗ id ∗ T ( ab ) = T ( ab ) , id ∗ T ∗ id ( ab ) = ab. Hence T is indeed define a weak antipode of w H n and w H n is a weak Hopf algebra, which isnon-commutative and non-cocommutative. (cid:3) Let J = Z n , it is easy to see that J and 1 − J are a pair of orthogonal central idempotents in w H n . Let w = w H n J , w = w H n (1 − J ). Proposition 3.2.
We have w H n = w ⊕ w as two-sided ideals. Moreover, w ∼ = H n as Hopfalgebras and w ∼ = k [ y ] / ( y ) as algebras.Proof. The first statement is easy to see. Let us prove the second one.Note that w is generated by Z , XJ and with J as the identity and the relations JZ = ZJ = Z, ( XJ ) = 0 , Z ( XJ ) = q ( XJ ) Z. Let ρ : H n → w be the map defined by ρ (1) = J, ρ ( z ) = Z, ρ ( z − ) = Z n − ρ ( x ) = XJ. J. Chen, S. Yang, D. Wang, Y. Xu
It is straightforward to see that ρ is well defined surjective algebraic homomorphism. Let φ : w H n → H n be the map given by φ (1) = 1 , φ ( X ) = x, φ ( Z ) = z. It is obvious that φ is a well defined algebra homomorphism. If we consider the restricted homo-morphism φ | w , then we have φ | w ◦ ρ = id H n . Hence, ρ is injective and w ∼ = H n as algebras.Furthermore, w is a Hopf algebra with comultiplication, counit and the antipode S as follows∆( Z ) = Z ⊗ Z + a (1 − q − ) Z n +1 XJ ⊗ ZXJ, ∆( XJ ) = XJ ⊗ Z n ⊗ XJ ; ǫ ( Z ) = 1 , ǫ ( XJ ) = 0 ,S ( Z ) = Z n − , S ( XJ ) = − Z n XJ.
It is clear that ρ is a Hopf algebra isomorphism. Now we prove that w ∼ = k [ y ] / ( y ). We first claimthat X (1 − J ) = 0. Let N be the w H n -module with the basis { w , w } . The action of w H n on N is given by Z · w i = 0 , i = 1 , .X · w i = ( w , i = 1 , , i = 2 . It follows that Jw i = 0 for i = 1 , X (1 − J )] w = w . Therefore, we have X (1 − J ) = 0 and[ X (1 − J )] = 0.Let φ : k [ y ] / ( y ) → w be the map defined by φ ( y ) = X (1 − J ) , φ (1) = 1 − J. It is easy to show that φ is an algebraic isomorphism, and we have w ∼ = k [ y ] / ( y ) . (cid:3) Indecomposable representations of w H n By Proposition 3.2, w H n = H n ⊕ k [ y ] / ( y ). Hence the indecomposable modules of H n and k [ y ] / ( y ) constitute all the indecomposable w H n -modules up to isomorphism.For any i ∈ Z n , let S i be the 1-dimensional cyclic w H n -module with base { v i } , withc theaction X · v i = 0 , Z · v i = q i v i , and M i be the 2-dimensional cyclic w H n -module with bases { v i , v i } . The module structures are as follows: X ( v i , v i ) = ( v i , v i ) ! ,Z ( v i , v i ) = ( v i , v i ) q i q i +1 ! . In fact, S i and M i are just indecomposable w H n -modules corresponding to those of H n -modules.Let N be the k-vector space with a basis w , the actions of w H n on N are defined by Z · w = 0 , X · w = 0. Let N be the 2-dimensional w H n -module with bases { w , w } . The reen ring of the nonsemisimple (weak) Hopf Algebras module structures are as follows: X ( w , w ) = ( w , w ) ! ,Z ( w , w ) = ( w , w ) ! . It is noted that N and N are just indecomposable w H n -modules corresponding to those of k [ y ] / ( y )-modules. Therefore, we have Proposition 4.1.
