Structure of semisimple Hopf algebras of dimension p 2 q 2 , II
aa r X i v : . [ m a t h . R A ] J a n STRUCTURE OF SEMISIMPLE HOPF ALGEBRAS OFDIMENSION p q , II JINGCHENG DONG
Abstract.
Let k be an algebraically closed field of characteristic 0. In thispaper, we obtain the structure theorems for semisimple Hopf algebras of di-mension p q over k , where p, q are prime numbers with p < q . As an appli-cation, we also obtain the structure theorems for semisimple Hopf algebras ofdimension 9 p and 25 q for all primes 3 = p and 5 = q . Introduction
Throughout this paper, we will work over an algebraically closed field k of char-acteristic 0.Quite recently, an outstanding classification result was obtained for semisimpleHopf algebras over k . That is, Etingof et al [6] completed the classification ofsemisimple Hopf algebras of dimension pq and pqr , where p, q, r are distinct primenumbers. The results in [6] showed that all these Hopf algebras can be constructedfrom group algebras and their duals by means of extensions. Up to now, besidesthose mentioned above, semisimple Hopf algebras of dimension p, p , p and pq havebeen completely classified. See [5, 8, 12, 13, 14, 23] for details.Recall that a semisimple Hopf algebra H is called of Frobenius type if the dimen-sions of the simple H -modules divide the dimension of H . Kaplansky conjecturedthat every finite-dimensional semisimple Hopf algebra is of Frobenius type [9, Ap-pendix 2]. It is still an open problem. Many examples show that a positive answerto Kaplansky’s conjecture would be very helpful in the classification of semisimpleHopf algebras. See [3] and the examples mentioned above for details.In a previous paper [4], we studied the structure of semisimple Hopf algebras ofdimension p q , where p, q are prime numbers with p < q . As an application, wealso studied the structure of semisimple Hopf algebras of dimension 4 q , where q is a prime number. In the present paper, we shall continue our investigation andprove that the main results in [4] can be extended to the case p < q . Moreover,the structure theorems for semisimple Hopf algebras of dimension 9 p and 25 q willalso be given in this paper, where 3 = p and 5 = q are prime numbers.The paper is organized as follows. In Section 2, we recall the definitions and ba-sic properties of semisolvability, characters and Radford’s biproducts, respectively.Some useful lemmas are also obtained in this section. In particular, we give anpartial answer to Kaplansky’s conjecture. We prove that if dim H is odd and H hasa simple module of dimension 3 then 3 divides dim H . Under the assumption that H does not have simple modules of dimension 3 and 7, we also prove that if dim H is odd and H has a simple module of dimension 5 then 5 divides dim H . Mathematics Subject Classification.
Key words and phrases. semisimple Hopf algebra, semisolvability, character, biproduct.
We begin our main work in Section 3. Let H be a semisimple Hopf algebras ofdimension p q , where p < q is a prime number. We first prove that if | G ( H ∗ ) | = q then H is upper semisolvable, in the sense of [15]. It is a generalization of [4, Lemma3.4]. We then present our main result. We prove that if p < q then H is eithersemisolvable or isomorphic to a Radford’s biproduct R kG , where kG is the groupalgebra of group G of order p , R is a semisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension q . Our approach is mainly based on looking for normal Hopfsubalgebras of H of dimension pq . In Section 4, we shall study the structure ofsemisimple Hopf algebras of dimension 9 p and 25 q .Throughout this paper, all modules and comodules are left modules and leftcomodules, and moreover they are finite-dimensional over k . ⊗ , dim mean ⊗ k ,dim k , respectively. Our references for the theory of Hopf algebras are [16] or [22].The notation for Hopf algebras is standard. For example, the group of group-likeelements in H is denoted by G ( H ).2. Preliminaries
Characters.
