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Pointed Hopf Algebras of Dimensionp3
209, 622]634 Ž1998. JA987504
JOURNAL OF ALGEBRA ARTICLE NO.
Pointed Hopf Algebras of Dimension p 3 S. Caenepeel Faculty of Applied Sciences, Uni¨ ersity of Brussels, VUB, B-1050 Brussels, Belgium
and S. Dascalescu* ˘ ˘ Uni¨ ersity of Bucharest, Faculty of Mathematics, RO-70109 Bucharest 1, Romania Communicated by Susan Montgomery Received November 6, 1997
We give a structure theorem for pointed Hopf algebras of dimension p 3 , having coradical kC p , where k is an algebraically closed field of characteristic zero. Combining this with previous results, we obtain the complete classification of all pointed Hopf algebras of dimension p 3. Q 1998 Academic Press
0. INTRODUCTION AND PRELIMINARIES Recent publications have been devoted to classifying finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero. Nevertheless, except for certain small dimensions, there is only one situation in which the classification was completed, that is in the case of prime dimension p w12x. While the classification of finite-dimensional Hopf algebras of a given dimension seems to be a very difficult task, some subclasses, like semisimple or pointed Hopf algebras, have been classified for certain dimensions. For dimension p 2 it was proved in w7x that all semisimple Hopf algebras are group algebras, and it seems to be ‘‘folklore’’ that all pointed Hopf algebras are group algebras and Taft Hopf algebras Žthe idea for proving * Research supported by the FWO research network WO.011.96N and the project Hopf algebras and co-Galois theory of the Flemish Community. 622 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
POINTED HOPF ALGEBRAS OF DIMENSION
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this appeared for the first time in w9, Sect. 4.3x.. Semisimple Hopf algebras of dimension p 3 were classified in w6x by using extensions of Hopf algebras. Pointed Hopf algebras of dimension p n with a coradical that is the group algebra of an abelian group of order p ny 1 were classified in w3x, and the proof was essentially based on the famous Taft]Wilson Theorem. In particular, we know all Hopf algebras of dimension p 3 with coradical kC p 2 or k Ž C p = C p .. These can be produced by starting with one of these group algebras, adding an indeterminate by an Ore extension, and then factoring by a Hopf ideal. In this paper we complete the classification of pointed Hopf algebras of dimension p 3. The case in which the coradical is kC p is more difficult, and it turns out that there are two classes of such Hopf algebras. Both of them can be produced by Ore extension construction, using nonzero derivations for one of the two classes. For p s 2 the two classes are equal; in fact, they contain only one object. Basically, our technique consists of proving that such a Hopf algebra contains two Taft Hopf algebras, with kC p as their intersection. The examples with nonzero derivations have appeared in the literature w10, 4x as quantum groups constructed for the purpose of computing knots and 3-manifold invariants. For p s 2, the Hopf algebra that occurs was used in w4x to compute invariants of lens spaces. After we submitted this paper, we learned that the classification was also done independently in w1x and w11x. However, it seems that the three approaches are very different. Throughout, k is an algebraically closed field of characteristic 0. We use the notation of w8x. If H is a Hopf algebra, then GŽ H . will denote the group of grouplike elements of H, and H0 , H1 , . . . will denote the coradical filtration of H. H is called pointed if H0 s kGŽ H .. If g, h g GŽ H ., then Pg, h s x g H < DŽ x . s x m g q h m x 4 is the set of Ž g, h.-primitive elements. If H is finite-dimensional, there are no nonzero Ž1, 1.-primitive elements. For g, h g GŽ H . let Pg,X h be a complement of k Ž g y h. in Pg, h . We will need the version of the Taft]Wilson Theorem proved in w8, Theorem 5.4.1x. THEOREM 0.1.
Let H be a pointed Hopf algebra. Then
Ž1. H1 s H0 [ Ž[g, h g GŽ H . Pg,X h .. Ž2. For any n G 1 and c g Hn , there exist Ž c g, h . g, h g GŽ H . in H and w g Hny 1 m Hny1 such that c s Ý g, h g GŽ H . c g, h and DŽ c g, h . s c g, h m g q h m c g, h q w. The first part of the theorem shows that GŽ H . is not trivial if H is pointed, nontrivial, and finite-dimensional.
CAENEPEEL AND DASCALESCU ˘ ˘
624
For l a primitive pth root of 1, denote by Tl the Taft Hopf algebra of dimension p 2 with generators c and x such that c p s 1, D Ž c . s c m c,
xc s l cx
x p s 0,
D Ž x . s c m x q x m 1,
« Ž c . s 1,
« Ž x . s 0.
There are p y 1 such nonisomorphic Taft Hopf algebras. We will use the q-binomial coefficients and the q-binomial formula as in w5, Chap. 4x.
1. CORADICAL kC p Throughout this section H will be a Hopf algebra of dimension p 3 with CoradŽ H . s kC p . LEMMA 1.1. There exist c g GŽ H ., x g H, and l, a primiti¨ e pth root of 1, such that xc s l cx and DŽ x . s c m x q x m 1. The Hopf subalgebra T generated by c and x is a Taft Hopf algebra. Proof. The Taft]Wilson theorem ensures the existence of some c g GŽ H . such that P1, c / k Ž1 y c .. If f : P1, c ª P1, c is the map defined by f Ž a. s cy1 ac for every a g H, then f p s Id, so P1, c has a basis of eigenvectors for f . Let x be such an eigenvector, which is not in k Ž1 y c ., and let l be the corresponding eigenvalue. If l s 1, then the Hopf subalgebra T generated by c and x is commutative, and hence involutory. Now the Larson]Radford Theorem states that a Hopf algebra over an algebraically closed field of characteristic zero is cosemisimple if and only if it is involutory. So T is cosemisimple of dimension bigger than p, and this is in contradiction to CoradŽ H . s kC p . We conclude that l / 1, which ends the proof of the first statement. From the fact that DŽ x . s c m x q x m 1, it easily follows that x p is Ž1, 1.-primitive, and x p s 0. This shows that T is a Taft Hopf algebra. In this section T will be the Hopf subalgebra generated by c and x. The Ž n q 1.th term Tn in the coradical filtration of T is the subspace spanned by all c i x j with j F n. In particular, Tpy 1 s T. In the sequel, we will assume that the sum over an empty family is zero. LEMMA 1.2.
Let a g H be such that py1 ny1
D Ž a. s g m a q a m 1 q
Ý Ý ¨i, j m ci x j is0
js0
Ž 1.
POINTED HOPF ALGEBRAS OF DIMENSION
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625
for some g g GŽ H ., n F p, and ¨ i, j g H. Then ny1
D Ž a q ¨ 0, 0 . s g m Ž a q ¨ 0, 0 . q Ž a q ¨ 0, 0 . m 1 q
Ý ¨ 0, j m x j , Ž 2. js1
and for all 1 F r F n y 1, D Ž ¨ 0, nyr . s g m ¨ 0 , nyr q ¨ 0, nyr m c ny r ry1
q
Ý is1
ž
nyrqi i
/
l
¨ 0 , nyrqi m c ny r x i
Ž 3.
