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On Finite-Dimensional Pointed Hopf Algebras Over Simple Groups On ¯nite-dimensional pointed Hopf algebras over simple groups Nicol¶asAndruskiewitsch Universidad de C¶ordoba,Argentina VII workshop in Lie theory and its appplications C¶ordoba,November 30th, 2009. Joint work with Fernando Fantino, Mat¶³asGra~naand Leandro Vendramin 1 Plan of the talk. I. The problem. II. Main results. III. The schemes of the proofs. References: Finite-dimensional pointed Hopf algebras with alternating groups are trivial. (N. A., F. Fantino, M. Gra~na,L. Vendramin). arXiv:00812.4628v4. 26 pages. Submitted. Pointed Hopf algebras over the sporadic simple groups. (N. A., F. Fantino, M. Gra~na,L. Vendramin). In preparation. 2 I. The problem. C alg. closed ¯eld char. 0 De¯nition. (H; m; ¢), Hopf algebra: ² (H; m) alg. with unit 1, ² ¢ : H ! H­H morphism of algebras (coproduct), ² ¢ coasociative with counit ", ² 9S : H ! H "antipode" s. t. m(S­ id)¢ = idH = m(id ­S)¢. 3 Examples. ² ¡ group, C¡ = group algebra = vector space with basis eg (g 2 ¡ ) and product egeh = egh. It becames a Hopf algebra with coproduct ¢(eg) = eg ­ eg and ¡1 antipode S(eg) = eg ² g Lie algebra, U(g) = universal algebra enveloping of g It becames a Hopf algebra with coproduct ¢(x) = x ­ 1 + 1 ­ x and antipode S(x) = ¡x, x 2 g 4 Suppose now that the group ¡ acts on a Lie algebra g by Lie algebra automorphisms. Let U(g)#C¡ = U(g)­C¡ as vector space, with tensor prod- uct comultiplication and semi-direct product multiplication. This is a Hopf algebra. Theorem. (Cartier, Kostant, Milnor-Moore). If H is a cocommutative Hopf algebra, H ' U(g)#C¡ . 5 De¯nition. A Hopf algebra H is pointed if any irreducible co- module (= representation of the dual Hopf algebra) has dimen- sion 1. For any Hopf algebra H, G(H) = fg 2 H ¡ 0 : ¢(g) = g­gg is a group. Then `pointed' means CG(H) ' the coradical of H. ² H = U(g)#C¡ is pointed with G(H) ' ¡ ; in particular, the group algebra C¡ is pointed; ² the quantum groups of Drinfeld-Jimbo and the ¯nite-dimensio- nal variations of Lusztig are pointed. 6 g simple Lie algebra, Cartan matrix (aij)1·i;j·θ; q root of 1, order N ¯nite quantum group uq(g) = Chk1; : : : ; kθ; e1; : : : ; eθ; f1; : : : ; fθi relations: N kikj = kjki; ki = 1; ¡1 diaij ¡1 ¡diaij kiejki = q ej; kifjki = q fj; 1¡a adc(ei) ij (ej) = 0; i 6= j 1¡a adc(fi) ij (fj) = 0; i 6= j ¡diaij 2 eifj ¡ q fjei = ±ij(1 ¡ ki ); i < j; i j N N e® = 0; f® = 0; ¢(g) = g­g; ¢(xi) = gi­xi + xi­1: dim g uq(g) is a pointed Hopf algebra, dim. N . Here adc(xi)(xj) = xixj ¡ qijxjxi: 7 An important part of the classi¯cation of all ¯nite-dimensional Hopf algebras over C is the following. Problem I. Classify all ¯nite-dimensional pointed Hopf algebras. Approach group-by-group. For a given ¯nite group ¡ , classify all ¯n.-dim. pointed Hopf algebras H such that G(H) ' ¡ . ² If ¡ is abelian and the prime divisors of ¡ are > 5, then the classi¯cation is known. N. A. and H.-J. Schneider, On the classi¯cation of ¯nite-dimensional pointed Hopf algebras, Ann. Math., to appear. The outcome is that all are variations of the Lusztig's small quantum groups. For small prime divisors, variations of small quantum supergroups appear. 8 In this talk we shall consider ¡ simple. A landmark in mathematics is the classi¯cation of ¯nite simple groups: any ¯nite simple group is isomorphic to one of ² a cyclic group Zp, of prime order p; ² an alternating group An, n ¸ 5; ² a ¯nite group of Lie type; ² a sporadic group { there are 27 of them (including the Tits group); the most prominent is the Monster. 9 II. Main results. We shall say that a ¯nite group ¡ collapses if for any ¯n.-dim. pointed Hopf algebra H, with G(H) ' ¡ , then H ' C¡ . Theorem I. If ¡ is either ² the alternating group An, n ¸ 5; ² or else a sporadic simple group, di®erent from the Fischer group F i22, the Baby Monster B, or the Monster M; then ¡ collapses. 10 Comments. The groups F i22, B and M do not admit so far any ¯n.-dim. pointed Hopf algebra (except the group algebra) but the com- putations are very hard. We are working in the same problem for ¯nite groups of Lie type. The proof lies on reductions to questions on conjugacy classes; then these questions are checked (for sporadic groups) using GAP. In order to state our other main results, and also to explain these reductions, we need to present the notion of Nichols algebra. 