Throughout this subsection, H will be a semisimple Hopf al-gebra over k . As an algebra, H is isomorphic to a direct product of full matrixalgebras H ∼ = k ( n ) × s Y i =2 M d i ( k ) ( n i ) , where n = | G ( H ∗ ) | . In this case, we say H is of type ( d , n ; · · · ; d s , n s ) as analgebra, where d = 1. If H ∗ is of type ( d , n ; · · · ; d s , n s ) as an algebra, we shallsay that H is of type ( d , n ; · · · ; d s , n s ) as a coalgebra.Obviously, H is of type ( d , n ; · · · ; d s , n s ) as an algebra if and only if H has n non-isomorphic irreducible characters of degree d , n non-isomorphic irreduciblecharacters of degree d , etc. In this paper, we shall use the notation X t to denotethe set of all irreducible characters of H of degree t .Let V be an H -module. The character of V is the element χ = χ V ∈ H ∗ definedby h χ, h i = Tr V ( h ) for all h ∈ H . The degree of χ is defined to be the integerdeg χ = χ (1) = dim V . If U is another H -module, we have χ U ⊗ V = χ U χ V , χ V ∗ = S ( χ V ) , where S is the antipode of H ∗ .All irreducible characters of H span a subalgebra R ( H ) of H ∗ , which is calledthe character algebra of H . By [23, Lemma 2], R ( H ) is semisimple. The antipode S induces an anti-algebra involution ∗ : R ( H ) → R ( H ), given by χ χ ∗ := S ( χ ).The character of the trivial H -module is the counit ε .Let χ U , χ V ∈ R ( H ) be the characters of the H -modules U and V , respectively.The integer m ( χ U , χ V ) = dimHom H ( U, V ) is defined to be the multiplicity of U in V . This can be extended to a bilinear form m : R ( H ) × R ( H ) → k .Let Irr( H ) denote the set of irreducible characters of H . Then Irr( H ) is a basisof R ( H ). If χ ∈ R ( H ), we may write χ = P α ∈ Irr( H ) m ( α, χ ) α . Let χ, ψ, ω ∈ R ( H ).Then m ( χ, ψω ) = m ( ψ ∗ , ωχ ∗ ) = m ( ψ, χω ∗ ) and m ( χ, ψ ) = m ( χ ∗ , ψ ∗ ). See [19,Theorem 9].For each group-like element g in G ( H ∗ ), we have m ( g, χψ ) = 1, if ψ = χ ∗ g and0 otherwise for all χ, ψ ∈ Irr( H ). In particular, m ( g, χψ ) = 0 if deg χ = deg ψ . Let χ ∈ Irr( H ). Then for any group-like element g in G ( H ∗ ), m ( g, χχ ∗ ) > TRUCTURE OF SEMISIMPLE HOPF ALGEBRAS OF DIMENSION p q , II 3 if m ( g, χχ ∗ ) = 1 if and only if gχ = χ . The set of such group-like elements forms asubgroup of G ( H ∗ ), of order at most (deg χ ) . See [19, Theorem 10]. Denote thissubgroup by G [ χ ]. In particular, we have χχ ∗ = X g ∈ G [ χ ] g + X α ∈ Irr( H ) , deg α> m ( α, χχ ∗ ) α. A subalgebra A of R ( H ) is called a standard subalgebra if A is spanned byirreducible characters of H . Let X be a subset of Irr( H ). Then X spans a stan-dard subalgebra of R ( H ) if and only if the product of characters in X decomposesas a sum of characters in X . There is a bijection between ∗ -invariant standardsubalgebras of R ( H ) and quotient Hopf algebras of H . See [19, Theorem 6].In the rest of this subsection, we shall present some results on irreducible char-acters and algebra types. Lemma 2.1.
Let χ ∈ Irr( H ) be an irreducible character of H . Then(1) The order of G [ χ ] divides (deg χ ) .(2) The order of G ( H ∗ ) divides n (deg χ ) , where n is the number of non-isomorphicirreducible characters of degree deg χ .Proof. It follows from Nichols-Zoeller Theorem [20]. See also [18, Lemma 2.2.2]. (cid:3)
Lemma 2.2.
Assume that dim H is odd and H is of type (1 , n ; · · · ; d s , n s ) as analgebra. Then d i is odd and n i is even for all ≤ i ≤ s .Proof. It follows from [10, Theorem 5] that d i is odd.If there exists i ∈ { , · · · , s } such that n i is odd, then there is at least oneirreducible character of degree d i such that it is self-dual. This contradicts [10,Theorem 4]. (cid:3) Lemma 2.3.
Assume that dim H is odd. If H has a simple module of dimension , then divides the order of G ( H ∗ ) . In particular, divides dim H .Proof. Let χ be an irreducible character of degree 3. By Lemma 2.2, H doesnot have irreducible characters of even degree. Therefore, if G [ χ ] is trivial then χ χ ∗ = ε + χ ′ + χ for some χ ′ ∈ X , χ ∈ X . Since χ χ ∗ is self-dual, χ ′ and χ are self-dual. It contradicts the assumption and [10, Theorem 4]. Hence, G [ χ ] isnot trivial for every χ ∈ X . By Lemma 2.1 (1), the order of G [ χ ] is 3 or 9. Thus,3 divides | G ( H ∗ ) | since G [ χ ] is a subgroup of G ( H ∗ ) for every χ ∈ X .The second statement can be obtained by the Nichols–Zoeller Theorem. (cid:3) Remark 2.4.