The set { S i , M i | i ∈ Z n } ∪ { N j | j = 0 , } forms a complete list of non-isomorphic indecomposable w H n -modules. Now we establish the decomposition formulas of the tensor product of two indecomposable w H n -modules. Theorem 4.2.
Let i, j ∈ Z n , then as w H n -modules, we have (1) S i ⊗ S j ∼ = S i + j (mod2 n ) ∼ = S j ⊗ S i . (2) S i ⊗ M j ∼ = M i + j (mod2 n ) ∼ = M j ⊗ S i . (3) M i ⊗ M j ∼ = M i + j (mod2 n ) ⊕ M i + j +1(mod2 n ) ∼ = M j ⊗ M i . (4) N ⊗ N ∼ = N ∼ = N ⊗ S i ∼ = S i ⊗ N . (5) N ⊗ N ∼ = N ⊕ N ∼ = N ⊗ M i . (6) N ⊗ N ∼ = N ∼ = M i ⊗ N ∼ = N ⊗ S i ∼ = S i ⊗ N . (7) N ⊗ N ∼ = N ⊕ N ∼ = N ⊗ M i ∼ = M i ⊗ N . Proof.
Recall that ∆( X ) = X ⊗ Z n ⊗ X , ∆( Z ) = Z ⊗ Z + a (1 − q − ) Z n +1 X ⊗ ZX . For i ∈ Z n ,let v i be the basis of S i , { v i , v i } be the basis of M i , { w } be the basis of N and { w , w } be thebasis of N . Note that (1)-(3) can be obtained as 2.1.(4). It is clear since for i ∈ Z n , we have X · w ⊗ w = 0 = X · w ⊗ v i = X · v i ⊗ w and X · w ⊗ w = 0 = X · w ⊗ v i = X · v i ⊗ w .(5). Note that for j, k ∈ { , } and i ∈ Z n , X · w ⊗ w j = 0 = X · w ⊗ v ik , and Z · w ⊗ w j =0 = X · w ⊗ v ik , so we have N ⊗ N ∼ = N ⊕ N ∼ = N ⊗ M i . (6). Since for j ∈ { , } and i ∈ Z n , X · w ⊗ w = w ⊗ w , X · w ⊗ w = 0 , Z · w j ⊗ w = 0; X · v i ⊗ w = v i ⊗ w , X · v i ⊗ w = 0 , Z · v ij ⊗ w = 0; X · w ⊗ v i = w ⊗ v i , X · w ⊗ v i = 0 , Z · w j ⊗ v i = 0; X · v i ⊗ w j = ( − i v i ⊗ w , X · v i ⊗ w = 0 , Z · v i ⊗ w j = 0it follows that N ⊗ N ∼ = N ∼ = M i ⊗ N ∼ = N ⊗ S i ∼ = S i ⊗ N . J. Chen, S. Yang, D. Wang, Y. Xu (7). For j, k ∈ { , } and i ∈ Z n , let σ ( i ) = ( − i . X · v i ⊗ w j = ( v i ⊗ w + σ ( i ) v i ⊗ w , j = 1; v i ⊗ w , j = 2 .X · v i ⊗ w j = ( σ ( i + 1) v i ⊗ w , j = 1;0 , j = 2 .Z · v ik ⊗ w j = 0 . Therefore, if we set ̟ = v i ⊗ w , ̟ = v i ⊗ w + σ ( i ) v i ⊗ w , ̟ = v i ⊗ w , ̟ = σ ( i + 1) v i ⊗ w .Then we have X · ̟ = ̟ , X · ̟ = 0 , X · ̟ = ̟ , X · ̟ = 0 , Z · ̟ l = 0( l = 1 , , , , and we obtain M i ⊗ N ∼ = N ⊕ N . Besides, note that X · w j ⊗ v i = ( w ⊗ v i , j = 1;0 , j = 2 .X · w j ⊗ v i = ( w ⊗ v i , j = 1;0 , j = 2 .Z · v ik ⊗ w j = 0 . wo we have N ⊗ M i ∼ = N ⊕ N . Furthermore, take w ′ k , k = 1 , N , then X · w j ⊗ w ′ k = ( w ⊗ w ′ k , j = 1;0 , j = 2 .Z · w j ⊗ w ′ k = 0 . wo we have N ⊗ N ∼ = N ⊕ N . (cid:3) Without confusion, we denote [ S ] = b , [ M ] = c , and [ N ] = d Corollary 4.3.