Proof. We have that
Ž D m I . D Ž a. s g m g m a q g m a m 1 q a m 1 m 1 py1 ny1
q
Ý Ý is0
py1 ny1
¨i, j m ci x j m 1 q
js0
Ý Ý DŽ ¨i, j . m ci x j is0
js0
and
Ž I m D . D Ž a. s g m g m a q g m a m 1 q a m 1 m 1 py1 ny1
q
Ý Ý is0
js0
py1 ny1
q
g m ¨i, j m ci x j j
Ý Ý Ý is0
js0 ss0
j s
ž/
l
¨ i , j m c iqjys x s m c i x jys .
Looking at the terms with 1 on the third tensor position, we find py1 ny1
ny1
Ý Ý ¨ i , j m c i x j q D Ž ¨ 0, 0 . s g m ¨ 0, 0 q Ý ¨ 0, j m x j q ¨ 0, 0 m 1, is0
js0
js1
and now we obtain Ž2., since py1 ny1
Ý Ý ¨ i , j m c i x j s D Ž a. y g m a y a m 1. is0
js0
Looking at the terms with x ny r on the third position, we find Ž3.. The key step is to show that there exist skew-primitives that are not in T.
CAENEPEEL AND DASCALESCU ˘ ˘
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LEMMA 1.3. dimŽ H1 . ) 2 p. Proof. Suppose dimŽ H1 . F 2 p. Since T1 : H1 , we have H1 s T1. In particular, dimŽ Pu, ¨ . s 2 only for ¨ s cu. Step 1. We prove by induction on n F p y 1 that Hn s Tn . Assume that Hny 1 s Tny1 and Hn / Tn , and pick some h g Hn y Tn . Write h s Ý u, ¨ g GŽ H . h u, ¨ as in the Taft]Wilson Theorem, and pick some h u, ¨ g Hn y Tn . Denoting g s uy1 ¨ , we have that a s uy1 h u, ¨ g Hn y Tn and py1 ny1
D Ž a. s g m a q a m 1 q
Ý Ý ¨i, j m ci x j, is0
js0
with ¨ i, j g Tny1. Let b s a q ¨ 0, 0 g Hn y Tn . Lemma 1.2 shows that DŽ ¨ 0, ny1 . s g m ¨ 0, ny1 q ¨ 0, ny1 m c ny 1. If g / c n, we have ¨ 0, ny1 g H0 , and then DŽ b . g H0 m H q H m Hny2 , which is a contradiction, since b f Hny 1. Hence g s c n and ¨ 0, ny1 s a Ž c n y c ny1 . q b c ny 1 for some a , b g k, b / 0. We have that D Ž b . y c n m b y b m 1 y a Ž c n m x ny 1 y c ny 1 m x ny 1 . y b c ny 1 x m x ny 1 g H m Hny 2 q H0 m H.
Ž 4.
Since
Ž D Ž x ny 1 . y c n m x ny 1 y x ny 1 m 1 . q Ž c n m x ny 1 y c ny 1 m x ny 1 . g H m Hny 2 q H0 m H and
Ž D Ž x n . y c n m x n y x n m 1 . y n1 c ny1 x m x ny1 g H m Hny 2 ,
ž /
l
relation Ž4. implies that b9 s b q a x ny 1 y Ž n1 .ly1b x n satisfies DŽ b9. y c n m b9 y b9 m 1 g H m Hny 2 q H0 m H. Therefore b9 g Hny1 s Tny1 and b g T l Hn s Tn , providing a contradiction. Step 2. We have from Step 1 that H py 1 s Tpy1 s T / Hp . By using the Taft]Wilson Theorem and Lemma 1.2 as in Step 1, we find some py 1 b g H p y T with DŽ b . s 1 m b q b m 1 q Ý js1 ¨ j m x j for some ¨ j g T p Žnote that we need here c s 1.. We use induction to show that for any 1 F m F p there exists bm g Hp y T such that pym
D Ž bm . s 1 m bm q bm m 1 q
Ý js1
for some wj g T, a j g k.
py1
wj m x j q
Ý jspym
a j c j x pyj m x j Ž 5 .
POINTED HOPF ALGEBRAS OF DIMENSION
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627
For m s 1, we see again as in Step 1 that ¨ py 1 s a Ž1 y c py 1 . q b c py 1 x for some a , b g k, b / 0. Observe that
Ž 1 y c py 1 . m x py 1 q Ž D Ž x py 1 . y 1 m x py 1 y x py 1 m 1 . g
T m Tj .
Ý jFpy2
Applying Lemma 1.2 to a s b q a x py 1 Žin the case n s p y 1., we obtain a b1 as wanted. Assume that we have found bm for some 1 F m F p y 1 satisfying Ž5.. Applying relation Ž3. to bm and r s m, we obtain D Ž ¨ 0 , pym . s D Ž a pym c py m x m . m
s a py m
Ý is0
m i
ž /
l
c py i x i m c pym x myi
s 1 m a py m c py m x m q a pym c py m x m m 1 my1
q
Ý is1
pymqi i
ž
/
l
a py mqi c py mqi x myi m c pym x i
and my1
Ý is1
m i
ž /a l
py m c
my1
s
Ý is1
ž
py mqi
x myi m c pym x i
pymqi i
/
l
a py mqi c py mqi x myi m c pym x i ,
which implies that m i
ž /a l
py m
s
pymqi i
ž
/
l
a pymqi
for every 1 F i F m y 1. For r s m q 1 relation Ž3. gives D Ž wpy my1 . s 1 m wpymy1 q wpymy1 m c py my1 m
qÝ is1
ž
mc
pymqiy1 i
py my1
i
x.
/
l
a py mqiy1 c py mqiy1 x myi
Ž 6.
CAENEPEEL AND DASCALESCU ˘ ˘
628
On the other hand, we have D Ž c py my1 x mq 1 . s 1 m c py my1 x mq 1 q c py my1 x mq 1 m c py my1 m
qÝ is1
ž
mq1 i
/
l
c py mqiy1 x myi m c pymy1 x i .
We obtain the following identities after we apply Ž6. with i replaced by i y 1 Žfirst equality. and some elementary computation with l-factorials Žsecond equality.:
ž
pymqiy1 i s s
ž ž
/
l
a py mqiy1
pymqiy1 i pym 1
/ž l
m iy1
/ž /ž l
mq1 1
y1
/ ž l
l
pymqiy1 iy1
mq1 i
/a l
y1
/
l
a py m
py m
We obtain that D wpy my1 y
ž
ž
pym 1
/ž l
mq1 1
y1
/
l
c py my1a pym x mq1 g Pc py my 1 , 1 .
/
As c py m / 1, we find wpy my1 y
ž
pym 1
/ž l
mq1 1
y1
/
l
c py my1 x mq1 s a Ž 1 y c pymy1 .
for some a g k. Since
Ž 1 y c py my1 . m x py my1 q Ž D Ž x py my1 . y 1 m x py my1 y x py my1 m 1 . g
T m Tj ,
Ý j-pymy1
we can apply Lemma 1.2 to bm q a x py my1 and get a bmq1 satisfying Ž5.. Step 3. Take a b s bp satisfying Ž5.: py1
DŽ b. s 1 m b q b m 1 q
Ý
a j c j x pyj m x j .
Ž 7.
js1
It follows easily that DŽ bc y cb . s c m Ž bc y cb . q Ž bc y cb . m c, and bc s cb. Furthermore D Ž bx y xb . q Ž a py 2 q l py 1a py1 y a py1 y l2a py2 . x 2 m x py 1 g H0 m H q H m Hpy2 .