The above lemma has appeared in [2, Corollary] and [11, Theorem4.4] , respectively. In the first paper, Burciu does not assume that the characteristicof the base field is zero, but adds the assumption that H has no even-dimensionalsimple modules. Accordingly, his proof is rather different from ours. The authorlearned the result in the second paper after he finished this paper. Our proof hereis slightly different from that in the second paper. So we give the proof for the sakeof completeness. Corollary 2.5.
Assume that dim H is odd and H is of type (1 , n ; 3 , m ; · · · ) as analgebra. If(1) H does not have irreducible characters of degree , or(2) there exists a non-trivial subgroup G of G ( H ∗ ) such that G [ χ ] = G for all χ ∈ X , JINGCHENG DONG then H has a quotient Hopf algebra of dimension n + 9 m .Proof. Let χ, ψ be irreducible characters of degree 3. By assumption and [4, Lemma2.5], χψ is not irreducible. If there exists χ ∈ X such that m ( χ , χψ ) > χψ = χ + χ + g for some χ ∈ X and g ∈ G ( H ∗ ), by Lemma 2.2. From m ( g, χψ ) = m ( χ, gψ ∗ ) = 1, we get χ = gψ ∗ . Then χψ = gψ ∗ ψ = χ + χ + g shows that ψ ∗ ψ = g − χ + g − χ + ε . This contradicts Lemma 2.3. Similarly, wecan show that there does not exist χ ∈ X such that m ( χ , χψ ) >
0. Therefore, χψ is a sum of irreducible characters of degree 1 or 3. It follows that irreduciblecharacters of degree 1 and 3 span a standard subalgebra of R ( H ) and H has aquotient Hopf algebra of dimension n + 9 m . (cid:3) Lemma 2.6.
Assume that dim H is odd and H does not have simple modules ofdimension and . If H has a simple module of dimension , then divides theorder of G ( H ∗ ) . In particular, divides dim H .Proof. Let χ be an irreducible character of degree 5. By assumption and Lemma2.2, if G [ χ ] is trivial then there are four possible decomposition of χχ ∗ : χχ ∗ = ε + χ + χ ; χχ ∗ = ε + χ + χ ; χχ ∗ = ε + χ + χ ; χχ ∗ = ε + χ + χ + χ + χ , where χ i , χ kj are irreducible characters of degree i, j . In all cases, there exists atleast one irreducible character such that it is self-dual, since χχ ∗ is self-dual. Itcontradicts the assumption and [10, Theorem 4]. Therefore, G [ χ ] is not trivial forevery χ ∈ X . Hence, 5 divides the order of G ( H ∗ ) by Lemma 2.1 (1). (cid:3) Semisolvability.
Let B be a finite-dimensional Hopf algebra over k . A Hopfsubalgebra A ⊆ B is called normal if h AS ( h ) ⊆ A and S ( h ) Ah ⊆ A , for all h ∈ B . If B does not contain proper normal Hopf subalgebras then it is calledsimple. The notion of simplicity is self-dual, that is, B is simple if and only if B ∗ is simple.The notions of upper and lower semisolvability for finite-dimensional Hopf alge-bras have been introduced in [15], as generalizations of the notion of solvability forfinite groups. By definition, H is called lower semisolvable if there exists a chain ofHopf subalgebras H n +1 = k ⊆ H n ⊆ · · · ⊆ H = H such that H i +1 is a normal Hopf subalgebra of H i , for all i , and all quotients H i /H i H + i +1 are trivial. That is, they are isomorphic to a group algebra or a dualgroup algebra. Dually, H is called upper semisolvable if there exists a chain ofquotient Hopf algebras H (0) = H π −→ H (1) π −→ · · · π n −−→ H ( n ) = k such that H coπ i ( i − = { h ∈ H ( i − | ( id ⊗ π i )∆( h ) = h ⊗ } is a normal Hopf subalgebraof H ( i − , and all H coπ i ( i − are trivial.In analogy with the situations for finite groups, it is enough for many applicationsto know that a Hopf algebra is semisolvable.By [15, Corollary 3.3], we have that H is upper semisolvable if and only if H ∗ islower semisolvable. If this is the case, then H can be obtained from group algebrasand their duals by means of (a finite number of) extensions. TRUCTURE OF SEMISIMPLE HOPF ALGEBRAS OF DIMENSION p q , II 5 Radford’s biproduct.
