Hochschild Cohomology and the Coradical Filtration of Pointed Coalgebras: Applications

Ž .JOURNAL OF ALGEBRA 210, 535]556 1998ARTICLE NO. JA987594

Hochschild Cohomology and the Coradical Filtration ofPointed Coalgebras: Applications

Dragos Stefan

Faculty of Mathematics, Uni ersity of Bucharest, RO-70109 Bucharest 1, Romania

and

Freddy Van Oystaeyen

Department of Mathematics, Uni ersity of Antwerp, UIA, B-2610 Wilrijk, Belgium

Communicated by Wilberd ¨an der Kallen

Received November 9, 1997

In this paper we show that there is a close connection between the coradicalfiltration of a pointed coalgebra and the Hochschild cohomology of that coalgebrawith coefficients in some one-dimensional bicomodules. As an application, for agiven prime number p and an algebraically closed field k of characteristic 0, weclassify all pointed Hopf algebras of dimension p3 over k. Q 1998 Academic Press

INTRODUCTION

w xIt was noted in S3 that the Hochschild cohomology of a pointedcoalgebra and its coradical filtration are strongly related. No details aboutthis relation are proved there, so the main purpose of this paper is to studythis connection and then to apply it to the classification of pointed Hopfalgebras of dimension p3.

Ž .Let CC, D, « be a coalgebra over a field k. We recall that a nonzeroŽ .element s g CC is called a group-like element if D s s s m s . The set

Ž . Ž .of group-like elements will be denoted by GG CC . Let s , t be in GG CC .Ž . Ž .If x g CC and D x s x m s q t m x, then x will be called s , t skew-

Ž .primitive. The set of all s , t skew-primitive elements will be denoted byŽ . Ž . X Ž .PP CC . For each pair s , t g GG CC we choose a subspace PP CC suchs , t s , t

Ž . X Ž . Ž .that k s y t [ PP CC s PP CC .s , t s , t

535

0021-8693r98 $25.00Copyright Q 1998 by Academic Press

All rights of reproduction in any form reserved.

STEFAN AND VAN OYSTAEYEN536

Ž .The coradical filtration CC is defined inductively by taking CC ton ng N 0y1Ž .be the coradical of CC and CC s D CC m CC q CC m CC , for all n g N.nq1 n 0

w Ž .x Ž .If CC is pointed, then CC s k GG CC , the vector space spanned by GG CC ,0

and for any n G 1 we have a nice description of CC due to E. G. Taft andnŽ w x.R. L. Wilson see TW .

Ž .THEOREM. Let CC be a pointed coalgebra and let CC be the coradi-n ng N

cal filtration of CC.

Ž . w Ž .x Ž X Ž .a CC s k GG CC [ PP CC .[1 s , t g GG ŽCC . s , t

Ž . s , t � Ž .b If n G 2 and CC s x g CC N D x s x m s q t m x q CC mn ny14 s , tCC , then CC s Ý CC .ny1 n s , t g GG ŽCC . n

Ž . Ž .Our aim is to describe the quotient vector spaces PP CC rk s y ts , ts , t 1Žt s . 2Žt s .and CC rCC by using the cohomology spaces H k , CC and H k , CC ,nq1 n

Ž . v Žt s .respectively see Theorem 1.2 . The cohomology H k , CC is called theHochschild cohomology of CC with coefficients in t k s and was defined by

w x Ž . t sY. Doi in D . Here s , t g GG CC , and k is the bicomodule defined byr : t k s ª CC mtk s, r : t k s ªtk s m CC, mapping a ¬ s m a and a ªl r

a m t , respectively. Furthermore, if DD is a subcoalgebra of CC, thisdescription will allow us to compare the coradical filtrations of CC and DD

Ž .see Corollary 1.3 . The first section is completed by proving that theHochschild cohomology of a finite-dimensional coalgebra CC with coeffi-cients in a finite-dimensional bicomodule M and the Hochschild cohomol-ogy of the dual algebra CC* with coefficients in the transposed bimoduleM* are isomorphic. Moreover, if AA is a finite-dimensional Hopf algebra

v Ž .and M is an AA y AA bimodule, then the Hochschild cohomology H AA, Mv Ž ad . adis isomorphic to Ext k, M , where M denotes the right AA-moduleAA

Ž .structure on M induced by the adjoint action Theorem 1.5 . In the secondv Žt s .section we shall compute the Hochschild cohomology H k , TT ofp, v

Ž .Taft’s Hopf algebras TT Theorem 2.8 . This technical part of the paperp, v

will be used to prove Theorem 3.2, the key step for classifying pointed3 < Ž . <p -dimensional Hopf algebras AA with GG AA s p. The main idea is the

same as the one used in the classification of pointed Hopf algebras of2 Žw x w x.dimension 8 and p , cases that are settled S2 and S3 . That is, we shall

prove that a pointed Hopf algebra AA of dimension p3 is always generatedas an algebra by group-like and skew-primitive elements. The types ofthese Hopf algebras are described in Theorem 3.5. Finally, pointed Hopf

3 < Ž . < 2algebras of dimension p with GG AA s p are classified in Corollaries 3.7and 3.8.

HOCHSCHILD COHOMOLOGY AND CORADICAL FILTRATION 537

1. THE CORADICAL FILTRATION VIAHOCHSCHILD COHOMOLOGY

Ž . Ž .Let CC, D, « be a pointed coalgebra over a field k and let GG CC be theset of group-like elements of CC. We denote the subspace generated by the

w Ž .x Ž .set of group-like elements by k GG CC . For s , t g GG CC we defines , t s , t Ž . Ž . s , td : CC ª CC m CC, d x s t m x y D x q x m s . The kernel of dŽ . Ž . Ž .is by definition the set of s , t -skew primitive elements PP CC . Lets , t

Ž .CC be the coradical filtration of CC. Recall that for any n G 2 andn ng N

Ž .s , t g GG CC we defined

y1s , t s , tCC s d CC m CC .Ž . Ž .n ny1 ny1

It is easy to see that CC s , t is a subcoalgebra and CC : CC s , t. By then ny1 nTaft]Wilson theorem CC s Ý CC s , t.n s , t g GG ŽCC . n

Ž . Ž .In this section we prove that the quotients PP CC rk s y t ands , t

CC s , trCC can be described by using the Hochschild cohomology of CCn ny1Ž .with coefficients in some CC-bicomodules. Let M, r , r be a CC-bicomod-l r

ule via the structure maps r : M ª CC m M and r : M ª M m CC. Wel rdenote the identity map of CCmn by I . The Hochschild cohomologyn

v Ž .H M,CC of CC with coefficients in M is by definition the homology of theŽ nŽ . n. nŽ . Ž mn.complex C M, CC , d , where C M, CC s Hom M, CC andng N k

nnd f s I m f ( r y D m I ( f q ??? q y1 I m D ( fŽ . Ž . Ž . Ž . Ž .l ny1 ny1

nq1q y1 f m I ( r .Ž . Ž . r

If f : CC ª DD is a morphism of coalgebras, then M can be viewed as aŽ . Ž .DD-bicomodule via f m I ( r and I m f ( r , and f induces a naturall r

v v Ž . v Ž .map f : H M, CC ª H M, DD .Let s , t g CC be two group-like elements. We denote by t k s the

CC-bimodule, which as a vector space is k, and the structure maps areŽ . Ž .defined by r a s t m a and r a s a m s , for all a g k.l r

v Žt s .LEMMA 1.1. H k , CC can be computed as the homology of the complexŽ mn n. 0 Ž . mnCC , d , where d sends any x g k to x t y s and for x g CC ,ng N

n G 1, we take

nnd x s t m x y D m I x q ??? q y1 I m D xŽ . Ž . Ž . Ž . Ž . Ž .ny1 ny1

nq1q y1 x m s .Ž .

