Monomial Hopf algebras

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1 Journa of Agebra Monomia Hopf agebras Xiao-Wu Chen, a Hua-Lin Huang, a Yu Ye, a and Pu Zhang b, a Department of Mathematics, University of Science and Technoogy of China, Hefei , Anhui, PR China b Department of Mathematics, Shanghai Jiao Tong University, Shanghai , PR China Received 28 January 2003 Communicated by Susan Montgomery Dedicated to Caus Michae Ringe on the occasion of his sixtieth birthday Abstract Let K be a fied of characteristic 0 containing a roots of unity. We cassified a the Hopf structures on monomia K-coagebras, or, in dua version, on monomia K-agebras Esevier Inc. A rights reserved. Keywords: Hopf structures; Monomia coagebras Introduction In the representation theory of agebras, one uses uivers and reations to construct agebras, and the resuted agebras are eementary, see Ausander, Reiten, and Smaø [1] and Ringe [15]. The construction of a path agebra has been duaized by Chin and Montgomery [4] to get a path coagebra. It is then natura to consider subcoagebras of a path coagebra, which are a pointed. There are aso severa works to construct neither commutative nor cocommutative Hopf agebras via uivers see, e.g., [5 7,9]. An advantage for this construction is that a natura basis consisting of paths is avaiabe, and one can reate the properties of a uiver to the ones of the corresponding Hopf structures. Supported in part by the Nationa Natura Science Foundation of China Grant No and No and the Europe Commission AsiaLink project Agebras and Representations in China and Europe ASI/B7-301/98/ * Corresponding author. E-mai addresses: xwchen@mai.ustc.edu.cn X.-W. Chen, huain@ustc.edu.cn H.-L. Huang, yeyu@ustc.edu.cn Y. Ye, pzhang@sjtu.edu.cn P. Zhang /$ see front matter 2004 Esevier Inc. A rights reserved. doi: /j.jagebra

2 X.-W. Chen et a. / Journa of Agebra In [5] Cibis determined a the graded Hopf structures with ength grading on the path agebra KZn a of basic cyce Z n of ength n; in [6], Cibis and Rosso studied graded Hopf structures on path agebras; in [9] E. Green and Soberg studied Hopf structures on some specia uadratic uotients of path agebras. More recenty, Cibis and Rosso [7] introduced the notion of the Hopf uiver of a group with ramification, and then cassified a the graded Hopf agebras with ength grading on path coagebras. It turns out that a path coagebra KQ c admits a graded Hopf structure with ength grading if and ony if Q is a Hopf uiver here a Hopf uiver is not necessariy finite. The cited works above stimuate us to ook for finite-dimensiona Hopf agebra structures, on more uotients of path agebras, or in dua version, on more subcoagebras of path coagebras. The aim of this paper is to cassify a the Hopf agebra structures on a monomia agebra, or euivaenty, on a monomia coagebra. Since a finite-dimensiona Hopf agebra is both Frobenius and cofrobenius, we first ook at the structure of monomia Frobenius agebras, or duay, the one of monomia cofrobenius coagebras. It turns out that each indecomposabe coagebra component of a non-semisimpe monomia cofrobenius coagebra is C d n with d 2, where C d n is the subcoagebra of path coagebra KZn c with basis the set of paths of ength stricty smaer than d. See Section 2. Then by a theorem of Montgomery Theorem 3.2 in [13], a non-semisimpe monomia Hopf agebra C is a crossed product of a Hopf structure on C d n with a group agebra. Thus, we turn to study the Hopf structures on C d n with d 2. It turns out that the coagebra C d n, d 2, admits a Hopf structure if and ony if d n Theorem 3.1. Moreover, when runs over primitive dth roots of unity, the generaized Taft agebras A n,d gives a the isocasses of graded Hopf structures on C d n with ength grading; whie the Hopf structures not necessariy graded with ength grading on C d n are exacty the agebras denoted by An,d,µ,, with a primitive dth root of unity and µ K. These agebras An,d,µ, have been studied by Radford [14], Andruskiewitsch and Schneider [2]. See Theorem 3.6. Note that agebra An,d,µ, is given by generators and reations. In Section 4, we prove that An,d,µ, is the product of KZd a/j d and n/d 1 copies of matrix agebra M d K when µ 0, and the product of n/d copies of KZd a/j d when µ = 0, see Theorem 4.3. Hence the Gabrie uiver and the Ausander Reiten uiver of An,d,µ, are known. Finay, we introduce the notion of a group datum. By using the uiver construction of C d n, the Hopf structure on it, and Montgomery s theorem Theorem 3.2 in [13], we get a one to one correspondence of Gaois type between the set of the isocasses of nonsemisimpe monomia Hopf K-agebras and the isocasses of group data over K.Thisgives a cassification of monomia Hopf agebras. 1. Preiminaries Throughout this paper, K denotes a fied of characteristic 0 containing a roots of unity. By an agebra we mean a finite-dimensiona associative K-agebra with identity eement.