Let A be a semisimple Hopf algebra and let AA YD de-note the braided category of Yetter-Drinfeld modules over A . Let R be a semisimpleYetter-Drinfeld Hopf algebra in AA YD . Denote by ρ : R → A ⊗ R , ρ ( a ) = a − ⊗ a ,and · : A ⊗ R → R , the coaction and action of A on R , respectively. We shall usethe notation ∆( a ) = a ⊗ a and S R for the comultiplication and the antipode of R , respectively.Since R is in particular a module algebra over A , we can form the smash product(see [15, Definition 4.1.3]). This is an algebra with underlying vector space R ⊗ A ,multiplication is given by( a ⊗ g )( b ⊗ h ) = a ( g · b ) ⊗ g h, for all g, h ∈ A, a, b ∈ R, and unit 1 = 1 R ⊗ A .Since R is also a comodule coalgebra over A , we can dually form the smashcoproduct. This is a coalgebra with underlying vector space R ⊗ A , comultiplicationis given by ∆( a ⊗ g ) = a ⊗ ( a ) − g ⊗ ( a ) ⊗ g , for all h ∈ A, a ∈ R, and counit ε R ⊗ ε A .As observed by D. E. Radford (see [21, Theorem 1]), the Yetter-Drinfeld condi-tion assures that R ⊗ A becomes a Hopf algebra with these structures. This Hopfalgebra is called the Radford’s biproduct of R and A . We denote this Hopf algebraby R A and write a g = a ⊗ g for all g ∈ A, a ∈ R . Its antipode is given by S ( a g ) = (1 S ( a − g ))( S R ( a ) , for all g ∈ A, a ∈ R. A biproduct R A as described above is characterized by the following prop-erty(see [21, Theorem 3]): suppose that H is a finite-dimensional Hopf algebraendowed with Hopf algebra maps ι : A → H and π : H → A such that πι : A → A is an isomorphism. Then the subalgebra R = H coπ has a natural structure ofYetter-Drinfeld Hopf algebra over A such that the multiplication map R A → H induces an isomorphism of Hopf algebras.The following lemma is a special case of [17, Lemma 4.1.9]. Lemma 2.7.
Let H be a semisimple Hopf algebra of dimension p q , where p, q are distinct prime numbers. If gcd ( | G ( H ) | , | G ( H ∗ ) | ) = p , then H ∼ = R kG is abiproduct, where kG is the group algebra of group G of order p , R is a semisimpleYetter-Drinfeld Hopf algebra in kGkG YD of dimension q . Semisimple Hopf algebras of dimension p q Let p, q be distinct prime numbers with p < q . Throughout this section, H will bea semisimple Hopf algebra of dimension p q , unless otherwise stated. By Nichols-Zoeller Theorem [20], the order of G ( H ∗ ) divides dim H . Moreover, | G ( H ∗ ) | 6 = 1by [6, Proposition 9.9]. By [4, Lemma 1], H is of Frobenius type. Therefore,the dimension of a simple H -module can only be 1 , p, p or q . Let a, b, c be thenumber of non-isomorphic simple H -modules of dimension p, p and q , respectively.It follows that we have an equation p q = | G ( H ∗ ) | + ap + bp + cq . In particular,if | G ( H ∗ ) | = p q then H is a dual group algebra; if | G ( H ∗ ) | = pq then H is uppersemisolvable by the following lemma, which is due to [4, Lemma 2.3]. Lemma 3.1. If H has a Hopf subalgebra K of dimension pq then H is lowersemisolvable. JINGCHENG DONG
The following lemma is a refinement of [4, Lemma 3.4].
Lemma 3.2.