The Green ring r ( w H n ) is a ring generated by b , c and d . The set { b i c j | ≤ i ≤ n − , j = 0 , } ∪ { c k d | k = 0 , } forms a Z -basis for r ( w H n ) .Proof. By Theorem 4.2, b n = 1 and { b i = [ S i ] | ≤ i ≤ n − } . Besides, for all 0 ≤ i ≤ n − S i ] c = [ M i ], hence [ M i ] = b i c and all the two-dimensional simple H n module { M i | ≤ i ≤ n − } are obtained. Note that [ N ] = d and N ∼ = M ⊗ N , we have [ N ] = cd . The result is obtained. (cid:3) Theorem 4.4.
The Green ring r ( w H n ) is isomorphic to the quotient ring of the ring Z h x , x , x i module the ideal I generated by the following elements x n − , x − x x − x , x x − x x ,x − x , x x − x , x x − x , x x − x . Proof.
By Corollary 2.2, r ( w H n ) is generated by b , c and d . Hence there is a unique ring epimor-phism Φ : Z h x , x , x i → r ( w H n ) reen ring of the nonsemisimple (weak) Hopf Algebras such that Φ( x ) = b = [ S ] , Φ( x ) = c = [ M ] , Φ( x ) = c = [ N ] . By Theorem 4.2 b n = 1 , c = bc + c, bc = cb,d = d, ad = da = d, dc = 2 d. Thus we have Φ( x n −
1) = 0 , Φ( x − x x − x ) = 0 , Φ( x x − x x ) = 0 , Φ( x − x ) = 0 , Φ( x x − x ) = 0 , Φ( x x − x ) = 0 , Φ( x x − x ) . It follows that Φ( I ) = 0 , and Φ induces a ring epimorphismΦ : Z h x , x , x i /I → r ( w H n ) . Comparing the rank of Z h x , x , x i /I and r ( w H n ) , it is easy to see that Φ is a ring isomorphism. (cid:3) The dual H ∗ n of H n and w H ∗ n In this section, we consider the dual Hopf algebra H ∗ n of H n and its weak Hopf algebra w H ∗ n ,we also describe the representation ring r ( w H ∗ n ) of w H ∗ n .Let α and η be the linear forms on H n defined on the basis { z i x j } ≤ i< n,j =0 , by h α, z i x j i = δ j, q i and h η, z i x j i = δ j, q i . It is easy to determine that H ∗ n is generated by α and η with the following relations α n = 1 , η = a (1 − α ) , αη = − ηα, ∆( α ) = α ⊗ α, ∆( η ) = η ⊗ α ⊗ η ; ǫ ( α ) = 1 , ǫ ( η ) = 0 ,S ( α ) = α − , S ( η ) = − α − η. Without lost of generality, we take a = 1 and we get η = 1 − α . The representations of H ∗ n andtheir tensor products decompositions have been described in[34], and the corresponding represen-tation ring are obtained in [31]. By Theorem 8.2([31]), the Green ring of H ∗ n is a commutativering generated by Y, Z, X , · · · , X n − with the relations Y = 1 , Z = Z + Y Z, Y X = X , ZX = 2 X ,X j = 2 j − X j f or ≤ j ≤ n − , X n = 2 n − Z Let w H ∗ n be the algebra generated by G, X with relations Z n +1 = Z, GX = − XG, X = 1 − G J. Chen, S. Yang, D. Wang, Y. Xu
Then w H ∗ n is a noncommutative and noncocommutative weak Hopf algebra with comultiplication,counit and the weak antipode T as follows∆( G ) = G ⊗ G, ∆( X ) = X ⊗ G ⊗ X ; ǫ ( G ) = 1 , ǫ ( X ) = 0 ,T ( Z ) = Z n − , T ( X ) = − Z n − X. Let J = Z n , it is easy to see that J and 1 − J are a pair of orthogonal central idempotents in w H ∗ n . Let w = w H ∗ n J , w = w H ∗ n (1 − J ). Proposition 5.1.
