- Email: [email protected]
Semisimple Hopf Algebras of DimensionpqAre Trivial
210, 664]669 Ž1998. JA987568
JOURNAL OF ALGEBRA ARTICLE NO.
Semisimple Hopf Algebras of Dimension pq Are Trivial Pavel Etingof and Shlomo Gelaki Department of Mathematics, Har¨ ard Uni¨ ersity, Cambridge, Massachusetts 02138 Communicated by Susan Montgomery Received February 6, 1998
This paper makes a contribution to the problem of classifying finitedimensional semisimple Hopf algebras H over an algebraically closed field k of characteristic 0. Specifically, we show that if H has dimension pq for primes p and q, then H is trivial; that is, H is either a group algebra or the dual of a group algebra. Previously known cases include dimension 2 p wM1x, dimension p 2 wM2x, and dimensions 3 p, 5 p, and 7p wGWx. Westreich and the second author also obtained the same result for H, which is, along with its dual H U , of Frobenius type Ži.e., the dimensions of their irreducible representations divide the dimension of H . wGW, Theorem 3.5x. They concluded with the conjecture that any semisimple Hopf algebra H of dimension pq over k is trivial. In this paper we use Theorem 1.4 in wEGx to prove that both H and H U are of Frobenius type and hence prove this conjecture. Throughout, k will denote an algebraically closed field of characteristic 0, and p and q will denote two prime numbers satisfying p - q. We let GŽ H . denote the group of grouplike elements of a Hopf algebra H, and let DŽ H . denote its Drinfel’d double. Recall that H and H U c o p are Hopf subalgebras of DŽ H ., and let i H : H ¨ DŽ H . and i H U c o p : H U c o p ¨ DŽ H . denote the corresponding inclusion maps. The following result will prove very useful in the sequel. THEOREM 1 wGW, Theorem 2.1x. Let H be a semisimple Hopf algebra of dimension pq, where p - q are two prime numbers. Then 1.
< GŽ H .< / q.
2.
If < GŽ H .< s p, then q s 1Žmod p .. 664
0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
SEMISIMPLE HOPF ALGEBRAS OF DIMENSION
pq
665
Suppose there exists a nontrivial semisimple Hopf algebra H of dimension pq. Then by Theorem 1, GŽ H . and GŽ H U . have either order 1 or p. By wRx, GŽ DŽ H .. s GŽ H U . = GŽ H . has therefore order 1, p, or p 2 . Since by wRx, the order of GŽ DŽ H .U . divides the order of GŽ DŽ H .., it follows that GŽ DŽ H .U . also has order 1, p, or p 2 . In the following we show that this leads to a contradiction and hence conclude that H is trivial. LEMMA 2. Let H be a semisimple Hopf algebra of dimension pq o¨ er k, and let V, U be p-dimensional irreducible representations of DŽ H .. If V m U contains a 1-dimensional representation x , then it must contain another 1-dimensional representation x X / x . Proof. Without loss of generality we can assume that x is trivial and hence that U s V U . Otherwise we can replace U with U m xy1 . Suppose, on the contrary, that V m V U does not contain a nontrivial 1-dimensional representation. By wEGx, the dimension of any irreducible representation of DŽ H . divides pq, and hence V m V U is a direct sum of the trivial representation k, p-dimensional irreducible representations of DŽ H ., and q-dimensional irreducible representations of DŽ H .. Therefore we have that p 2 s 1 q ap q bq. Clearly b ) 0. Let W be a q-dimensional irreducible representation of DŽ H . such that W ; V m V U . Since 0 / Hom DŽ H .Ž V m V U , W . s Hom DŽ H .