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Biperfect Hopf Algebras

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Biperfect Hopf Algebras
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  Ž . Journal of Algebra  232,  331  335 2000doi:10.1006   jabr.2000.8405, available online at http:   www.idealibrary.com on Biperfect Hopf Algebras Pavel Etingof  1  Department of Mathematics, Rm 2-165, MIT, Cambridge, Massachusetts 02139 E-mail: etingof@math.mit.edu Shlomo Gelaki 2  MSRI, 1000 Centennial Dri    e, Berkeley, California 94720 E-mail: shlomi@msri.org Robert Guralnick 1, 2  Department of Mathematics, USC, Los Angeles, California 90089-1113 E-mail: guralnic@math.usc.edu andJan Saxl  D.P.M.M.S.,Cambridge Uni    ersity, Cambridge CB2 1SB, United Kingdom E-mail: J.Saxl@dpmms.cam.ac.uk Communicated by Susan Montgomery Received December 8, 1999 1. INTRODUCTIONRecall that a finite group is called perfect if it does not have non-trivial Ž . one-dimensional representations over    . By analogy, let us say that afinite-dimensional Hopf algebra  H   over    is  perfect  if any one-dimen-sional  H  -module is trivial. Let us say that  H   is  biperfect  if both  H   and  H     are perfect. Note that by R ,  H   is biperfect if and only if its quantum Ž . double  D H   is biperfect. 1 Partially supported by the NSF. 2 The authors thank MSRI for its support.3310021-8693  00 $35.00 Copyright   2000 by Academic Press  All rights of reproduction in any form reserved.  ETINGOF ET AL  .332It is not easy to construct a biperfect Hopf algebra of dimension   1.The goal of this note is to describe the simplest such example we know.The biperfect Hopf algebra  H   we construct is semisimple. Therefore, it     yields a negative answer to EG, Question 7.5 . Namely, it shows that EG,  Corollary 7.4 , stating that a triangular semisimple Hopf algebra over   has a non-trivial group-like element, fails in the quasitriangular case. The Ž . counterexample is the quantum double  D H   .2. BICROSSPRODUCTSLet  G  be a finite group. If   G  and  G  are subgroups of   G  such that 1 2 G  G G  and  G   G   1, we say that  G  G G  is an  exact factoriza - 1 2 1 2 1 2 tion . In this case  G  can be identified with  G  G  , and  G  can be 1 2 2 identified with  G  G  as sets, so  G  is a  G  -set and  G  is a  G  -set. Note 1 1 2 2 1 that if   G  G G  is an exact factorization, then  G  G G  is also an 1 2 2 1 exact factorization by taking the inverse elements.     Following Kac K and Takeuchi T , one can construct a semisimpleHopf algebra from these data as follows. Consider the vector space       H     G     G  . Introduce a product on  H   by 2 1     a      b      a        ab  1 Ž . Ž . Ž . Ž .    for all    ,      G  and  a ,  b  G  . Here    denotes the associated action 2 1    Ž . of   G  on the algebra    G  , and     a      is the multiplication of      and 1 2     a      in the algebra    G  . 2 Identify the vector spaces           H    H      G     G      G     G Ž . Ž . 2 2 1 1          Hom    G     G  ,   G     G Ž .   2 2 1 1 in the usual way, and introduce a coproduct on  H   by       a b   c      bc a   b  1   a  2 Ž . Ž . Ž . Ž . Ž .    for all       G  ,  a  G  , and  b ,  c  G  . Here    denotes the action of  2 1 2 G  on  G  . 2 1   T HEOREM  2.1 K, T .  There exists a unique semisimple Hopf algebra       structure on the     ector space H     G     G with the multiplication 2 1 Ž . Ž .  and comultiplication described in  1  and  2 .The Hopf algebra  H   is called the  bicrossproduct  Hopf algebra associ- Ž . ated with  G , G  , G  and is denoted by  H G , G  , G  . 1 2 1 2  BIPERFECT HOPF ALGEBRAS  333    Ž . Ž .  T HEOREM  2.2 M .  H G , G  , G    H G , G  , G as Hopf algebras . 2 1 1 2 We are ready now to prove our first result. Ž . T HEOREM  2.3.  H G , G  , G is biperfect if and only if G  , G are self  - 1 2 1 2  normalizing perfect subgroups of G .  Proof.  It is well known that the category of finite-dimensional repre- Ž . sentations of   H G , G  , G  is equivalent to the category of   G  -equivariant 1 2 1  vector bundles on  G  , and hence that the irreducible representations of  2 Ž . Ž .  H G , G  , G  are indexed by pairs  V  ,  x  where  x  is a representative of a 1 2 Ž . G  -orbit in  G  , and  V   is an irreducible representation of   G  , where 1 2 1  x Ž . G  is the isotropy subgroup of   x . Moreover, the dimension of the 1  x Ž .    Ž .   corresponding irreducible representation is dim  V G    G  . Thus, the 1 1  x Ž . one-dimensional representations of   H G , G  , G  are indexed by pairs 1 2 Ž . Ž Ž . . V  ,  x  where  x  is a fixed point of   G  on  G   G  G  i.e.,  x   N G   G  , 1 2 1  G  1 1 and  V   is a one-dimensional representation of   G  . The result follows now 1 using Theorem 2.2.3. THE EXAMPLEBy Theorem 2.3, in order to construct an example of a biperfectsemisimple Hopf algebra, it remains to find a finite group  G  which admitsan exact factorization  G  G G  , where  G  , G  are self-normalizing per- 1 2 1 2 fect subgroups of   G . Amazingly the Mathieu group  G   M   of degree 24 24 provides such an example! Once the example is found, it is not hard to verify. Still for the reader’s convenience we will give a complete argumentbelow.We suspect that not only is  M   the smallest example but it may be the 24 only finite simple group with a factorization with all the needed properties. Ž . T HEOREM  3.1.  