Žt s mn. mnProof. Let u : Hom k , CC ª CC be the bijective map sending fn kŽ . Ž .to f 1 . It is easy to see that u is an isomorphism of complexes.n ng N


Remark 1. By definition we have ds , t s d1.

THEOREM 1.2. Let CC be a pointed coalgebra. Then

Ž . Ž . Ž . 1Žt s .a PP CC rk s y t , H k , CC .s , t

Ž . s , t w 2Žt s . 2Žt s .xb CC rCC , ker H k , CC ª H k , CC , for any n G 2.n ny1 ny1

1Žt s . Ž . 1Žt s . Ž .Proof. By definition, Z k , CC s PP CC and B k , CC s k s y t ,s , t

Ž . Ž . s , t Ž . 2Žt s .so a follows. To prove b let us note that d CC ; B k , CC , ands , t Ž s , t . s , t Ž . 2Žt s . s , td CC ; CC m CC , so d x g Z k , CC for any x g CC .n ny1 ny1 ny1 n

s , t 2Žt s . Ž . w s , t Ž .xHence we can define f : CC ª H k , CC by f x s d x . Byn ny1Ž . w 2Žt s . 2Žt s .xLemma 1.1 f x g ker H k , CC ª H k , CC ; hence we have tony1

Ž .prove that f is surjective and ker f s CC . If v belongs to CC m CCny1 ny1 ny1w x w 2Žt s . 2Žt s .x w x w s , t Ž .xsuch that v g ker H k , CC ª H k , CC , then v s d x ,ny1

for a certain x g CC. Since v g CC m CC , it results that x g CC s , t ;ny1 ny1 nq1w x Ž . Ž .therefore v s f x . Take x g ker f ; then there is y g CC such thatny1

s , t Ž . s , t Ž . Ž . Ž .d x s d y . We get x y y g PP CC , so x g CC q PP CC ;s , t ny1 s , t

CC .ny1

COROLLARY 1.3. Let CC be a pointed finite-dimensional coalgebra, DD aŽ .subcoalgebra in CC, and s , t g GG CC . Suppose that there is n G 1 such that

2Žt s . s , t s , tDD s CC and H k , DD s 0. Then DD s CC .n n nq1 nq1

Proof. We have the following diagram with commutative squares andexact rows:

6 s , t 6 2 t s 6 2 t sŽ . Ž .0 DD rDD H k , DD H k , DD s 0nq1 n n

6 6

t ts , t 2 s 2 s6 6 6Ž . Ž .0 CC rCC H k , CC H k , CC .nq1 n n

2Žt s . 2Žt s .It results that the map H k , CC ª H k , CC is 0, son

CC s , trCC , H2 t k s , CC s H2 t k s , DD , DDs , trDD s DDs , trCC .Ž . Ž .nq1 n n n nq1 n nq1 n

In conclusion, dim CC s , t s dim DDs , t . Obviously, DDs , t : CC s , t , so CC s , t snq1 nq1 nq1 nq1 nq1s , tDD .nq1

We now focus on the computation of the Hochschild cohomology of aŽ .finite-dimensional coalgebra CC. Let M, r , r be a finite-dimensionall r

CC-bicomodule, r : M ª CC m M and r : M ª M m CC being the struc-l rŽ .ture maps. We shall use Sweedler’s notation, r m s Ým m m andl y1 0

Ž .r m s Ým m m , for any m g M. Recall that M can be regarded as ar 0 1CC*-bimodule via © : CC* m M ª M and £ : M m CC* ª M, maps de-

Ž . Ž .fined by a © m s Ýa m m and m £ a s Ýa m m , respectively. It1 0 y1 0follows that the dual vector space M* may also be regarded as a CC*-


Ž .Ž . Ž .bimodule with the following structures: a . f m s f m £ a andŽ .Ž . Ž .f.a m s f a © m , for any f g M*, a g CC*, m g M. The connection

nŽ . nŽ .between H M, CC and the Hochschild homology H CC*, M* of the dualalgebra CC* with coefficients in M* is explained in the following proposi-tion.

PROPOSITION 1.4. For any finite-dimensional coalgebra CC and any finite-nŽ . nŽ .dimensional CC-bicomodule M, we ha¨e H M, CC , H CC*, M* .

v Ž .Proof. The Hochschild cohomology H CC*, M* is by definition theŽ nŽ . n. nŽ .homology of the complex C CC*, M* , d , where C CC*, M* sng N

ŽŽ .mn .Hom CC* , M* andknnd f s v ( I m f y f ( v m I q ??? q y1 f ( I m vŽ . Ž . Ž . Ž . Ž .l ny1 ny1

nq1q y1 f m I ( v .Ž . Ž . r

In this relation v, v , and v denote the multiplication in CC*, the leftl rCC*-module structure of M, and, respectively, the right CC*-module struc-ture of M.

nŽ . nŽ . Ž .Ž .We define F : C M, C ª C C*, M* by F g a m ??? m a sn n 1 nŽ . mn Ž .Ž .a m ??? m a ( g. Obviously, for x g C we have a m ??? m a x s1 n 1 n

Ž .0 for all a , . . . , a g C* if and only if x s 0, so F g s 0 if and only if1 n nnŽ . nŽ .g s 0. Since dim C M, C s dim C C*, M* , it results that F is bijec-n

Ž .tive. It remains to prove that F is a map of complexes. Indeed, forn ng NnŽ .m g M, a , . . . , a g C* and g g C M, C we have1 nq1

v ( I m F g a m ??? m a mŽ . Ž . Ž .� 4Ž .l n 0 n

s F g a m ??? m a a £ mŽ . Ž . Ž .n 1 n 0

s a m F g a m ??? m a mŽ . Ž . Ž . Ž .Ý 0 y1 n 1 n 0

s a m a m ??? m a g mŽ . Ž . Ž .Ž .Ý 0 y1 1 n 0

s a m ??? m a m m g mŽ . Ž .Ž .Ý0 nq1 y1 0

s a m ??? m a ( I m g ( r m .Ž . Ž . Ž .0 nq1 l

Thusv ( I m F g s F I m g ( r . 1Ž . Ž . Ž .Ž . Ž .l n nq1 l

A similar computation proves that

F g ( I m v m I s F I m D m I ( g 2Ž . Ž . Ž . Ž .Ž .n iy1 nyi nq1 iy1 nyi

v ( F g m I s F g m I ( r . 3Ž . Ž . Ž .Ž . Ž .r n nq1 r

Ž . Ž . Ž .The proposition now follows from relations 1 , 2 , and 3 and definitions.


When CC is the coalgebra structure of a Hopf algebra we can apply thew xresults of S1 for computing its Hochschild cohomology. Let AA be a Hopf

Ž .algebra. An algebra RR is called an AA comodule-algebra if RR is a right AA

comodule via r : RR ª RR m AA such that r is an algebra map. The subalge-� Ž . 4bra x g RR N r x s x m 1 is called the subalgebra of coinvariant ele-

ments, and it is denoted by RRcoŽ AA .. If SS s RRcoŽ AA . the extension SS ; RR iscalled Hopf Galois, or AA-Galois, if the canonical map b : RR RR ªm SS

Ž . Ž .RR m AA, b x m y s xr y is bijective.Let SS ; RR be an AA-Galois extension and M an RR-bimodule. ThennŽ . Ž w x.H SS , M is a right AA-module cf. S1 . For instance, this module struc-

0Ž .ture on H SS , M may be described as follows: if a g AA and m g0Ž . X XH SS , M take elements r , . . . , r , r , . . . , r g RR such that1 n 1 nŽ n X. n Xb Ý r r s 1 m a and then define m.a s Ý r mr . Moreover,mis1 i SS i is1 i i

v Ž . v Ž .the cohomologies H SS , M and H RR, M are connected by a spectralsequence generalizing Lyndon]Hochschild]Serre spectral sequence forcohomology of groups:

Ext p k , H q SS , M « H pqq RR, M .Ž . Ž .Ž .AA

In this paper we are interested in the particular case when RR s AA andAA is regarded as an AA comodule-algebra via D: AA ª AA m AA. Then AAcoŽ AA .