We have w H ∗ n = w ⊕ w as two-sided ideals. Moreover, w ∼ = H ∗ n as Hopfalgebras and w ∼ = k [ y ] / ( y − as algebras. The proof is similar to Proposition 3.2, and we omit here. By Proposition 5.1, w H ∗ n = H ∗ n ⊕ k [ y ] / ( y − H ∗ n -modules and k [ y ] / ( y − w H ∗ n -modules.For s = 0 or n , let M [1 , s ] be the 1-dimensional cyclic w H ∗ n -module with the base { v s } definedby X · v s = 0 , G · v s = ( − sn v s . Let M [2 , s ] be the 2-dimensional cyclic w H ∗ n -module with bases { v s , v s } defined as follows X ( v s , v s ) = ( v s , v s ) ! ,G ( v s , v s ) = ( v s , v s ) ( − sn
00 ( − sn +1 ! . For 1 ≤ j ≤ n −
1, let P j be the 2-dimensional w H ∗ n -module with bases { p j , p j } and modulestructures as follows: X ( p j , p j ) = ( p j , p j ) − q j ! ,G ( p j , p j ) = ( p j , p j ) q j − q j ! . In fact, M [ k, s ] , k = 1 , s = 0 , n and P j , ≤ j ≤ n − w H ∗ n -modulescorresponding to those of H ∗ n -modules.Let N i ( i = 0 ,
1) be the k-vector space with a basis w i , the actions of w H ∗ n on N i are defined by X · w i = ( − i w i , X · w i = 0. Let M be the 2-dimensional w H ∗ n -module with bases { m , m } and module structures as follows: X ( m , m ) = ( m , m ) ! ,G ( m , m ) = ( m , m ) ! . It is noted that N , N and M are just indecomposable w H ∗ n -modules corresponding to those of k [ y ] / ( y − Proposition 5.2.
The set { M [ k, s ] , P j | k = 1 , s = 0 , n ; 1 ≤ j ≤ n − } ∪ { N i , M | i = 0 , } reen ring of the nonsemisimple (weak) Hopf Algebras forms a complete list of non-isomorphic indecomposable w H ∗ n -modules. Now we establish the decomposition formulas of the tensor product of two indecomposable w H ∗ n -modules. Theorem 5.3. As w H ∗ n -modules, we have (1) For ≤ i, j ≤ n − , P i ⊗ P j ∼ = ( M [2 , ⊕ M [2 , n ] , n | i + j ;2 P i + j , n ∤ i + j. (2) For k ∈ { , } , s ∈ { , n } , ≤ j ≤ n − , M [ k, s ] ⊗ P j ∼ = kP j ∼ = P j ⊗ M [ k, s ] . (3) For k, l ∈ { , } , s, j ∈ { , n } , M [ k, s ] ⊗ M [ l, j ] ∼ = ( M [2 , ⊕ M [2 , n ] , k + l = 4; M [ k + l − , s + j (mod2 n )] , k + l < . (4) For i, j ∈ { , } , N i ⊗ N j ∼ = N i . (5) For k ∈ { , } , s ∈ { , n } , j ∈ { , } , M [ k, s ] ⊗ N j ∼ = ( N j + sn , k = 1; M , k = 2 . (6) For k ∈ { , } , s ∈ { , n } , j ∈ { , } , N j ⊗ M [ k, s ] ∼ = kN j . (7) For i ∈ { , } , ≤ j ≤ n − , N i ⊗ P j ∼ = 2 N i , P j ⊗ N i ∼ = M . (8) For i ∈ { , } , N i ⊗ M ∼ = 2 N i , M ⊗ N i ∼ = M . (9) M ⊗ M ∼ = 2 M . (10) For k ∈ { , } , s ∈ { , n } , M ⊗ M [ k, s ] ∼ = kM ∼ = M [ k, s ] ⊗ M . (11) For ≤ j ≤ n − , M ⊗ P j ∼ = 2 M ∼ = P j ⊗ M . Proof.