Ž 8.
POINTED HOPF ALGEBRAS OF DIMENSION
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Applying Ž6. with i s 1, m s 2, we obtain
Ž l q 1 . a py 2 s
l py 1 y 1 ly1
a py1
and
Ž l2 y 1 . a py 2 y Ž l py 1 y 1 . a py1 s 0. We see that the coefficient of x 2 m x py1 is 0 in Ž8. and bx y xb g H py 1 s T. Clearly, d s bx y xb g Tqs T l KerŽ « .. The relations bc s cb, bx y xb s d , and Ž7. show that the algebra generated by c, b, and x is a Hopf subalgebra of H, so it is the whole of H by the Nichols]Zoeller Theorem w8, Corollary 3.2.1x. They also show that Tq H s HTq, which means that T is a normal Hopf subalgebra. Hence we have an extension of Hopf algebras T ª H ª HrTq H. But in HrTq H we have ˆ c s ˆ1, ˆ x s 0, and ˆ b is Ž1, 1.-primitive, thus 0. We obtain HrTq H , k, and then dimŽ H . s dimŽT ., which provides a final contradiction. At this point we know that there are two different cases: 3 F dimŽ P1, c ., or there exists g / c such that 2 F dimŽ P1, g .. In the first case let us pick some y g P1, c y kx such that yc s m cy for some primitive pth root m of 1 Žrecall that P1, c has a basis of eigenvectors for the conjugation by c .. In the second case pick y g P1, g such that yg s m gy for some m / 1. Write g s c d Žin the first case we will take d s 1.. LEMMA 1.4. The set c q x i y j <0 F q, i, j F p y 14 is a basis of H. Proof. We prove by induction on 1 F n F 2 p y 2 that the set Bn s c q x i y j <0 F q, i, j F p y 1 and i q j F n4 is linearly independent. For n s 1 this follows from the Taft]Wilson Theorem. Suppose that Bn is linearly py 1 q i j independent and take Ý qs 0 Ý iqj F nq1 a q, i, j c x y s 0. Applying D, we find py1
Ý
i
Ý
j
Ý Ý aq, i, j
qs0 iqjFnq1 ss0 ts0
ž si /
l
j t
ž/
m
l s dŽ jyt . c qqiysqdŽ jyt . x s y t
m c q x iys y jyt s 0. Fix some triple Ž q0 , i 0 , j0 . with i 0 q j0 s n q 1 and assume i 0 / 0 Žotherwise j0 / 0 and we proceed in a similar way.. Take f g H * mapping c q 0qi 0y1 qd j0 x to 1 and any other element of Bn to 0, and c g H * mapping c q 0 x i 0y1 y j 0 to 1 and the rest of Bn to 0. Applying f m c , we find that a q 0 , i 0 , j 0 s 0. Now all of the a q, i, j are zero by the induction hypothesis.
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Bn spans Hn for e¨ ery 0 F n F 2 p y 2.
COROLLARY 1.5.
Proof. We prove by induction on 1 F n F 2 p y 2 that Bn y Bny1 : Hn y Hny1. This is clear for n s 1. For the induction step, pick x i y j g Bnq 1 y Bn . Then j
i
DŽ x i y j . s
Ý Ý ss0 ts0
ž si /
l
j t
ž/
m
l s dŽ jyt . c iysqdŽ jyt . x s y t m x iys y jyt
g H 0 m H q H m Hn , and x i y j g Hnq 1. Assuming again that i / 0, choose f , c g H * such that f Ž c iq d jy 1 x . s c Ž x iy 1 y j . s 1, f Ž B0 . s 0, and c Ž Bny 1 . s 0. Then Ž f m c .Ž H0 m H q H m Hny1 . s 0, while Ž f m c .Ž DŽ x i y j .. s 1i l d j / 0, and this shows that x i y j f Hn .
ž/
l
We now define some Hopf algebras. If l is a primitive pth root of 1 and 1 F i F p y 1 an integer, we denote by H Ž l, i . the Hopf algebra with generators c, x, y defined by c p s 1,
x p s y p s 0,
D Ž c . s c m c,
xc s l cx,
D Ž x . s c m x q x m 1,
yc s lyi cy,
yx s lyi xy
D Ž y . s c i m y q y m 1.
We also denote by Hd Ž l. the Hopf algebra with generators c, x, y defined by c p s 1, x p s y p s 0, xc s l cx, yc s ly1 cy, D Ž c . s c m c,
yx s ly1 xy q c 2 y 1
D Ž x . s c m x q x m 1,
D Ž y . s c m y q y m 1.
Note that for p s 2 the only Hopf algebra of the second type, Hd Žy1., is equal to H Žy1, 1.. Remark 1.6. Suppose that p is odd, and let q be a primitive pth root of unity. The referee kindly pointed out to us that Hd Ž q 2 . is the Frobenius]Lusztig kernel Uq Žsl 2 .9 with the canonical generators cy1 , c, xcy1 , and y. THEOREM 1.7. Let H be a Hopf algebra of dimension p 3 with CoradŽ H . s kC p . Then H is isomorphic either to some Hd Ž l. or to some H Ž l, i .. Proof. We first consider an odd prime p. We distinguish the following cases. Case 1.
x, y g P1, c . Then yx g H2 , and since B2 spans H2 , we have ny1
yx s
Ý Ž a i c i x 2 q bi c i xy q g i c i y 2 . q d is0
Ž 9.
POINTED HOPF ALGEBRAS OF DIMENSION
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631
for some a i , bi , g i g k, d g H1. Applying D and replacing everywhere yx by the right-hand side in Ž9., we find ny1
Ý Ž a i c 2 m c i x 2 q bi c 2 m c i xy q g i c 2 m c i y 2 q a i c i x 2 m 1 is0
qbi c i xy m 1 q g i c i y 2 m 1 . q c 2 m d q d m 1 q m cy m x q cx m y ny1
s
Ý ai
ž
is0
c iq2 m c i x 2 q c i x 2 m c i q
ny1
Ý gi
q
ž
is0
2 1
c iq1 x m c i x
ž/ ž/
c iq2 m c i y 2 q c i y 2 m c i q
l
2 1
m
/
c iq1 y m c i y
/
ny1
Ý bi Ž c iq2 m c i xy q c i xy m c i q c iq1 y m c i x q l c iq1 x m c i y .
q
is0
q DŽ d . Since B2 is linearly independent, we get a i s g i s 0 for all i Žlooking at the coefficients of c iq1 x m c i x and c iq1 y m c i y ., bi s 0 for any i / 0 Žlooking at the coefficient of c iq2 m c i xy ., b 0 l s 1 Žlooking at the coefficient of cx m y ., m s b 0 Žlooking at the coefficient of cy m x ., and DŽ d . s c 2 m d q d m 1. Thus m s ly1 and d g P1, c 2 s k Ž c 2 y 1.. If d s 0, then H , H Ž l, 1.. If d / 0, then H , Hd Ž i .. Case 2. x g P1, c , y g P1, g , where g s c d / c. Writing again yx as in Ž9. and applying D, we find ny1
Ý Ž a i cg m c i x 2 q bi cg m c i xy q g i cg m c i y 2 q a i c i x 2 m 1 is0
qbi c i xy m 1 q g i c i y 2 m 1 . q cg m d q d m 1 q m cy m x q gx m y ny1
s
Ý ai is0
ž
c iq2 m c i x 2 q c i x 2 m c i q
ny1
q
Ý gi is0
ž
2 1
c iq1 x m c i x
ž/ ž/
cig 2 m ci y2 q ci y2 m ci q
l
2 1
m
/
c i gy m c i y
/
ny1
q
Ý bi Ž c iq1 g m c i xy q c i xy m c i q c iq1 y m c i x q ld c i gx m c i y . is0
q DŽ d . .