Ž Ž . .s k and the extension k ; AA is AA-Galois, since b Ý xS y m y s x m y,1 2Ž .where D y s Ý y m y and S is the antipode of AA. Since k is the base1 2

qŽ .field, we have H k, M s 0 for q ) 0, so the above spectral sequencedegenerates. One gets

Ext p k , H0 k , M , H p AA, M ,Ž . Ž .Ž .AA

0Ž .for any p G 0. By definition we have H k, M s M and the right AA-module structure on M is the right adjoint action M ad, given by

Ž . Ž .m.a s Ý S a ma see the preceding paragraph . In conclusion,1 2

H p AA, M , Ext p k , M ad .Ž . Ž .AA

Let us suppose now that the right structure of M is trivial, that is,Ž . adm.a s « a m, where « is the counit of AA. In this case M is given byŽ .m.a s S a .m and will be denoted by M. Since a finite-dimensional HopfS

algebra has a bijective antipode it follows that M is an injective leftŽmodule if and only if M is an injective right module the functorS

transforming M to M is an equivalence from the category of leftS.AA-modules to the category of right AA-modules .

Throughout this paper a Hopf algebra AA will be regarded as anAA-bimodule with the left structure given by the ring multiplication and withtrivial right action. Moreover, we shall always consider AA-bimodules thatare trivial as right modules.


Since any finite-dimensional Hopf algebra has a nonzero integral, such aŽ w x.Hopf algebra is Frobenius as an algebra for details see D, Theorem 2 .

pŽ .Therefore, AA is self-injective. In particular, we get H AA, AA s 0, for anyp ) 0. Summarizing, we obtain

THEOREM 1.5. Let AA be a finite-dimensional Hopf algebra with antipodeS and let M be an AA-bimodule. Then

H p AA, M , Ext p k , M ad .Ž . Ž .AA

If the right structure of M is tri ial, M is an injecti e left AA-module and p ) 0,pŽ . pŽ .then H AA, M s 0. In particular, H AA, AA s 0, for any p ) 0.

2. HOCHSCHILD COHOMOLOGY OF TTp, v

Ž .Throughout this section we shall assume that char k s 0. Let p be aprime number and v a primitive root of order p. Recall that Taft’s Hopfalgebra TT is generated as an algebra by two elements g and x satisfyingp, v

the following relations: g p s 1, x p s 0, and xg s v gx. The coalgebrastructure of TT is defined such that g is a group-like element and x isp, v

Ž .g, 1 skew-primitive. To simplify the notation we shall write TT instead ofŽ .TT . The group GG TT is cyclic of order p, and the antipode S sends g top, v

y1 y1 � i j 4g and x to yxg . It is easy to see that the set g x N i, j s 0, p y 1 isa basis on TT, so dim TT s p2. It is well known that TT , TT * as Hopf

Ž i j. i Ž i j.algebras, since G: TT ª k, G g x s d v , and X : TT ª k, X g x s0, jd generate TT * as an algebra, G p s « , X p s 0, XG s vGX, G is a1, j

Ž .group-like element, and X is G, « skew-primitive.2Ž1 s .In this section we compute the Hochschild cohomology H k , TT ,

where s is an arbitrary group-like element in TT. This computation will beused in the next section for describing the types of pointed p3-dimensionalHopf algebras.

2Ž Ž1 s . .In view of Proposition 1.4 we have to compute H TT *, k * . TheŽ1 s .bimodule k * may be identified with k as a vector space, and it is easy

to see that, via this identification, the left and right TT *-structures areŽ . Ž .given by a © x s a s x and x £ a s a 1 x, respectively, for all x g k,

Ž1 s .a g TT *. The bimodule k * will for simplicity be denoted by k .s 1Ž . � 4Since GG TT is cyclic and generated by g there is d g 0, . . . , p y 1 such

that s s g d. Let N s v dÝ py1 vyd iGi g TT *. Some properties of thesed is0elements are studied in the next lemma.


� 4 w Ž .xLEMMA 2.1. Suppose that d, d9 g 0, . . . , p y 1 and B g k GG TT * .Then

Ž . da N N s d pv N .d d9 d, d9 d

Ž . w Ž .x Ž ydb N B s 0 if and only if there is A g k GG TT * such that v G yd.1 A s B.

Ž . Ž yd . w Ž .xc v G y 1 B s 0 if and only if there is A g k GG TT * such thatN A s B.d

Ž . py1 Ždyd9. jProof. a Since Ý v s d p we getjs0 d, d9

py1 py1dqd9 ydiyd9 j iqjN N s v v GÝ Ýd d9

is0 js0

py1 py1dqd9 Ždyd9. j ydk k ds v v v G s d pv N .Ý Ý d , d9 d

js0 ks0

Ž . py1 ydib N B s 0 m Ý b v s 0. Thus we may take A sd is0 iÝ py1 a Gi, where a is arbitrary in k and a s vyd ia y Ýi b vyd Ž jyi.,is0 i 0 i 0 js1 jfor i G 1.

Ž . Ž yd . py1 i Ž yd .c If v G y 1 B s 0 and B s Ý b G then B s v b N .is0 i 0 d

Let AA be a Hopf algebra with antipode S. If a g AA* and x g AA weŽ . Ž Ž ..define a g AA* by a y s a yS x . Recall that for any Hopf algebra AAx x

Ž .an element l g AA* is called a left integral if al s a 1 l, for all a g AA*.If AA is finite-dimensional, then there is a nonzero left integral l g AA*,

Ž .and the map u : AA ª AA*, u x s l is an isomorphism of left AA*-mod-l l xules. Here AA is regarded as a left AA*-module with the structure afforded

Ž . wby the right comodule AA, D . For details the reader is referred to Sw,xCorollary 5.1.6 .

LEMMA 2.2. If AA is an arbitrary finite-dimensional Hopf algebra andŽ .s g GG AA , then k can be embedded in AA* as a bimodule.s 1

Proof. The right structure of AA* as an AA*-bimodule is trivial byŽassumption recall that, throughout this section, a Hopf algebra is always

.regarded as a bimodule over itself with trivial right structure . Since theŽ . Žcounit « of AA* sends a to a 1 , it follows that AA* acts trivially on k tos 1

.the right . Hence it is enough to embed k into AA* as a left submodule.s 1Ž .Let u 9: k ª AA be the injective map given by u 9 a s as . We haves 1

Ž . Ž Ž . . Ž . Ž .u 9 a © a s u 9 a s a s a s as s a © u 9 a , so u 9 is AA*-linear. ThusŽ .Ž .u (u 9: k ª AA*, u (u 9 a s al is an embedding of left AA*-modules.l s 1 l s


d Ž . Ž .COROLLARY 2.3. If s s g g GG TT then is : k ª TT *, is a ss 1aN X py1 is an embedding of AA*-bimodules.d

Proof. We have to prove that l s N X py1, for a certain integrals d� py1 i. py1l g TT *. Indeed, it is easy to see that l s Ý G X is a left integralis0

Ž .of TT. Since ab s Ýa b for all a , b g TT * and a g TT, we geta a a2 1Ž .ab s a b . Thuss s s

py1i py1

l s G X .Ž . Ž .Ýs s sis0

Let a g TT. Since G is a morphism of algebras, we have

G a s G asy1 s G a G sy1 s G a G gyd s vyd G a .Ž . Ž . Ž . Ž . Ž . Ž .Ž .s

Ž . Ž . Ž . Ž .On the other hand, D X s X m G q « m X, so X ab s X a G b qŽ . Ž .« a X b , for a, b g TT. Thus

X a s X asy1 s X a G sy1 q « a X sy1 s vyd X a .Ž . Ž . Ž . Ž . Ž . Ž . Ž .s

yd yd py1In conclusion, G s v G and X s v X ; therefore l s N X .s s s d

PROPOSITION 2.4. Let s s g d be a group-like element in TT. If Q ss

Ž . 2Ž . 1Ž .TT *ris k , then H TT *, k , H TT *, Q .s 1 s 1 s

Proof. By Corollary 2.3 we have the following exact sequence of TT *-bimodules:

is0 ª k ª TT * ª Q ª 0.s 1 s

Writing the long exact sequence for Hochschild cohomology, we obtain

??? ª H1 TT *, TT * ª H1 TT *, QŽ . Ž .s

d 2 2ª H TT *, k ª H TT *, TT * ª ???Ž . Ž .s 1

1Ž . 2Ž .We finish the proof by remarking that H TT *, TT * s H TT *, TT * s 0Ž .Theorem 1.5 .