3 214 X.-W. Chen et a. / Journa of Agebra Quivers considered here are aways finite. Given a uiver Q = Q 0,Q 1 with Q 0 the set of vertices and Q 1 the set of arrows, denote by KQ, KQ a,andkq c,thek-space with basis the set of a paths in Q, the path agebra of Q, and the path coagebra of Q, respectivey. Note that they are a graded with respect to ength grading. For α Q 1,et sα and tα denote respectivey the starting and ending vertex of α. Reca that the comutipication of the path coagebra KQ c is defined by see [4] p = βα=p 1 β α = α α 1 sα 1 + α α i+1 α i α 1 + tα α α 1 i=1 for each path p = α α 1 with each α i Q 1 ;andεp = 0if 1, and 1 if = 0. This is a pointed coagebra. Let C be a coagebra. The set of group-ike eements is defined to be GC := { c C c = c c, c 0 }. It is cear εc = 1forc GC. Forx,y GC, denote by P x,y C := { c C c = c x + y c }, the set of x,y-primitive eements in C. It is cear that εc = 0forc P x,y C. Note that Kx y P x,y C. Aneementc P x,y C is non-trivia if c/ Kx y. Note that GKQ c = Q 0 ;and Lemma 1.1. For x,y Q 0, we have P x,y KQ c = ykq 1 x Kx y where ykq 1 x denotes the K-space spanned by a arrows from x to y. In particuar, there is a non-trivia x,y-primitive eement in KQ c if and ony if there is an arrow from x to y in Q. An idea I of KQ a is admissibe if J N I J 2 for some positive integer N 2, where J is the idea generated by a arrows. An agebra A is eementary if A/R = K n as agebras for some n, wherer is the Jacobson radica of A. For an eementary agebra A, there is a uniue uiver Q, and an admissibe idea I of KQ a, such that A = KQ a /I. See [1,15]. An agebra A is monomia if there exists an admissibe idea I generated by some paths in Q such that A = KQ a /I. Duay, we have Definition 1.2. A subcoagebra C of KQ c is caed monomia provided that the foowing conditions are satisfied: i C contains a vertices and arrows in Q;

4 X.-W. Chen et a. / Journa of Agebra ii C is contained in subcoagebra C d Q := d 1 i=0 KQi for some d 2, where Qi is the set of a paths of ength i in Q; iii C has a basis consisting of paths. It is cear by definition that both monomia agebras and monomia coagebras are finitedimensiona; and A is a monomia agebra if and ony if the inear dua A is a monomia coagebra. In the foowing, for convenience, we wi freuenty pass from a monomia agebra to a monomia coagebra by duaity. For this we wi use the foowing: Lemma 1.3. The path agebra KQ a is exacty the graded dua of the path coagebra KQ c, i.e., KQ a = KQ c gr ; and for each d 2 there is a graded agebra isomorphism: KQ a /J d = Cd Q Let K be an nth root of unity. For non-negative integers and m, the Gaussian binomia coefficient is defined to be where m + := + m!! m!! := 1, 0! := 1, := Observe that d = 0for1 d 1 if the order of is d Denote by Z n the basic cyce of ength n, i.e., an oriented graph with n vertices e 0,...,e n 1, and a uniue arrow α i from e i to e i+1 for each 0 i n 1. Take the indices moduo n. Denote by p i the path in Z n of ength starting at e i. Thus we have p 0 i = e i and p 1 i = α i. For each nth root K of unity, Cibis and Rosso [7] have defined a graded Hopf agebra structure KZ n with ength grading on the path coagebra KZ c n by m + pi pm j = j pi+j +m, with antipode S mapping p i to i p n i.

5 216 X.-W. Chen et a. / Journa of Agebra In the foowing, denote C d Z n by C d n. Thatis,C d n is the subcoagebra of KZn c with basis the set of a paths of ength stricty ess than d. Since m+ = 0form d 1, d 1, + m d, it foows that if the order of is d then C d n is a subhopfagebra of KZ n. Denote this graded Hopf structure on C d n by C d n,. Let d be the order of. Reca that by definition A n,d is an associative agebra generated by eements g and x, with reations g n = 1, x d = 0, xg = gx. Then A n,d is a Hopf agebra with comutipication, counit ε, and antipode S given by g = g g, εg = 1, x = x 1 + g x, εx = 0, Sg = g 1 = g n 1, Sx= xg 1 = 1 g n 1 x. In particuar, if is an nth primitive root of unity i.e., d = n, then A n,d is the n 2 -dimensiona Hopf agebra introduced by Taft [17]. For this reason A n,d is caed a generaized Taft agebra in [10]. Observe that C d n, is generated by e 1 and α 0 as an agebra. Mapping g to e 1 and x to α 0, we get a Hopf agebra isomorphism A n,d = C d n, Let K be an nth root of unity of order d. For each µ K, define a Hopf structure C d n,µ,on coagebra C d n by and with antipode m + pi pm j = j pi+j +m, pi pm j = + m d! µj p +m d i+j! m! if + m<d, S pi = i pn i, pi+j+d +m d, if + m d, where 0,m d 1, and 0 i, j n 1. This is indeed a Hopf agebra with identity eement p 0 0 = e 0 and of dimension nd. Note that this is in genera not graded with respect to the ength grading; and that C d n, 0,= C d n,.