Recall that ∆( G ) = G ⊗ G and ∆( X ) = X ⊗ G ⊗ X . (1)-(3) can be proved provedsimilarly as in [34, 31].(4). Note that G · w i = 0 , X · w i = ( − i w i , therefore N i ⊗ N j ∼ = N i . for i, j ∈ { , } .(5) and (6). Let k ∈ { , } , s ∈ { , n } , j ∈ { , } and v s be the basis of M [1 , s ], then X · v s = 0and G · v s = ( − sn v s , so we have G · ( v s ⊗ w j ) = 0 , X · ( v s ⊗ w j ) = ( − ( sn + j ) v s ⊗ w j ,G · ( w j ⊗ v s ) = 0 , X · ( w j ⊗ v s ) = ( − j w j ⊗ v s , hence M [1 , s ] ⊗ N j ∼ = N j + sn and N j ⊗ M [1 , s ] ∼ = N j . Let { v s , v s } be the basis of M [2 , s ], then X · ( v s ⊗ w j ) = v s ⊗ w j + ( − ( sn + j ) v s ⊗ w j ,G · ( v s ⊗ w j ) = 0 ,X · ( v s ⊗ w j + ( − ( sn + j ) v s ⊗ w j ) = v s ⊗ w j ,G · ( v s ⊗ w j + ( − ( sn + j ) v s ⊗ w j ) = 0 , therefore M [2 , s ] ⊗ N j ∼ = M . Besides, X · ( w j ⊗ v s ) = ( − j ( w j ⊗ v s ) , X · ( w j ⊗ v s ) = ( − j ( w j ⊗ v s ) , G · ( w j ⊗ v sk ) = 0 , therefore N j ⊗ M [2 , s ] ∼ = 2 N j . J. Chen, S. Yang, D. Wang, Y. Xu (7). Note that X · ( p j ⊗ w i ) = p j ⊗ w i + q j ( − i p j ⊗ w i ,X · ( p j ⊗ w i ) = (1 − q j ) p j ⊗ w i − q j ( − i p j ⊗ w i , let ω = p j ⊗ w i , ω = p j ⊗ w i + q j ( − i p j ⊗ w i , it follows that X · ω = ω , X · ω = ω and G · ω k = 0 , therefore P j ⊗ N i ∼ = M . Besides, X · ( w i ⊗ p jk ) = ( − i w i ⊗ p jk and G · ( w i ⊗ p jk ) = 0for k = 1 ,
2, therefore N i ⊗ P j ∼ = 2 N i . (8). Let i, j ∈ { , } . Since X · ( w i ⊗ m j ) = ( − i w i ⊗ m j , G · ( w i ⊗ m j ) = 0 , therefore N i ⊗ M ∼ = 2 N i . Note that X · ( m ⊗ w i ) = m ⊗ w i , X · ( m ⊗ w i ) = m ⊗ w i , G · ( m j ⊗ w i ) = 0 , we have M ⊗ N i ∼ = M .(9). Suppose that m , m and m ′ , m ′ are two basis of M respectively, then for i, j ∈ { , } , X · ( m i ⊗ m ′ j ) = m τ ( i ) ⊗ m ′ j , X · ( m τ ( i ) ⊗ m ′ j ) = m i ⊗ m ′ j , G · ( m i ⊗ m ′ j ) = 0 , thus we have M ⊗ M ∼ = 2 M . (10). Let τ : { , } → { , } be the permutation with τ (0) = 1 , τ (1) = 0. For i ∈ { , } , j ∈{ , } and s ∈ { , n } , we have X · ( m ⊗ v s ) = m ⊗ v s , X · ( m ⊗ v s ) = m ⊗ v s , G · ( m i ⊗ v s ) = 0 ,X · ( v s ⊗ m ) = ( − sn v s ⊗ m , X · ( v s ⊗ m ) = ( − sn v s ⊗ m , G · ( v s ⊗ m i ) = 0 , therefore M ⊗ M [1 , s ] ∼ = M ∼ = M [1 , s ] ⊗ M . Besides, X · ( m i ⊗ v sj ) = m τ ( i ) ⊗ v sj , X · ( m τ ( i ) ⊗ v sj ) = m i ⊗ v sj , G · ( m i ⊗ v sj ) = 0 , hence M ⊗ M [2 , s ] ∼ = 2 M . Furthermore, X · ( v s ⊗ m ) = v s ⊗ m + ( − sn v s ⊗ m ,X · ( v s ⊗ m ) = v s ⊗ m + ( − sn v s ⊗ m ,X · ( v s ⊗ m ) = ( − sn +1 v s ⊗ m ,X · ( v s ⊗ m ) = ( − sn +1 v s ⊗ m , let ω = v s ⊗ m , ω = v s ⊗ m + ( − sn v s ⊗ m , ω = v s ⊗ m , ω = ( − sn +1 v s ⊗ m , then wehave X · ω = ω , X · ω = ω , X · ω = ω , X · ω = ω and G · ω l = 0, for l = 1 , , , . Thereforewe get M [2 , s ] ⊗ M ∼ = 2 M . (11). For i ∈ { , } , k ∈ { , } and j ∈ { , , · · · n − } , since X · ( p j ⊗ m ) = p j ⊗ m + q j p j ⊗ m ,X · ( p j ⊗ m ) = p j ⊗ m + q j p j ⊗ m ,X · ( p j ⊗ m ) = (1 − q j ) p j ⊗ m − q j p j ⊗ m ,X · ( p j ⊗ m ) = (1 − q j ) p j ⊗ m − q j p j ⊗ m , reen ring of the nonsemisimple (weak) Hopf Algebras let ω = p j ⊗ m , ω = p j ⊗ m + q j p j ⊗ m , ω = p j ⊗ m , ω = p j ⊗ m + q j p j ⊗ m , then wehave X · ω = ω , X · ω = ω , X · ω = ω , X · ω = ω and G · ω l = 0, for l = 1 , , , . Thereforewe get P j ⊗ M ∼ = 2 M . Besides, Since X · ( m ⊗ p jk ) = m ⊗ p jk ,X · ( m ⊗ p jk ) = m ⊗ p jk , and G · ( m i ⊗ p jk ) = 0, it follows that M ⊗ P j ∼ = 2 M . (cid:3) Denote M [1 , n ] = b , M [2 ,
0] = c , P j = a j , j ∈ { , , · · · n − } , and N = d , then we have Corollary 5.4.
The Green ring r ( w H ∗ n ) is a ring generated by b , c , d and a j . The set { a j , b i c k | ≤ j ≤ n − , i, k = 0 , } ∪ { b i d, c k d, | i, k = 0 , } forms a Z -basis for r ( w H ∗ n ) .Proof. By Theorem 5.3, b = 1 , bc = cb = M [2 , n ] and c = c + bc . Therefore, the set { a j , b i c k | ≤ j ≤ n − , i, k = 0 , } has a one to one correspondence with the modules { M [ k, s ] , P j } . Besides,note that d = d , [ N ] = bd and [ M ] = cd , the result is obtained. (cid:3) Theorem 5.5.
The Green ring r ( w H ∗ n ) is isomorphic to the quotient ring of the ring Z h Y, Z, X j , W i module the ideal I generated by the following elements Y − , Z − Z − Y Z, Y Z − ZY, Y X − X , ZX − X , (5.1) X j − j − X j (1 ≤ j ≤ n − , X n − n − Z (5.2) W − W, W Y − W, W Z − W, W X − W, X W − W. (5.3) Proof.
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