Let M be a TT *-bimodule. We recall that a k-linear map f : TT * ª M isŽ . Ž . Ž .called a k-derivation if f ab s a . f b q f a .b for all a , b g TT *. The

Ž .derivation f is called inner if there is m g M such that f a s a .m ym.a , for all a g TT *. We shall denote the space of all derivation by

Ž . Ž .Der TT *, M and the space of inner derivations by Inn TT *, M . It is wellk kknown that

Der TT *, MŽ .k1H TT *, M , .Ž .Inn TT *, MŽ .k


1Ž .Therefore, to compute H TT *, M , we have to describe the spacesŽ . Ž . Ž .Der TT *, M and Inn TT *, M . We denote by ZZ M the set of all pairsk k

Ž .m , m g M [ M satisfying the following relations:1 2

N .m s 0 4Ž .0 1

X py1.m s 0 5Ž .2

vG y 1 .m s X .m , 6Ž . Ž .2 1

Ž . Ž . Ž .and by BB M the set of pairs m , m g ZZ M such that1 2

m s G y 1 .m 7Ž . Ž .1

m s X .m. 8Ž .2

PROPOSITION 2.5. Let M be a TT *-bimodule with tri ial right action. Then1Ž . Ž . Ž .H TT *, M , ZZ M rBB M .

Ž .Proof. We shall prove that there is a k-isomorphism w : Der TT *, MkŽ . Ž Ž .. Ž .ª ZZ M such that w Inn TT *, M s BB M . Indeed, if f gkŽ .Der TT *, M , it follows easily thatk

0, if i s j s 0;¡i jy1G X . f X , if 0 F i F p y 1,Ž . Ž .i j ~f G X sŽ . 1 F j F p y 1;¢ iy1G q ??? qG q 1 . f G , if 1 F i F p y 1, j s 0.Ž . Ž .

Ž Ž . Ž .. Ž .Therefore, f G , f X g ZZ M and f is uniquely determined byŽ Ž . Ž .. Ž . Ž Ž . Ž ..f G , f X . Hence we can define the map w by w f s f G , f X .

Ž .We have already remarked that w is injective, so let m , m be a pair in1 2Ž .ZZ M . Define f : TT * ª M by

0, if i s j s 0;¡i jy1G X .m , if 0 F i F p y 1,Ž . 2i j ~f G X sŽ .

1 F j F p y 1;iy1¢ G q ??? qG q 1 .m , if 1 F i F p y 1, j s 0.Ž . 1

Ž .By a straightforward computation one shows that f g Der TT *, M andkŽ . Ž .w f s m , m .1 2

Ž .Finally, a derivation f g Der TT *, M is inner if and only if there iskŽ . Ž Ž ..m g M such that f A s A.m y m. A s A y « A .m, for any A g TT *.

Ž . Ž . Ž . Ž Ž ..In particular, f G s G y 1 .m and f X s X.m, so w Inn TT *, M skŽ .BB M .


We recall that for a primitive root of unity v g k of order n and for twonatural numbers 0 F j F i F n the v-binomial coefficients are defined as

Ž . Ž . iy1 Ž .follows: set 0 v s 1 and i v s 1 q ??? qv , for i ) 0. Define 0 v!sn n nŽ . Ž . Ž . Ž . Ž . Ž .1, and for i ) 0 take i v!s 1 v ??? i v. Then v s v s 1, v s0 n i

iŽ . Ž . Ž . Ž .0, for 0 - i - n and v s i v!r j v! i y j v!, whenever 0 F j F i - n.j

LEMMA 2.6. Let z g TT m TT be gi en by z s Ý py1 a x py j m g py j x j,js1 jp y 1Ž . Ž Ž . .Ž .where a are defined by a s 1r j v .j y 1j js0, py1 j v

Ž . Ž mn n. Ža z is a 2-cocycle in the complex TT , d see Lemma 1.1 forng N

.the definition of this complex.Ž . Ž . Ž . Ž . Ž .b g m g z s z g m g and D x z s zD x .Ž . 2Ž1 .c H ks , TT / 0.

Proof. The first two assertions may be checked by direct computation;this will be left to the reader.

Ž .c Let 0 F j F p y 1 and let TT be the k vector space spanned byj� i j 4 w xg x N i s 0, p y 1 . By R, Proposition 1 we have

d1 g i x j s g i x j m 1 q 1 m g i x jŽ .j j

j i jyr iqjyr ry g x m g x g TT m TT ,Ý Ý jyr rž /r vrs0 rs0

1Ž . py1 j � i j 4so d TT ; Ý Ý TT m TT . Since g x N i, j s 0, p y 1 is a basis onjs1 rs0 jyr rTT, the sum Ý py1 TT is direct. Hence Ý py1 Ý py1 TT m TT is direct, too.js0 j is0 js0 i j

1Ž . py1 jSuppose that z g d TT . Then, on one hand, z g Ý Ý TT m TT ,js0 rs0 jyr rpy1and on the other hand, by definition, z g Ý TT m TT , contradiction.js1 pyj j

LEMMA 2.7. Let s s g d be a group-like element in TT, where d g� 40, 1, . . . , p y 1 . Let Q denote the TT *-bimodule TT *r k .s s 1

$ $Ž . Ž . Ž .a Let 0 - d F p y 1 and q , q g TT *. If q , q g ZZ Q , then1 2 1 2 s

X X Ž . Xthere are q , q g TT * and b g k such that q s G y 1 q , q s bv N q1 2 1 1 2 dy1$ $X XˆŽ . Ž . Ž .Xq , and q , q s 0, q q BB Q , where q s q y Xq .ˆ2 1 2 s 2 1

Ž . Ž . py1b Let q , q be as in a . If 0 - d - p y 1, then X q s 0.1 2 2

Ž . Ž . Ž .c If d s p y 1, then ZZ Q s BB Q .s s

Ž . Ž dq1.y1d If 0 F d - p y 1 and A s N q 1 y v N , thend py1 d$py1ˆŽ . Ž .0, A X g BB Q .d s $ $ˆ ˆŽ . Ž . Ž . Ž . Ž .e If d s 0, then there is 0, q g ZZ Q such that 0, q f BB Q ,0 s 0 s$ˆ ˆ ˆŽ . Ž . Ž . Ž .and for any 0, q g ZZ Q there exists a g k such that 0, q s a 0, q qˆ ˆs 0

Ž .BB Q .s


$ $Ž . Ž . Ž .Proof. a By definition, q , q g ZZ Q if and only if there exist1 2 s

a, b, c g k such that

N q s aN X py1 9Ž .0 1 d

X py1q s bN X py1 10Ž .2 d

vG y 1 q s Xq q cN X py1. 11Ž . Ž .2 1 d

Ž . Ž . py1Multiplying 9 to the left by G y 1, we get a G y 1 N X s 0. SincedŽ . py1 py1 1 id / 0, it follows G y 1 N X / 0, so a s 0. Write q s Ý q X ,d 1 is0 i

