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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Generalized cocommutativity of some Hopf algebras and their relationship with finite fields

Author: S. Yu. Spiridonova
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 25 (2013), nomer 5.
Journal: St. Petersburg Math. J. 25 (2014), 855-868
MSC (2010): Primary 16T05
Published electronically: July 18, 2014
MathSciNet review: 3184611
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Abstract: Semisimple finite-dimensional Hopf algebras with only one summand of dimension not equal to one are considered. The group of group-like elements in the dual Hopf algebra is assumed to have minimal order and to be cyclic. Under these restrictions it is proved that the Hopf algebra is cocommutative up to numerical coefficients in the comultiplication and the antipode. A natural relationship is established between such Hopf algebras and finite fields, and it is proved that these Hopf algebras exist only for $ n=p^k-1$, where $ n$ is the order of the group of group-like elements in the dual Hopf algebra, $ p$ is prime, and $ k$ is a positive integer.

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  • 1. V. A. Artamonov, On semisimple finite-dimensional Hopf algebras, Mat. Sb. 198 (2007), no. 9, 3-28; English transl., Sb. Math. 198 (2007), no. 9-10, 1221-1245. MR 2360805 (2008i:16049)
  • 2. V. A. Artamonov and I. A. Chubarov, Properties of some semisimple Hopf algebras, Algebras, Representations and Applications, Contemp. Math., vol. 483, Amer. Math. Soc., Providence, RI, 2009, pp. 23-26. MR 2497948 (2010k:16045)
  • 3. V. A. Artamonov, On semisimple Hopf algebras with few representations of dimension greater than one, Rev. Un. Mat. Argentina 51 (2010), no. 2, 91-105. MR 2840164 (2012h:16061)
  • 4. S. Montgomery, Hopf algebras and their actions on rings, CBMS Reg. Conf. Ser. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1993. MR 1243637 (94i:16019)
  • 5. S. Natale and J. Y. Plavnik, On fusion categories with few irreducible degrees, Algebra Number Theory 6 (2012), no. 6, 1171-1197. MR 2968637
  • 6. S. Yu. Spiridonova, On finite-dimensional semisimple Hopf algebras of dimension $ n(n+1)$, Mat. Zametki 91 (2012), no. 2, 253-269; English transl., Math. Notes 91 (2012), no. 1-2, 243-258.
  • 7. H. Zassenhaus, Über endliche Fastkörper, Abh. Math. Semin. Hamburg Univ. 11 (1935), 187-220. MR 3069653
  • 8. J. L. Zemmer, The additive group of an infinite near-field is Abelian, J. London Math. Soc. 44 (1969), 65-67. MR 0231902 (38:228)

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Additional Information

S. Yu. Spiridonova
Affiliation: Department of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie gory, GSP-1, Moscow 119991, Russia

Keywords: Semisimple Hopf algebras, group of group-like elements, cocommutativity in the wide sense finite fields
Received by editor(s): July 7, 2012
Published electronically: July 18, 2014
Additional Notes: Partially supported by RFBR (grant no. 12-01-00070)
Article copyright: © Copyright 2014 American Mathematical Society

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