Let H be a non-semisimple Hopf algebra with antipode S of dimension pq over an algebraically closed field of characteristic 0 where p ≤ q are odd primes. We prove that Tr(S2p) = p2d where d ≡ pq (mod 4). As a consequence, if p, q are twin primes, then any Hopf algebra of dimension pq is semisimple. © 2004 Elsevier Inc. All rights reserved

Ng, Siu Hung

English

arXiv

AbstractLet H be a non-semisimple Hopf algebra with antipode S of dimension pq over an algebraically closed field of characteristic 0 where p⩽q are odd primes. We prove that Tr(S2p)=p2d where d≡pq (mod4). As a consequence, if p, q are twin primes, then any Hopf algebra of dimension pq is semisimple

Ng, Siu-Hung

Elsevier - Publisher Connector 

Journal of Algebra 276 (2004) 399–406
www.elsevier.com/locate/jalgebra
Hopf algebras of dimension pq
Siu-Hung Ng a,b
a Mathematics Department, Iowa State University, Ames, IA 50011, USA
b Mathematics Department, Towson University, Baltimore, MD 21252, USA
Received 12 May 2003
Communicated by Susan Montgomery
Abstract
Let H be a non-semisimple Hopf algebra with antipode S of dimension pq over an algebraically
closed field of characteristic 0 where p  q are odd primes. We prove that Tr(S2p) = p2d where
d ≡ pq (mod 4). As a consequence, if p, q are twin primes, then any Hopf algebra of dimension pq
is semisimple.
 2004 Elsevier Inc. All rights reserved.
1. Introduction
Let p be an odd prime and k an algebraically closed field of characteristic 0. If H is
a semisimple Hopf algebra of dimension p2 over k, then H is isomorphic to k[Zp2] or
k[Zp × Zp] by [8]. A more general result for semisimple Hopf algebras of dimension
pq , where p, q are primes, was obtained by [4]. In [10], the author proved that non-
semisimple Hopf algebras of dimension p2 over k are Taft algebras and hence completed
the classification of Hopf algebras of dimension p2. However, if p, q are distinct primes,
there is still no example of non-semisimple Hopf algebras of dimension pq . In fact, it was
shown in [1] and [3] that there is no non-semisimple Hopf algebras over k of dimension
14, 15, 21, 35, 55, 77, 65, 91 or 143.
By [10], if p  q are odd primes and H is a non-semisimple Hopf algebra with antipode
S of dimension pq , then S4p = idH and Tr(S2p) = p2d for some odd integer d . In
this paper, we prove that d ≡ pq (mod 4). As a consequence, we prove that if p, q are
twin primes, any Hopf algebra of dimension pq over k is semisimple. Recently, Etingof
E-mail addresses: rng@math.iastate.edu, rng@towson.edu.0021-8693/$ – see front matter  2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jalgebra.2003.11.008
400 S.-H. Ng / Journal of Algebra 276 (2004) 399–406and Gelaki also announce an even more general result [5] which covers the cases when
p < q  2p + 1.
2. Notation and preliminaries
Throughout this paper, p  q are odd primes, k denotes an algebraically closed
field of characteristic 0, and H denotes a finite-dimensional Hopf algebra over k with
antipode S. Its comultiplication and counit are, respectively, denoted by ∆ and ε. We will
use Sweedler’s notation [16]:
∆(x) =
∑
x(1) ⊗ x(2).
A non-zero element a ∈ H is called group-like if ∆(a) = a ⊗ a. The set of all group-
like elements G(H) of H is a linearly independent set, and it forms a group under the
multiplication of H . For the details of elementary aspects for finite-dimensional Hopf
algebras, readers are referred to [9,16].