If the order of G ( H ∗ ) is q then H is upper semisolvable.Proof. If p = 2 and q = 3 then it is the case discussed in [17, Chapter 8]. Hence, H is upper semisolvable. Throughout the remainder of the proof, we assume that p ≥ a = 0 then ap ≥ p q , a contradiction. Hence, a = 0.Similarly, b = 0. If follows that H is of type (1 , q ; q, p −
1) as an algebra.The group G ( H ∗ ) acts by left multiplication on the set X q . The set X q is a unionof orbits which have length 1 , q or q . Since q > p ≥ q does not divides p − χ q ∈ X q such that G [ χ q ] = G ( H ∗ ). This means that gχ q = χ q = χ q g forall g ∈ G ( H ∗ ).Let C be a q -dimensional simple subcoalgebra of H ∗ , corresponding to χ q . Then gC = C = Cg for all g ∈ G ( H ∗ ). By [17, Proposition 3.2.6], G ( H ∗ ) is normal in k [ C ], where k [ C ] denotes the subalgebra generated by C . It is a Hopf subalgebraof H ∗ containing G ( H ∗ ). Counting dimension, we know dim k [ C ] ≥ q . Sincedim k [ C ] divides dim H , we know dim k [ C ] = pq or p q . If dim k [ C ] = pq thenLemma 3.1 shows that H ∗ is lower semisolvable. If dim k [ C ] = p q then k [ C ] = H ∗ .Since kG ( H ∗ ) is a group algebra and the quotient H ∗ /H ∗ ( kG ( H ∗ )) + is trivial (see[13]), H ∗ is lower semisolvable. Hence, H is upper semisolvable. This completesthe proof. (cid:3) Theorem 3.3. If q > p then H is either semisolvable or isomorphic to a Radford’sbiproduct R kG , where kG is the group algebra of group G of order p , R is asemisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension q .Proof. By [1, Proposition 1.1], H has a quotient Hopf algebra H of dimension | G ( H ∗ ) | + ap + bp . In particular, | G ( H ∗ ) | divides dim H and | G ( H ∗ ) | + ap + bp divides dim H .We first prove that the order of G ( H ∗ ) can not be q . Suppose on the contrarythat | G ( H ∗ ) | = q . We first note that c = 0, since otherwise we get the contradiction p | q . Since q divides dim H and c = 0, we have that dim H < p q . Therefore,dim H = q, pq, p q, pq or q . If dim H = q then ( H ) ∗ ⊆ kG ( H ∗ ) by [13]. It isimpossible since q = dim H does not divide | G ( H ∗ ) | = q . If dim H = q, pq or p q then we have p q = q + cq , p q = pq + cq or p q = p q + cq . They allimpossible. Hence, dim H = p q . That is q + ap + bp = pq . It is impossible, too.We then prove that if | G ( H ∗ ) | = p or pq then H is upper semisolvable. We firstnote that c = 0, since otherwise we get the contradiction p | p . Then p | dim H anddim H < p q . Therefore dim H = p, pq, p q, pq or p . Moreover, dim H = p , sinceotherwise ( H ) ∗ ⊆ kG ( H ∗ ) by [13], but p = dim H does not divide | G ( H ∗ ) | = p or pq . The possibilities dim H = p, pq or p q lead, respectively to the contradictions p q = p + cq , p q = pq + cq and p q = p q + cq . Hence these are also discarded,and therefore dim H = pq . This implies that H is upper semisolvable, by Lemma3.1.Finally, the theorem follows from Lemma 2.7, 3.1 and 3.2. (cid:3) As an immediate consequence of Theorem 3.3, we have the following corollary.
TRUCTURE OF SEMISIMPLE HOPF ALGEBRAS OF DIMENSION p q , II 7 Corollary 3.4. If p < q and H is simple as a Hopf algebra then H is isomorphicto a Radford’s biproduct R kG , where kG is the group algebra of group G of order p , R is a semisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension q . In fact, examples of nontrivial semisimple Hopf algebras of dimension p q whichare Radford’s biproducts in such a way, and are simple as Hopf algebras do exists.A construction of such examples as twisting deformations of certain groups appearsin [7, Remark 4.6]. 4. Applications
Semisimple Hopf algebras of dimension q . In this subsection, we shallprove the following theorem.
Theorem 4.1. If H is a semisimple Hopf algebra of dimension q then H is eithersemisolvable or isomorphic to a Radford’s biproduct R kG , where kG is the groupalgebra of group G of order , R is a semisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension q . By Theorem 3.3, it suffices to consider the case q = 5 and 7. Lemma 4.2. If q = 5 then H is either semisolvable or isomorphic to a Radford’sbiproduct R kG , where kG is the group algebra of group G of order , R is asemisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension .Proof. By Lemma 2.1, 2.2 and 2.3, if dim H = 3 × then H is of one of thefollowing types as an algebra:(1 ,
25; 5 , , (1 ,
75; 5 , , (1 ,
3; 3 ,
8; 5 , , (1 ,
9; 3 ,
6; 9 , , (1 ,
9; 3 , , (1 ,
45; 3 , . If H is of type (1 ,
25; 5 ,
8) as an algebra then Lemma 3.2 shows that H is uppersemisolvable. If H is of type (1 ,
75; 5 ,
6) as an algebra then Lemma 3.1 shows that H is upper semisolvable. If H is of type (1 ,
3; 3 ,
8; 5 ,
6) as an algebra then Corollary2.5 shows that H has a quotient Hopf algebra of dimension 75. Hence, Lemma 3.1shows that H is upper semisolvable. The lemma then follows from Lemma 2.7. (cid:3) Remark 4.3.