6 X.-W. Chen et a. / Journa of Agebra In [14] and [2] Radford and Andruskiewitsch Schneider have considered the foowing Hopf agebra An,d,µ,, which as an associative agebra is generated by two eements g and x with reations g n = 1, x d = µ 1 g d, xg= gx, with comutipication, counit ε, andthe antipodes givenas in 1.6. It is cear that An, d, 0,= A n,d ; and if d = n then An,d,µ, is the n 2 -dimensiona Taft agebra. Observe that C d n,,µis generated by e 1 and α 0. By sending g to e 1 and x to α 0 we obtain a Hopf agebra isomorphism An,d,µ, = C d n,µ,. 2. Monomia Frobenius agebras and cofrobenius coagebras The aim of this section is to determine the form of monomia Frobenius, or duay, monomia cofrobenius coagebras, for ater appication. This is we-known, but it seems that there are no exact references. Let A be a monomia agebra. Thus, A = KQ a /I for a finite uiver Q, wherei is an admissibe idea generated by some paths of engths 2. For p KQ a,et p be the image of p in A. Then the finite set { p A p does not beong to I} forms a basis of A. It is easy to see the foowing Lemma 2.1. Let A be a monomia agebra. Then i The K-dimension of socae i is the number of the maxima paths starting at vertex i, which do not beong to I. ii The K-dimension of soce i A is the number of the maxima paths ending at vertex i, which do not beong to I. Lemma 2.2. Let A be an indecomposabe, monomia agebra. Then A is Frobenius if and ony if A = k, ora = KZ a n /J d for some positive integers n and d, with d 2. Proof. The sufficiency is straightforward. If A is Frobenius i.e., there is an isomorphism A = A as eft A-modues, or euivaenty, as right A-modues, then the soce of an indecomposabe projective eft A-modue is simpe see, e.g., [8]. It foows from Lemma 2.1 that there is at most one arrow starting

7 218 X.-W. Chen et a. / Journa of Agebra at each vertex i. Repacing eft by right we observe that there is at most one arrow ending at each vertex i. On the other hand, the uiver of an indecomposabe Frobenius agebra is a singe vertex, or has no sources and sinks a source is a vertex at which there are no arrows ending; simiary for a sink, see, e.g., [8]. It foows that if A k then the uiver of A is a basic cyce Z n for some n. However it is we-known that an agebra KZn a /I with I admissibe is Frobenius if and ony if I = J d for some d 2. The dua version of Lemma 2.2 gives the foowing: Lemma 2.3. Let A be an indecomposabe, monomia coagebra. Then A is cofrobenius i.e., A is Frobenius if and ony if A = k, ora = C d n for some positive integers n and d, with d 2. An agebra A is caed Nakayama, if each indecomposabe projective eft and right modue has a uniue composition series. It is we known that an indecomposabe eementary agebra is Nakayama if and ony if its uiver is a basic cyce or a inear uiver A n see [8]. Note that a finite-dimensiona Hopf agebra is Frobenius and cofrobenius see, e.g., [12, p. 18]. Coroary 2.4. An agebra is a monomia Frobenius agebra if and ony if it is eementary Nakayama Frobenius. Hence, a Hopf agebra is monomia if and ony if it is eementary and Nakayama. 3. Hopf structures on coagebra C d n The aim of this section is to give a numerica description such that coagebra C d n admits Hopf structures Theorem 3.1, and then cassify a the graded, or not necessariy graded Hopf structures on C d n Theorem 3.6. Theorem 3.1. Let K be a fied of characteristic 0, containing an nth primitive root of unity. Let d 2 be a positive integer. Then coagebra C d n admits a Hopf agebra structure if and ony if d n. The sufficiency foows from 1.6, or 1.7. In order to prove the necessity we need some preparations. Lemma 3.2. Suppose that the coagebra C d n admits a Hopf agebra structure. Then i The set {e 0,...,e n 1 } of the vertices in C d n forms a cycic group, say, with identity eement 1 = e 0.Thene 1 is a generator of the group. ii Set g := e 1. Then up to a Hopf agebra isomorphism we have for any i such that 0 i n 1 α i g = α i+1 + κ i+1 g i+1 g i+2

8 X.-W. Chen et a. / Journa of Agebra and where,λ i,κ i K, with n = 1. g α i = α i+1 + λ i+1 g i+1 g i+2, Proof. i Since C d n is a Hopf agebra, it foows that GC d n ={e 0,...,e n 1 } is a group, say with identity eement e 0.Sinceα 0 is a non-trivia e 0,e 1 -primitive eement, it foows that α 0 e 1 is a non-trivia e 1,e1 2-primitive eement, i.e., there is an arrow in C dn from e 1 to e1 2. Thus e2 1 = e 2. A simiar argument shows that e i = e1 i for any i. ii Since both α i g and gα i are non-trivia g i+1, g i+2 -primitive eements, it foows that and α i g = w i+1 α i+1 + κ i+1 g i+1 g i+2 g α i = y i+1 α i+1 + λ i+1 g i+1 g i+2 with w i,κ i,y i,λ i K. Since g n α 0 = α 0, it foows that y 1 y n = 1. Set θ j := y j+1 y n, 1 j n 1, and θ n := 1. Define a inear isomorphism Θ : C d n C d n by p i θ i θ i+ 1 p i. In particuar Θe i = e i and Θα i = θ i α i.thenθ : C d n C d n is a coagebra map. Endow C d n = ΘC d n with the Hopf agebra structure via the given Hopf agebra structure of C d n and Θ. TheninΘC d n we have g θ i α i = Θg Θα i = Θg α i = y i+1 Θα i+1 + λ i+1 g i+1 g i+2 = y i+1 θ i+1 α i+1 + λ i+1 g i+1 g i+2. Since θ i = y i+1 θ i+1, it foows that in ΘC d we have g α i = α i+1 + λ i+1 g i+1 g i+2 with λ i+1 = λ i+1 /θ i. Assume that now in ΘC d n we have α i g = i+1 α i+1 + κ i+1 g i+1 g i+2. Since α 0 g n = α 0, it foows that 1 n = 1. However, g α i g = g α i g impies i = i+1 for each i. Write i =.Then n = 1. This competes the proof.