Let λ ∈ H ∗ be a non-zero right integral of H ∗ and Λ ∈ H a non-zero left integral
of H . There exists α ∈ Alg(H, k) = G(H ∗), independent of the choice of Λ, such that
Λa = α(a)Λ for a ∈ H . Likewise, there is a group-like element g ∈ H , independent of
the choice of λ, such that βλ = β(g)λ for β ∈ H ∗. We call g the distinguished group-like
element of H and α the distinguished group-like element of H ∗. Then we have a formula
for S4 in terms of α and g [11]:
S4(a) = g(α ⇀ a ↼ α−1)g−1 for a ∈ H, (2.1)
where ⇀ and ↼ denote the natural actions of the Hopf algebra H ∗ on H described by
β ⇀ a =
∑
a(1)β(a(2)) and a ↼ β =
∑
β(a(1))a(2)
for β ∈ H ∗ and a ∈ H . There are some useful formulae for the trace of a linear
endomorphism on H in terms of λ and Λ.
Theorem 2.1 [13, Theorem 1]. Let H be a finite-dimensional Hopf algebra with antipode
S over the field k. Suppose that λ is a right integral of H ∗, and that Λ is a left integral of
H such that λ(Λ) = 1. Then for any f ∈ Endk(H),
Tr(f ) =
∑
λ
(
S(Λ(2))f (Λ(1))
)=∑λ((S ◦ f )(Λ(2))Λ(1))=∑λ((f ◦ S)(Λ(2))Λ(1)).
Following [10, Section 2], the index of H is the least positive integer n such that
S4n = idH and gn = 1.
S.-H. Ng / Journal of Algebra 276 (2004) 399–406 401Suppose that H is a finite-dimensional non-semisimple Hopf algebra of odd index n, and
that ω ∈ k is a primitive nth root of unity. Since gn = 1 and α is an algebra map, α(g) is a
nth root of unity. There exists a unique element x(ω,H) ∈ Zn such that
α(g) = ωx(ω,H).
Following the notation in [10], we let
Hωa,i,j =
{
u ∈ H | S2(u) = (−1)aωiu and ug = ωju}
for any (a, i, j) ∈ Z2 × Zn × Zn. Since r(g) ∈ Endk(H), defined by r(g)(a) = ag for
a ∈ H , commutes with S2, we have
H =
⊕
a∈Kn
Hωa , (2.2)
where Kn denotes the group Z2 ×Zn ×Zn.
Using the eigenspace decomposition of H in (2.2), the diagonalization of a left integral
Λ of H admits the following form (cf. [10]),
∆(Λ) =
∑
a∈Kn
(∑
ua ⊗ v−a+x
)
, (2.3)
where
∑
ua ⊗ v−a+x ∈ Hωa ⊗ Hω−a+x and x = (0,−x(ω,H), x(ω,H)) in Kn.
In the sequel, we will call the expression in Eq. (2.3) the normal form of ∆(Λ)
associated with ω. We will simply write ua ⊗ v−a+x for the sum ∑ua ⊗ v−a+x in the
normal form of ∆(Λ).
Let Eωa , a ∈ Kn, be the set of orthogonal projections associated with the decomposi-
tion (2.2). Then
dim
(
Hωa
)= Tr(Eωa )
and we have the following lemma.
Lemma 2.2. Let H be a finite-dimensional non-semisimple Hopf, algebra with antipode S
of odd index n over k, and ω ∈ k a primitive nth root of unity. Then we have
dim
(
Hωa
)= dim(Hωx−a)
for all a ∈Kn where x = (0,−x(ω,H), x(ω,H)).
Proof. Let Λ be a left integral of H and let λ be a right integral of H ∗ such that λ(Λ) = 1.
Using the normal form of ∆(Λ) associated with ω in (2.3) and Theorem 2.1, we have
Tr
(
Eωa
)= ∑ λ(S(v−b+x)Eωa (ub))= λ(S(v−a+x)ua).b∈Kn
402 S.-H. Ng / Journal of Algebra 276 (2004) 399–406By Theorem 2.1 again, we also have
Tr
(
Eω−a+x
)= ∑
b∈Kn
λ
(
S
(
Eω−a+x(v−b+x)
)
ub
)= λ(S(v−a+x)ua).