1 w Ž .x 1where q g k GG TT * . Hence N q s 0 for any i, 0 F i F p y 1. It resultsi 0 i1 Ž . w Ž .x Ž Ž .in q s G y 1 q , for some q g k GG TT * see Lemma 2.1 b . Thisi i i

Ž . X X py1 iimplies that q s G y 1 q , where q s Ý q X .1 1 1 is0 ipy1 2 i Ž . py1 2Similarly, if q s Ý q X , then q satisfies 10 if and only if X q2 is0 i 2 0

s bN X py1. The last relation is equivalent to q2 s v bN ; therefored 0 dy1

py1X2 iq s v bN q q X s v bN q Xq ,Ý2 dy1 i dy1 2

is1

2 2 w Ž .x Xwhere q , . . . , q are arbitrary in k GG TT * and q g TT * is uniquely1 py1 2determined such that XqX s Ý py1 q2 X i. Obviously, if q s q y XqX , then2 is1 i 2 1$ $ $ $

X XˆŽ . Ž . ŽŽ . .q , q s 0, q q G y 1 .q , X.q , so this part of the lemma is proved.ˆ1 2 1 1

Ž . Ž .b We keep notation from the proof of a . It is enough to show thatŽ .b s 0. Then, by replacing q and q in 11 , we obtain1 2

py1X2 i py1v b vG y 1 N q vG y 1 q X s X G y 1 q q cN X .Ž . Ž . Ž .Ýdy1 i 1 d

is1

w Ž .x � py14Recall that TT * is a free left k GG TT * -module with basis 1, X, . . . , X .Since the right-hand side of the above relation can be written as a sum

py1 i w Ž .x Ž .Ý r X , r g k GG TT * , we obtain v b vG y 1 N s 0. By hypothe-is1 i i dy1Ž .sis d / p y 1, so vG y 1 N / 0. Hence b s 0.dy1 $ $

Ž . Ž . Ž .c Let q , q be elements in TT * such that q , q g ZZ Q . By the1 2 1 2 s

Ž . X X Ž . Xproof of a there are q , q g TT * and b, c g k such that q s G y 1 .q ,1 2 1 1py1 X Ž . py1q s bv N X q Xq , and vG y 1 q s Xq q cN X . Hence2 py2 2 2 1 py1

Ž . Ž . X py 1 Ž . Xv G y 1 q s X G y 1 q q cN X s v G y 1 Xq q2 1 py 1 1cN X py1. We obtainpy1

vG y 1 q y XqX s cN X py1 , 12Ž . Ž . Ž .2 1 py1

Ž py1. 2 y2so N cN X s 0. Since N s v pN / 0, it follows thatpy1 py1 py1 py1c s 0. On the other hand, by replacing q s bv N X py1 q XqX and2 py2 2


Ž .c s 0 in 12 , we obtain

bv vG y 1 N q vG y 1 X q y qX s 0.Ž . Ž . Ž .py2 2 1

Ž . � py14Thus b s 0, as v vG y 1 N / 0 and 1, X, . . . , X is a basis onpy2w Ž .xTT * as a left k GG TT * -module. Therefore we have proved that q and q1 2

Ž . Ž . Ž . Ž . Ž .satisfy relations 9 , 10 , 11 with a s b s c s 0, so q , q g ZZ TT * .1 2Ž . Ž . 1Ž . ŽSince ZZ TT * rBB TT * , H TT *, TT * s 0 see Proposition 2.5 and Theo-$ $. Ž . Ž . Ž . Ž .rem 2.4 , it results in q , q g BB TT * and then q , q g BB Q .1 2 1 2 s

Ž . py1d Multiplying both sides of the relation that defines A by X ,dwe obtain

y1py1 py1 dq1 py1A X s N X q 1 y v N XŽ .d py1 d

y1y1 py2 dq1 py1s X v N X q 1 y v N X .Ž .Ž .0 d

Ž .Ž y1 py2 .Since G y 1 v N X s 0, one obtains0

$ $ $py1 y1 py2 y1 py20, A X s G y 1 . v N X , X . v N X g BB Q .Ž . Ž .Ž . Ž .ž /ž /d 0 0 s

Ž . Ž .e By Proposition 2.5, Proposition 2.4, and Lemma 2.6 c , we get

ZZ Q rBB Q , H1 TT *, Q , H2 TT *, k , H2 1k1 , TT / 0.Ž . Ž . Ž . Ž . Ž .s s s 1 1

$ $Ž . Ž . Ž .Hence there exists q , q g ZZ Q , which does not belong to BB Q . The1 2 s s$ $ $ $

X XˆŽ . Ž .first assertion of the lemma yields a q g Q such that q , q s 0, q q0 s 1 2 0$ $X Xˆ ˆŽ . Ž . Ž . Ž . Ž .BB Q . Then 0, q g ZZ Q , 0, q f BB Q , and for some a , b g ks 0 s 0 s 0 0

we have

X py1qX s a N X py1 ,0 0 0

vG y 1 qX s b N X py1.Ž . 0 0 0

Ž .Since vG y 1 A s N , we obtain0 0

X py1 qX y b A X py1 s a N X py1 , 13Ž .Ž .0 0 0 0 0

vG y 1 qX y b A X py1 s 0. 14Ž . Ž .Ž .0 0 0

Ž X py1. Ž .If a s 0, then 0, q y b A X g ZZ TT * . By Theorem 1.5 we have0 0 0 0Ž . Ž . Ž X py1. Ž .ZZ TT * s BB TT * ; therefore 0, q y b A X g BB TT * . This implies0 0 0$$ $

X Xpy1ˆ ˆ ˆŽ . Ž . Ž . Ž . Ž . Ž .0, q y b 0, A X g BB Q , and by d it results that 0, q g BB Q ,0 0 0 s 0 sy1Ž X py1.contradiction. Thus a / 0, so take q s a q y b A X . It follows0 0 0 0 0 0$ $ $

Xy1ˆ ˆ ˆŽ . Ž . Ž . Ž . Ž . Ž .that 0, q s a 0, q q BB Q . Thus 0, q g ZZ Q _ BB Q , and we0 0 0 s 0 s spy1 py1 Ž .find that X q s N X , vG y 1 q s 0.0 0 0


ˆŽ . Ž .Let us now take 0, q to be an element in ZZ Q . A computation asˆ s

before proves that there are a, b g k such that

X py1 q y bA X py1 s aN X py1 ,Ž .0 0

vG y 1 q y bA X py1 s 0.Ž . Ž .0

py1 ˆŽ . Ž . Ž . ŽThus 0, q y aq y bA X g ZZ TT * s BB TT * . This implies 0,0 0 $$ $py1 py1ˆ ˆ ˆ. Ž . Ž . Ž . Ž .q y aq y bA X g BB Q . Finally, 0, q s a 0, q q b 0, A Xˆ0 0 s 0 0$ˆŽ . Ž . Ž .q BB Q s a 0, q q BB Q .s 0 s

THEOREM 2.8. Let s s g d be a group-like element in TT, where d g� 40, 1, . . . , p y 1 .

Ž . 2Ž .a If d / 0, then H TT *, k s 0.s 1

Ž . 2Ž .b H TT *, k is a ¨ector space of dimension 1.1 1

Ž .Proof. a In case d s p y 1 the assertion follows immediately fromŽ .Proposition 2.5, Proposition 2.4, and Lemma 2.7 c . So assume 0 - d -

Ž . Ž .p y 1. We have to prove that ZZ Q s BB Q . Indeed, if q , q g TT *s s 1 2$ $XŽ . Ž .such that q , q g ZZ Q , then, by Lemma 2.7, there is q g TT * such1 2 s 1

Ž . X py1 Xthat q s G y 1 .q and X q s 0. Furthermore, if q s q y Xq ,1 1 2 2 1$ $ ˆ ˆŽ . Ž . Ž . Ž . Ž .then q , q s 0, q q BB Q , so it is enough to prove that 0, q g BB Q .ˆ ˆ1 2 s s

Ž . Ž .By the definition of ZZ Q there is c g k such that vG y 1 q s Xq qs 2 1cN X py1. Henced

cN X py1 s vG y 1 q y Xq s vG y 1 q y vG y 1 XqXŽ . Ž . Ž .d 2 1 2 1

s vG y 1 q.Ž .