The computation in the proof of Lemma 4.2 is partly handled by acomputer. For example, it is easy to write a computer program by which one findsout all non-negative integers n , n , n , n such that
225 = n + 9 n + 81 n + 25 n ,and then one can eliminate those which can not be algebra types of H by usingLemma 2.1, 2.2 and 2.3. The computations in the followings are handled similarly. Lemma 4.4. If q = 7 then H is either semisolvable or isomorphic to a Radford’sbiproduct R kG , where kG is the group algebra of group G of order , R is asemisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension .Proof. By Lemma 2.1, 2.2 and 2.3, if dim H = 3 × then H is of one of thefollowing types as an algebra:(1 ,
3; 3 ,
14; 5 ,
6; 9 , , (1 ,
3; 3 ,
32; 5 , , (1 ,
3; 3 ,
16; 7 , , (1 ,
21; 3 ,
14; 7 , , (1 ,
49; 7 , , (1 , , , (1 ,
9; 3 ,
12; 9 , , (1 ,
9; 3 ,
30; 9 , , (1 ,
9; 3 , , (1 ,
63; 3 , . Corollary 2.5 shows that H can not be of type (1 ,
3; 3 ,
14; 5 ,
6; 9 , , (1 ,
3; 3 ,
32; 5 , (cid:3) JINGCHENG DONG
Corollary 4.5. If H is a semisimple Hopf algebra of dimension q and is simpleas a Hopf algebra then H is isomorphic to a Radford’s biproduct R kG , where kG is the group algebra of group G of order , R is a semisimple Yetter-Drinfeld Hopfalgebra in kGkG YD of dimension q . Semisimple Hopf algebras of dimension q . In this subsection, we shallprove the following theorem.
Theorem 4.6. If H is a semisimple Hopf algebra of dimension q then H iseither semisolvable or isomorphic to a Radford’s biproduct R kG , where kG isthe group algebra of group G of order , R is a semisimple Yetter-Drinfeld Hopfalgebra in kGkG YD of dimension q . By Theorem 3.3, it suffices to consider the case 7 ≤ q ≤ Lemma 4.7. If q = 7 then H is either semisolvable or isomorphic to a Radford’sbiproduct R kG , where kG is the group algebra of group G of order , R is asemisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension .Proof. By Lemma 2.1, 2.2 and 2.6, if dim H = 5 × then H is of one of thefollowing types as an algebra:(1 ,
35; 5 ,
28; 7 , , (1 ,
49; 7 , , (1 , , , (1 , , , (1 ,
25; 5 , . We shall prove that H can not be of type (1 ,
35; 5 ,
28; 7 ,
10) as an algebra. Thelemma then will follow from Lemma 2.7, 3.1 and 3.2.Suppose on the contrary that H is of type (1 ,
35; 5 ,
28; 7 ,
10) as an algebra. Thegroup G ( H ∗ ) acts by left multiplication on the set X . The set X is a union oforbits which have length 1 , G [ χ ] is a proper subgroupof G ( H ∗ ) for every χ ∈ X . Hence, there does not exist orbits with length 1.Accordingly, every orbit has length 7 and the order of G [ χ ] is 5 for every χ ∈ X .In particular, the decomposition of χχ ∗ does not contain irreducible characters ofdegree 7.Let χ ′ , χ be distinct irreducible characters of degree 5. Suppose that there exists χ ∈ X such that m ( χ , χ ′ χ ∗ ) >
0. Then there must exist ε = g ∈ G ( H ∗ ) suchthat m ( g, χ ′ χ ∗ ) = 1. From this observation, we know χ ′ = gχ and χ ′ χ ∗ = gχχ ∗ .Since χχ ∗ does not contain irreducible characters of degree 7, χ ′ χ ∗ does not containsuch characters, too. This contradicts the assumption. Therefore, χ ′ χ ∗ is a sum ofirreducible characters of degree 1 or 5. It follows that G ( H ∗ ) ∪ X spans a standardsubalgebra of R ( H ), and H has a quotient Hopf algebra of dimension 735. Thiscontradicts the Nichols-Zoeller Theorem [20]. (cid:3) Lemma 4.8.