Therefore, Tr(Eωa ) = Tr(Eωx−a). Since dim(Hωa ) = Tr(Eωa ) for any a ∈ Kn, the result
follows. 
Theorem 2.3 [10]. Let H be a non-semisimple Hopf algebra of dimension pq over k with
antipode S, where p  q are odd primes. Then the index of H and the order of S4 are
equal to p, and Tr(S2p) = p2d for some odd integer d .
Lemma 2.4. Suppose that H is a non-semisimple Hopf algebra of dimension pq over k
where p q are odd primes, and that ω ∈ k is a primitive pth root of unity. Let g and α be
the distinguished group-like elements of H and H ∗, respectively. If g is non-trivial, then
the integer d in Theorem 2.3 is given by
dim
(
Hω0,i,j
)− dim(Hω1,i,j )= d
for all i, j ∈ Zp . Moreover, if both g and α are not trivial, then
dim
(
Hωa,i,j ′
)= dim(Hωa,i,j )
for any a ∈ Z2 and i, j, j ′ ∈ Zp .
Proof. If α is trivial and g = 1, then by [10, Lemma 4.3],
dim
(
Hω0,i,j
)− dim(Hω1,i,j)= d.
If both g and α are non-trivial, then by the proof of [10, Proposition 5.3], H is isomorphic
to the biproduct
R × B (2.4)
as Hopf algebras (cf. [12]) where B = k[g] and R is a left B-comodule subalgebra of H .
It was shown in [2, Section 4] that R is invariant under S2. Moreover, in the identification
H ∼= R ⊗ B given by multiplication, one has
S2 = T ⊗ idB, (2.5)
where T is the restriction of S2 on R. Let
Ra,i =
{
x ∈ R | S2(x) = (−1)aωix}
for any (a, i) ∈ Z2 × Zp . It follows from the proof of [10, Proposition 5.3] that
dim(R0,i) − dim(R1,i) = d.
S.-H. Ng / Journal of Algebra 276 (2004) 399–406 403By (2.4),
Hωa,i,j = Ra,i ⊗ ej
for all (a, i, j) ∈Kp where ej is the central idempotent of B such that ejg = ωjej . Thus,
dim
(
Hωa,i,j
)= Ra,i
for all (a, i, j) ∈Kp and hence
dim
(
Hω0,i,j
)− dim(Hω1,i,j )= d. 
3. Proofs of main results
Lemma 3.1. Let H be a finite-dimensional non-semisimple Hopf algebra with antipode
S of odd index n over k, and ω ∈ k a primitive nth root of unity. Let 
 ∈ Zn such that
2
 = x(ω,H). Then
dim
(
Hω1,−
,

)
is even.
Proof. Let V be the space of all f ∈ H ∗ such that f (Hωa,i,j ) = {0} whenever (a, i, j) =
(1,−
, 
). Obviously, V is isomorphic to (Hω1,−
,
)∗ and so dim(V ) = dim(Hω1,−
,
). Let
Λ be a non-zero left integral of H and
∆(Λ) =
∑
a∈Kn
ua ⊗ v−a+x
the normal form of ∆(Λ) associated with ω where x = (0,−2
,2
). Then
(f,h) = (f ⊗ h)∆(Λ)
defines a non-degenerate bilinear form on H ∗. Let f ∈ V such that (f,h) = 0 for all h ∈ V .
For any h′ ∈ H ∗, there exists h ∈ V such that h′(u) = h(u) for all u ∈ Hω1,−
,
. Thus
(
f,h′
)= ∑
a∈Kn
f (ua)h
′(v−a+x) = f (u1,−
,
)h′(v1,−
,
) = (f,h) = 0.
By the non-degeneracy of (· , ·), f = 0. Therefore, (· , ·) induces a non-degenerate bilinear
form on V . Using [14, Theorem 3(d)], we have
∆op(Λ) =
∑
(−1)aω−i−j
(∑
ua,i,j ⊗ va,−2
−i,2
−j
)
.(a,i,j)∈Kn
404 S.-H. Ng / Journal of Algebra 276 (2004) 399–406Therefore, for any f,h ∈ V ,
(h,f ) = (f ⊗ h)∆op(Λ) = −f (u1,−
,
)h(v1,−
,
) = −(f,h).