9 220 X.-W. Chen et a. / Journa of Agebra Lemma 3.3. Suppose that there is a Hopf agebra structure on C d n. Then up to a Hopf agebra isomorphism we have m + pi pm j j pi+j +m modc+m n for 0 i, j n 1, and for,m d 1,where K is an nth root of unity. Proof. Use induction on N := +m.forn = 0 or 1, the formua foows from Lemma 3.2. Assume that the formua hods for N N 0 1. Then for N = N 0 1wehave pi pm j = p i p m j m = pi+r r i pj+s m s j N 0 r=0 = s=0 k=0 r+s=k,0 r,0 s m p r i+r pm s j+s pr i ps j = p i pm j gi+j + g i+j+n 0 p i pm j N k=1 r+s=k,0 r,0 s m p r i+r pm s j+s pr i ps j. By the induction hypothesis for each r and s with 1 k := r + s N 0 1wehave p r i ps j jr k r pi+j k modck n and pi+r r pm s j+s N0 k j+s r p N 0 k i+j+k r mod CN0 kn. It foows that pi pm j p i pj m gi+j + g i+j+n 0 pi pm j + Σ mod C N0 kn C k n 1 k N 0 1 where

10 N 0 1 Σ = j N 0 1 = j X.-W. Chen et a. / Journa of Agebra k=1 r+s=k,0 r,0 s m N0 k=1 p N 0 k i+j+k pk i+j. s sr k r N0 k r p N 0 k i+j+k pk i+j Note that in the ast euaity the foowing identity has been used see, e.g., PropositionIV.2.3in[11]: k s sr N0 k N0 =, 0 <k<n 0. r r r+s=k Now, put X := pi pm j j N 0 pn 0 i+j. Then by the computation above we have X X g i+j + g i+j+n 0 X mod 1 k N 0 1 C N0 kn C k n. Let X = v 0 c v,wherec v is the vth homogeneous component with respect to the ength grading. Then we have c v cv g i+j + g i+j+n 0 c v v v mod 1 k N 0 1 C N0 kn C k n. Since the eements in C N0 kn C k n are of degrees stricty smaer than N 0, it foows that for v N 0 we have c v = c v g i+j + g i+j+n 0 c v. Now for each v N 0 1, note that in the right hand side of the above euaity the terms are of degree v, 0 or 0,v; but in the eft hand side if c v 0 then it reay contains a term of degree which is neither v, 0 nor 0,v. This forces c v = 0forv N 0. It foows that pi pm j = N0 j p N 0 i+j + X N0 j p N 0 i+j modcn0 n. This competes the proof. By a direct anaysis from the definition of the Gaussian binomia coefficients we have Lemma 3.4. Let 1 K be an nth root of unity of order d.then [ ] [ ] [ ] m + m + m = 0 if and ony if > 0, d d d

11 222 X.-W. Chen et a. / Journa of Agebra where [x] means the integer part of x Proof of Theorem 3.1 Assume that C d n admits a Hopf agebra structure. Let be the nth root of unity as appeared in Lemma 3.3 with order d 0. It suffices to prove d = d 0.SinceC d n has a basis pi with d 1and0 i n 1, it foows from Lemma 3.3 that m + = 0 for,m d 1, + m d. Whie by Lemma 3.4 m + = 0 if and ony if [ m + d 0 ] [ m d 0 ] [ ] > 0. d 0 Note that here we have used the assumption that K is of characteristic 0: since K is of characteristic zero, it foows that m+ can never be zero. Thus 1, and then 1 Lemma 3.4 can be appied. Take = 1andm = d 1. Then we have [d/d 0 ] [d 1/d 0 ] > 0. This means d 0 d. Let d = kd 0 with k a positive integer. If k>1, then by taking = d 0 and m = k 1d 0 we get a desired contradiction +m 0. Theorem 3.6. Assume that K is a fied of characteristic 0, containingan nth primitive root of unity. Let d n with d 2.Then i Any graded Hopf structure with ength grading on C d n is isomorphic to as a Hopf agebra some C d n, = A n,d,wherec d n, and A n,d are given as in 1.6. ii Any Hopf structure not necessariy graded on C d n is isomorphic to as a Hopf agebra some C d n,µ, = An,d,µ,,whereC d n,µ, and An,d,µ, are given as in 1.7. iii If An 1,d 1,µ 1, 1 An 2,d 2,µ 2, 2 as Hopf agebras, then n 1 = n 2, d 1 = d 2, 1 = 2. If d n,thenan, d, µ 1, An, d, µ 2,as Hopf agebras if and ony if µ 1 = δ d µ 2 for some 0 δ K, and An, n, µ 1, An, n, µ 2,for any µ 1,µ 2 K. In particuar, for each n, C d n, 1 is isomorphic to C d n, 2 if and ony if 1 = 2. Proof. i By Lemma 3.3 and by the proof of Theorem 3.1 we see that any graded Hopf agebra on C d n is isomorphic to C d n, for some root of unity of order d. ii Assume that C d n is a Hopf agebra. By Lemma 3.2 we have α 0 e 1 = e 1 α 0 + κ e 1 e1 2

