Pointed Hopf algebras over some sporadic simple groups

N. Andruskiewitsch; F. Fantino; M. Graña; L. Vendramin

doi:10.1016/j.crma.2010.04.023

Algebra/Group Theory

Pointed Hopf algebras over some sporadic simple groups
[Algèbres de Hopf pointées sur quelques groupes simples sporadiques]

Présenté par : Jean-Pierre Serre

N. Andruskiewitsch ¹ ; F. Fantino ¹ ; M. Graña ² ; L. Vendramin ^2,³

¹ Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba. CIEM – CONICET. Medina Allende s/n (5000) Ciudad Universitaria, Córdoba, Argentina
² Departamento de Matemática – FCEyN, Universidad de Buenos Aires, Pab. I – Ciudad Universitaria (1428) Buenos Aires, Argentina
³ Instituto de Ciencias, Universidad de Gral. Sarmiento, J.M. Gutierrez 1150, Los Polvorines (1653), Buenos Aires, Argentina

Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 605-608.

Résumé
Abstract

Soit G un groupe sporadique différent du groupe de Fischer ${Fi}_{22}$ , du bébé monstre B et du monstre M. Soit H une algèbre de Hopf complexe pointée de dimension finie dont le groupe des éléments dont le co-produit est égal au carré tensoriel est isomorphisme à G, alors H est isomorphe a l'algèbre de groupe de G.

Any finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group, with the possible exception of the Fischer group ${Fi}_{22}$ , the Baby Monster B and the Monster M, is a group algebra.

Reçu le : 2009-02-11
Accepté le : 2010-04-06
Publié le : 2010-05-10

DOI : 10.1016/j.crma.2010.04.023

Affiliations des auteurs :

N. Andruskiewitsch ¹ ; F. Fantino ¹ ; M. Graña ² ; L. Vendramin ^{2, 3}

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     author = {N. Andruskiewitsch and F. Fantino and M. Gra\~na and L. Vendramin},
     title = {Pointed {Hopf} algebras over some sporadic simple groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {605--608},
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     volume = {348},
     number = {11-12},
     year = {2010},
     doi = {10.1016/j.crma.2010.04.023},
     language = {en},
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N. Andruskiewitsch; F. Fantino; M. Graña; L. Vendramin. Pointed Hopf algebras over some sporadic simple groups. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 605-608. doi : 10.1016/j.crma.2010.04.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.023/

Bibliographie
Cité par

[1] N. Andruskiewitsch; F. Fantino New techniques for pointed Hopf algebras, New developments in Lie theory and geometry, Contemp. Math., Volume 491 (2009), pp. 323-348

[2] N. Andruskiewitsch; F. Fantino; M. Graña; L. Vendramin Finite-dimensional pointed Hopf algebras with alternating groups are trivial | arXiv

[3] N. Andruskiewitsch; F. Fantino; M. Graña; L. Vendramin Pointed Hopf algebras over the sporadic groups | arXiv

[4] N. Andruskiewitsch; F. Fantino; S. Zhang On pointed Hopf algebras associated to symmetric groups, Manuscripta Math., Volume 128 (2009), pp. 359-371

[5] N. Andruskiewitsch; M. Graña From racks to pointed Hopf algebras, Adv. Math., Volume 178 (2003), pp. 177-243

[6] N. Andruskiewitsch; I. Heckenberger; H.-J. Schneider The Nichols algebra of a semisimple Yetter-Drinfeld module | arXiv

[7] N. Andruskiewitsch; H.-J. Schneider Pointed hopf algebras, New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., vol. 43, Univ. Press, Cambridge, 2002, pp. 1-68

[8] N. Andruskiewitsch; H.-J. Schneider On the classification of finite-dimensional pointed Hopf algebras, Ann. Math., Volume 171 (2010), pp. 375-417

[9] F. Fantino On pointed Hopf algebras associated with Mathieu groups, J. Algebra Appl., Volume 8 (2009), pp. 633-672

[10] S. Freyre; M. Graña; L. Vendramin On Nichols algebras over $GL (2, F_{q})$ and $SL (2, F_{q})$ , J. Math. Phys., Volume 48 (2007) no. 123513, pp. 1-11

[11] S. Freyre; M. Graña; L. Vendramin On Nichols algebras over $PSL (2, q)$ and $PGL (2, q)$ , J. Algebra Appl., Volume 9 (2010) no. 2, pp. 195-208

[12] The GAP Group GAP – Groups, Algorithms, and Programming, 2008 http://www.gap-system.org (Version 4.4.12)

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

[14] I. Heckenberger; H.-J. Schneider Root systems and Weyl groupoids for semisimple Nichols algebras (Proc. London Math. Soc.) | arXiv

[15] R.A. Wilson; S.J. Nickerson; J.N. Bray Atlas of finite group representations http://brauer.maths.qmul.ac.uk/Atlas/v3/ (Version 3 2005/6/7)

[16] R.A. Wilson; R.A. Parker; S.J. Nickerson; J.N. Bray; T. Breuer AtlasRep, A GAP Interface to the Atlas of Group Representations http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep (Version 1.4 2008, Refereed GAP package)

Cité par Sources :

^☆ Some of the results presented here are part of the PhD theses of F.F. and L.V., work under the supervision of N.A. and M.G., respectively. This work was partially supported by ANPCyT-Foncyt, CONICET, Ministerio de Ciencia y Tecnología (Córdoba) and Secyt (UNC).

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