Ž . Ž .Ž py1. ŽSince vG y 1 A s N we have vG y 1 q y cA X s 0, so 0,d d d $$py1 py1ˆ ˆ ˆ. Ž . Ž . Ž . Ž .q y cA X g BB TT * . As 0, q s 0, q y cA Xpy1 q c 0, A X ,ˆd d d$ˆŽ .we conclude this part of the theorem by remarking that 0, q y cA Xpy1d

Ž . Ž .g BB Q and by using Lemma 2.7 d .s$ $ $ $

Ž . Ž . Ž . Ž .b If q , q g ZZ Q , then there is q g TT * such that q , q s1 2 s 1 2ˆŽ . Ž . Ž .0, q q BB Q . The theorem now follows by Lemma 2.7 e .ˆ s

3. POINTED HOPF ALGEBRAS OF DIMENSION p3

Let p be a prime number. In this section we shall describe the types ofpointed Hopf algebras of dimension p3 over an algebraically closed field ofcharacteristic 0. We start by recalling the following result, which can be

w xfound in S2 .


THEOREM 3.1. Let AA be a finite-dimensional Hopf algebra o¨er analgebraically closed field of characteristic 0. Then there are two naturalnumbers n, m g N, two elements g, x g AA and v g k such that

Ž .1 m ) 1 and m N n.Ž . Ž .2 g is a group-like of order n and x g PP AA .g , 1

Ž .3 v is a primiti e root of unity of order m.Ž . m � m 44 xg s v gx and x g 0, g y 1 .

Ž . � 4In particular, GG AA / 1 , and the subalgebra BB generated by g and xis a Hopf subalgebra. Let now suppose that dim AA s p3. The classificationof cosemisimple pointed Hopf algebras is trivial, because for such a Hopf

w x Ž . 3algebra AA we have AA , k GG , where GG s GG AA is a group of order p .Hence there are five types of pointed cosemisimple Hopf algebras ofdimension p3. In this section we shall always consider p3-dimensionalpointed Hopf algebras that are not cosemisimple. Let AA be such a Hopf

< Ž . < Ž .algebra. Then GG AA divides dim AA by the Nichols]Zoeller theorem and< Ž . < 3 < Ž . <GG AA / p . In conclusion, there are two possibilities: either GG AA s p

< Ž . < 2or GG AA s p .< Ž . <If GG AA s p, then n s m s p, so BB is one of Taft’s Hopf algebras

TT that we studied in the preceding section. Our main aim is to provep, v

Ž .that AA is generated as an algebra by AA , where AA is the coradical1 n ng N

filtration of AA.

THEOREM 3.2. Let AA be a pointed Hopf algebra of dimension p3. If< Ž . <GG AA s p and TT s TT is a Taft Hopf subalgebra, then TT / AA .p, v 1 1

Proof. Let us suppose that TT s AA . Let r G 1 be a natural number1 1Ž .such that TT s AA and TT / AA there is such a number as TT / AA .r r rq1 rq1

s , t Ž .Since AA s Ý AA , it follows that there are s , t g GG AA suchrq1 s , t g GG Ž AA . rq1s , t s , t Ž 1, sy1t . Ž .that AA TT . Furthermore, AA s s AA and TT s s TT ,rq1 rq1 rq1 rq1 rq1 rq1

Ž . 1, tso we may suppose that there is a t g GG AA such that AA TT . Ifrq1 rq12Ž1 t . Ž .t / 1, then H k , TT s 0 see Theorem 2.8 ; therefore, by Corollary 1.3,

we get AA1, t ; TT , contradiction. Hence AA1, 1 TT , so we may chooserq1 rq1 rq1 rq1an element y9 g AA1, 1 _ TT . Thenrq1 rq1

D y9 s y9 m 1 q 1 m y9 q u ,Ž .

2Ž1 1 . 2Ž1 1 .where u g AA m AA s TT m TT is a two cocycle. If i: H k , TT ª H k , TTr r r r rŽw x. w x 1, 1is the canonical map and i u s 0, then u g TT , contradiction. Byrq1

2Ž1 1 .Theorem 2.8 and Lemma 2.6, the space H k , TT is of dimension 1 and zŽ mn n.is a nonzero 2-cocycle of the complex TT , d , which computes theng N

Hochschild cohomology of TT with coefficients in 1k1. Hence there are1Ž . y1Ž .a g k* and y0 g TT such that u s az q d y0 . If y s a y9 q y0 , then


Ž . 1Ž .y f TT since y9 f TT and z s yd y ; that is,

D y s y m 1 q 1 m y q z .Ž .Ž . ŽTT q ky is a subcoalgebra of AA and S TT q ky s TT q ky S is the antipode

.of AA . Thus the subalgebra AA9 generated by TT q ky is a Hopf subalgebrathat strictly contains TT. It follows that dim AA9 divides p3 and is greaterthan p2. Hence AA s AA9, so AA is generated by g, x, and y, where g and xare as in Theorem 3.1.

Let us prove that TT is a normal Hopf subalgebra of AA. Let TT q be thekernel of « , the counit of TT. By definition, we have to check that TT q AA is aHopf ideal. It is easy to see that TT q AA is a Hopf ideal if and only if

q q Ž .TT AA s AATT if and only if TT is invariant with respect to the left or rightadjoint action of AA on itself, that is, defined by

ad a b s a bS a .Ž . Ž . Ž .Ý 1 2

Ž .Since AA respectively, TT is generated as an algebra by g, x, and yŽ . Ž .Ž . Ž .Ž .Ž .Ž .respectively, by g and x and ad a b9b0 s Ý ad a b9 ad a b0 , it is1 2enough to prove that

ad y g g TT 15Ž . Ž . Ž .ad y x g TT. 16Ž . Ž . Ž .

Ž . Ž . Ž . Ž .As z g m g s g m g z see Lemma 2.6 , it results that D gy y yg s 0,Ž .so gy y yg s 0. Then 15 follows from the following computation:

py1py i pyi iad y g s S y g q gy q a S x gg x s yyg q gy s 0,Ž . Ž . Ž . Ž .Ý i

is1

where we used the definitions of z and adjoint action. Analogously, by theŽ . Ž .same lemma as before, we have D x z s zD x . Hence

D xy y yx s xy y yx m g q 1 m xy y yx .Ž . Ž . Ž .Ž . Ž . Ž .Thus xy y yx g PP AA s PP TT ; TT. This implies 16 , sinceg , 1 g , 1

py1py i pyi iad y x s S y x q gx q a S x xg x s yyx q xy g TT.Ž . Ž . Ž . Ž .Ý i

is1

q wSince TT is normal, AA s AArTT AA is a Hopf algebra, and by M, Theoremx3.5 we have AA , TT m AA, isomorphisms of left TT-modules and right AA-

comodules. In particular, dim AA s p. On the other hand, AA is generatedˆ ˆ Žas an algebra by the images of g, x, and y in AA that are either 1 or 0 the

Ž . w x.image of y belongs to PP AA s 0; see S1, Proposition 1 , so AA s k,ˆ ˆ1, 1contradiction.


< Ž . <The case GG AA s p

We start by proving the following result, which holds for an arbitraryfinite-dimensional Hopf algebra AA.

ŽLEMMA 3.3. Let AA be a finite-dimensional Hopf algebra not necessarily. Ž . Ž .pointed , and let BB be a Hopf subalgebra such that GG BB s GG AA . IfŽ . Ž .g g GG AA , y g PP AA _ BB, and j is a primiti e root of unity of order ng , 1

� ny14such that yg s j gy, then the set 1, y, . . . , y is linearly independento¨er BB.