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CAENEPEEL AND DASCALESCU ˘ ˘
Since B2 is linearly independent, we obtain that a i s g i s 0 for any i, bi s 0 for any i / 0, b 0 s m , l db 0 s 1, and DŽ d . s d m 1 q cg m d . Thus m s lyd and d g P1, c dq 1 s k Ž c dq 1 y 1.. If d s 0, then H , H Ž l, i .. If d / 0, then yxc s Ž b 0 xy q d . c s b 0 m xcy q d c s lb 0 m cxy q d c, and yxc s l ycx s lm cyx s lm c Ž b 0 xy q d . s lmb 0 cxy q lm c d . These show that lm s 1, and this implies d s 1, which is impossible. If p s 2, then x 2 s y 2 s 0 and B2 s c q, c q x, c q y, c q xy <0 F q F p y 14 , so in Eq. Ž9. we consider from the beginning that a i s g i s 0, and the proof works with the same computations. The number of types of Hopf algebras of the form Hd Ž l. and H Ž l, i . has been evaluated in w2, Sect. 6x. PROPOSITION 1.8. Ž1. H Ž l, i . , H Ž m , j . if and only if either Ž l, i . s Ž m , j . or m s ly1 and ij ' 1 Žmod p .. Ž2. Hd Ž l. , Hd Ž m . if and only if l s m. Ž3. If p is odd, then no one of the Hd Ž m . is isomorphic to any H Ž l, i .. COROLLARY 1.9. If p is odd, then there exist Ž p y 1. 2r2 types of Hopf algebras of the form H Ž l, i . and p y 1 types of the form Hd Ž l.. If p s 2, then there is only one Hopf algebra of the form H Ž l, i ., namely, H Žy1, 1., and it is equal to Hd Žy1.. 2. THE CLASSIFICATION Let H be a pointed Hopf algebra of dimension p 3. We have seen that the coradical of H cannot be of dimension 1. By the Nichols]Zoeller Theorem we have dimŽCoradŽ H .. g p, p 2 , p 34 . If dimŽCoradŽ H .. s p, then CoradŽ H . s kC p , and this case was discussed in Section 1. If dimŽCoradŽ H .. s p 2 , then CoradŽ H . s kC p 2 or CoradŽ H . s k Ž C p = C p ., and the classification in these cases was made in w3x, where pointed Hopf algebras H of dimension p n with abelian GŽ H . of order p ny 1 are classified. To present this classification we first introduce some Hopf algebras. 2 For every l / 1 and i such that l p s 1 and li is a primitive pth root of 1, Hp 2 Ž l, i . is the Hopf algebra with generators c and x and 2
c p s 1, D Ž c . s c m c,
xc s l cx
x p s 0,
D Ž x . s c m x q x m 1. i
POINTED HOPF ALGEBRAS OF DIMENSION
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633
Two such Hopf algebras H p 2 Ž l, i . and H p 2 Ž m , j . are isomorphic if and only if there exists h not divisible by p such that m s l h and i ' hj Žmod p 2 . w3, Lemma 5x. If l is a primitive pth root of 1, we denote by H Ž l. the Hopf algebra with generators c and x defined by 2
c p s 1, D Ž c . s c m c,
x p s c p y 1,
xc s l cx
D Ž x . s c m x q x m 1.
Then H˜Ž l. , H˜Ž m . if and only if l s m w3, Lemma 6x. The classification in this case is now a consequence of w3, Theorem 2x. PROPOSITION 2.1.
Let H be a pointed Hopf algebra of dimension p 3.
Ži. If CoradŽ H . s k Ž C p = C p ., then H , Tl m kC p , where Tl is one of the Taft Hopf algebras, and there exist p y 1 types of such Hopf algebras. Žii. If CoradŽ H . s kC p 2 , then H is isomorphic either to some Hp 2 Ž l, i . or to a H˜Ž l., and there are 3Ž p y 1. types of such Hopf algebras. Finally, if dimŽCoradŽ H .. s p 3 , then H is the group algebra of one of the following groups: C p = C p = C p , C p 2 = C p , C p 3 , G1 s C p 2 i C p , G 2 s C p i C p 2 , where G1 , G 2 are the two types of nontrivial semidirect products. We now have the complete classification of pointed Hopf algebras of dimension p 3. THEOREM 2.2. Let H be a pointed Hopf algebra of dimension p 3. Then H is isomorphic to one of the following: Hd Ž l., H Ž l, i ., H˜Ž l., Tl m kC p , where l is a primiti¨ e pth root of 1, Hp 2 Ž l, i . for some l / 1 and i such that 2 l p s 1, and li is a primiti¨ e pth root of 1, k Ž C p = C p = C p ., k Ž C p 2 = C p ., k Ž C p 3 ., k Ž G1 ., k Ž G 2 .. If p is odd, then there are Ž p y 1.Ž p q 9.r2 q 5 such types. If p s 2, then there are 10 types. Note added in proof: Combining our results with the results in w6x, w7x, and w12x, we find the complete classification of all Hopf algebras of dimension p 3 that have a Hopf algebra as coradical Ž p odd.. Indeed, if the coradical of H has dimension p or p 2 , then it is a group algebra, because it is semisimple, and because of the classification results in w12x and w7x, and it follows that H is pointed. If the coradical of H has dimension p 3 , then H is semisimple. In w6, Theorem 3.1x it is proved that there are p q 8 semisimple Hopf algebras of dimension p 3 . The intersection of the two classifications consists of the five group algebras, and we conclude that there are in total Ž p 2 q 10 p q 7.r2 Hopf algebras of dimension p 3 with a Hopf algebra filtration. This does not give the complete classification of all Hopf algebras of dimension p 3. To see this, we consider the Hopf algebra H˜Ž l. introduced at the beginning of this section. In w2, Example 3.4x it is shown that H * is not pointed, and it is clear that it is not semisimple, since that would imply that the coradical of H would be the whole of H. Thus the coradical of H is not a Hopf algebra.
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ACKNOWLEDGMENT The authors thank the referee for his or her interesting remarks and suggestions.
