Monomial Hopf algebras over fields of positive characteristic

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1 Monomia Hopf agebras over fieds of positive characteristic Gong-xiang Liu Department of Mathematics Zhejiang University Hangzhou, Zhejiang , China Yu Ye Department of Mathematics University of Science and Technoogy of China Hefei, Anhui , China September 10, 2004 Abstract This paper can be ooked as a continues work of [3 where the authors cassified a the Hopf structures on monomia coagebras over a fied of characteristic zero and containing a roots of unity. Let k be a fied of characteristic p. We give a necessary and sufficient condition for the monomia coagebra C d (n) to admitting Hopf structure. We give a graded Hopf structure on C d (n) and construct a Hopf agebras fitration on it which wi hep us to discuss Andruskiewitsch-Schneider conjecture. At ast, we give a description of a monomia Hopf agebras. 1 Introduction There are severa works to construct neither commutative nor cocommutative Hopf agebras via uivers (see [3[4[7). 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. In [3, the authors have cassified a the finite-dimensiona Hopf structures on a monomia agebra, or euivaenty, on a monomia coagebra over a fied of characteristic zero and containing a roots of unity. As a continuous work of [3, we want to cassify Hopf structures on a monomia coagebra when the characteristic of the fied is positive. On the one hand, we note that there do exist Hopf structures on monomia coagebra when the characteristic of the fied is not zero (see Exampe 2.1 ). On the other hand, Project(No ) supported by the Natura Science Foundation of Zhejiang Province of China gxiu7969@yahoo.com.cn corresponding author yeyu@ustc.edu.cn

2 2 PRELIMINARIES 2 we note that there exists essentia difference on the monomia Hopf structures when the characteristic of the base fied is different. For exampe, we can get exampes of finitedimensiona monomia (of course pointed) Hopf agebras which can not generated by group-ike and primitive eements when characteristic of the base fied is p. But, if the characteristic is zero, we can not get such exampes (see Section 3). These facts stimuate us to write this paper. Just ike in [3, our main task is to study the Hopf structures on C d (n), where C d (n) is the sub-coagebra of path coagebra kzn c with basis the set of paths of ength stricty smaer than d (see Section 2). It turns out that the coagebra C d (n) admits a Hopf structure if and ony if there exists a d 0 -th primitive root of unity k with d 0 n and r 0 such that d = p r d 0, where p is the characteristic of k(theorem 3.4). By this concusion, we can get a Hopf agebras fitration for C d (n) which wi hep us to discuss Andruskiewitsch-Schneider conjecture. We give a the graded (with ength grading) Hopf structures on C d (n) (see Theorem 3.5). As for non-graded case, unfortunatey, we can not give them a. But we show there dose exist non-graded structures on C d (n) (see Exampe 3.1). Then, we discuss any monomia Hopf agebra H, it shows that its every indecomposabe component as coagebras is isomorphic to C d (n) (d 2) or the fied k simutaneous (see Lemma 4.1). At ast, by a theorem of Montgomery (Theorem 3.2 in [9), we can describe the structure of monomia Hopf agebras. 2 Preiminaries Throughout this paper, k denotes a fied of characteristic p. By an agebra we mean a finite-dimensiona associative k-agebra with identity eement. For competeness, we reca some definitions, notations and resuts in [3. 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, and kq c, the k-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 respective to ength grading. For α Q 1, et s(α) and t(α) denote respectivey the starting and ending vertex of α. Reca that the comutipication of the path coagebra kq c is defined by (p) = 1 β α = α α 1 s(α 1 ) α α i1 α i α 1 t(α ) α α 1 βα=p i=1 for each path p = α α 1 with each α i Q 1 ; and ε(p) = 0 for 1 and 1 if = 0 ( Note that = 0 means p is a vertex). This is a pointed coagebra. Let C be a coagebra. The set of group-ike eements is defined to be G(C) := {c C (c) = c c, c 0}