Hence, V admits a non-degenerate alternating form and so dim(V ) is even. 
If H is a finite-dimensional Hopf algebra of index n, we define
H− :=
{
u ∈ H | S2n(u) = −u},
H+ :=
{
u ∈ H | S2n(u) = u}.
Corollary 3.2. Suppose H is a finite-dimensional non-semisimple Hopf algebra with
antipode S of odd index n over k. Then, the subspace H− is of even dimension.
Proof. Let ω ∈ k be a nth root of unity and 
 ∈ Zn such that 2
 = x(ω,H). We then have
H− =
⊕
i,j∈Zn
Hω1,i,j = Hω1,−
,
 ⊕
( ⊕
some i,j∈Zn
(i,j) =(−
,
)
Hω1,i,j ⊕ Hω1,−2
−i,2
−j
)
.
It follows from Lemmas 2.2 and 3.1 that dim(H−) is even. 
Theorem 3.3. Let H be a non-semisimple Hopf algebra with antipode S of dimension pq
where p  q are odd primes. Then
Tr
(
S2p
)= p2d and d ≡ pq (mod 4).
Proof. By Theorem 2.3, H is of index p and Tr(S2p) = p2d for some odd integer d . Since
dim(H+) + dim(H−) = pq
and
Tr
(
S2p
)= dim(H+) − dim(H−) = p2d,
we have
dim(H−) = p(q − pd)/2.
By Corollary 3.2, p(q − pd) ≡ 0 (mod 4) or d ≡ pq (mod 4). 
Theorem 3.4. For any pair of twin primes p < q , if H is a Hopf algebra of dimension pq ,
then H is semisimple.
S.-H. Ng / Journal of Algebra 276 (2004) 399–406 405Proof. Suppose there is a non-semisimple Hopf algebra H of dimension pq . By [6], H ∗ is
also non-semisimple. Since dim(H) is odd, by [7, Theorem 2.1], H and H ∗ can not be both
unimodular. By duality, we may simply assume that H ∗ is not unimodular. It follows from
Theorem 2.3 that |G(H)| = p and so
dim(C) p,
where C is the coradical of H . If dim(C) = p, then H is pointed and hence, by [15, Co-
rollary 4], H is semisimple. Therefore, dim(C) > p and so we have
Tr
(
S2p
∣∣
H/C
)
−(pq − dim(C))> −pq + p = −p2 − p.
It follows from [6, Lemma 3.2] that
Tr
(
S2p
∣∣
C
)
 p.
Thus, we have
Tr
(
S2p
)= Tr(S2p∣∣
C
)+ Tr(S2p∣∣
H/C
)
> −p2. (3.1)
Since pq ≡ −1 (mod 4), by Theorem 3.3,
Tr
(
S2p
)= −p2
but this contradicts (3.1). 
Acknowledgment
The author thanks P. Etingof for his useful suggestions for Theorem 3.4 and bringing
the author’s attention to his recent work [5] with S. Gelaki.
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Hopf algebras of dimension pq 

LSU Scholarly Repository (Louisiana State Univ.)