Ž V, W m V ., we have that V ; W mV. Since W m V has no linear constituent Žbecause dim V / dim W ., dimŽW m V . s pq, and W m V contains a p-dimensional irreducible representation of DŽ H ., it follows that W m V s V1 [ ??? [ Vq , where Vi is a p-dimensional irreducible representation of DŽ H . with V1 s V. We wish to show that for any i s 1, . . . , q there exists a 1-dimensional representation x i such that Vi s V m x i . Suppose, on the contrary, that this is not true for some i. Then V m ViU has no linear constituent; hence it must be a direct sum of p p-dimensional irreducible representations of DŽ H .. Therefore, W m Ž V m ViU . has no linear constituent. But ŽW m V . m ViU s Ž Vi [ ??? . m ViU s k [ ??? , which is a contradiction. Therefore, W m V s V m Ž x 1 [ ??? [ x q .. Note that for all i, x i is uniquely determined. Indeed, if there exists x iX / x i such that V m x i s V m x iX , then x iX xy1 is a nontrivial linear constituent of V m V U , which is a contrai diction. We now consider two possible cases. First, suppose that any 1-dimensional representation x such that x m W s W is trivial. Then for any q-dimensional irreducible representation W X of DŽ H ., W m W X contains no more than one 1-dimensional representation. Therefore, since V m V U contains b G 1 q-dimensional irreducible representations of DŽ H ., it follows that W m V m V U contains at most b 1-dimensional representations. Therefore, b G q, and p 2 s dimŽ V m V U . G bq G q 2 , which is a contradiction.
666
ETINGOF AND GELAKI
Second, suppose there exists a nontrivial 1-dimensional representation x such that x m W s W. Then x m W m V s W m V, which is equivalent to x m V m Ž x 1 [ ??? [ x q . s V m Ž x 1 [ ??? [ x q .. Write x 1 [ ??? [ x q s Ý a g G Ž D Ž H .U . m a a . Then V m Ý a g G Ž D Ž H .U . m a x y 1 a s V m Ý a g GŽ DŽ H .U . ma ax s V m Ý a g GŽ DŽ H .U . ma a . But V m a 1 s V m a 2 implies a 1 s a 2 , so ma s ma xy1 . Since the order of x in GŽ DŽ H .U . is divisible by p, this implies that q s Ý a ma is divisible by p, which is a contradiction. This concludes the proof of the lemma. LEMMA 3. Let H be a semisimple Hopf algebra of dimension pq o¨ er k. Then GŽ DŽ H .U . is nontri¨ ial. Proof. Suppose, on the contrary, that < GŽ DŽ H .U .< s 1. By wEGx, the dimension of any irreducible representation of DŽ H . divides pq; hence p 2 q 2 s 1 q ap 2 q bq 2 for some integers a, b ) 0. In particular, DŽ H . has a p-dimensional irreducible representation V. But then by Lemma 2, DŽ H . must have a nontrivial 1-dimensional representation, which is a contradiction. LEMMA 4. Let H be a nontri¨ ial semisimple Hopf algebra of dimension pq o¨ er k. Then the order of GŽ DŽ H .U . is not p 2 . Proof. Suppose, on the contrary, that < GŽ DŽ H .U .< s p 2 . By wRx, any x g GŽ DŽ H .U . is of the form x s g m a , where g g GŽ H . and a g GŽ H U .. In particular, either GŽ H . or GŽ H U . is nontrivial, and hence by Theorem 1, either GŽ H . or GŽ H U . has order p. Therefore we must have that both GŽ H . and GŽ H U . are of order p, and hence that GŽ DŽ H .U . s GŽ H . = GŽ H U .. This implies that there exists a nontrivial g g GŽ H . so that g m 1 g GŽ DŽ H .U .. By wRx, 1 m g g GŽ DŽ H .. is central in DŽ H .. In particular, g is central in H, and hence H possesses a normal Hopf subalgebra isomorphic to kC p , which implies that H is commutative Žsee the proof of Theorem 2.1 in wGWx.. This is a contradiction, and the result