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Pointed Hopf algebras over some sporadic simple groups. (English) Zbl 1200.16046

Summary: 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.

MSC:

16T05 Hopf algebras and their applications
16S34 Group rings

References:

[1] Andruskiewitsch, N.; Fantino, F., New techniques for pointed Hopf algebras, New developments in Lie theory and geometry, Contemp. Math., 491, 323-348 (2009) · Zbl 1194.16024
[2] Andruskiewitsch, N.; Fantino, F.; Graña, M.; Vendramin, L., Finite-dimensional pointed Hopf algebras with alternating groups are trivial · Zbl 1234.16019
[3] Andruskiewitsch, N.; Fantino, F.; Graña, M.; Vendramin, L., Pointed Hopf algebras over the sporadic groups · Zbl 1217.16026
[4] Andruskiewitsch, N.; Fantino, F.; Zhang, S., On pointed Hopf algebras associated to symmetric groups, Manuscripta Math., 128, 359-371 (2009) · Zbl 1169.16022
[5] Andruskiewitsch, N.; Graña, M., From racks to pointed Hopf algebras, Adv. Math., 178, 177-243 (2003) · Zbl 1032.16028
[6] Andruskiewitsch, N.; Heckenberger, I.; Schneider, H.-J., The Nichols algebra of a semisimple Yetter-Drinfeld module · Zbl 1214.16024
[7] Andruskiewitsch, N.; Schneider, H.-J., Pointed hopf algebras, (New Directions in Hopf Algebras. New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., vol. 43 (2002), Univ. Press: Univ. Press Cambridge), 1-68 · Zbl 1011.16025
[8] Andruskiewitsch, N.; Schneider, H.-J., On the classification of finite-dimensional pointed Hopf algebras, Ann. Math., 171, 375-417 (2010) · Zbl 1208.16028
[9] Fantino, F., On pointed Hopf algebras associated with Mathieu groups, J. Algebra Appl., 8, 633-672 (2009) · Zbl 1215.16019
[10] Freyre, S.; Graña, M.; Vendramin, L., On Nichols algebras over \(GL(2, F_q)\) and \(SL(2, F_q)\), J. Math. Phys., 48, 123513, 1-11 (2007) · Zbl 1153.81361
[11] Freyre, S.; Graña, M.; Vendramin, L., On Nichols algebras over \(PSL(2, q)\) and \(PGL(2, q)\), J. Algebra Appl., 9, 2, 195-208 (2010) · Zbl 1196.16029
[12] GAP - Groups, Algorithms, and Programming (2008), Version 4.4.12
[13] Heckenberger, I., Classification of arithmetic root systems, Adv. Math., 220, 59-124 (2009) · Zbl 1176.17011
[14] Heckenberger, I.; Schneider, H.-J., Root systems and Weyl groupoids for semisimple Nichols algebras, Proc. London Math. Soc. · Zbl 1210.17014
[15] Wilson, R. A.; Nickerson, S. J.; Bray, J. N., Atlas of finite group representations, 2005/6/7
[16] Wilson, R. A.; Parker, R. A.; Nickerson, S. J.; Bray, J. N.; Breuer, T., AtlasRep, A GAP Interface to the Atlas of Group Representations, Version 1.4 2008, Refereed GAP package
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