12 X.-W. Chen et a. / Journa of Agebra for some primitive dth root. SetX := α κ 1 e 1. ThenXe 1 = e 1 X.Since X = e 1 X + X 1, it foows that X d = X d = d i=0 where in the ast euaity we have used the fact that d e d i X i X d i = e d X d + X d 1, i d = 0 for1 i d 1. i Since there is no non-trivia 1,e d -primitive eement in C d n, it foows that X d = µ1 e1 d for some µ K. Hence we obtain an agebra map F : An,d,µ, C d n such that Fg= e 1 and Fx= X.SinceC d n is generated by e 1 and α 0 by Lemma 3.3, it foows that F is surjective, and hence an agebra isomorphism by comparing the K- dimensions. It is cear that F is aso a coagebra map, hence a biagebra isomorphism, which is certainy a Hopf isomorphism [16]. iii If C d1 n 1,µ 1, 1 = C d2 n 2,µ 2, 2, then their groups of the group-ike eements are isomorphic. Thus n 1 = n 2, and hence d 1 = d 2 by comparing the K-dimensions. The remaining assertions can be easiy deduced. We omit the detais. Remark 3.7. The foowing exampe shows that, the assumption K is of characteristic 0 is reay needed in Theorem 3.1. Let K be a fied of characteristic 2, and et n 2 be an arbitrary integer. Then each graded Hopf agebra structure on C 2 n is given by up to a Hopf agebra isomorphism: g j α i = α i g j = α i+j, α i α j = 0, Sα i = α n i 1, S g j = g n j for a 0 i, j n 1. In fact, consider the Hopf agebra structure KZ n 1 on Z n. Its subcoagebra C 2 n is aso a subagebra, which is exacty the given Hopf agebra. On the other hand, for each graded Hopf agebra over C 2 n, the corresponding in Lemma 3.3 must satisfy 2 1 = 1 + = 0, and hence = 1. Then the assertion foows from Lemma 3.3. Remark 3.8. It is easy to determine the automorphism group of the Hopf agebra An,d,µ,:itisK {0} if µ = 0ord = n, andz d otherwise.

13 224 X.-W. Chen et a. / Journa of Agebra The Gabrie uiver and the Ausander Reiten uiver of An, d, µ, The aim of this section is to determine the Gabrie uiver and the Ausander Reiten uiver of agebra An,d,µ, = C d n,µ,,where is an nth root of unity of order d. We start from the centra idempotent decomposition of A := An,d,µ,. Lemma 4.1. The center of A has a inear basis {1,g d,g 2d,...,g n d }. Let ω K be a root of unity of order n/d. Then we have the centra idempotent decomposition 1 = c 0 + c 1 + +c t with c i = d/n t j=0 ω i g d j for a 0 i t, where t = n/d 1. Proof. By 1.7 the dimension of A is nd, thus {g i x j 0 i n 1, 0 j d 1} is a basis of A. Aneementc = a ij g i x j is in the center of A if and ony if xc = cx and gc = cg. By comparing the coefficients, we get a ij = 0 uness j = 0andd i. Obviousy, g d is in the center. It foows that the center of A has a basis {1,g d,g 2d,...,g n d }. Since t i=0 ω j i = 0 for each 1 j t, it foows that c 0 + c 1 + +c t = d n t j=0 g dj t ω j t i d = 1 + n i=0 i=0 t j=1 g dj = d t + 1 = 1; n and c i c i = d2 n 2 = d2 n 2 0 j,j t 2t g dk ω i k k=0 t = d2 n 2 g dk ω i k k=0 t = d2 n 2 g dk ω i k k=0 = d2 n 2 g dt ω i t g dj+j ω ij +i j 0 j min{k,t},0 k j t 0 j t 0 j k 0 j k ω i i j + ω i i j 2t k=t+1 t 1 g dk ω i k ω i i j + g dk ω i k t 1 k =0 ω i i j + g dk ω i k k=0 0 j t = d2 t 1 n 2 g dt ω i t δ i,i t g dk ω i k δ i,i t + 1 = t + 1 d2 n 2 δ i,i k=0 t g dk ω i k = δ i,i c i k=0 k t j t 1+k j t ω i i j ω i i j ω i i j where δ i,i is the Kronecker symbo. This competes the proof.