follows. LEMMA 5. Let H be a nontri¨ ial semisimple Hopf algebra of dimension pq o¨ er k. Then there exists a quotient Hopf algebra A of DŽ H . of dimension pq 2 that contains H and H U c o p as Hopf subalgebras, and H l H U c o p s kGŽ A. s kC p . Proof. By Lemmas 3 and 4, the order of GŽ DŽ H .U . is p. If either GŽ H . or GŽ H U . is trivial, then either H U or H, respectively, contains a central grouplike element. Hence, as in the proof of Lemma 4, H is trivial. Therefore, GŽ H . and GŽ H U . both have order p. Since DŽ H . is of Frobenius type wEGx, and since by Theorem 1, q s 1Žmod p ., we must have that p 2 q 2 s p q ap 2 q bq 2 , where a s
SEMISIMPLE HOPF ALGEBRAS OF DIMENSION
pq
667
Ž q 2 y 1.rp and b s pŽ p y 1.. In particular, DŽ H . has irreducible representations of dimensions p and q. We wish to show that if V and U are two irreducible representations of DŽ H . of dimension p, then V m U is a direct sum of 1-dimensional representations of DŽ H . and p-dimensional irreducible representations of DŽ H . only. Indeed, by Lemma 2, either V m U does not contain any 1-dimensional representation or it must contain at least two different 1-dimensional representations. But if it contains two different 1-dimensional representations, then since the 1-dimensional representations of DŽ H . form a cyclic group of order p, it follows that V m U contains all of the p 1-dimensional representations of DŽ H .. We conclude that either p 2 s ap q bq or p 2 s p q ap q bq. At any rate b s 0, and the result follows. Therefore, the subcategory C of RepŽ H . generated by the 1- and p-dimensional representations of DŽ H . gives rise to a quotient Hopf algebra A of DŽ H . of dimension pq 2 whose irreducible representations have either dimension 1 or p Ž A is the quotient of DŽ H . over the Hopf ideal I s ÝV f C End k Ž V ... Let f denote the corresponding surjective homomorphism, and consider the sequences iH
f
H ¨ D Ž H . ª A and
i HU c o p
f
H U c o p ¨ D Ž H . ª A.
Since f Ž H ., f Ž H U c o p . are Hopf subalgebras of A, it follows that they have dimension 1, p, q, or pq wNZx. If, say dimŽ f Ž H .. s q, then f Ž H . s kC q is a group algebra wZx. Since Ž kC q .U ( kC q , the surjection of Hopf algebras f : H ª kC q gives rise to an inclusion of Hopf algebras f U : kCq ª H U . Since f U Ž kC q . ; kGŽ H U ., it follows that q divides < GŽ H U .< s p, which is impossible. Similarly, dimŽ f Ž H U c o p .. / q, and we conclude that f Ž H ., f Ž H U c o p . have dimension 1, p, or pq. Since A s f Ž H . f Ž H U c o p . is of dimension pq 2 , it follows that the dimensions must equal pq and hence that H and H U c o p can be considered as Hopf subalgebras of A. Furthermore, GŽ H . and GŽ H U c o p . are mapped onto GŽ A.; otherwise < GŽ A.< is divisible by p 2 , which is impossible. Therefore GŽ H . l GŽ H U c o p . s GŽ A. in A. Since H / H U c o p in A, this implies that H l H U c o p s kGŽ A. s kC p . We are ready to prove our main result. THEOREM 6. Let H be a semisimple Hopf algebra of dimension pq o¨ er k, where p and q are distinct prime numbers. Then H is tri¨ ial. Proof. Suppose, on the contrary, that H is nontrivial. We wish to prove that H and H U are of Frobenius type. Indeed, consider the Hopf algebra quotient A, which exists by Lemma 5. For any 1-dimensional representa-
668
ETINGOF AND GELAKI
tion x of A, we define the induced representations Vqx s A mH x and
Vyx s A mH U x .