Let H be a semisimple Hopf algebra of dimension q , where q =11 , , . If | G ( H ∗ ) | = 5 or q then H has a quotient Hopf algebra of dimension | G ( H ∗ ) | + 25 a , where a is the cardinal number of X .Proof. In fact, it can be checked directly that G [ χ ] = 5 for every χ ∈ X . Then thelemma follows from a similar argument as in the proof of Lemma 4.7. (cid:3) Lemma 4.9. If q = 11 then H is either semisolvable or isomorphic to a Radford’sbiproduct R kG , where kG is the group algebra of group G of order , R is asemisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension . TRUCTURE OF SEMISIMPLE HOPF ALGEBRAS OF DIMENSION p q , II 9 Proof.
By Lemma 2.1, 2.2 and 2.6, if dim H = 5 × then H is of one of thefollowing types as an algebra:(1 ,
5; 5 ,
24; 11 , , (1 ,
55; 5 ,
22; 11 , , (1 , , , (1 , , , (1 , , , (1 ,
25; 5 ,
20; 25 , , (1 ,
25; 5 ,
70; 25 , , (1 ,
25; 5 , . By Lemma 4.8, if H is of type (1 ,
5; 5 ,
24; 11 ,
20) or (1 ,
55; 5 ,
22; 11 ,
20) as an algebrathen H has a quotient Hopf algebra of dimension 605. Then H is upper semisolvableby Lemma 3.1. The lemma then follows from Lemma 2.7, 3.1 and 3.2. (cid:3) Lemma 4.10. If q = 13 then H is either semisolvable or isomorphic to a Radford’sbiproduct R kG , where kG is the group algebra of group G of order , R is asemisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension .Proof. By Lemma 2.1, 2.2 and 2.6, if dim H = 5 × then H is of one of thefollowing types as an algebra:(1 , , , (1 , , , (1 , , , (1 ,
25; 5 ,
18; 25 , , (1 ,
25; 5 , , (1 ,
25; 5 ,
68; 25 , , (1 ,
25; 5 , , . The lemma then follows directly from Lemma 2.7, 3.1 and 3.2. (cid:3)
Lemma 4.11. If q = 17 then H is either semisolvable or isomorphic to a Radford’sbiproduct R kG , where kG is the group algebra of group G of order , R is asemisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension .Proof. By Lemma 2.1, 2.2 and 2.6, if dim H = 5 × then H is of one of thefollowing types as an algebra: (1 ,
85; 5 , , , (1 , , , (1 , , , (1 , , , (1 ,
25; 5 , , (1 ,
25; 5 , , ,
25; 5 ,
38; 25 , , (1 ,
25; 5 ,
88; 25 , , (1 ,
25; 5 , , , (1 ,
25; 5 , , . By Lemma 4.8, if H is of type (1 ,
85; 5 , ,
10) as an algebra then H has a quo-tient Hopf algebra of dimension 4335. This contradicts the Nichols-Zoeller Theorem[20]. The lemma then follows from Lemma 2.7, 3.1 and 3.2. (cid:3) Lemma 4.12. If q = 19 then H is either semisolvable or isomorphic to a Radford’sbiproduct R kG , where kG is the group algebra of group G of order , R is asemisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension .Proof. By Lemma 2.1, 2.2 and 2.6, if dim H = 5 × then H is of one of thefollowing types as an algebra:(1 ,
5; 5 ,
22; 19 ,
20; 25 , , (1 ,
5; 5 ,
72; 19 , , (1 , , , (1 , , , (1 , , , (1 ,
25; 5 ,
10; 25 , , (1 ,
25; 5 ,
60; 25 , , (1 ,
25; 5 , , , (1 ,
25; 5 , , (1 ,
25; 5 , , , (1 ,
25; 5 , , , (1 ,
25; 5 , , , (1 ,
25; 5 , , . By Lemma 4.8, if H is of type (1 ,
5; 5 ,
22; 19 ,
20; 25 ,
2) as an algebra then H has aquotient Hopf algebra of dimension 555. This contradicts the Nichols-Zoeller Theo-rem [20]. Again by Lemma 4.8, if H is of type (1 ,
5; 5 ,
72; 19 ,
20) as an algebra then H has a quotient Hopf algebra of dimension 1805. Then H is upper semisolvableby Lemma 3.1. The lemma then follows from Lemma 2.7, 3.1 and 3.2. (cid:3) Lemma 4.13. If q = 23 then H is either semisolvable or isomorphic to a Radford’sbiproduct R kG , where kG is the group algebra of group G of order , R is asemisimple Yetter-Drinfeld Hopf algebra in kGkG YD of dimension .Proof. By Lemma 2.1, 2.2 and 2.6, if dim H = 5 × then H is of one of thefollowing types as an algebra:(1 , , , (1 , , , (1 , , , (1 ,