REFERENCES 1. N. Andruskiewitsch and H.-J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of order p 3 , preprint. 2. M. Beattie, S. Dascalescu, and L. Grunenfelder, Constructing pointed Hopf algebras by ˘ ˘ ¨ Ore extensions, preprint. 3. M. Beattie, S. Dascalescu, and L. Grunenfelder, On pointed Hopf algebras of dimension ˘ ˘ ¨ p n , Proc. Amer. Math. Soc. to appear. 4. S. Gelaki, On pointed ribbon Hopf algebras, J. Algebra 181 Ž1996., 760]786. 5. C. Kassel, ‘‘Quantum Groups,’’ Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, Berlin, 1995. 6. A. Masuoka, Selfdual Hopf algebras of dimension p 3 obtained by extension, J. Algebra 178 Ž1995., 791]806. 7. A. Masuoka, The p n theorem for semisimple Hopf algebras, Proc. Amer. Math. Soc. 124 Ž1996., 735]737. 8. S. Montgomery, Hopf algebras and their actions on rings, in ‘‘CBMS Reg. Conf. Series,’’ Vol. 82, Am. Math. Soc., Providence, 1993. 9. W. D. Nichols, Bialgebras of type one, Comm. Algebra 6 Ž1978., 1521]1552. 10. D. E. Radford, On Kauffman’s knot invariants arising from finite-dimensional Hopf algebras; Advances in Hopf algebra, Lecture Notes in Pure and Appl. Math. 158 Ž1994., 205]266. 11. D. Stefan and F. Van Oystaeyen, Hochschild cohomology and coradical filtration of pointed coalgebras. Applications, J. Algebra, to appear. 12. Y. Zhu, Hopf algebras of prime dimension, Internat. Math. Res. Notices 1 Ž1994., 53]59.
JOURNAL OF ALGEBRA ARTICLE NO.
Pointed Hopf Algebras of Dimension p 3 S. Caenepeel Faculty of Applied Sciences, Uni¨ ersity of Brussels, VUB, B-1050 Brussels, Belgium
and S. Dascalescu* ˘ ˘ Uni¨ ersity of Bucharest, Faculty of Mathematics, RO-70109 Bucharest 1, Romania Communicated by Susan Montgomery Received November 6, 1997
We give a structure theorem for pointed Hopf algebras of dimension p 3 , having coradical kC p , where k is an algebraically closed field of characteristic zero. Combining this with previous results, we obtain the complete classification of all pointed Hopf algebras of dimension p 3. Q 1998 Academic Press
0. INTRODUCTION AND PRELIMINARIES Recent publications have been devoted to classifying finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero. Nevertheless, except for certain small dimensions, there is only one situation in which the classification was completed, that is in the case of prime dimension p w12x. While the classification of finite-dimensional Hopf algebras of a given dimension seems to be a very difficult task, some subclasses, like semisimple or pointed Hopf algebras, have been classified for certain dimensions. For dimension p 2 it was proved in w7x that all semisimple Hopf algebras are group algebras, and it seems to be ‘‘folklore’’ that all pointed Hopf algebras are group algebras and Taft Hopf algebras Žthe idea for proving * Research supported by the FWO research network WO.011.96N and the project Hopf algebras and co-Galois theory of the Flemish Community. 622 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
POINTED HOPF ALGEBRAS OF DIMENSION
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this appeared for the first time in w9, Sect. 4.3x.. Semisimple Hopf algebras of dimension p 3 were classified in w6x by using extensions of Hopf algebras. Pointed Hopf algebras of dimension p n with a coradical that is the group algebra of an abelian group of order p ny 1 were classified in w3x, and the proof was essentially based on the famous Taft]Wilson Theorem. In particular, we know all Hopf algebras of dimension p 3 with coradical kC p 2 or k Ž C p = C p .. These can be produced by starting with one of these group algebras, adding an indeterminate by an Ore extension, and then factoring by a Hopf ideal. In this paper we complete the classification of pointed Hopf algebras of dimension p 3. The case in which the coradical is kC p is more difficult, and it turns out that there are two classes of such Hopf algebras. Both of them can be produced by Ore extension construction, using nonzero derivations for one of the two classes. For p s 2 the two classes are equal; in fact, they contain only one object. Basically, our technique consists of proving that such a Hopf algebra contains two Taft Hopf algebras, with kC p as their intersection. The examples with nonzero derivations have appeared in the literature w10, 4x as quantum groups constructed for the purpose of computing knots and 3-manifold invariants. For p s 2, the Hopf algebra that occurs was used in w4x to compute invariants of lens spaces. After we submitted this paper, we learned that the classification was also done independently in w1x and w11x. However, it seems that the three approaches are very different. Throughout, k is an algebraically closed field of characteristic 0. We use the notation of w8x. If H is a Hopf algebra, then GŽ H . will denote the group of grouplike elements of H, and H0 , H1 , . . . will denote the coradical filtration of H. H is called pointed if H0 s kGŽ H .. If g, h g GŽ H ., then Pg, h s x g H < DŽ x . s x m g q h m x 4 is the set of Ž g, h.-primitive elements. If H is finite-dimensional, there are no nonzero Ž1, 1.-primitive elements. For g, h g GŽ H . let Pg,X h be a complement of k Ž g y h. in Pg, h . We will need the version of the Taft]Wilson Theorem proved in w8, Theorem 5.4.1x. THEOREM 0.1.
Let H be a pointed Hopf algebra. Then
Ž1. H1 s H0 [ Ž[g, h g GŽ H . Pg,X h .. Ž2. For any n G 1 and c g Hn , there exist Ž c g, h . g, h g GŽ H . in H and w g Hny 1 m Hny1 such that c s Ý g, h g GŽ H . c g, h and DŽ c g, h . s c g, h m g q h m c g, h q w. The first part of the theorem shows that GŽ H . is not trivial if H is pointed, nontrivial, and finite-dimensional.
CAENEPEEL AND DASCALESCU ˘ ˘
624
For l a primitive pth root of 1, denote by Tl the Taft Hopf algebra of dimension p 2 with generators c and x such that c p s 1, D Ž c . s c m c,
xc s l cx
x p s 0,
D Ž x . s c m x q x m 1,
« Ž c . s 1,
« Ž x . s 0.
There are p y 1 such nonisomorphic Taft Hopf algebras. We will use the q-binomial coefficients and the q-binomial formula as in w5, Chap. 4x.
1. CORADICAL kC p Throughout this section H will be a Hopf algebra of dimension p 3 with CoradŽ H . s kC p . LEMMA 1.1. There exist c g GŽ H ., x g H, and l, a primiti¨ e pth root of 1, such that xc s l cx and DŽ x . s c m x q x m 1. The Hopf subalgebra T generated by c and x is a Taft Hopf algebra. Proof. The Taft]Wilson theorem ensures the existence of some c g GŽ H . such that P1, c / k Ž1 y c .. If f : P1, c ª P1, c is the map defined by f Ž a. s cy1 ac for every a g H, then f p s Id, so P1, c has a basis of eigenvectors for f . Let x be such an eigenvector, which is not in k Ž1 y c ., and let l be the corresponding eigenvalue. If l s 1, then the Hopf subalgebra T generated by c and x is commutative, and hence involutory. Now the Larson]Radford Theorem states that a Hopf algebra over an algebraically closed field of characteristic zero is cosemisimple if and only if it is involutory. So T is cosemisimple of dimension bigger than p, and this is in contradiction to CoradŽ H . s kC p . We conclude that l / 1, which ends the proof of the first statement. From the fact that DŽ x . s c m x q x m 1, it easily follows that x p is Ž1, 1.-primitive, and x p s 0. This shows that T is a Taft Hopf algebra. In this section T will be the Hopf subalgebra generated by c and x. The Ž n q 1.th term Tn in the coradical filtration of T is the subspace spanned by all c i x j with j F n. In particular, Tpy 1 s T. In the sequel, we will assume that the sum over an empty family is zero. LEMMA 1.2.
Let a g H be such that py1 ny1
D Ž a. s g m a q a m 1 q
Ý Ý ¨i, j m ci x j is0
js0
Ž 1.