Louisiana State University LSU Scholarly Repository Faculty Publications Department of Mathematics 6-1-2004 Hopf algebras of dimension pq Siu Hung Ng Iowa State University Follow this and additional works at: https://repository.lsu.edu/mathematics_pubs Recommended Citation Ng, S. (2004). Hopf algebras of dimension pq. Journal of Algebra, 276 (1), 399-406. https://doi.org/10.1016/j.jalgebra.2003.11.008 This Article is brought to you for free and open access by the Department of Mathematics at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. Journal of Algebra 276 (2004) 399–406www.elsevier.com/locate/jalgebraHopf algebras of dimensionpqSiu-Hung Nga,ba Mathematics Department, Iowa State University, Ames, IA 50011, USAb Mathematics Department, Towson University, Baltimore, MD 21252, USAReceived 12 May 2003Communicated by Susan MontgomeryAbstractLet H be a non-semisimple Hopf algebra with antipodeS of dimensionpq over an algebraicallyclosed field of characteristic 0 wherep  q are odd primes. We prove that Tr(S2p) = p2d whered ≡ pq (mod 4). As a consequence, ifp, q are twin primes, then any Hopf algebra of dimensionpqis semisimple. 2004 Elsevier Inc. All rights reserved.1. IntroductionLet p be an odd prime andk an algebraically closed field of characteristic 0. IfH isa semisimple Hopf algebra of dimensionp2 over k, thenH is isomorphic tok[Zp2] ork[Zp × Zp] by [8]. A more general result for semisimple Hopf algebras of dimensionpq , wherep, q are primes, was obtained by [4]. In [10], the author proved that non-semisimple Hopf algebras of dimensionp2 overk are Taft algebras and hence completedthe classification of Hopf algebras of dimensionp2. However, ifp, q are distinct primes,there is still no example of non-semisimple Hopf algebras of dimensionpq . In fact, it wasshown in [1] and [3] that there is no non-semisimple Hopf algebras overk of dimension14, 15, 21, 35, 55, 77, 65, 91 or 143.By [10], if p  q are odd primes andH is a non-semisimple Hopf algebra with antipodeS of dimensionpq , then S4p = idH and Tr(S2p) = p2d for some odd integerd . Inthis paper, we prove thatd ≡ pq (mod 4). As a consequence, we prove that ifp, q aretwin primes, any Hopf algebra of dimensionpq overk is semisimple. Recently, EtingofE-mail addresses:rng@math.iastate.edu, rng@towson.edu.0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2003.11.008400 S.-H. Ng / Journal of Algebra 276 (2004) 399–406and Gelaki also announce an even more general result [5] which covers the cases whenp < q  2p + 1.2. Notation and preliminariesThroughout this paper,p  q are odd primes,k denotes an algebraically closedfield of characteristic 0, andH denotes a finite-dimensional Hopf algebra overk withantipodeS. Its comultiplication and counit are, respectively, denoted by∆ andε. We willuse Sweedler’s notation [16]:∆(x) =∑x(1) ⊗ x(2).A non-zero elementa ∈ H is called group-like if∆(a) = a ⊗ a. The set of all group-like elementsG(H) of H is a linearly independent set, and it forms a group under themultiplication of H . For the details of elementary aspects for finite-dimensional Hopfalgebras, readers are referred to [9,16].Let λ ∈ H ∗ be a non-zero