Note that since finite-dimensional Hopf algebras are free over any of their Hopf subalgebras wNZx, it follows that Vqx s H U mH l H U x s H U mk C p x , and Vyx s H mH l H U x s H mk C p x . We first show that if x is nontrivial, then x is nontrivial on H and H U as well. Indeed, for any nontrivial 1-dimensional representation x ˜ , we have by Frobenius reciprocity that Hom AŽ Vqx , x ˜ . s Hom H Ž x , x˜ .. Therefore, if x is trivial on H, then x ˜ is also trivial on H Žsince x˜ s x l as x generates the group of 1-dimensional representations of A., and Hom AŽ Vqx , x ˜ . s k. But then Vqx is a sum of p 1-dimensional representations and p-dimensional irreducible representations, which contradicts the fact that p does not divide q. Similarly, x is nontrivial on H U . Next, since dimŽHom AŽ Vqx , x ˜ .. s dimŽHom H Ž x , x˜ .. s dx , x˜ for any 1dimensional representation x ˜ , and dim Vqx s q, it follows that Vqx s x [ V1 [ ??? [ VŽ qy1.r p , where Vi is a p-dimensional irreducible representation of A for all 1 F i F Ž q y 1.rp. We wish to show that the Vi ’s do not depend on x . Indeed, let x be a nontrivial 1-dimensional representation of A. Then any 1-dimensional representation of A is of the form x l for some 0 F l F p y l 1, and Vqx s Vqx m x ly1 s x l [ Ž V1 m x ly1 . [ ??? [ Ž VŽ qy1.r p m x ly1 .. But by Lemma 2, x ly1 ; Vi m ViU for all i, and hence Vi m x ly1 ( Vi , which proves the claim. Similarly, Vyx s x [ W1 [ ??? [ WŽ qy1.r p , where Wi is a p-dimensional irreducible representation of A for all 1 F i F Ž q y 1.rp. Finally, we wish to show that the set C s x < H U < x is a 1-dimensional representation of A4 j Vi < H U < 1 F i F Ž q y 1 . rp 4 is a full set of representatives of the irreducible representations of H U . Indeed, consider the regular representation of H U : H U s H U mk C p kC p s s
[H x
[ x [ [ pV x
i
i < HU .
U
mk C p x s
[V x
x q< H U
SEMISIMPLE HOPF ALGEBRAS OF DIMENSION
pq
669
Since in the regular representation any representation can occur no more times than its dimension, we get that Vi < H U are irreducible. By summation of degrees, C is a full set of representatives, and since dimŽ Vi < H U . s p, we have that H U is of Frobenius type. Similarly, H is of Frobenius type. But Theorem 3.5 in wGWx states that if this is the case, then H is trivial. This concludes the proof of the theorem.
ACKNOWLEDGMENTS The first author was supported by a National Science Foundation postdoctoral fellowship. The second author was supported by a Rothschild postdoctoral fellowship. We thank Sonia Natale, who noticed an incompleteness in the proof of Lemma 2 in the original version of the paper.
REFERENCES wEGx wGWx wM1x wM2x wNZx wRx wZx
P. Etingof and S. Gelaki, Some properties of finite-dimensional semisimple Hopf algebras, Math. Res. Lett. 5 Ž1998., 191]197. S. Gelaki and S. Westreich, On semisimple Hopf algebras of dimension pq, Proc. Amer. Math. Soc. to appear. A. Masuoka, Semisimple Hopf algebras of dimension 2 p, Comm. Algebra 23Ž5. Ž1995., 1931]1940. A. Masuoka, The p n theorem for semisimple Hopf Algebras, Proc. Amer. Math. Soc. 124 Ž1996., 735]737. W. D. Nichols and M. B. Zoeller, A Hopf algebra freeness theorem, Amer. J. Math. 111 Ž1989., 381]385. D. E. Radford, Minimal quasitriangular Hopf algebras, J. Algebra 157 Ž1993., 281]315. Y. Zhu, Hopf algebras of prime dimension, Internat. Math. Res. Notices 1 Ž1994., 53]59.