3 2 PRELIMINARIES 3 It is cear ε(c) = 1 for c G(C). For x, y G(C), 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 ε(c) = 0 for c P x,y (C). Note that k(x y) P x,y (C). An eement c P x,y (C) in non-trivia if c / k(x y). Ceary, G(kQ c ) = Q 0 and (Lemma 1.1 in [3) Lemma 2.1 For x, y Q 0, we have P x,y (C) = y(kq 1 )x k(x y) where y(kq 1 )x denotes the k-space spanned by a arrows from x to y. In particuar, there is a non-trivia 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, where R 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 [2). 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, the authors of [3 gave the definition of monomia coagebras. Definition 2.1 A subcoagebra C of kq c is caed monomia provided that the foowing conditions are satisfied: (1) C contains a vertices and arrows in Q; (2) C is contained in subcoagebra C d (Q) := d 1 i=0 kq(i) for some d 2, where Q(i) is the set of a paths of ength i in Q; (3) C has a basis consisting of paths. Consider the foowing uiver. α e n 1 0 α 0 e n 1 e 1 α n 2 e n 2 e 2 α n 3 e n 3 α 1 We denote this uiver by Z n and ca it the basic cyce of ength 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.

4 2 PRELIMINARIES 4 For each n-th root k of unity, Cibis and Rosso [4 have defined a graded Hopf agebra structure kz n () (with ength grading ) on the path coagebra kzn c by p i p m j = j m p m ij with antipode S mapping p i to ( 1) (1) 2 i p n i, where m binomia coefficient defined by 1. m is the Gaussian := (m)!! m! where! = 1, := 1 In the foowing, denote C d (Z n ) by C d (n). That is, C d (n) is the subcoagebra of kzn c with basis theset of a paths of ength stricty ess than d. Ceary, if m 0 for a 0 < m, < d and m d, then C d (n) wi be a graded sub-hopf agebra of kz n (). Exampe 2.1 Let be a d 0 -th primitive root of unity with d 0 n. Assume k. In next section (Proposition 3.3), we wi prove that if d = p t d 0 for some nonnegative integer t, then d = 0 for a 0 < < d. By a standard identity about Gaussian binomia coefficients (See [8), that is, we have m n k = n 1 k 1 k n 1 k = 0 for a 0 < m, < d and m d. Therefore, by discussion above, C p t d 0 (n) is a graded sub-hopf agebra of kz n (). We denote this Hopf agebra by C(d 0, t, n, ) Next concusion (Lemma 2.3 in [3) shows the importance of C d (n). Lemma 2.2 Let A be an indecomposabe monomia coagebra. Then A is cofrobenius (i.e. A is Frobenius) if and ony if A = k or A = C d (n) for some positive integers n and d, with d 2. The foowing emma (Lemma 3.3 in [3) is needed in our proof of Theorem 3.4. Lemma 2.3 Suppose that there is a Hopf agebra structure on C d (n). Then up to a Hopf agebra isomorphism we have p i p m j j m p m ij (mod C m (n)) for 0 i, j n 1, and for, m d 1, where k is an n-th root of unity.

5 3 HOPF STRUCTURES ON C D (N) AND ANDRUSKIEWITSCH-SCHNEIDER CONJECTURE5 3 Hopf structures on C d (n) and Andruskiewitsch-Schneider conjecture The aim of section is to give an euivaent condition for C d (n) to admitting Hopf structures (Theorem 3.4), and then cassify a the graded Hopf structures on C d (n). At ast, we wi construct a Hopf agebras fitration of C d (n) which wi hep us to discuss Andruskiewitsch- Schneider conjecture. By a direct anaysis from the definition of the Gaussian binomia coefficients we have Lemma 3.1 Let be above. Then [ [ m m d 0 pd 0 [ [ m m ( d0 pd 0 [ [ ( d 0 pd 0 m [ m [ m p 2 d 0 p 2 d 0 [ p 2 d 0 = 0 if and ony if [ m [ m p i d 0 [ p i d 0 p i d 0 ) ) > 0 Lemma 3.2 Let m > 1 be a positive integer. Then [ [ [ m m m [m p p 2 p i [ [ [ n n n ([n p p 2 p i ) [ m n ([m n p for a 0 < n < m if and ony if m = p t for t 1 p 2 p i ) > 0 Proof: For simpicity, denote [ m [m p [ [ m m p 2 p i [ [ [ n n n ([n p p 2 p i ) [ m n ([m n p p 2 p i ) by I m,n [ [ [ If Part: Ceary, m n m n 0 for a i N. So, in order to prove the p i p i p i [ [ [ concusion, it is enough to find one j N such that m n m n > 0. In fact, et [ [ [ p j p j p j j = t, 1 = p t p > n t p m n t p = 0 for a 0 < n < m. Thus, we proved the sufficiency. t