Proof. Let 1 F t F n and AA s Ýty1 BBy i. Obviously, the assertion oft is0the lemma is equivalent to the fact that dim AA s t dim BB, for any 1 F t Ftn. If t s 1, then AA s BB, so dim AA s dim BB. Let us suppose that1 1

Ž .dim AA s t dim BB. One can see easily that AA is a BB, AA -Hopf module,t tq1which in this case means that AA is a left BB-submodule and a righttq1

Ž .AA-subcomodule. By the Nichols]Zoeller theorem, BB, AA -Hopf modulesare free BB-modules, so there is r g N such that AA , BBŽ r .. Thentq1dim AA s r dim BB, and by hypothesis, dim AA s t dim BB. Since AA :tq1 t t

Ž .AA , we get r G t. If we suppose that dim AA - t q 1 dim BB, thentq1 tq1r F t. It follows that r s t and AA s AA . In particular, we deduce thattq1 t

t Ž .y g AA and t G 2 otherwise y g BB . Thust

ty1t iy s a y , 17Ž .Ý i

is0

w xwhere the coefficients a are in BB. By R, Proposition 1 we findi

s ss syj syj jD y s y m g yŽ . Ý ž /jjjs0

Ž .for 0 F s F n. Therefore, applying D to 17 , we obtain

ty1 ty1 ty1 ti t i tyj tyj ja y m g q 1 m a y q y m g yÝ Ý Ýi i ž /jž / ž / jis0 is0 js1

nty1 i i i iy j iyj js b y m c g y ,Ý Ý Ý i s i sž /jjis0 js0 ss1

Ž . niwhere D a s Ý b m c , and on the left-hand side we have alreadyi ss1 i s i sreplaced y t by Ýty1 a y i. Note that, by the induction hypothesis, Ýty1 BBy i

is0 i is0ty1 ty1 Ž i j.is direct in AA . Then Ý Ý BBy m BBy is so in AA m AA . It follows thatt is0 js0 t t

any z g AA m AA can be uniquely written as z s Ýty1 Ýty1 z , with z gt t is0 js0 i, j i, ji j Ž Ž t.. ty1 1BBy m BBy . Computing the element D y g BBy m BBy in twoty1, 1


different ways, we obtain

t ty1 ty1y m g y s 0,ž /1 j

i iy j iyj j iyj jŽ . Ž . Ž .since b y m c g y g BBy m BBy and i y j, j / t y 1, 1 forj j i s i s

tŽ .every 0 F i F t y 1, 0 F j F i. This leads to a contradiction, as / 0.j1

We are now able to classify all p3-dimensional pointed Hopf algebras AA

Ž .with GG AA , Z . For such a Hopf algebra AA we fix a Taft]Hopf subalge-pbra TT as in Theorem 3.1. Throughout this subsection g will denote a

Ž . Ž . Ž .generator of GG TT s GG AA and x will be a nonzero g, 1 skew-primitiveelement in TT such that x p s 0, g p s 1, xg s v gx, for a certain primitivepth root of unity.

LEMMA 3.4. Let AA be a pointed Hopf algebra of dimension p3. Let TT be aTaft]Hopf subalgebra generated as an algebra by g and x. There are n g� 4 Ž . yn1, . . . , p y 1 and y g PP AA _ TT such that yg s v gy.gn , 1

Proof. From Theorem 3.2 it follows that AA / TT . One can show, as in1 1w x Ž . Ž .the proof of S1, Theorem 1 , that there are s g GG AA and y g PP AAs , 1

such that y f TT and y is an eigenvector of the inner automorphism f .gn Ž .Write s s g , for 0 F n F p y 1. Since TT AA s 0, it follows that n / 0.s , 1

Suppose that j is the eigenvalue corresponding to y and that j / vyn .Ž n .y1Ž n .Take z [ 1 y jv xy y v yx . Then

D z s z m g nq1 q 1 m z q x m gy. 18Ž . Ž .

We apply the foregoing lemma to get a contradiction. The set� ny14 py1 i1, y, . . . , y is linearly independent over TT, so the sum Ý TT y isis0

Ž 3. py1 py1Ž idirect and equals AA both are of dimension p . Hence Ý Ý TT y mis0 js0j.TT y is direct, too, and equals AA m AA. It results that there are unique

i j Ž . py1 py1elements z g TT y m TT y such that D z s Ý Ý z . Let z si, j is0 js0 i, jÝt a y i, with a g TT and a / 0. If we suppose that t G 2, then we canis0 i i t

t ty1Ž . Ž .nproceed as in the proof of Lemma 3.3 to show that z s D a yty1, 1 j t1

m g ty1 y is not zero. On the other hand, the component in TT y ty1 m TT y ofz m g n q 1 m z q x m gy is zero for t G 2, contradiction. It follows thatz s a q by, a, b g TT. Computing the component in TT y m TT of both sides

Ž . Ž . n nq1 � i j 4of 18 , we get D b y m g s by m g . The set g x N i, j s 0, p y 1 is aŽ . py1 i jbasis of TT; hence D b s Ý b m g x , where b g TT. Thusi, js0 i j i j

py1n j iqn j nq1v b y m g x s by m g ,Ý i j

i , js0


j Ž � py14which implies v b s d d b recall that 1, y, . . . , y is a basis on AAi j i, 1 j, 0. Ž .over TT . It results in D b s b m g, so b s a g, where a g k. Hence the

Ž . n Ž .component in TT m TT y of D z y z m g y 1 m z is a g y 1 m gy / x mgy, a contradiction.

We can state the theorem on the classification of pointed p3-dimen-Ž . ² :sional Hopf algebras AA with GG AA , Z . Let k G, X, Y be the freep

² :algebra in the indeterminates G, X, Y. We endow k G, X, Y with aŽ .bialgebra structure, taking G to be group-like, X to be G, 1 skew-

Ž n . Ž . Ž .primitive, and Y to be G , 1 skew-primitive, D G s G m G, D X sŽ . nX m G q 1 m X, and D Y s Y m G q 1 m Y, where 0 - n - p is a

natural number. The counit « is the unique algebra morphism mappingG to 1 and X, Y to 0.

� 4If l g 0, 1 and v is a primitive root of unity of order p, we define² :AA to be the quotient bialgebra of k G, X, Y modulo the two-sidedZ , v , n , lp

� p p pideal generated by the set G y 1, X , Y , XG y vGX , YG yyn yn Ž nq1 .4v GY, YX y v XY y l G y 1 .Actually, it is easy to see that AA is a Hopf algebra with antipodeZ , v , n , lp

ˆ ˆ py1 ˆ ˆˆ py1 ˆ ˆˆn Ž py1.Ž . Ž . Ž .S given by S G s G , S X s yXG , S Y s yYG . More-over, by the Diamond lemma we have dim AA s p3.Z , v , n , lp

THEOREM 3.5. Let AA be a pointed Hopf algebra of dimension p3 suchŽ .that GG AA is cyclic of order p. Then there are v, l g k and n g N such that

� 4v is a primiti e root of unity of order p, l g 0, 1 , 0 - n - p, and AA ,AA .Z , v , n , lp

Proof. We choose the largest natural number 1 F n F p y 1 such thatŽ .there is an element y g PP AA _ TT that is an eigenvector of the innergn , 1

automorphism f . If the eigenvalue corresponding to y is j , then j s vyn ,gby the preceding lemma. On the other hand, it is easy to see that

Ž .nq1xy y vn yx g PP AA , and this element is an eigenvector of f . Ifg , 1 gŽ .n s p y 1, then xy y vn yx g PP AA s 0. If n - p y 1, by the assump-1, 1

Ž .nq1tion on n , it follows that xy y vn yx g PP TT . Thus xy y vn yx sg , 1Ž nq1 . yna g y 1 , a g k. By rescaling y we may suppose that yx y v xy sŽ nq1 . � 4l g y 1 , where l g 0, 1 . Moreover, AA is generated as an algebra by