JOURNAL OF ALGEBRA ARTICLE NO.
Semisimple Hopf Algebras of Dimension pq Are Trivial Pavel Etingof and Shlomo Gelaki Department of Mathematics, Har¨ ard Uni¨ ersity, Cambridge, Massachusetts 02138 Communicated by Susan Montgomery Received February 6, 1998
This paper makes a contribution to the problem of classifying finitedimensional semisimple Hopf algebras H over an algebraically closed field k of characteristic 0. Specifically, we show that if H has dimension pq for primes p and q, then H is trivial; that is, H is either a group algebra or the dual of a group algebra. Previously known cases include dimension 2 p wM1x, dimension p 2 wM2x, and dimensions 3 p, 5 p, and 7p wGWx. Westreich and the second author also obtained the same result for H, which is, along with its dual H U , of Frobenius type Ži.e., the dimensions of their irreducible representations divide the dimension of H . wGW, Theorem 3.5x. They concluded with the conjecture that any semisimple Hopf algebra H of dimension pq over k is trivial. In this paper we use Theorem 1.4 in wEGx to prove that both H and H U are of Frobenius type and hence prove this conjecture. Throughout, k will denote an algebraically closed field of characteristic 0, and p and q will denote two prime numbers satisfying p - q. We let GŽ H . denote the group of grouplike elements of a Hopf algebra H, and let DŽ H . denote its Drinfel’d double. Recall that H and H U c o p are Hopf subalgebras of DŽ H ., and let i H : H ¨ DŽ H . and i H U c o p : H U c o p ¨ DŽ H . denote the corresponding inclusion maps. The following result will prove very useful in the sequel. THEOREM 1 wGW, Theorem 2.1x. Let H be a semisimple Hopf algebra of dimension pq, where p - q are two prime numbers. Then 1.
< GŽ H .< / q.
2.
If < GŽ H .< s p, then q s 1Žmod p .. 664
0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
SEMISIMPLE HOPF ALGEBRAS OF DIMENSION
pq
665
Suppose there exists a nontrivial semisimple Hopf algebra H of dimension pq. Then by Theorem 1, GŽ H . and GŽ H U . have either order 1 or p. By wRx, GŽ DŽ H .. s GŽ H U . = GŽ H . has therefore order 1, p, or p 2 . Since by wRx, the order of GŽ DŽ H .U . divides the order of GŽ DŽ H .., it follows that GŽ DŽ H .U . also has order 1, p, or p 2 . In the following we show that this leads to a contradiction and hence conclude that H is trivial. LEMMA 2. Let H be a semisimple Hopf algebra of dimension pq o¨ er k, and let V, U be p-dimensional irreducible representations of DŽ H .. If V m U contains a 1-dimensional representation x , then it must contain another 1-dimensional representation x X / x . Proof. Without loss of generality we can assume that x is trivial and hence that U s V U . Otherwise we can replace U with U m xy1 . Suppose, on the contrary, that V m V U does not contain a nontrivial 1-dimensional representation. By wEGx, the dimension of any irreducible representation of DŽ H . divides pq, and hence V m V U is a direct sum of the trivial representation k, p-dimensional irreducible representations of DŽ H ., and q-dimensional irreducible representations of DŽ H .. Therefore we have that p 2 s 1 q ap q bq. Clearly b ) 0. Let W be a q-dimensional irreducible representation of DŽ H . such that W ; V m V U . Since 0 / Hom DŽ H .Ž V m V U , W . s Hom DŽ H .