right integral ofH ∗ and Λ ∈ H a non-zero left integralof H . There existsα ∈ Alg(H, k) = G(H ∗), independent of the choice ofΛ, such thatΛa = α(a)Λ for a ∈ H . Likewise, there is a group-like elementg ∈ H , independent ofthe choice ofλ, such thatβλ = β(g)λ for β ∈ H ∗. We callg the distinguished group-likeelement ofH andα the distinguished group-like element ofH ∗. Then we have a formulafor S4 in terms ofα andg [11]:S4(a) = g(α ⇀ a ↼ α−1)g−1 for a ∈ H, (2.1)where⇀ and↼ denote the natural actions of the Hopf algebraH ∗ onH described byβ ⇀ a =∑a(1)β(a(2)) and a ↼ β =∑β(a(1))a(2)for β ∈ H ∗ and a ∈ H . There are some useful formulae for the trace of a linearendomorphism onH in terms ofλ andΛ.Theorem 2.1 [13, Theorem 1].Let H be a finite-dimensional Hopf algebra with antipodeS over the fieldk. Suppose thatλ is a right integral ofH ∗, and thatΛ is a left integral ofH such thatλ(Λ) = 1. Then for anyf ∈ Endk(H),Tr(f ) =∑λ(S(Λ(2))f (Λ(1))) = ∑λ((S ◦ f )(Λ(2))Λ(1)) = ∑λ((f ◦ S)(Λ(2))Λ(1)).Following [10, Section 2], the index ofH is the least positive integern such thatS4n = idH and gn = 1.S.-H. Ng / Journal of Algebra 276 (2004) 399–406 401Suppose thatH is a finite-dimensional non-semisimple Hopf algebra of odd indexn, andthatω ∈ k is a primitiventh root of unity. Sincegn = 1 andα is an algebra map,α(g) is anth root of unity. There exists a unique elementx(ω,H) ∈ Zn such thatα(g) = ωx(ω,H).Following the notation in [10], we letHωa,i,j ={u ∈ H | S2(u) = (−1)aωiu andug = ωju}for any (a, i, j) ∈ Z2 × Zn × Zn. Sincer(g) ∈ Endk(H), defined byr(g)(a) = ag fora ∈ H , commutes withS2, we haveH =⊕a∈KnHωa , (2.2)whereKn denotes the groupZ2 × Zn × Zn.Using the eigenspace decomposition ofH in (2.2), the diagonalization of a left integralΛ of H admits the following form (cf. [10]),∆(Λ) =∑a∈Kn(∑ua ⊗ v−a+x), (2.3)where∑ua ⊗ v−a+x ∈ Hωa ⊗ Hω−a+x andx = (0,−x(ω,H), x(ω,H)) in Kn.In the sequel, we will call the expression in Eq. (2.3) thenormal form of ∆(Λ)associated withω. We will simply write ua ⊗ v−a+x for the sum∑ua ⊗ v−a+x in thenormal form of∆(Λ).Let Eωa , a ∈ Kn, be the set of orthogonal projections associated with the decomposi-tion (2.2). Thendim(Hωa) = Tr(Eωa )and we have the following lemma.Lemma 2.2. LetH be a finite-dimensional non-semisimple Hopf, algebra with antipodeSof odd indexn overk, andω ∈ k a primitiventh root of unity. Then we havedim(Hωa) = dim(Hωx−a)for all a ∈Kn wherex = (0,−x(ω,H), x(ω,H)).Proof. Let Λ be a left integral ofH and letλ be a right integral ofH ∗ such thatλ(Λ) = 1.Using the normal form of∆(Λ) associated withω in (2.3) and Theorem 2.1, we haveTr(Eωa) = ∑b∈Knλ(S(v−b+x)Eωa (ub)) = λ(S(v−a+x)ua).402 S.-H. Ng / Journal of Algebra 276 (2004) 399–406By Theorem 2.1 again, we also haveTr(Eω−a+x) = ∑b∈Knλ(S(Eω−a+x(v−b+x))ub) = λ(S(v−a+x)ua).Therefore, Tr(Eωa ) = Tr(Eωx−a). Since dim(Hωa ) = Tr(Eωa ) for any a ∈ Kn, the resultfollows. Theorem 2.3 [10]. LetH be a non-semisimple Hopf algebra of dimensionpq overk withantipodeS, wherep  q are odd primes. Then the index ofH and the order ofS4 areequal top, andTr(S2p) = p2d for some odd integerd .Lemma 2.4. Suppose thatH is a non-semisimple Hopf algebra of