14 X.-W. Chen et a. / Journa of Agebra Lemma 4.2. Let B = Bd,λ, be an agebra generated by g and x with reations {g d = 1,x d = λ,xg = gx},whereλ, K, and is a root of unity of order d. i If λ = 0,thenB KZ a d /J d. ii If λ 0,thenB M d K. Proof. i Note that if λ = 0, then B Ad,d,0, = C d d, 0,,whichisad 2 -dimensiona Taft agebra. By the sef-duaity of the Taft agebras see [5, Proposition 3.8] we have agebra isomorphisms B = Ad,d,0, Ad,d,0, C d d, 0, KZ a d /J d. ii If λ 0, then define an agebra homomorphism φ : B M d K: 1 φg = 2... d 1 and φx= λ 0 Note that φ is we-defined. It is easy to check that φg and φx generate the agebra M d K. Thus φ is a surjective map. However, the dimension of B is at most d 2, thus φ is an agebra isomorphism. Now we are ready to prove the main resut of this section. Theorem 4.3. Write A = An,d,µ, and t = n/d 1. i If µ 0,thenA KZd a/j d M d K M d K with t copies of M d K. ii If µ = 0, thena KZd a/j d KZd a/j d KZd a/j d with n/d copies of KZd a/j d. Proof. By Lemma 4.1 we have A = c 0 A c 1 A c t A as agebras. Write A i = c i A. Note that c i g d = ω i c i for a 0 i t. It foows that {c i g k x j 0 k d 1, 0 j.

15 226 X.-W. Chen et a. / Journa of Agebra d 1} is a inear basis of A i.letω 0 K be an nth primitive root of unity such that ω d 0 = ω. Obviousy, as an agebra each A i is generated by ω i 0 c ig and c i x, satisfying ω i 0 c i g d = ci, c i x d = c i µ 1 g d = c i µ 1 ω i and c i x ω i 0 c ig = ω i 0 c ig c i x. Note that c i is the identity of A i. Thus we have an agebra homomorphism θ i : B d,µ 1 ω i, A i such that θ i g = ω0 i c ig and θ i x = c i x. A simpe dimension argument shows that θ i is an agebra isomorphism. Note that µ1 ω i = 0 if and ony if µ = 0ori = 0. Then the assertion foows from Lemma 4.2. Coroary 4.4. The Gabrie uiver of agebra An,d,µ, is the disjoint union of a basic d-cyce and t isoated vertices if µ 0, and the disjoint union of n/d basic d-cyces if µ = 0. Since the Ausander Reiten uiver ΓKZ a d /J d is we-known see, e.g., [1, p. 111], it foows that the Ausander Reiten uiver of An,d,µ, is cear. 5. Hopf structures on monomia agebras and coagebras The aim of is section is to cassify non-semisimpe monomia Hopf K-agebras, by estabishing a one-to-one correspondence between the set of the isocasses of nonsemisimpe monomia Hopf K-agebras and the isocasses of group data over K. Theorem 5.1. i Let A be a monomia agebra. Then A admits a Hopf agebra structure if and ony if A = k k as an agebra, or A = KZ a n /J d KZ a n /J d as an agebra, for some d 2 dividing n. ii Let C be a monomia coagebra. Then C admits a Hopf agebra structure if and ony if C = k k as a coagebra, or C = C d n C d n as a coagebra, for some d 2 dividing n.

16 X.-W. Chen et a. / Journa of Agebra Proof. By duaity it suffices to prove one of them. We prove ii. If C = C 1 C as a coagebra, where each C i = C1 as coagebras, and C 1 admits Hopf structure H 1,thenH 1 KG is a Hopf structure on C, whereg is any group of order. This gives the sufficiency. Let C be a monomia coagebra admitting a Hopf structure. Since a finite-dimensiona Hopf agebra is cofrobenius, it foows from Lemma 2.3 that as a coagebra C has the form C = C 1 C with each C i indecomposabe as coagebra, and C i = k or C i = C di n i for some n i and d i 2. We caim that if there exists a C i = k, thenc j = k for a j. Thus,ifC k k, then C is of the form C = C d1 n 1 C d n as a coagebra, with each d i 2. Otherwise, et C j = C d n for some j. Letα be an arrow in C j from x to y. Leth be the uniue group-ike eement in C i = k. Since the set GC of the group-ike eements of C forms a group, it foows that there exists an eement k GC such that h = kx. Then kα is a h, ky-primitive eement in C. But according to the coagebra decomposition C = C 1 C with C i = Kh, C has no h, ky-primitive eements. A contradiction. Assume that the identity eement 1 of GC is contained in C 1 = C d1 n. It foows from a theorem of Montgomery [13, Theorem 3.2] that C 1 is a subhopfagebra of C, and that gi 1 C di n i = C di n i gi 1 = C d1 n 1 for any g i GC di n i and for each i. By comparing the numbers of group-ike eements in gi 1 C di n i and in C d1 n 1 we have n i = n 1 = n for each i. Whie by comparing the K-dimensions we see that d i = d 1 = d for each i. Now,sinceC 1 = C d n is a Hopf agebra, it foows from Theorem 3.1 that d divides n For convenience, we ca a Hopf structure on a monomia coagebra C a monomia Hopf agebra. Note that a monomia Hopf agebra is not necessariy graded with ength grading, by Lemma iii beow. Lemma. Let C be a non-semisimpe, monomia Hopf agebra. i Let C 1 be the indecomposabe coagebra component containing the identity eement 1. Then GC 1 is a cycic group contained in the center of GC. ii There exists a uniue eement g C such that there is a non-trivia 1,g-primitive eement in C. The eement g is a generator of GC 1. iii As an agebra, C is generated by GC and a non-trivia 1,g-primitive eement x, satisfying x d = µ g d 1 for some µ K, whered = dim K C 1 /og, og is the order of g.