11 Nichols algebras. Suppose that the group ¡ acts linearly on a vector space V , hence on the free Lie algebra L(V ) by Lie algebra automorphisms. Since U(L(V )) ' T (V ), we have the Hopf algebra T (V )#C¡ . Variation. What else is needed to have Hopf algebra T (V )#C¡ = T (V )­C¡ as vector space, but with semi-direct product comultiplication and semi-direct product multiplication? 12 De¯nition. A Yetter-Drinfeld module over ¡ is a vector space V provided with ² a linear action of ¡ , ² a ¡ -grading V = ©g2¡ Vg, such that h ¢ Vg = Vhgh¡1, for all g; h 2 ¡ . Then T (V )#C¡ = T (V )­C¡ as vector space, but with semi- direct product comultiplication and semi-direct product mul- tiplication is a Hopf algebra. Here ¢(v) = v­1 + g­v, g 2 ¡ , v 2 Vg. 13 If V is a Yetter-Drinfeld module over ¡ , then H = T (V )#C¡ is pointed with G(H) ' ¡ . n Fact. There exists a homogeneous ideal J = ©n¸2J of T (V ) such that ² the quotient T (V )#C¡=J #C¡ is a Hopf algebra, ² J is maximal with respect to this property. The Nichols algebra is B(V ) = T (V )=J . If ¡ is ¯nite and dim B(V ) < 1, then T (V )#C¡=J #C¡ ' B(V )#C¡ is a ¯n.-dim. pointed Hopf algebra over ¡ . 14 Conversely, if H is a ¯n.-dim. pointed Hopf algebra over a ¯nite group ¡ , then there exists a Yetter-Drinfeld module V canonically attached to H s. t. dim B(V ) < 1. Problem II. For a given ¯nite group ¡ , classify all Yetter- Drinfeld modules V s. t. dim B(V ) < 1. Since any Yetter-Drinfeld module is semisimple, the question splits into two cases: (i) V irreducible, (ii) V direct sum of (at least 2) irreducibles. 15 It turns out that irreducible Yetter-Drinfeld modules are param- eterized by pairs (O; ½), where ² O a conjugacy class of G, ² ½ an irreducible repr. of the centralizer CG(σ) of σ 2 O ¯xed. If M(O; ½) denotes the irreducible Yetter-Drinfeld module cor- responding to a pair (O; ½) and V is the vector space a®ording the representation ½, then M(O; ½) ' IndG ½ with the grading CG(σ) given by the identi¯cation IndG ½ = CG ­ V ' CO ­ V . CG(σ) CG(σ) C 16 The Nichols algebra of M(O; ½) is denoted B(O; ½). Problem II bis. For a given ¯nite group ¡ , ¯nd all (O; ½) s. t. dim B(O; ½) < 1. Warning. Nichols algebras are very di±cult to compute; it is not known if the ideal of relations is ¯nitely-generated or not; in all known cases it is, but the degrees of the de¯ning relations can be arbitrarily high. 17 n n n Theorem II. Let m ¸ 5. Let σ 2 Sm be of type (1 1; 2 2; : : : ; m m), \ O the conjugacy class of σ; let ½ 2 CSm(σ). If dim B(O; ½) < 1, then the type of σ and ½ are in the following list: n ² (1 1; 2), ½1 = sgn or ², ½2 = sgn. ¡! ² (2; 3) in S5, ½2 = sgn, ½3 = Â0. 3 ¡! ¡! ² (2 ) in S6, ½2 = Â1 ­ ² or Â1 ­ sgn. 4 Actually, the orbit of type (1 ; 2) in S6 is isomorphic to that of 3 type (2 ), because of the outer automorphism of S6. The Nichols algebras corresponding to types (1n1; 2), and these representations, were considered by Fomin and Kirillov in relation with the quantum cohomology of the flag variety. They can not be treated by our methods. 18 Example: O = transpositions in G = Sn, s = (12), ½ = sgn n rk Relations dim B(V ) top 3 3 5 relations in degree 2 12 = 3.22 4 = 22 4 6 16 relations in degree 2 576 12 5 10 45 relations in degree 2 8294400 40 S3, S4: A. Milinski & H. Schneider, Contemp. Math. 267 (2000), 215{236. S. Fomin & K. Kirillov, Progr. Math. 172, Birkhauser, (1999), pp. 146{182. S5: [FK], plus web page of M. Gra~na. http://mate.dm.uba.ar/~matiasg/ Sn, n ¸ 6: open! 19 It is convenient to restate Problem II in terms of racks and co- cycles. For this, recall ¯rst that a braided vector space is a pair (V; c), where V is a vector space and c 2 GL(V ­ V ) is a solution of the braid equation: (c ­ id)(id ­c)(c ­ id) = (id ­c)(c ­ id)(id ­c): Any super vector space V = V0 © V1 is braided: c(v ­ w) = (¡1)jvjjwjw ­ v; for v; w 2 V . Any Yetter-Drinfeld module V is naturally a braided vector space: c(v ­ w) = g ¢ w ­ v; v 2 Vg; (g 2 G); w 2 V: 20 If V is a Yetter-Drinfeld module, then the Nichols algebra B(V ) depends only on the braiding c. This leads us to the consideration of a class of braided vector spaces where the study of the corresponding Nichols algebras is performed in a uni¯ed way.
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