The group G contains a subgroup G    PSL  2,23 ,  and a 1 Ž . 4 Ž . 4  subgroup G        A where A acts on      ia the embedding  2 2 7 7 2 Ž . ŽŽ . 4 .  A    A   SL  4,2    Aut    .  These subgroups are perfect self  -  normaliz - 7 8 2 Ž ing and G admits an exact factorization G  G G  .  In particular  ,  H G , G  , 1 2 1 . G is biperfect . 2  Proof.  The order of   G  is 2 10   3 3   5    7    11    23, and  G  has a transitivepermutation representation of degree 24 with point stabilizer  C   M   . It 23 Ž    . Ž . is known see AT that  G  contains a maximal subgroup  G    PSL  2,23 1 Ž Ž . the elements of   PSL  2,23 are regarded as fractional linear transforma- 1 Ž .. tions on the projective line    F   and that  G  is transitive in the degree 23 1 24 representation. Thus,  G  G C . 1  ETINGOF ET AL  .334L  EMMA   1.  G is perfect and self  -  normalizing  . 1  Proof.  This is clear, since  G  is maximal and not normal in the simple 1 group  G . Ž . 4 Ž It is known that  C  contains a maximal subgroup  G        A  see 2 2 7   .  AT .L  EMMA   2.  G is perfect . 2 Ž . 4  Proof.  Note that  E     is the unique minimal normal subgroup of  2 G  ,  E  is noncentral, and  G   E  is simple. Thus,  G  is perfect. 2 2 2 L  EMMA   3.  G is self  -  normalizing  . 2  Proof.  We note that  G  is a subgroup of   F    E   A  which is a 2 8 Ž    . maximal subgroup of   G  see AT . Since  E  is the unique minimal normal Ž . Ž . subgroup of   G  , it follows that  N G  is contained in  N E  . Since  F  2  G  2  G Ž . normalizes  E  and is maximal,  F    N E  . Since  G  is a maximal subgroup G  2 of   F   and is not normal in  F  ,  G  is self-normalizing. 2 L  EMMA   4.  G  G G is an exact factorization . 1 2       Proof.  Since  G   G G  , it suffices to show that  G  G G  . Let  T  1 2 1 2 Ž be the normalizer of a Sylow 23-subgroup. So  T   has order 11    23  T   is atleast this large since this is the normalizer of a Sylow 23-subgroup of   G  ; 1 on the other hand, this is also the normalizer of a Sylow 23-subgroup in .  A  which contains  G  . The subgroup of order 23 has a unique fixed point 24  which must be  T  -invariant in the degree 24 permutation representation of  Ž G . Moreover,  T   is also contained in some conjugate of   G  since the 1 normalizer of a Sylow 23-subgroup of   G  has the same form and all Sylow 1 . 23-subgroups are conjugate . So replacing  G  and  C  by conjugates, we 1 may assume that  T   G   C . 1      Since  T   and  G  have relatively prime orders and  C   T G  , it 2 2 follows that  C  TG  . Thus,  G  G C  G TG   G G  , as required. 2 1 1 2 1 2 Ž . Finally, by Theorem 2.3,  H G , G  , G  is biperfect. 1 2  Remark  3.2. One characterization of the Mathieu group is that it is theautomorphism group of a certain Steiner system. The group  G  is the 2 stabilizer of a flag in the Steiner system.  Remark  3.3. Given an example of a biperfect Hopf algebra  H  , one hasalso an example of a self-dual biperfect Hopf algebra. Indeed,  H    H   issuch a Hopf algebra. Ž . Q UESTION  3.4. 1  Does there exist a biperfect Hopf algebra which is not semisimple ?  Which has odd dimension ? Ž .    2  Do there exist biperfect Hopf algebras of dimension less than M   ? 24  BIPERFECT HOPF ALGEBRAS  335 Ž . 3  Does there exist a nonzero finite -  dimensional biperfect Lie bialgebra Ž     .  see ,  e .  g  .,  ES ,  Sects . 2, 3  for the theory of Lie bialgebras  ,  i .  e .,  a Lie bialgebra  g  such that both  g  and  g   are perfect Lie algebras ? Ž . 4  Does there exist a nonzero quasitriangular Lie bialgebra for whichthe cocommutator is injecti    e ? Ž .  Remark  3.5. 1 A non-semisimple biperfect Hopf algebra  H   must have 4 Ž  2 . even dimension, since  S    I   and tr  S   0. Note that an odd-dimen- Ž . sional biperfect Hopf algebra cannot be of the form  H G , G  , G  since 1 2 groups of odd order are solvable. Ž . Ž . 2 A positive answer to question 3 implies a positive answer to Ž . question 4 by the double construction. Ž . Ž . Ž . 3 Questions 3 and 4 are equivalent to the same questions about   QUE algebras, by the results of EK .REFERENCES    AT J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘Atlas of Finite Groups,’’ Clarendon, Oxford, 1985.   EG P. Etingof and S. Gelaki, The classification of triangular semisimple and cosemisimple Ž . Hopf algebras over an algebraically closed field,  Internat .  Math .  Res .  Notices  5  2000 ,223  234.    Ž . EK P. Etingof and D. Kazhdan, Quantization of Lie bialgebras, II,  Selecta Math .  4  1998 ,213  231.   ES P. Etingof and O. Schiffmann, ‘‘Lectures on Quantum Groups, Lectures in Mathemat-ical Physics,’’ International Press, Boston, MA, 1998.    Ž . K G. I. 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