Biperfect Hopf Algebras
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1 Ž. Journal of Algebra 3, doi:0.006 jabr , available online at http: on Biperfect Hopf Algebras Pavel Etingof Department of Mathematics, Rm -65, MIT, Cambridge, Massachusetts Shlomo Gelaki MSRI, 000 Centennial Dri e, Berkeley, California Robert Guralnick, Department of Mathematics, USC, Los Angeles, California and Jan Saxl D.P.M.M.S., Cambridge Uni ersity, Cambridge CB SB, United Kingdom Communicated by Susan Montgomery Received December 8, 999. INTRODUCTION Recall that a finite group is called perfect if it does not have non-trivial one-dimensional representations Ž over.. By analogy, let us say that a finite-dimensional Hopf algebra H over is perfect if any one-dimensional 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 double DŽ H. is biperfect. Partially supported by the NSF. The authors thank MSRI for its support $35.00 Copyright 000 by Academic Press All rights of reproduction in any form reserved.
2 33 ETINGOF ET AL. It is not easy to construct a biperfect Hopf algebra of dimension. 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 counterexample is the quantum double DŽ H... BICROSSPRODUCTS Let G be a finite group. If G and G are subgroups of G such that G GG and G G, we say that G GG is an exact factoriza- tion. In this case G can be identified with G G, and G can be identified with G G as sets, so G is a G -set and G is a G -set. Note that if G GG is an exact factorization, then G GG is also an exact factorization by taking the inverse elements. Following Kac K and Takeuchi T, one can construct a semisimple Hopf algebra from these data as follows. Consider the vector space H G G. Introduce a product on H by Ž a.ž b. Ž a. ab Ž. for all, G and a, b G. Here denotes the associated action of G on the algebra G, and Ž a. is the multiplication of and a in the algebra G. Identify the vector spaces Ž. Ž. H H G G G G Hom Ž G G, G G. in the usual way, and introduce a coproduct on H by for all G G on G. Ž Ž a.. Ž b c. Ž bc. a b a Ž., a G, and b, c G. Here denotes the action of THEOREM. K, T. There exists a unique semisimple Hopf algebra structure on the ector space H G G with the multiplication and comultiplication described in Ž. and Ž.. The Hopf algebra H is called the bicrossproduct Hopf algebra associated with G, G, G and is denoted by HG, Ž G, G..
3 BIPERFECT HOPF ALGEBRAS 333 Ž. Ž. THEOREM. M. HG, G, G HG, G, G as Hopf algebras. We are ready now to prove our first result. THEOREM.3. HG, Ž G, G. is biperfect if and only if G, G are selfnormalizing perfect subgroups of G. Proof. It is well known that the category of finite-dimensional representations of HG, Ž G, G. is equivalent to the category of G-equivariant vector bundles on G, and hence that the irreducible representations of HG, Ž G, G. are indexed by pairs Ž V, x. where x is a representative of a G -orbit in G, and V is an irreducible representation of Ž G. x, where Ž G. x is the isotropy subgroup of x. Moreover, the dimension of the corresponding irreducible representation is dimž V. G Ž G. x. Thus, the one-dimensional representations of HŽ G, G, G. are indexed by pairs Ž V, x. where x is a fixed point of G on G G G Ži.e., x N Ž G. G. G, and V is a one-dimensional representation of G. The result follows now using Theorem.. 3. THE EXAMPLE By Theorem.3, in order to construct an example of a biperfect semisimple Hopf algebra, it remains to find a finite group G which admits an exact factorization G GG, where G, G are self-normalizing perfect subgroups of G. Amazingly the Mathieu group G M4 of degree 4 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 argument below. We suspect that not only is M4 the smallest example but it may be the only finite simple group with a factorization with all the needed properties. THEOREM 3.. The group G contains a subgroup G PSLŽ, 3., and a subgroup G Ž. 4 Ž. A7 where A7 acts on 4 ia the embedding A A SLŽ 4,. AutŽŽ These subgroups are perfect self-normalizing and G admits an exact factorization G GG. In particular, HG, Ž G, G. is biperfect. Proof. The order of G is , and G has a transitive permutation representation of degree 4 with point stabilizer C M 3.It is known Žsee AT. that G contains a maximal subgroup G PSLŽ, 3. Žthe elements of PSLŽ, 3. are regarded as fractional linear transformations on the projective line Ž F.. 3 and that G is transitive in the degree 4 representation. Thus, G GC.