17 228 X.-W. Chen et a. / Journa of Agebra iv There exists a one-dimensiona K-representation χ of G such that x h = χhh x, h G, and µ = 0 if og = d note that d = oχg; and χ d = 1 if µ 0 and g d 1. Proof. i Note that C 1 is a subhopfagebra of C by Theorem 3.2 in [13]. By Theorem 5.1ii we have C 1 = Cd n as a coagebra. It foows from Lemma 3.3 that GC 1 is a cycic group. By Theorem 5.1ii we can identify each indecomposabe coagebra component C i of C with C d n. Foranyh GC with h C i, note that hα 0 is a nontrivia h, he 1 -primitive eement in C i,andα 0 h is a non-trivia h, e 1 h-primitive eement in C i. This impies that there is an arrow in C i = C d n from h to he 1, and that there is an arrow in C i from h to e 1 h. Thus by the structure of a basic cyce we have he 1 = e 1 h. Whie e 1 is a generator of GC 1. Thus, GC 1 is contained in the center of GC. ii One can see this assertion from Theorem 5.1ii by identifying C 1 with C d n, and the caimed g is exacty e 1 in C d n. iii By Theorem 3.2 in [13], as an agebra, C is generated by C 1 and GC. By the proof of Theorem 3.1ii C 1 is generated by g = e 1 and a non-trivia 1,e 1 -primitive eement x, satisfying the given reation, together with xe 1 = e 1 x with a primitive dth root of unity. iv For any h G, since both x h and h x are non-trivia h, gh-primitive eements in C note gh = hg, it foows that there exists K-functions χ and χ on G such that and x h = χhh x + χ h1 gh. We caim that χ is a one-dimensiona representation of G and χ = 0. By x h 1 h 2 = x h 1 h 2, one infers that χh 1 h 2 = χh 1 χh 2 χ h 1 h 2 = χh 1 χ h 2 + χ h 1. Since χg= and χ g = 0, it foows that χ h g = χ h for a h G. Thus, we have χ h = χ h g = χ g h = χgχ h, which impies χ = 0. Since x d = µ1 g d, it foows that one can make a choice such that µ = 0ifd = n. By x d h = χ d hh x d and x d = µg d 1 we see χ d = 1ifµ 0andg d 1.

18 X.-W. Chen et a. / Journa of Agebra In order to cassify non-semisimpe monomia Hopf K-agebras, we introduce the notion of group data. Definition 5.3. A group datum α = G,g,χ,µ over K consists of i a finite group G, with an eement g in its center; ii a one-dimensionak-representation χ of G; and iii an eement µ K, such that µ = 0ifog = oχg, and that if µ 0then χ oχg = 1. Definition 5.4. Two group data α = G,g,χ,µ and α = G,g,χ,µ are said to be isomorphic, if there exist a group isomorphism f : G G and some 0 δ K such that fg= g, χ = χ f and µ = δ d µ. Lemma 5.2 permits us to introduce the foowing notion. Definition 5.5. Let C be a non-semisimpe monomia Hopf agebra. A group datum α = G,g,χ,µ is caed an induced group datum of C provided that i G = GC; ii there exists a non-trivia 1,g-primitive eement x in C such that x d = µ 1 g d, xh= χhhx, h G, where d is the mutipicative order of χg. For exampe, Z n, 1,χ,µ with χ 1 = is an induced group datum of An,d,µ, asdefinedin1.7. Lemma 5.6. i Let C,C be non-semisimpe monomia Hopf agebras, f : C C a Hopf agebra isomorphism, and α = G,g,χ,µ an induced group datum of C. Thenfα = f G, f g, χf 1,µis an induced group datum of C. ii If α = G,g,χ,µ and β = G,g,χ,µ both are induced group data of a nonsemisimpe monomia Hopf agebra C, thenα is isomorphic to β. Thus, we have a map α from the set of the isocasses of non-semisimpe monomia Hopf K-agebras to the set of the isocasses of group data over K, by assigning each non-semisimpe monomia Hopf agebra C to its induced group datum αc. Proof. The assertion i is cear by definition. ii By definition we have G = GC = G. By definition there exists a non-trivia 1, g-eement x, and aso a non-trivia 1,g -eement x. But according to Theorem 5.1ii

19 230 X.-W. Chen et a. / Journa of Agebra such g and g turn out to be uniue, i.e., g = g = e 1 if we identify C 1 with C d n. And according to the coagebra structure of C, and of C 1 = Cd n, wehave for some δ 0,κ K. It foows that and x = δx + κ1 g x h = χhh x = χhδh x + χhκh 1 g x h = δx + κ1 g h = δχ hh x + κh 1 g and hence χ = χ and κ = 0. Thus µ 1 g d = x d = δx d = δ d µ 1 g d, i.e., µ = δ d µ, which impies that α and β are isomorphic For a group datum α = G,g,χ,µ over K, defineaα to be an associative agebra with generators x and a h G, with reations x d = µ 1 g d, xh= χhhx, h G, where d = oχg. One can check that dim K Aα = G d by Bergman s diamond emma in [3] here the condition χ d = 1ifµ 0 is needed. Endow Aα with comutipication, counit ε, and antipode S by x = g x + x 1, εx= 0, h = h h, εh = 1, h G, Sx = g 1 x, Sh = h 1, h G. It is straightforward to verify that Aα is indeed a Hopf agebra. Lemma 5.8. i For each group datum α = G,g,χ,µ over K, Aα is a non-semisimpe monomia Hopf K-agebra, with the induced group datum α. ii If α and β are isomorphic group data, then Aα and Aβ are isomorphic as Hopf agebras. Thus, we have a map A from the set of the isocasses of group data over K to the set of the isocasses of non-semisimpe monomia Hopf K-agebras, by assigning each group datum α to Aα.