ˆ ˆ ˆg, x, and y, and these elements have the same properties as G, X, Y fromthe very definition of AA . It follows that AA is a quotient HopfZ , v , n , lp

Ž 3.algebra of AA , so they are isomorphic both are of dimension p .Z , v , n , lp

< Ž . < 2The case GG AA s p

In this subsection we classify all pointed Hopf algebras of dimension p3

< Ž . < 2having many group-like elements, that is GG AA s p . As a matter of fact,for a prime number p, we describe the types of all pointed Hopf algebras


Ž . < Ž . <AA such that GG AA is an abelian group and dim AA s p GG AA . We firstconstruct a class of pointed Hopf algebras depending on an abelian group

� 4GG, a group-like g g GG, a character x : GG ª k*, and an element l g 0, 1 .Ž .We shall assume that x g is a primitive root of unity of order m. If

² : ² : t1 t rGG , g = ??? = g , g s g ??? g , ord g s n , and ord g s n, then we1 r 1 r i i

define AA to be the quotient algebra of the free k-algebraGG , g , x , l

² :k G , . . . , G , X modulo the two-sided ideal I spanned by the set of all1 rŽ . Ž . Ž . mG G y G G 1 F i - j F r , XG y x g G X 1 F i F r , and X yj i i j i i i

Ž m . t1 t rl G y 1 , where G s G ??? G . We endow the algebra1 r² :k G , . . . , G , X with a bialgebra structure by demanding G , . . . , G to1 r 1 r

Ž .be group-like elements and X to be G, 1 skew-primitive. It may easily bechecked that I is a two-sided coideal too, so AA is a bialgebra withGG , g , x , l

Ž .GG AA , GG. Actually, the unique antihomomorphism of algebrasGG , g , x , l

mapping G ¬ Gniy1 and X ¬ yXGny1 induces a map S: AA ªi i GG , g , x , l

AA , which is an antipode for AA . Furthermore, by the DiamondGG , g , x , l GG , g , x , l

< <lemma, dim AA s m GG .GG , g , x , l

Ž .THEOREM 3.6. Let AA be a pointed Hopf algebra. Suppose that GG AA is< Ž . <an abelian group and p is a prime number. If dim AA s p GG AA , then there

Ž . � 4 Ž .are g g GG AA , l g 0, 1 , and a character x : GG AA ª k* such that AA ,AA .GG Ž AA ., g , x , l

Ž . ² : ² :Proof. Write GG AA as a product of cyclic groups g = ??? = g ,1 r

where ord g s n . Proceeding as in the proof of Theorem 3.1, we can findi iŽ . Ž .a group-like element g g GG AA and a g, 1 skew-primitive element x that

w Ž .x Ž .is not in k GG AA . Since GG AA is abelian, we may choose x to be anŽ .eigenvector for all inner automorphisms f , s g GG AA . This means thats

Ž . Ž .there is a character x : GG AA ª k* such that xs s x s s x, for allŽ . Ž .s g GG AA . Let m be the order of x g in k*. Then, by applying Lemma

w Ž .x Ž . Ž .3.3 with BB s k GG AA , we get ord x g s p. Finally, since x g is ap Ž p .primitive root of unity of order p, it follows that x is g , 1 skew-

primitive. On the other hand, x p and g p commute, so the algebra spannedby these two elements is a commutative Hopf subalgebra of AA. We get

p w Ž .x p Ž p .x g k GG AA , hence x s a g y 1 , for a certain a g k. If a / 0, let bsatisfy the relation ab p s 1. Then, by replacing x by b x, we may choosex such that x p s g p y 1. Therefore, AA is a quotient Hopf algebra of

< Ž . <AA . Since both Hopf algebras are p GG AA -dimensional, it followsGG Ž AA ., g , x , l

that they are isomorphic.

In particular, suppose that AA is a pointed Hopf algebra of dimension p3

Ž . ² : 2such that GG AA s g is cyclic of order p . In this case the character x is0


Ž .uniquely determined by its value x g s v. Let g s g n and x g0 0Ž .PP AA be elements as in the proof of the preceding theorem. Theng n , 10

v p2 s 1 and ord vn s p, so the Hopf algebra AA is the quotientGG Ž AA ., g , x , l

² : ² p2 pof the free algebra k G, X through the two-sided ideal G y 1, X yŽ p . :l G y 1 , XG y vn GX . We denote this Hopf algebra by AA . WeZ 2, v , n , lp

have proved the following result.

COROLLARY 3.7. Let AA be a pointed Hopf algebra of dimension p3 suchŽ . � 4 2

2that GG AA , Z . Then there are l g 0, 1 , v g k*, and 1 F n F p y 1psuch that v p2 s 1, ord vn s p, and AA , AA .Z 2, v , n , lp

Finally, let us consider a pointed Hopf algebra AA such that dim AA s p3

Ž . Ž .and GG AA , Z = Z . Take x to be a g, 1 skew-primitive element that isp pan eigenvalue for all inner automorphisms f , as in the proof of Theoremg

Ž .3.6. If h g GG AA and h does not belong to the subgroup generated by g,Ž . ² : ² : Ž . Ž .then GG AA , g = h and the elements v s x g and z s x h

uniquely define the character x . Obviously, v p s z p s 1, and by theassumption on x we have v / 1. Hence, the Hopf algebra AA isGG Ž AA ., g , x , l

² : ² pthe quotient of the free algebra k G, H, X modulo the ideal G yp p :1, H y 1, X , GH y HG, XG y vGX, XH y z HX . If we denote this

Hopf algebra by AA , we haveZ =Z , v , zp p

COROLLARY 3.8. Let AA be a pointed Hopf such that dim AA s p3 andŽ . p pGG AA , Z = Z . Then there are v, z g k* such that v s z s 1, v / 1,p p

and AA , AA .AA =Z , v , zZ pp

Remark 2. Pointed Hopf algebras of dimension p3 were independentlyw x w xclassified, using different methods, in AS and CD .

REFERENCES

w xAS N. Andruskievitsch and H.-J. Schneider, Lifting of quantum linear spaces and pointedHopf algebras of dimension p3, preprint.

w x Ž .D Y. Doi, Homological coalgebra, J. Math. Soc. Japan 33 1981 , 31]50.w x 3CD S. Caenepeel and S. Dascalescu, Pointed Hopf algebras of dimension p , preprint.˘ ˘w xLR R. G. Larson and D. E. Radford, Finite dimensional cosemisimple Hopf algebras in

Ž .characteristic 0 are semisimple, J. Algebra 117 1988 , 267]289.w xM A. Masuoka, ‘‘Quotient Theory of Hopf Algebras,’’ Lecture Notes in Pure and

Applied Mathematics, Vol. 158, Marcel Dekker, New York, 1994.w xR D. E. Radford, ‘‘On Kauffman’s Knots Invariants Arising from Finite Dimensional

Hopf Algebras,’’ Lecture Notes in Pure and Applied Mathematics, Vol. 158, MarcelDekker, New York, 1994.

w xS1 D. Stefan, Hochschild cohomology of Hopf Galois extensions, J. Pure Appl. AlgebraŽ .103 1995 , 221]233.


w xS2 D. Stefan, Hopf subalgebras of pointed Hopf algebras and applications, Proc. 125Ž .1997 , 3191]3193.

w xS3 D. Stefan, Hopf algebras of low dimension, J. Algebra, to appear.w xSw M. Sweedler, ‘‘Hopf Algebras,’’ Benjamin, New York, 1969.w xTW E. J. Taft and R. L. Wilson, On antipodes in pointed Hopf algebras, J. Algebra 29

Ž .1974 , 27]32.w x Ž .Z Y. Zhu, Hopf algebras of prime dimension, Internat. Math. Res. Notices 1 1994 ,

53]59.

Hochschild Cohomology and the Coradical Filtration of Pointed Coalgebras: Applications

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