25; 5 ,
28; 25 , , (1 ,
25; 5 , , (1 ,
25; 5 , , ,
25; 5 , , , (1 ,
25; 5 , , , (1 ,
25; 5 , , , (1 ,
25; 5 , , , (1 ,
25; 5 ,
78; 25 , , (1 ,
25; 5 , , , (1 ,
25; 5 , , , (1 ,
25; 5 , , . The lemma then follows directly from Lemma 2.7, 3.1 and 3.2. (cid:3)
Corollary 4.14. If H is a semisimple Hopf algebra of dimension q and is simpleas a Hopf algebra then H is isomorphic to a Radford’s biproduct R kG , where kG is the group algebra of group G of order , R is a semisimple Yetter-Drinfeld Hopfalgebra in kGkG YD of dimension q . References [1] V. A. Artamonov, Semisimple finite-dimensional Hopf algebras, Sbornik: Mathematics,198(9), 1221C1245 (2007).[2] S. Burciu, Representations of degree three for semisimple Hopf algebras, J. Pure Appl. Alge-bra, 194, 85–93 (2004).[3] J. Dong, Structure of a class of semisimple Hopf algebra, Acta Mathematica Sinica, ChineseSeries, 54(2):1–8 (2011).[4] J. Dong, Structure of semisimple Hopf algebras of dimension p q , arXiv:1009.3541, to appearin Communications in Algebra.[5] P. Etingof and S. Gelaki, Semisimple Hopf algebras of dimension pq are trivial, J. Algebra,210(2), 664–669 (1998).[6] P. Etingof, D. Nikshych and V. Ostrik, Weakly group-theoretical and solvable fusion cate-gories, Adv. Math., 226 (1), 176–505 (2011).[7] Galindo, C., Natale, S.: Simple Hopf algebras and deformations of finite groups. Math. Res.Lett. 14 (6), 943–954 (2007).[8] S. Gelaki and S. Westreich, On semisimple Hopf algebras of dimension pq , Proc. Amer. Math.Soc., 128(1), 39–47 (2000).[9] I. Kaplansky, Bialgebras. University of Chicago Press, Chicago (1975).[10] Y. Kashina, Y. Sommerhauser, Y. Zhu, Self-dual modules of semisimple Hopf algebras, J.Algebra, 257, 88–96 (2002).[11] Y. Kashina, Y. Sommerhauser, Y. Zhu, On higher Frobenius-Schur indicators.[12] A. Masuoka, Semisimple Hopf algebras of dimension 2 p , Comm. Algebra, 23, 1931–1940(1995).[13] A. Masuoka, The p n theorem for semisimple Hopf algebras, Proc. Amer. Math. Soc., 124,735–737 (1996).[14] A. Masuoka, Self-dual Hopf algebras of dimension p obtained by extension, J. Algebra, 178,791-806 (1995).[15] S. Montgomery and S. Whiterspoon, Irreducible representations of crossed products, J. PureAppl. Algebra, 129, 315–326 (1998).[16] S. Montgomery, Hopf algebras and their actions on rings. CBMS Reg. Conf. Ser. Math. 82.Amer. Math. Soc., Providence 1993.[17] S. Natale, Semisolvability of semisimple Hopf algebras of low dimension, Mem. Amer. Math.Soc., 186 (2007).[18] S. Natale, On semisimple Hopf algebras of dimension pq , J. Algebra, 221(2), 242–278 (1999).[19] W. D. Nichols and M. B. Richmond, The Grothendieck group of a Hopf algebra, J. PureAppl. Algebra, 106, 297–306 (1996). TRUCTURE OF SEMISIMPLE HOPF ALGEBRAS OF DIMENSION p q , II 11 [20] W. D. Nichols and M. B. Zoelle, A Hopf algebra freeness theorem, Amer. J. Math., 111(2),381–385 (1989).[21] D. Radford, The structure of Hopf algebras with a projection, J. Algebra, 92, 322–347 (1985).[22] M. E. Sweedler, Hopf Algebras. Benjamin, New York(1969).[23] Y. Zhu, Hopf algebras of prime dimension, Internat. Math. Res. Notices, 1, 53–59 (1994). College of Engineering, Nanjing Agricultural University, Nanjing 210031, Jiangsu,People’s Republic of China
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