6 3 HOPF STRUCTURES ON C D (N) AND ANDRUSKIEWITSCH-SCHNEIDER CONJECTURE6 Ony if Part: Ceary, p m. At first, we caim that p m. Otherwise, assume m = kp r with k 1 and 0 < r < p. Let n = kp, then it is easy to see that I m,n = 0 now. It is contradict to the assumption. Thus, generay, et m = p r (a p a 1 p a 0 ) where r 1 and a i < p for i = 1, 2,...,. Let n = a 0 p r and then m n = p r (a p a 1 p). Then, for any 0 j r, [ [ [ m = p r j (a p j p a 1 p a 0 ), m n = p r j (a p j p a 1 p) and n = a p j 0 p r j. This [ [ [ impies m = m n n when j r. If j > r, then m = p j (a p j p j p j p (j r) a j r ) a j r 1 p j 1 a 0 p r. But, a j r 1 p j 1 a 0 p r (p 1)p j 1 (p 1)p r = [ [ p j p r < p j. Thus m = a p j p (j r) a j r, m n = a p j p (j r) a j r and [ [ [ [ n = 0. This impies m = m n n for j > r. Summarizing above discussion, p j [ [ [ p j p j p j we have m = n m n for a j and thus I p j p j p j m,n = 0. It is contradict to assumption. Therefore we know that a 0 = 0 or a i = 0 for a i 1. We caim there is ony one a i 0. In fact, if a 0 0, then above concusion asserts a i = 0 for a i 1. If a 0 = 0, then repeat above discussion shows that a 1 = 0 or a i = 0 for a i 2. So, at ast, we have a uniue a i such that m = a i p ri. If a i > 1, then we can write a i = 1 2 with Let n = 1 p ri, then it is easy to see that I m,n = 0. It is aso contradict to the assumption. Thus m = p ri and we get the desire concusion. With these preparations, we can give the foowing concusion which wi hep us to give our main resuts (Theorem 3.4) in this section. Proposition 3.3 Let k be a d 0 -th primitive root of unity. Then d = 0 for a n 0 < n < d if and ony if d = p r d 0 for some nonnegative integer r. Proof: by I m,n,. For simpicity, denote [ [ [ [ m m m m d0 pd 0 p 2 d 0 p i d 0 [ n ( d0 ( d 0 [ n pd 0 pd 0 [ n p 2 d 0 p 2 d 0 [ n If Part Simiary to the proof of Lemma 3.2, 1 = ) p i d 0 p i d 0 ) [ [ p r d 0 p r d 0 > n p r d 0 [ p r d 0 n p t = 0 for a 0 < n < p r d 0. That s to say, I d,n, = 0 for a n < d and thus d = 0 for a n 0 < n < d according to Lemma 3.1.

7 3 HOPF STRUCTURES ON C D (N) AND ANDRUSKIEWITSCH-SCHNEIDER CONJECTURE7 Ony if Part Ceary, d d 0. We caim d 0 d. If not, d = kd 0 r with k 1 and 0 < r < d 0. Let n = kd 0, then it is easy to see that I d,n, = 0 and thus d 0 by n Lemma 3.1. It is contradict to the assumption. So, we now have d d 0 is an positive integer and denote it by m. If m = 1, then d = p 0 d 0.If m > 1, then Lemma 3.1 and Lemma 3.2 assert that m = p r and thus d = md 0 = p r d 0 for some r 1. Therefore, d = p r d 0 for some r 0. Theorem 3.4 C d (n) admits a Hopf agebra structure if and ony if there exist a d 0 -th primitive root of unity k with d 0 n such that d = p r d 0 for some r 0. Proof: If Part By Proposition 3.3, d = 0 for a 0 < n < d. So, Exampe 2.1 n impies the sufficiency. Ony if Part If there is a Hopf structure on C d (n), then Lemma 2.3 impies there is a n-th root of unity k such that p i p m j j m p m ij (mod C m (n)) There is no harm to assume that is ad 0 -th primitive root of unity. Since the ength of a paths in C d (n) is stricty ess than d, m = 0 for a 0 <, m < d. In particuar, d n = 0 for a 0 < n < d. Thus, by Proposition 3.3, d = p r d 0 for r 0. Theorem 3.5 Any graded Hopf structure (with ength grading) on C d (n) is isomorphic to some C(d 0, t, n, ), where C(d 0, t, n, ) is given in Exampe 2.1. Proof: By Lemma 2.3 and the proof of Theorem 3.4 we see that any graded Hopf structure (with ength grading) is isomorphic to C(d 0, t, n, ) for some d 0 -th primitive root of unity with d 0 n and d = p t d 0. The foowing exampe wi show that there exist non-graded Hopf structures on C d (n). But, unfortunatey, we can not give a compete cassification in this case. Exampe 3.1 We give a non-graded Hopf structure on C pd0 (n). Let k be a d 0 -th primitive root of unity and d 0 n. We aso denote p i the path in Z n of ength staring at e i. Define, for s 1 d 0 r 1 pd 0 and s 2 d 0 r 2 pd 0, p s 1d 0 r 1 i p s 2d 0 r 2 j = 0 if r 1 r 2 d 0