POINTED HOPF ALGEBRAS OF DIMENSION
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625
for some g g GŽ H ., n F p, and ¨ i, j g H. Then ny1
D Ž a q ¨ 0, 0 . s g m Ž a q ¨ 0, 0 . q Ž a q ¨ 0, 0 . m 1 q
Ý ¨ 0, j m x j , Ž 2. js1
and for all 1 F r F n y 1, D Ž ¨ 0, nyr . s g m ¨ 0 , nyr q ¨ 0, nyr m c ny r ry1
q
Ý is1
ž
nyrqi i
/
l
¨ 0 , nyrqi m c ny r x i
Ž 3.
Proof. We have that
Ž D m I . D Ž a. s g m g m a q g m a m 1 q a m 1 m 1 py1 ny1
q
Ý Ý is0
py1 ny1
¨i, j m ci x j m 1 q
js0
Ý Ý DŽ ¨i, j . m ci x j is0
js0
and
Ž I m D . D Ž a. s g m g m a q g m a m 1 q a m 1 m 1 py1 ny1
q
Ý Ý is0
js0
py1 ny1
q
g m ¨i, j m ci x j j
Ý Ý Ý is0
js0 ss0
j s
ž/
l
¨ i , j m c iqjys x s m c i x jys .
Looking at the terms with 1 on the third tensor position, we find py1 ny1
ny1
Ý Ý ¨ i , j m c i x j q D Ž ¨ 0, 0 . s g m ¨ 0, 0 q Ý ¨ 0, j m x j q ¨ 0, 0 m 1, is0
js0
js1
and now we obtain Ž2., since py1 ny1
Ý Ý ¨ i , j m c i x j s D Ž a. y g m a y a m 1. is0
js0
Looking at the terms with x ny r on the third position, we find Ž3.. The key step is to show that there exist skew-primitives that are not in T.
CAENEPEEL AND DASCALESCU ˘ ˘
626
LEMMA 1.3. dimŽ H1 . ) 2 p. Proof. Suppose dimŽ H1 . F 2 p. Since T1 : H1 , we have H1 s T1. In particular, dimŽ Pu, ¨ . s 2 only for ¨ s cu. Step 1. We prove by induction on n F p y 1 that Hn s Tn . Assume that Hny 1 s Tny1 and Hn / Tn , and pick some h g Hn y Tn . Write h s Ý u, ¨ g GŽ H . h u, ¨ as in the Taft]Wilson Theorem, and pick some h u, ¨ g Hn y Tn . Denoting g s uy1 ¨ , we have that a s uy1 h u, ¨ g Hn y Tn and py1 ny1
D Ž a. s g m a q a m 1 q
Ý Ý ¨i, j m ci x j, is0
js0
with ¨ i, j g Tny1. Let b s a q ¨ 0, 0 g Hn y Tn . Lemma 1.2 shows that DŽ ¨ 0, ny1 . s g m ¨ 0, ny1 q ¨ 0, ny1 m c ny 1. If g / c n, we have ¨ 0, ny1 g H0 , and then DŽ b . g H0 m H q H m Hny2 , which is a contradiction, since b f Hny 1. Hence g s c n and ¨ 0, ny1 s a Ž c n y c ny1 . q b c ny 1 for some a , b g k, b / 0. We have that D Ž b . y c n m b y b m 1 y a Ž c n m x ny 1 y c ny 1 m x ny 1 . y b c ny 1 x m x ny 1 g H m Hny 2 q H0 m H.
Ž 4.
Since
Ž D Ž x ny 1 . y c n m x ny 1 y x ny 1 m 1 . q Ž c n m x ny 1 y c ny 1 m x ny 1 . g H m Hny 2 q H0 m H and
Ž D Ž x n . y c n m x n y x n m 1 . y n1 c ny1 x m x ny1 g H m Hny 2 ,
ž /
l
relation Ž4. implies that b9 s b q a x ny 1 y Ž n1 .ly1b x n satisfies DŽ b9. y c n m b9 y b9 m 1 g H m Hny 2 q H0 m H. Therefore b9 g Hny1 s Tny1 and b g T l Hn s Tn , providing a contradiction. Step 2. We have from Step 1 that H py 1 s Tpy1 s T / Hp . By using the Taft]Wilson Theorem and Lemma 1.2 as in Step 1, we find some py 1 b g H p y T with DŽ b . s 1 m b q b m 1 q Ý js1 ¨ j m x j for some ¨ j g T p Žnote that we need here c s 1.. We use induction to show that for any 1 F m F p there exists bm g Hp y T such that pym
D Ž bm . s 1 m bm q bm m 1 q
Ý js1
for some wj g T, a j g k.
py1
wj m x j q
Ý jspym
a j c j x pyj m x j Ž 5 .
POINTED HOPF ALGEBRAS OF DIMENSION
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627
For m s 1, we see again as in Step 1 that ¨ py 1 s a Ž1 y c py 1 . q b c py 1 x for some a , b g k, b / 0. Observe that
Ž 1 y c py 1 . m x py 1 q Ž D Ž x py 1 . y 1 m x py 1 y x py 1 m 1 . g
T m Tj .
Ý jFpy2
Applying Lemma 1.2 to a s b q a x py 1 Žin the case n s p y 1., we obtain a b1 as wanted. Assume that we have found bm for some 1 F m F p y 1 satisfying Ž5.. Applying relation Ž3. to bm and r s m, we obtain D Ž ¨ 0 , pym . s D Ž a pym c py m x m . m
s a py m
Ý is0
m i
ž /
l
c py i x i m c pym x myi
s 1 m a py m c py m x m q a pym c py m x m m 1 my1
q
Ý is1
pymqi i
ž
/
l
a py mqi c py mqi x myi m c pym x i
and my1
Ý is1
m i
ž /a l
py m c
my1
s
Ý is1
ž
py mqi
x myi m c pym x i
pymqi i
/
l
a py mqi c py mqi x myi m c pym x i ,
which implies that m i
ž /a l
py m
s
pymqi i
ž
/
l
a pymqi
for every 1 F i F m y 1. For r s m q 1 relation Ž3. gives D Ž wpy my1 . s 1 m wpymy1 q wpymy1 m c py my1 m
qÝ is1
ž
mc
pymqiy1 i
py my1
i
x.
/
l
a py mqiy1 c py mqiy1 x myi
Ž 6.
CAENEPEEL AND DASCALESCU ˘ ˘
628
On the other hand, we have D Ž c py my1 x mq 1 . s 1 m c py my1 x mq 1 q c py my1 x mq 1 m c py my1 m
qÝ is1
ž
mq1 i
/
l
c py mqiy1 x myi m c pymy1 x i .
We obtain the following identities after we apply Ž6. with i replaced by i y 1 Žfirst equality. and some elementary computation with l-factorials Žsecond equality.:
ž
pymqiy1 i s s
ž ž
/
l
a py mqiy1
pymqiy1 i pym 1
/ž l
m iy1
/ž /ž l
mq1 1
y1
/ ž l
l
pymqiy1 iy1
mq1 i
/a l
y1
/
l
a py m
py m
We obtain that D wpy my1 y
ž
ž
pym 1
/ž l
mq1 1
y1
/
l
c py my1a pym x mq1 g Pc py my 1 , 1 .