Ž V, W m V ., we have that V ; W mV. Since W m V has no linear constituent Žbecause dim V / dim W ., dimŽW m V . s pq, and W m V contains a p-dimensional irreducible representation of DŽ H ., it follows that W m V s V1 [ ??? [ Vq , where Vi is a p-dimensional irreducible representation of DŽ H . with V1 s V. We wish to show that for any i s 1, . . . , q there exists a 1-dimensional representation x i such that Vi s V m x i . Suppose, on the contrary, that this is not true for some i. Then V m ViU has no linear constituent; hence it must be a direct sum of p p-dimensional irreducible representations of DŽ H .. Therefore, W m Ž V m ViU . has no linear constituent. But ŽW m V . m ViU s Ž Vi [ ??? . m ViU s k [ ??? , which is a contradiction. Therefore, W m V s V m Ž x 1 [ ??? [ x q .. Note that for all i, x i is uniquely determined. Indeed, if there exists x iX / x i such that V m x i s V m x iX , then x iX xy1 is a nontrivial linear constituent of V m V U , which is a contrai diction. We now consider two possible cases. First, suppose that any 1-dimensional representation x such that x m W s W is trivial. Then for any q-dimensional irreducible representation W X of DŽ H ., W m W X contains no more than one 1-dimensional representation. Therefore, since V m V U contains b G 1 q-dimensional irreducible representations of DŽ H ., it follows that W m V m V U contains at most b 1-dimensional representations. Therefore, b G q, and p 2 s dimŽ V m V U . G bq G q 2 , which is a contradiction.
666
ETINGOF AND GELAKI
Second, suppose there exists a nontrivial 1-dimensional representation x such that x m W s W. Then x m W m V s W m V, which is equivalent to x m V m Ž x 1 [ ??? [ x q . s V m Ž x 1 [ ??? [ x q .. Write x 1 [ ??? [ x q s Ý a g G Ž D Ž H .U . m a a . Then V m Ý a g G Ž D Ž H .U . m a x y 1 a s V m Ý a g GŽ DŽ H .U . ma ax s V m Ý a g GŽ DŽ H .U . ma a . But V m a 1 s V m a 2 implies a 1 s a 2 , so ma s ma xy1 . Since the order of x in GŽ DŽ H .U . is divisible by p, this implies that q s Ý a ma is divisible by p, which is a contradiction. This concludes the proof of the lemma. LEMMA 3. Let H be a semisimple Hopf algebra of dimension pq o¨ er k. Then GŽ DŽ H .U . is nontri¨ ial. Proof. Suppose, on the contrary, that < GŽ DŽ H .U .< s 1. By wEGx, the dimension of any irreducible representation of DŽ H . divides pq; hence p 2 q 2 s 1 q ap 2 q bq 2 for some integers a, b ) 0. In particular, DŽ H . has a p-dimensional irreducible representation V. But then by Lemma 2, DŽ H . must have a nontrivial 1-dimensional representation, which is a contradiction. LEMMA 4. Let H be a nontri¨ ial semisimple Hopf algebra of dimension pq o¨ er k. Then the order of GŽ DŽ H .U . is not p 2 . Proof. Suppose, on the contrary, that < GŽ DŽ H .U .< s p 2 . By wRx, any x g GŽ DŽ H .U . is of the form x s g m a , where g g GŽ H . and a g GŽ H U .. In particular, either GŽ H . or GŽ H U . is nontrivial, and hence by Theorem 1, either GŽ H . or GŽ H U . has order p. Therefore we must have that both GŽ H . and GŽ H U . are of order p, and hence that GŽ DŽ H .U . s GŽ H . = GŽ H U .. This implies that there exists a nontrivial g g GŽ H . so that g m 1 g GŽ DŽ H .U .. By wRx, 1 m g g GŽ DŽ H .. is central in DŽ H .. In particular, g is central in H, and hence H possesses a normal Hopf subalgebra isomorphic to kC p , which implies that H is commutative Žsee the proof of Theorem 2.1 in wGWx.. This is a contradiction, and the result follows. LEMMA 5. Let H be a nontri¨ ial