POINTED HOPF ALGEBRAS OVER SOME SPORADIC SIMPLE GROUPS

N. ANDRUSKIEWITSCH, F. FANTINO, M. GRAÑA, L. VENDRAMIN

Abstract. Any finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group G, where G is either the Mathieu groups M_{22} or M_{24}, the Janko groups J_{1}, J_{2} or J_{3}, the Suzuki or the Held group, is a group algebra. Résumé. Soit G un des groupes de Mathieu M_{22} ou M_{24}, ou un des groupes de Janko J_{1}, J_{2} ou J_{3}, ou le groupe de Suzuki ou le groupe de Held. Soit H une algèbre de Hopf pointée de dimension finie dont le group des grouplikes est isomorphe a G. Alors H est isomorphe a l’algèbre de groupe de G.

1. Introduction

Let 𝕜 be a field of characteristic 0. In this Note, we announce a new contribution to the classification of finite-dimensional Hopf algebras over 𝕜. As is known, different classes of finite-dimensional Hopf algebras have to be studied separately because the pertaining methods are radically different. There is a method for pointed Hopf algebras (those whose coradical is a group algebra 𝕜G) that has been applied with satisfactory results when G is abelian [8]; an exposition of the method can be found in [7]. Recently, it appeared that many finite simple (or almost simple) groups G admit very few finite-dimensional, pointed non-semisimple Hopf algebras with coradical isomorphic to 𝕜G:

• Any finite-dimensional complex pointed Hopf algebra with group of grouplikes isomorphic to 𝔸_{m}, m ≥ 5, m ≠ 6, is a group algebra [2].

• Same for the groups SL(2, 2^{n}), n > 1 [10] and M_{20}, M_{21} = P SL(3, 4) [11].

• Most of the pointed Hopf algebras over the symmetric groups have infinite dimension, with the exception of a short list of open possibilities, see [4] and references therein (see 2.1 below).

We are presently studying finite-dimensional pointed Hopf algebras over sporadic simple groups. As part of our results, we have the following.

Theorem 1. Let G be any of the Mathieu groups M_{22}, M_{24}, the Janko groups J_{1}, J_{2}, J_{3}, the Suzuki group Suz, or the Held group He. If H is a finite-dimensional pointed Hopf algebra with G(H) ≃ G, then H ≃ 𝕜G.

2. Outline of the proof

A complete proof of Theorem 1 for the groups M_{22} and M_{24} is contained in [9]; the proof for the other groups will be included in [3].

We sketch now the proof in two main reductions. The first one has been explained in several places, with detail in [7], but we include a brief summary for completeness. We assume the reader familiar with the important notion of the Nichols algebra of a braided vector space, discussed at length in [7]. We remind that if U is a braided vector subspace of V, then 𝔅(U) ↪ 𝔅(V).

Date: February 4, 2009.

2000 Mathematics Subject Classification. 16W30; 17B37.

This work was partially supported by ANPCyT-Foncyt, CONICET, Ministerio de Ciencia y Tecnología (Córdoba) and Secyt (UNC).

2.1. A general reduction. Let G be a finite group, H a pointed Hopf algebra with G(H) ≃ G. Then there are two basic invariants of H, a Yetter-Drinfeld module V over 𝕜G (called the infinitesimal braiding of H) and its Nichols algebra 𝔅(V). We have |G| dim 𝔅(V) ≤ dim H. Therefore, the following statements are equivalent:

(1) If H is a finite-dimensional pointed Hopf algebra with G(H) ≃ G, then H ≃ 𝕜G.

(2) If V ≠ 0 is a Yetter-Drinfeld module over 𝕜G, then dim 𝔅(V) = ∞.

(3) If V is an irreducible Yetter-Drinfeld module over 𝕜G, then dim 𝔅(V) = ∞.

2.2. Looking at subracks. We focus on (3) above. The second reduction has been the basis of our recent papers. It starts from the well-known classification of irreducible Yetter-Drinfeld modules over 𝕜G by pairs (O, ρ), where O is a conjugacy class in G and ρ is an irreducible representation of the stabilizer G^{s} of a fixed point s ∈ O. Now, the definition of the Nichols algebra 𝔅(O, ρ) of the corresponding Yetter-Drinfeld module M(O, ρ) just depends on the braiding. If dim ρ = 1, then this braiding depends only on the rack O and a 2-cocycle q : O × O → 𝕜^{×} [5]. Namely, O is a rack with the product x ▷ y := xyx^{−1}, M(O, ρ) has a natural basis (e_{x})_{x∈O} and the braiding is given by c(e_{x} ⊗ e_{y}) = q_{xy}e_{x▷y} ⊗ e_{x}. If there exists a subrack X of O such that the Nichols algebra of the braided vector space defined by X and the restriction of q is infinite dimensional, then dim 𝔅(O, ρ) = ∞.

We recall some examples of racks which are relevant in this work.

(i) Abelian racks: those racks X such that x ▷ y = y for all x, y ∈ X.

(ii) D_{p}: the class of involutions in the dihedral group 𝔻_{p}, p a prime.

(iii) 𝔒: the class of 4-cycles in 𝕊_{4}.

(iv) Doubles of racks: if X is a rack, then X^{(2)} denotes the disjoint union of two copies of X each acting on the other by left multiplication.