/
As c py m / 1, we find wpy my1 y
ž
pym 1
/ž l
mq1 1
y1
/
l
c py my1 x mq1 s a Ž 1 y c pymy1 .
for some a g k. Since
Ž 1 y c py my1 . m x py my1 q Ž D Ž x py my1 . y 1 m x py my1 y x py my1 m 1 . g
T m Tj ,
Ý j-pymy1
we can apply Lemma 1.2 to bm q a x py my1 and get a bmq1 satisfying Ž5.. Step 3. Take a b s bp satisfying Ž5.: py1
DŽ b. s 1 m b q b m 1 q
Ý
a j c j x pyj m x j .
Ž 7.
js1
It follows easily that DŽ bc y cb . s c m Ž bc y cb . q Ž bc y cb . m c, and bc s cb. Furthermore D Ž bx y xb . q Ž a py 2 q l py 1a py1 y a py1 y l2a py2 . x 2 m x py 1 g H0 m H q H m Hpy2 .
Ž 8.
POINTED HOPF ALGEBRAS OF DIMENSION
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Applying Ž6. with i s 1, m s 2, we obtain
Ž l q 1 . a py 2 s
l py 1 y 1 ly1
a py1
and
Ž l2 y 1 . a py 2 y Ž l py 1 y 1 . a py1 s 0. We see that the coefficient of x 2 m x py1 is 0 in Ž8. and bx y xb g H py 1 s T. Clearly, d s bx y xb g Tqs T l KerŽ « .. The relations bc s cb, bx y xb s d , and Ž7. show that the algebra generated by c, b, and x is a Hopf subalgebra of H, so it is the whole of H by the Nichols]Zoeller Theorem w8, Corollary 3.2.1x. They also show that Tq H s HTq, which means that T is a normal Hopf subalgebra. Hence we have an extension of Hopf algebras T ª H ª HrTq H. But in HrTq H we have ˆ c s ˆ1, ˆ x s 0, and ˆ b is Ž1, 1.-primitive, thus 0. We obtain HrTq H , k, and then dimŽ H . s dimŽT ., which provides a final contradiction. At this point we know that there are two different cases: 3 F dimŽ P1, c ., or there exists g / c such that 2 F dimŽ P1, g .. In the first case let us pick some y g P1, c y kx such that yc s m cy for some primitive pth root m of 1 Žrecall that P1, c has a basis of eigenvectors for the conjugation by c .. In the second case pick y g P1, g such that yg s m gy for some m / 1. Write g s c d Žin the first case we will take d s 1.. LEMMA 1.4. The set c q x i y j <0 F q, i, j F p y 14 is a basis of H. Proof. We prove by induction on 1 F n F 2 p y 2 that the set Bn s c q x i y j <0 F q, i, j F p y 1 and i q j F n4 is linearly independent. For n s 1 this follows from the Taft]Wilson Theorem. Suppose that Bn is linearly py 1 q i j independent and take Ý qs 0 Ý iqj F nq1 a q, i, j c x y s 0. Applying D, we find py1
Ý
i
Ý
j
Ý Ý aq, i, j
qs0 iqjFnq1 ss0 ts0
ž si /
l
j t
ž/
m
l s dŽ jyt . c qqiysqdŽ jyt . x s y t
m c q x iys y jyt s 0. Fix some triple Ž q0 , i 0 , j0 . with i 0 q j0 s n q 1 and assume i 0 / 0 Žotherwise j0 / 0 and we proceed in a similar way.. Take f g H * mapping c q 0qi 0y1 qd j0 x to 1 and any other element of Bn to 0, and c g H * mapping c q 0 x i 0y1 y j 0 to 1 and the rest of Bn to 0. Applying f m c , we find that a q 0 , i 0 , j 0 s 0. Now all of the a q, i, j are zero by the induction hypothesis.
CAENEPEEL AND DASCALESCU ˘ ˘
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Bn spans Hn for e¨ ery 0 F n F 2 p y 2.
COROLLARY 1.5.
Proof. We prove by induction on 1 F n F 2 p y 2 that Bn y Bny1 : Hn y Hny1. This is clear for n s 1. For the induction step, pick x i y j g Bnq 1 y Bn . Then j
i
DŽ x i y j . s
Ý Ý ss0 ts0
ž si /
l
j t
ž/
m
l s dŽ jyt . c iysqdŽ jyt . x s y t m x iys y jyt
g H 0 m H q H m Hn , and x i y j g Hnq 1. Assuming again that i / 0, choose f , c g H * such that f Ž c iq d jy 1 x . s c Ž x iy 1 y j . s 1, f Ž B0 . s 0, and c Ž Bny 1 . s 0. Then Ž f m c .Ž H0 m H q H m Hny1 . s 0, while Ž f m c .Ž DŽ x i y j .. s 1i l d j / 0, and this shows that x i y j f Hn .
ž/
l
We now define some Hopf algebras. If l is a primitive pth root of 1 and 1 F i F p y 1 an integer, we denote by H Ž l, i . the Hopf algebra with generators c, x, y defined by c p s 1,
x p s y p s 0,
D Ž c . s c m c,
xc s l cx,
D Ž x . s c m x q x m 1,
yc s lyi cy,
yx s lyi xy
D Ž y . s c i m y q y m 1.
We also denote by Hd Ž l. the Hopf algebra with generators c, x, y defined by c p s 1, x p s y p s 0, xc s l cx, yc s ly1 cy, D Ž c . s c m c,
yx s ly1 xy q c 2 y 1
D Ž x . s c m x q x m 1,
D Ž y . s c m y q y m 1.
Note that for p s 2 the only Hopf algebra of the second type, Hd Žy1., is equal to H Žy1, 1.. Remark 1.6. Suppose that p is odd, and let q be a primitive pth root of unity. The referee kindly pointed out to us that Hd Ž q 2 . is the Frobenius]Lusztig kernel Uq Žsl 2 .9 with the canonical generators cy1 , c, xcy1 , and y. THEOREM 1.7. Let H be a Hopf algebra of dimension p 3 with CoradŽ H . s kC p . Then H is isomorphic either to some Hd Ž l. or to some H Ž l, i .. Proof. We first consider an odd prime p. We distinguish the following cases. Case 1.
x, y g P1, c . Then yx g H2 , and since B2 spans H2 , we have ny1
yx s
Ý Ž a i c i x 2 q bi c i xy q g i c i y 2 . q d is0
Ž 9.