semisimple Hopf algebra of dimension pq o¨ er k. Then there exists a quotient Hopf algebra A of DŽ H . of dimension pq 2 that contains H and H U c o p as Hopf subalgebras, and H l H U c o p s kGŽ A. s kC p . Proof. By Lemmas 3 and 4, the order of GŽ DŽ H .U . is p. If either GŽ H . or GŽ H U . is trivial, then either H U or H, respectively, contains a central grouplike element. Hence, as in the proof of Lemma 4, H is trivial. Therefore, GŽ H . and GŽ H U . both have order p. Since DŽ H . is of Frobenius type wEGx, and since by Theorem 1, q s 1Žmod p ., we must have that p 2 q 2 s p q ap 2 q bq 2 , where a s
SEMISIMPLE HOPF ALGEBRAS OF DIMENSION
pq
667
Ž q 2 y 1.rp and b s pŽ p y 1.. In particular, DŽ H . has irreducible representations of dimensions p and q. We wish to show that if V and U are two irreducible representations of DŽ H . of dimension p, then V m U is a direct sum of 1-dimensional representations of DŽ H . and p-dimensional irreducible representations of DŽ H . only. Indeed, by Lemma 2, either V m U does not contain any 1-dimensional representation or it must contain at least two different 1-dimensional representations. But if it contains two different 1-dimensional representations, then since the 1-dimensional representations of DŽ H . form a cyclic group of order p, it follows that V m U contains all of the p 1-dimensional representations of DŽ H .. We conclude that either p 2 s ap q bq or p 2 s p q ap q bq. At any rate b s 0, and the result follows. Therefore, the subcategory C of RepŽ H . generated by the 1- and p-dimensional representations of DŽ H . gives rise to a quotient Hopf algebra A of DŽ H . of dimension pq 2 whose irreducible representations have either dimension 1 or p Ž A is the quotient of DŽ H . over the Hopf ideal I s ÝV f C End k Ž V ... Let f denote the corresponding surjective homomorphism, and consider the sequences iH
f
H ¨ D Ž H . ª A and
i HU c o p
f
H U c o p ¨ D Ž H . ª A.
Since f Ž H ., f Ž H U c o p . are Hopf subalgebras of A, it follows that they have dimension 1, p, q, or pq wNZx. If, say dimŽ f Ž H .. s q, then f Ž H . s kC q is a group algebra wZx. Since Ž kC q .U ( kC q , the surjection of Hopf algebras f : H ª kC q gives rise to an inclusion of Hopf algebras f U : kCq ª H U . Since f U Ž kC q . ; kGŽ H U ., it follows that q divides < GŽ H U .< s p, which is impossible. Similarly, dimŽ f Ž H U c o p .. / q, and we conclude that f Ž H ., f Ž H U c o p . have dimension 1, p, or pq. Since A s f Ž H . f Ž H U c o p . is of dimension pq 2 , it follows that the dimensions must equal pq and hence that H and H U c o p can be considered as Hopf subalgebras of A. Furthermore, GŽ H . and GŽ H U c o p . are mapped onto GŽ A.; otherwise < GŽ A.< is divisible by p 2 , which is impossible. Therefore GŽ H . l GŽ H U c o p . s GŽ A. in A. Since H / H U c o p in A, this implies that H l H U c o p s kGŽ A. s kC p . We are ready to prove our main result. THEOREM 6. Let H be a semisimple Hopf algebra of dimension pq o¨ er k, where p and q are distinct prime numbers. Then H is tri¨ ial. Proof. Suppose, on the contrary, that H is nontrivial. We wish to prove that H and H U are of Frobenius type. Indeed, consider the Hopf algebra quotient A, which exists by Lemma 5. For any 1-dimensional representa-
668
ETINGOF AND GELAKI
tion x of A, we define the induced representations Vqx s A mH x and
Vyx s A mH U x .