We are interested in finding subracks which are abelian, or isomorphic to D_{p}^{(2)} or to 𝔒^{(2)}, by the following reasons:

(A) If X is abelian, then the corresponding braided vector space is of diagonal type. Braided vector spaces of diagonal type with finite-dimensional Nichols algebra where classified in [13]; thus, we just need to check if the matrix (q_{xy}) belongs or not to the list in [13].

(B) If X is isomorphic either to D_{p}^{(2)} or to 𝔒^{(2)}, then for some specific cocycles, the related Nichols algebras have infinite dimension [6, Ths. 4.7, 4.8].

Variations.

(a) If dim ρ > 1, similar arguments apply.

(b) Sometimes the rack X is not abelian, but the braided vector space produced by X and the 2-cocycle can be realized with an abelian rack, by a suitable change of basis.

(c) Let F < G be a subgroup, s ∈ F, O^{F}, resp. O^{G} the conjugacy class of s in F, resp. in G. If dim 𝔅(O^{F} , τ) = ∞ for any irreducible representation τ of F^{s}, then dim 𝔅(O^{G}, ρ) = ∞ for any irreducible representation ρ of G^{s}.

(d) A conjugacy class O is real if O = O^{−1}. It is quasireal if O = O^{m} for some integer m, 1 < m < N, where N is the order of the elements in O. The search of subracks isomorphic to D_{p}^{(2)} or to 𝔒^{(2)}, as well as the verification that the restriction of the cocycle q is as needed in (B), is greatly simplified in a real (quasireal) conjugacy class [1].

2.3. Computations. We now fix a sporadic group G.

• We extracted relevant information from the ATLAS [14] by using GAP [12] and the AtlasRep package [15].

• We checked with GAP [12] when a conjugacy class is real; the correspondence between conjugacy classes in a group G and in a subgroup H. We wrote GAP functions to find subracks of types (i),…,(iv).

These tools allow to apply the techniques sketched above to all pairs (O, ρ) and establish the validity of (3).

2.4. Final remarks. Some of the results presented here are part of the PhD theses of FF and LV.

References

[1] N. Andruskiewitsch and F. Fantino, New techniques for pointed Hopf algebras, Contemp.

Math. (to appear); arXiv:0803.3486 [math.QA].

[2] N. Andruskiewitsch and F. Fantino and M. Graña and L. Vendramin, Finitedimensional pointed Hopf algebras with alternating groups are trivial, arXiv:00812.4628

[math.QA].

[3] , Pointed Hopf algebras over the sporadic simple groups, in preparation.

[4] N. Andruskiewitsch and F. Fantino and S. Zhang, On pointed Hopf algebras associated to symmetric groups, Manuscripta Math. (to appear); arXiv:0807.2406 [math.QA].

[5] N. Andruskiewitsch and M. Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), 177 – 243.

[6] N. Andruskiewitsch and I. Heckenberger and H.-J. Schneider, The Nichols algebra of a semisimple Yetter-Drinfeld module, arXiv:0803.2430 [math.QA].

[7] N. Andruskiewitsch and H.-J. Schneider, Pointed Hopf Algebras, in “New directions in Hopf algebras”, 1–68, Math. Sci. Res. Inst. Publ. 43, Cambridge Univ. Press, 2002.

[8] , On the classification of finite-dimensional pointed Hopf algebras, Ann. Math. (to

appear); arXiv:math/0502157 [math.QA].

[9] F. Fantino, On pointed Hopf algebras associated with Mathieu groups. J. Algebra Appl. (to

appear); arXiv:0711.3142 [math.QA].

[10] S. Freyre and M. Graña and L. Vendramin, On Nichols algebras over GL(2, 𝔽_{q}) and SL(2, 𝔽_{q}), J. Math. Phys. 48 (2007), 123513, 1–11.

[11] On Nichols algebras over PSL(2, 𝔽_{q}) and PGL(2, 𝔽_{q}), in preparation.

[12] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.12 ; 2008,

(http://www.gap-system.org).

[13] I. Heckenberger, Classification of arithmetic root systems, Adv. Math. 220 (2009) 59–124.

[14] R.A. Wilson and S.J. Nickerson and J.N. Bray Atlas of finite group representations,

Version 3, http://brauer.maths.qmul.ac.uk/Atlas/v3/, 2005/6/7.

[15] R.A. Wilson and R.A. Parker and S.J. Nickerson and J.N. Bray and T. Breuer, AtlasRep, A GAP Interface to the Atlas of Group Representations, Version 1.4,

http://www.math.rwth-aachen.de/˜ Thomas.Breuer/atlasrep 2008, Refereed GAP package.

aFacultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba. CIEM – CONICET. Medina Allende s/n (5000) Ciudad Universitaria, Córdoba, Argentina

bDepartamento de Matemática – FCEyN, Universidad de Buenos Aires, Pab. I – Ciudad Universitaria (1428) Buenos Aires – Argentina

cInstituto de Ciencias, Universidad de Gral. Sarmiento, J.M. Gutierrez 1150, Los Polvorines (1653), Buenos Aires – Argentina

E-mail address: andrus@famaf.unc.edu.ar ^{a}
E-mail address: fantino@famaf.unc.edu.ar ^{a}
E-mail address: matiasg@dm.uba.ar ^{b}
E-mail address: lvendramin@dm.uba.ar ^{b,c}