8 3 HOPF STRUCTURES ON C D (N) AND ANDRUSKIEWITSCH-SCHNEIDER CONJECTURE8 and p s 1d 0 r 1 i p s 2d 0 r 2 j = r 1j (s 1 s 2 )d 0 r 1 r 2 s 1 d 0 r 1 p (s 1s 2 )d 0 r 1 r 2 ij if r 1 r 2 < d 0 and (s 1 s 2 )d 0 r 1 r 2 < pd 0 and p s 1d 0 r 1 i p s 2d 0 r 2 j = r 1j ((d 0)! ) p ((s 1 s 2 )d 0 r 1 r 2 )! (p (s 1s 2 p)d 0 r 1 r 2 ij p (s 1s 2 p)d 0 r 1 r 2 ijpd (s 1 d 0 r 1 )! (s 2 d 0 r 2 )! 0 ) if r 1 r 2 < d 0 and (s 1 s 2 )d 0 r 1 r 2 pd 0 with S(p i) := ( 1) (1) 2 i p n i for pd 0. This is indeed a Hopf agebra with identity eement p 0 0 = e 0 and note that it is not graded with respect to ength grading. We can see that, as an agebra, it is generated by p 0 1, p1 0 and pd 0 0. An advantage of this construction is that we have a natura basis. We can aso get this Hopf agebra through generators and reations. Let n, d 0, p, be above. We define A(n, d 0, p, ) as foows. As an agebra, it is generated by g, x, y with reations g n = 1, x d 0 = 0, y p = 1 g pd 0, xg = gx, yg = gy, yx = xy Its comutipication, counit ε and the antipode defined by (g) = g g, (x) = x 1g x, (y) = y 1g d 0 yσ d 0 1 i=1 ε(g) = 1, ε(x) = ε(y) = 0 S(g) = g n 1, S(x) = g n 1 x, S(y) = g n d 0 y 1 g i x d0 i x i (d 0 i)! (i)! Through tedious but straightforward computation, we can prove A(n, d 0, p, ) is indeed a Hopf agebra. p 0 1, x p1 0 and y pd 0 0. We can aso see that A(n, d 0, p, ) = C pd0 (n) as Hopf agebras by g Let k be a d 0 -th primitive root of unity with d 0 n, then Theorem 3.4 impies that we have a series of Hopf agebras C d0 (n) C pd0 (n) C p 2 d 0 (n) C p i d 0 (n) ( ) Noth that, if d 0 = 1, C 1 (n) is indeed not a monomia coagebra since it does not contain any arrow. But it is a Hopf agebra and ceary isomorphic to a group agebra. If d 0 2, then C d0 (n) contains a vertices and arrows. By Lemma 2.1, a group-ike and primitive eements of kz n () ie in C d0 (n). Thus any path β whose ength is not ess than d 0 can not be generated by group-ike and primitive eements since β / C d0 (n). Therefore, if d 0 2, then for any t 1, C p t d 0 can not be generated by group-ike and