20 X.-W. Chen et a. / Journa of Agebra Proof. i Since dim k Aα = G d, it foows that {hx i h G, i d} is a basis for Aα. Let {a 1 = 1,...,a } be a set of representatives of cosets of G respect to G 1. For each 1 i, eta i be the K-span of the set {a i g j x k 0 j n 1, 0 k d 1}, where n = G 1. It is straightforward to verify that Aα = A 1 A as a coagebra, and A i = Aj as coagebras for a 1 i, j. Note that there is a coagebra isomorphism A 1 = Cd n, by sending g i x j to j! p j i,wherepj i is the path starting at e i and of ength j. This proves that Aα = C d n C d n as coagebras. ii Let α = G,g,χf,δ d µ = β = f G, f g, χ, µ with a group isomorphism f : G G.ThenF: Aα Aβ given by Fx = δx, Fh = fh, h G, isa surjective agebra map, and hence an isomorphism by comparing the K-dimensions. This is aso a coagebra map, and hence a Hopf agebra isomorphism. The foowing theorem gives a cassification of non-semisimpe, monomia Hopf K-agebras via group data over K. Theorem 5.9. The maps α and A above gives a one to one correspondence between sets and {the isocasses of non-semisimpe monomia Hopf K-agebras} {the isocasses of group data over K}. Proof. By Lemmas 5.6 and 5.8, it remains to prove that C = AαC as Hopf agebras, which are straightforward by constructions A group datum α = G,g,χ,µ is said to be trivia, if G = g N, andthe restriction of χ to N is trivia. Coroary. Let α = G,g,χ,µ be a group datum over K. ThenAα is isomorphic to Aog, oχg, µ, χg KN as Hopf agebras, if and ony if α is trivia with G = g N, whereaog, oχg, µ, χg is as defined in 1.7. Proof. If α is trivia with G = g N, then α A og,o χg,µ,χg KN = α,

21 232 X.-W. Chen et a. / Journa of Agebra it foows from Theorem 5.9 that Aα = A og,o χg,µ,χg KN. Conversey, we then have α = α Aα = α A og,o χg,µ,χg KN is trivia. Remark It is easy to determine the automorphism group of Aα with α = G,g,χ,µ:itisK Γ if µ = 0, and Z d Γ if µ 0, where Γ := {f AutG fg= g,χf = χ}. Acknowedgment We thank the referee for the hepfu suggestions. References [1] M. Ausander, I. Reiten, S.O. Smaø, Representation Theory of Artin Agebras, in: Cambridge Stud. Adv. Math, vo. 36, Cambridge Univ. Press, Cambridge, [2] N. Andruskiewitsch, H.J. Schneider, Lifting of uantum inear spaces and pointed Hopf agebras of order p 3, J. Agebra [3] G.M. Bergman, The diamond emma for ring theory, Adv. Math [4] W. Chin, S. Montgomery, Basic coagebras, in: Moduar Interfaces Riverside, CA, 1995, in: AMS/IP Stud. Adv. Math., vo. 4, Amer. Math. Soc., Providence, RI, 1997, pp [5] C. Cibis, A uiver uantum group, Comm. Math. Phys [6] C. Cibis, M. Rosso, Agebres des chemins uantiue, Adv. Math [7] C. Cibis, M. Rosso, Hopf uivers, J. Agebra [8] Yu.A. Drozd, V.V. Kirichenko, Finite Dimensiona Agebras, Springer-Verag, Berin Heideberg New York Tokyo, [9] E.L. Green, Ø. Soberg, Basic Hopf agebras and uantum groups, Math. Z [10] H.L. Huang, H.X. Chen, P. Zhang, Generaized Taft agebras, Agebra Coo., in press. [11] C. Kasse, Quantum Groups, in: Grad. Texts in Math., vo. 155, Springer-Verag, New York, [12] S. Montgomery, Hopf Agebras and Their Actions on Rings, in: CBMS Reg. Conf. Ser. Math., vo. 82, Amer. Math. Soc., Providence, RI, [13] S. Montgomery, Indecomposabe coagebras, simpe comodues and pointed Hopf agebras, Proc. Amer. Math. Soc [14] D.E. Radford, On the coradica of a finite-dimensiona Hopf agebra, Proc. Amer. Math. Soc [15] C.M. Ringe, Tame Agebras and Integra Quadratic Forms, in: Lecture Notes in Math., vo. 1099, Springer- Verag, [16] M.E. Sweeder, Hopf Agebras, Benjamin, New York, [17] E.J. Taft, The order of the antipode of finite dimensiona Hopf agebras, Proc. Nat. Acad. Sci. USA