4 334 ETINGOF ET AL. LEMMA. G is perfect and self-normalizing. Proof. This is clear, since G is maximal and not normal in the simple group G. It is known that C contains a maximal subgroup G Ž. 4 A Ž 7 see AT.. LEMMA. G is perfect. Proof. Note that E Ž. 4 is the unique minimal normal subgroup of G, E is noncentral, and G E is simple. Thus, G is perfect. LEMMA 3. G is self-normalizing. Proof. We note that G is a subgroup of F E A8 which is a maximal subgroup of G Žsee AT.. Since E is the unique minimal normal subgroup of G, it follows that N Ž G. is contained in N Ž E. G G. Since F normalizes E and is maximal, F N Ž E. G. Since G is a maximal subgroup of F and is not normal in F, G is self-normalizing. LEMMA 4. G G G is an exact factorization. Proof. Since G G G, it suffices to show that G GG. Let T be the normalizer of a Sylow 3-subgroup. So T has order 3 Ž T is at least this large since this is the normalizer of a Sylow 3-subgroup of G ; on the other hand, this is also the normalizer of a Sylow 3-subgroup in A which contains G. 4. The subgroup of order 3 has a unique fixed point which must be T-invariant in the degree 4 permutation representation of G. Moreover, T is also contained in some conjugate of G Ž since the normalizer of a Sylow 3-subgroup of G has the same form and all Sylow 3-subgroups are conjugate.. So replacing G and C by conjugates, we may assume that T G C. Since T and G have relatively prime orders and C T G, it follows that C TG. Thus, G GC GTG GG, as required. Finally, by Theorem.3, HG, Ž G, G. is biperfect. Remark 3.. One characterization of the Mathieu group is that it is the automorphism group of a certain Steiner system. The group G is the stabilizer of a flag in the Steiner system. Remark 3.3. Given an example of a biperfect Hopf algebra H, one has also an example of a self-dual biperfect Hopf algebra. Indeed, H H is such a Hopf algebra. Ž. QUESTION 3.4. Does there exist a biperfect Hopf algebra which is not semisimple? Which has odd dimension? Ž. Do there exist biperfect Hopf algebras of dimension less than M? 4
5 BIPERFECT HOPF ALGEBRAS 335 Ž. 3 Does there exist a nonzero finite-dimensional biperfect Lie bialgebra Žsee, e. g., ES, Sects., 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 which the cocommutator is injecti e? Remark 3.5. Ž. A non-semisimple biperfect Hopf algebra H must have 4 even dimension, since S I and tržs. 0. Note that an odd-dimensional biperfect Hopf algebra cannot be of the form HG, Ž G, G. since groups of odd order are solvable. Ž. A positive answer to question Ž. 3 implies a positive answer to question Ž. 4 by the double construction. Ž. 3 Questions Ž. 3 and Ž. 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, 985. EG P. Etingof and S. Gelaki, The classification of triangular semisimple and cosemisimple Hopf algebras over an algebraically closed field, Internat. Math. Res. Notices 5 Ž 000., EK P. Etingof and D. Kazhdan, Quantization of Lie bialgebras, II, Selecta Math. 4 Ž 998., 3 3. ES P. Etingof and O. Schiffmann, Lectures on Quantum Groups, Lectures in Mathematical Physics, International Press, Boston, MA, 998. K G. I. Kac, Extensions of groups to ring groups, Math. USSR. Sb. 5, No. 3 Ž M S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 30 Ž 990., R D. E. Radford, Minimal quasitriangular Hopf algebras, J. Algebra 58 Ž 993., T M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9, No. 8 Ž 98.,
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