dimensionpq over kwherep  q are odd primes, and thatω ∈ k is a primitivepth root of unity. Letg andα bethe distinguished group-like elements ofH andH ∗, respectively. Ifg is non-trivial, thenthe integerd in Theorem2.3 is given bydim(Hω0,i,j) − dim(Hω1,i,j ) = dfor all i, j ∈ Zp . Moreover, if bothg andα are not trivial, thendim(Hωa,i,j ′) = dim(Hωa,i,j )for anya ∈ Z2 andi, j, j ′ ∈ Zp .Proof. If α is trivial andg = 1, then by [10, Lemma 4.3],dim(Hω0,i,j) − dim(Hω1,i,j) = d.If both g andα are non-trivial, then by the proof of [10, Proposition 5.3],H is isomorphicto the biproductR × B (2.4)as Hopf algebras (cf. [12]) whereB = k[g] andR is a leftB-comodule subalgebra ofH .It was shown in [2, Section 4] thatR is invariant underS2. Moreover, in the identificationH ∼= R ⊗ B given by multiplication, one hasS2 = T ⊗ idB, (2.5)whereT is the restriction ofS2 onR. LetRa,i ={x ∈ R | S2(x) = (−1)aωix}for any(a, i) ∈ Z2 × Zp . It follows from the proof of [10, Proposition 5.3] thatdim(R0,i) − dim(R1,i) = d.S.-H. Ng / Journal of Algebra 276 (2004) 399–406 403By (2.4),Hωa,i,j = Ra,i ⊗ ejfor all (a, i, j) ∈ Kp whereej is the central idempotent ofB such thatejg = ωjej . Thus,dim(Hωa,i,j) = Ra,ifor all (a, i, j) ∈ Kp and hencedim(Hω0,i,j) − dim(Hω1,i,j ) = d. 3. Proofs of main resultsLemma 3.1. Let H be a finite-dimensional non-semisimple Hopf algebra with antipodeS of odd indexn over k, and ω ∈ k a primitive nth root of unity. Let ∈ Zn such that2 = x(ω,H). Thendim(Hω1,−,)is even.Proof. Let V be the space of allf ∈ H ∗ such thatf (Hωa,i,j ) = {0} whenever(a, i, j) =(1,−, ). Obviously,V is isomorphic to(Hω1,−,)∗ and so dim(V ) = dim(Hω1,−,). LetΛ be a non-zero left integral ofH and∆(Λ) =∑a∈Knua ⊗ v−a+xthe normal form of∆(Λ) associated withω wherex = (0,−2,2). Then(f,h) = (f ⊗ h)∆(Λ)defines a non-degenerate bilinear form onH ∗. Letf ∈ V such that(f,h) = 0 for allh ∈ V .For anyh′ ∈ H ∗, there existsh ∈ V such thath′(u) = h(u) for all u ∈ Hω1,−,. Thus(f,h′) = ∑a∈Knf (ua)h′(v−a+x) = f (u1,−,)h′(v1,−,) = (f,h) = 0.By the non-degeneracy of(· , ·), f = 0. Therefore,(· , ·) induces a non-degenerate bilinearform onV . Using [14, Theorem 3(d)], we have∆op(Λ) =∑(a,i,j)∈Kn(−1)aω−i−j(∑ua,i,j ⊗ va,−2−i,2−j).404 S.-H. Ng / Journal of Algebra 276 (2004) 399–406Therefore, for anyf,h ∈ V ,(h,f ) = (f ⊗ h)∆op(Λ) = −f (u1,−,)h(v1,−,) = −(f,h).Hence,V admits a non-degenerate alternating form and so dim(V ) is even. If H is a finite-dimensional Hopf algebra of indexn, we defineH− :={u ∈ H | S2n(u) = −u},H+ :={u ∈ H | S2n(u) = u}.Corollary 3.2. SupposeH is a finite-dimensional non-semisimple Hopf algebra withantipodeS of odd indexn overk. Then, the subspaceH− is of even dimension.Proof. Let ω ∈ k be anth root of unity and ∈ Zn such that 2 = x(ω,H). We then haveH− =⊕i,j∈ZnHω1,i,j = Hω1,−, ⊕( ⊕somei,j∈Zn(i,j) =(−,)Hω1,i,j ⊕ Hω1,−2−i,2−j).It follows from Lemmas 2.2 and 3.1 that dim(H−) is even. Theorem 3.3. Let H be a non-semisimple Hopf algebra with antipodeS of dimensionpqwherep  q are odd primes. ThenTr(S2p) = p2d and d ≡ pq (mod 4).Proof. By Theorem 2.3,H is of indexp and Tr(S2p) = p2d for some odd integerd . Sincedim(H+) + dim(H−) = pqandTr(S2p) = dim(H+) − dim(H−) = p2d,we havedim(H−) = p(q − pd)/2.By Corollary 3.2,p(q − pd) ≡ 0 (mod 4) or d ≡ pq (mod 4). Theorem 3.4. For any pair of twin primesp < q , if H is a Hopf algebra of dimensionpq ,thenH is semisimple.S.