9 4 ON MONOMIAL HOPF ALGEBRAS 9 primitive eements as Hopf agebras. This suppies many counter-exampes for the foowing Andruskiewitsch-Schneider Conjecture (see [1) when the characteristic of fied is positive. Andruskiewitsch-Schneider Conjecture Let H be a finite-dimensiona pointed Hopf agebra over a agebraicay cosed fied of characteristic zero, then it is generated by groupike and primitive eements. When the characteristic of k is zero, above Hopf agebras serious ( ) wi not happen. In fact, in [3, the authors have shown (See the proof of Theorem 3.1 of [3) that d = d 0, that is C d (n) = C d0 (n) now. In this case, we can not deny above conjecture since C d0 (n) is indeed generated by group-ike and primitive eements (See Theorem 3.6 in [3). 4 On monomia Hopf agebras The main aim of this section is to discuss the structures of monomia Hopf agebras. Reca that a Hopf agebra is monomia if it is monomia as coagebra. We firsty prove a resut which is simiar to Theorem 5.1 in [3. Lemma 4.1 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 = p r d 0 2 with d 0 n and there exists a d 0 -th primitive root of unity k. The proof of this emma is aso simiar to that of Theorem 5.1 in [3. For competeness, we write it out. Proof: If Part By assumption, we have C = C 1 C as a coagebra, where each C i = C1 as coagebras for 1 i and C 1 admits a Hopf structure H 1 by Theorem 3.4. Then H 1 kg is a Hopf structure on C, where G is any group of order. This gives the sufficiency. Ony if Part Let C be a monomia coagebra admitting a Hopf structure. Since a finite-dimensiona Hopf agebra is cofrobenius, it foows foows from Lemma 2.2 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, then C j = k for a j. In fact, otherwise, et C j = C d (n) for some j. Let α be an arrow in C j from x to y. Let h be the uniue groupike eement in C i = k. Since the set G(C) of the group-ike eements of C forms a group,

10 REFERENCES 10 it foows that there exists an eement g G(C) such that h = gx. Then gα if a h, gyprimitive eement in C. But according to the coagebra decomposition C = C 1 C with C i = kh, C has no h, gy-primitive eements. A contradiction. Thus, by caim above, if C k k, then C is of the form C = C d1 (n 1 ) C d (n ) as coagebras, with each d i 2. Assume that the identity eement 1 of G(C) is contained in C 1 = C d1 (n 1 ). It foows from a theorem of Montgomery (Theorem 3.2 in [9) that C 1 is a sub-hopfagebra of C, and that gi 1 C di (n i ) = C di (n i )gi 1 = C d1 (n 1 ) for any g i G(C di (n i )) and for each i. By comparing the numbers of group-ike eements in 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, since C 1 = C d (n) is a Hopf agebra, it foows Theorem 3.4, there exist a d 0 -th primitive root of unity k with d 0 n and r 0 such that d = p r d 0. Theorem 4.2 Let H be a non-semisimpe monomia Hopf agebra over k. Then there exist a d 0 -th primitive root of unity k with d 0 n, r 0 and d = p r d 0 2 such that H = C d (n) C d (n) as coagebras and H = C d (n)# σ k(g/n) as Hopf agebras, where G = G(H) and N = G(C d (n)). Proof: By Theorem 3.2 in [9 and Lemma 4.1 above, we can get this concusion directy. Acknowedgment This work was beginning when the second author was visiting at The Center of Mathematica Science of Zhejiang University. He is gratefu for the support and hep from the center. References [1 N.Andruskiewitsch,; H-J.Schneider, Pointed Hopf agebras, 2001, QA/ v1 [2 M.Ausander,; I.Reiten,; Smaφ, Representation Theory of Artin Agebras. Cambridge University Press, 1995

11 REFERENCES 11 [3 Xiao-Wu Chen, Hua-Lin Huang, Yu Ye, and Pu Zhang, Monomia Hopf agerbas, J.Agerba. 275(2004), [4 C.Cibis and M.Rosso, Hopf uivers, J.Agebra 254(2)(2002), [5 Yu.A.Drozd, V.V.Kirichenko, Finite dimensiona agebras, Springer-Verag, Berin- Heideberg-New York-Tokyo, 1993 [6 E.Green, Graphs with reations, coerings and group-graded agebras, Trans.American Math. Soc., 279(1983), [7 E.Green,.Soberg, Basic Hopf agebras and uantum groups, Math.Z. 229(1998), [8 C.Kasse, Quantum groups, GTM 155, Springer-Verag, New York, 1995 [9 S.Montgmery, Indecomposabe coagebras, simpe comodues and pointed Hopf agebras, Proc.Amer.Math.Soc.123(1995),