POINTED HOPF ALGEBRAS OF DIMENSION
p3
631
for some a i , bi , g i g k, d g H1. Applying D and replacing everywhere yx by the right-hand side in Ž9., we find ny1
Ý Ž a i c 2 m c i x 2 q bi c 2 m c i xy q g i c 2 m c i y 2 q a i c i x 2 m 1 is0
qbi c i xy m 1 q g i c i y 2 m 1 . q c 2 m d q d m 1 q m cy m x q cx m y ny1
s
Ý ai
ž
is0
c iq2 m c i x 2 q c i x 2 m c i q
ny1
Ý gi
q
ž
is0
2 1
c iq1 x m c i x
ž/ ž/
c iq2 m c i y 2 q c i y 2 m c i q
l
2 1
m
/
c iq1 y m c i y
/
ny1
Ý bi Ž c iq2 m c i xy q c i xy m c i q c iq1 y m c i x q l c iq1 x m c i y .
q
is0
q DŽ d . Since B2 is linearly independent, we get a i s g i s 0 for all i Žlooking at the coefficients of c iq1 x m c i x and c iq1 y m c i y ., bi s 0 for any i / 0 Žlooking at the coefficient of c iq2 m c i xy ., b 0 l s 1 Žlooking at the coefficient of cx m y ., m s b 0 Žlooking at the coefficient of cy m x ., and DŽ d . s c 2 m d q d m 1. Thus m s ly1 and d g P1, c 2 s k Ž c 2 y 1.. If d s 0, then H , H Ž l, 1.. If d / 0, then H , Hd Ž i .. Case 2. x g P1, c , y g P1, g , where g s c d / c. Writing again yx as in Ž9. and applying D, we find ny1
Ý Ž a i cg m c i x 2 q bi cg m c i xy q g i cg m c i y 2 q a i c i x 2 m 1 is0
qbi c i xy m 1 q g i c i y 2 m 1 . q cg m d q d m 1 q m cy m x q gx m y ny1
s
Ý ai is0
ž
c iq2 m c i x 2 q c i x 2 m c i q
ny1
q
Ý gi is0
ž
2 1
c iq1 x m c i x
ž/ ž/
cig 2 m ci y2 q ci y2 m ci q
l
2 1
m
/
c i gy m c i y
/
ny1
q
Ý bi Ž c iq1 g m c i xy q c i xy m c i q c iq1 y m c i x q ld c i gx m c i y . is0
q DŽ d . .
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Since B2 is linearly independent, we obtain that a i s g i s 0 for any i, bi s 0 for any i / 0, b 0 s m , l db 0 s 1, and DŽ d . s d m 1 q cg m d . Thus m s lyd and d g P1, c dq 1 s k Ž c dq 1 y 1.. If d s 0, then H , H Ž l, i .. If d / 0, then yxc s Ž b 0 xy q d . c s b 0 m xcy q d c s lb 0 m cxy q d c, and yxc s l ycx s lm cyx s lm c Ž b 0 xy q d . s lmb 0 cxy q lm c d . These show that lm s 1, and this implies d s 1, which is impossible. If p s 2, then x 2 s y 2 s 0 and B2 s c q, c q x, c q y, c q xy <0 F q F p y 14 , so in Eq. Ž9. we consider from the beginning that a i s g i s 0, and the proof works with the same computations. The number of types of Hopf algebras of the form Hd Ž l. and H Ž l, i . has been evaluated in w2, Sect. 6x. PROPOSITION 1.8. Ž1. H Ž l, i . , H Ž m , j . if and only if either Ž l, i . s Ž m , j . or m s ly1 and ij ' 1 Žmod p .. Ž2. Hd Ž l. , Hd Ž m . if and only if l s m. Ž3. If p is odd, then no one of the Hd Ž m . is isomorphic to any H Ž l, i .. COROLLARY 1.9. If p is odd, then there exist Ž p y 1. 2r2 types of Hopf algebras of the form H Ž l, i . and p y 1 types of the form Hd Ž l.. If p s 2, then there is only one Hopf algebra of the form H Ž l, i ., namely, H Žy1, 1., and it is equal to Hd Žy1.. 2. THE CLASSIFICATION Let H be a pointed Hopf algebra of dimension p 3. We have seen that the coradical of H cannot be of dimension 1. By the Nichols]Zoeller Theorem we have dimŽCoradŽ H .. g p, p 2 , p 34 . If dimŽCoradŽ H .. s p, then CoradŽ H . s kC p , and this case was discussed in Section 1. If dimŽCoradŽ H .. s p 2 , then CoradŽ H . s kC p 2 or CoradŽ H . s k Ž C p = C p ., and the classification in these cases was made in w3x, where pointed Hopf algebras H of dimension p n with abelian GŽ H . of order p ny 1 are classified. To present this classification we first introduce some Hopf algebras. 2 For every l / 1 and i such that l p s 1 and li is a primitive pth root of 1, Hp 2 Ž l, i . is the Hopf algebra with generators c and x and 2
c p s 1, D Ž c . s c m c,
xc s l cx
x p s 0,
D Ž x . s c m x q x m 1. i
POINTED HOPF ALGEBRAS OF DIMENSION
p3
633
Two such Hopf algebras H p 2 Ž l, i . and H p 2 Ž m , j . are isomorphic if and only if there exists h not divisible by p such that m s l h and i ' hj Žmod p 2 . w3, Lemma 5x. If l is a primitive pth root of 1, we denote by H Ž l. the Hopf algebra with generators c and x defined by 2
c p s 1, D Ž c . s c m c,
x p s c p y 1,
xc s l cx
D Ž x . s c m x q x m 1.
Then H˜Ž l. , H˜Ž m . if and only if l s m w3, Lemma 6x. The classification in this case is now a consequence of w3, Theorem 2x. PROPOSITION 2.1.
Let H be a pointed Hopf algebra of dimension p 3.
Ži. If CoradŽ H . s k Ž C p = C p ., then H , Tl m kC p , where Tl is one of the Taft Hopf algebras, and there exist p y 1 types of such Hopf algebras. Žii. If CoradŽ H . s kC p 2 , then H is isomorphic either to some Hp 2 Ž l, i . or to a H˜Ž l., and there are 3Ž p y 1. types of such Hopf algebras. Finally, if dimŽCoradŽ H .. s p 3 , then H is the group algebra of one of the following groups: C p = C p = C p , C p 2 = C p , C p 3 , G1 s C p 2 i C p , G 2 s C p i C p 2 , where G1 , G 2 are the two types of nontrivial semidirect products. We now have the complete classification of pointed Hopf algebras of dimension p 3. THEOREM 2.2. Let H be a pointed Hopf algebra of dimension p 3. Then H is isomorphic to one of the following: Hd Ž l., H Ž l, i ., H˜Ž l., Tl m kC p , where l is a primiti¨ e pth root of 1, Hp 2 Ž l, i . for some l / 1 and i such that 2 l p s 1, and li is a primiti¨ e pth root of 1, k Ž C p = C p = C p ., k Ž C p 2 = C p ., k Ž C p 3 ., k Ž G1 ., k Ž G 2 .. If p is odd, then there are Ž p y 1.Ž p q 9.r2 q 5 such types. If p s 2, then there are 10 types. Note added in proof: Combining our results with the results in w6x, w7x, and w12x, we find the complete classification of all Hopf algebras of dimension p 3 that have a Hopf algebra as coradical Ž p odd.. Indeed, if the coradical of H has dimension p or p 2 , then it is a group algebra, because it is semisimple, and because of the classification results in w12x and w7x, and it follows that H is pointed. If the coradical of H has dimension p 3 , then H is semisimple. In w6, Theorem 3.1x it is proved that there are p q 8 semisimple Hopf algebras of dimension p 3 . The intersection of the two classifications consists of the five group algebras, and we conclude that there are in total Ž p 2 q 10 p q 7.r2 Hopf algebras of dimension p 3 with a Hopf algebra filtration. This does not give the complete classification of all Hopf algebras of dimension p 3. To see this, we consider the Hopf algebra H˜Ž l. introduced at the beginning of this section. In w2, Example 3.4x it is shown that H * is not pointed, and it is clear that it is not semisimple, since that would imply that the coradical of H would be the whole of H. Thus the coradical of H is not a Hopf algebra.
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ACKNOWLEDGMENT The authors thank the referee for his or her interesting remarks and suggestions.
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