-H. Ng / Journal of Algebra 276 (2004) 399–406 405Proof. Suppose there is a non-semisimple Hopf algebraH of dimensionpq . By [6], H ∗ isalso non-semisimple. Since dim(H) is odd, by [7, Theorem 2.1],H andH ∗ can not be bothunimodular. By duality, we may simply assume thatH ∗ is not unimodular. It follows fromTheorem 2.3 that|G(H)| = p and sodim(C)  p,whereC is the coradical ofH . If dim(C) = p, thenH is pointed and hence, by [15, Co-rollary 4], H is semisimple. Therefore, dim(C) > p and so we haveTr(S2p∣∣H/C) −(pq − dim(C)) > −pq + p = −p2 − p.It follows from [6, Lemma 3.2] thatTr(S2p∣∣C) p.Thus, we haveTr(S2p) = Tr(S2p∣∣C) + Tr(S2p∣∣H/C)> −p2. (3.1)Sincepq ≡ −1 (mod 4), by Theorem 3.3,Tr(S2p) = −p2but this contradicts (3.1).AcknowledgmentThe author thanks P. Etingof for his useful suggestions for Theorem 3.4 and bringingthe author’s attention to his recent work [5] with S. Gelaki.References[1] N. Andruskiewitsch, S. Natale, Counting arguments for Hopf algebras of low dimension, Tsukuba J.Math. 25 (1) (2001) 187–201.[2] N. Andruskiewitsch, H.-J. Schneider, Hopf algebras of orderp2 and braided Hopf algebras of orderp,J. Algebra 199 (2) (1998) 430–454.[3] M. Beattie, S. Dascalescu, Hopf algebras of dimension 14, prepint, math.QA/0205243.[4] P. Etingof, S. Gelaki, Semisimple Hopf algebras of dimensionpq are trivial, J. Algebra 210 (2) (1998)664–669.[5] P. Etingof, S. Gelaki, On Hopf algebras of dimensionpq, preprint, math.QA/0303359.[6] R.G. Larson, D.E. Radford, Finite-dimensional cosemisimple Hopf algebras in characteristic 0 aresemisimple, J. Algebra 117 (2) (1988) 267–289.[7] R.G. Larson, D.E. Radford, Semisimple Hopf algebras, J. Algebra 171 (1) (1995) 5–35.[8] A. Masuoka, Thepn theorem for semisimple Hopf algebras, Proc. Amer. Math. Soc. 124 (3) (1996) 735–737.406 S.-H. Ng / Journal of Algebra 276 (2004) 399–406[9] S. Montgomery, Hopf Algebras and Their Actionson Rings, in: CBMS Reg. Conf. Ser. Math., vol. 82,Conference Board of the Mathematical Sciences, Washington, DC, 1993.[10] S.-H. Ng, Non-semisimple Hopf algebras of dimensionp2, J. Algebra 255 (1) (2002) 182–197.[11] D.E. Radford, The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math. 98 (2)(1976) 333–355.[12] D.E. Radford, The structure of Hopf algebras with a projection, J. Algebra 92 (2) (1985) 322–347.[13] D.E. Radford, The group of automorphisms of a semisi ple Hopf algebra over a field of characteristic 0 isfinite, Amer. J. Math. 112 (2) (1990) 331–357.[14] D.E. Radford, The trace function and Hopf algebras, J. Algebra 163 (3) (1994) 583–622.[15] D. Ştefan, Hopf subalgebras of pointed Hopf algebrasand applications, Proc. Amer. Math. Soc. 125 (11)(1997) 3191–3193.[16] M.E. Sweedler, Hopf Algebras, in: Math. Lecture Note Ser., Benjamin, New York, 1969.

Hopf algebras of dimension pq

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Hopf algebras of dimension pq

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