Note that since finite-dimensional Hopf algebras are free over any of their Hopf subalgebras wNZx, it follows that Vqx s H U mH l H U x s H U mk C p x , and Vyx s H mH l H U x s H mk C p x . We first show that if x is nontrivial, then x is nontrivial on H and H U as well. Indeed, for any nontrivial 1-dimensional representation x ˜ , we have by Frobenius reciprocity that Hom AŽ Vqx , x ˜ . s Hom H Ž x , x˜ .. Therefore, if x is trivial on H, then x ˜ is also trivial on H Žsince x˜ s x l as x generates the group of 1-dimensional representations of A., and Hom AŽ Vqx , x ˜ . s k. But then Vqx is a sum of p 1-dimensional representations and p-dimensional irreducible representations, which contradicts the fact that p does not divide q. Similarly, x is nontrivial on H U . Next, since dimŽHom AŽ Vqx , x ˜ .. s dimŽHom H Ž x , x˜ .. s dx , x˜ for any 1dimensional representation x ˜ , and dim Vqx s q, it follows that Vqx s x [ V1 [ ??? [ VŽ qy1.r p , where Vi is a p-dimensional irreducible representation of A for all 1 F i F Ž q y 1.rp. We wish to show that the Vi ’s do not depend on x . Indeed, let x be a nontrivial 1-dimensional representation of A. Then any 1-dimensional representation of A is of the form x l for some 0 F l F p y l 1, and Vqx s Vqx m x ly1 s x l [ Ž V1 m x ly1 . [ ??? [ Ž VŽ qy1.r p m x ly1 .. But by Lemma 2, x ly1 ; Vi m ViU for all i, and hence Vi m x ly1 ( Vi , which proves the claim. Similarly, Vyx s x [ W1 [ ??? [ WŽ qy1.r p , where Wi is a p-dimensional irreducible representation of A for all 1 F i F Ž q y 1.rp. Finally, we wish to show that the set C s x < H U < x is a 1-dimensional representation of A4 j Vi < H U < 1 F i F Ž q y 1 . rp 4 is a full set of representatives of the irreducible representations of H U . Indeed, consider the regular representation of H U : H U s H U mk C p kC p s s
[H x
[ x [ [ pV x
i
i < HU .
U
mk C p x s
[V x
x q< H U
SEMISIMPLE HOPF ALGEBRAS OF DIMENSION
pq
669
Since in the regular representation any representation can occur no more times than its dimension, we get that Vi < H U are irreducible. By summation of degrees, C is a full set of representatives, and since dimŽ Vi < H U . s p, we have that H U is of Frobenius type. Similarly, H is of Frobenius type. But Theorem 3.5 in wGWx states that if this is the case, then H is trivial. This concludes the proof of the theorem.
ACKNOWLEDGMENTS The first author was supported by a National Science Foundation postdoctoral fellowship. The second author was supported by a Rothschild postdoctoral fellowship. We thank Sonia Natale, who noticed an incompleteness in the proof of Lemma 2 in the original version of the paper.
REFERENCES wEGx wGWx wM1x wM2x wNZx wRx wZx
P. Etingof and S. Gelaki, Some properties of finite-dimensional semisimple Hopf algebras, Math. Res. Lett. 5 Ž1998., 191]197. S. Gelaki and S. Westreich, On semisimple Hopf algebras of dimension pq, Proc. Amer. Math. Soc. to appear. A. Masuoka, Semisimple Hopf algebras of dimension 2 p, Comm. Algebra 23Ž5. Ž1995., 1931]1940. A. Masuoka, The p n theorem for semisimple Hopf Algebras, Proc. Amer. Math. Soc. 124 Ž1996., 735]737. W. D. Nichols and M. B. Zoeller, A Hopf algebra freeness theorem, Amer. J. Math. 111 Ž1989., 381]385. D. E. Radford, Minimal quasitriangular Hopf algebras, J. Algebra 157 Ž1993., 281]315. Y. Zhu, Hopf algebras of prime dimension, Internat. Math. Res. Notices 1 Ž1994., 53]59.