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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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Additional Information

S. Yu. Spiridonova
Affiliation: Department of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie gory, GSP-1, Moscow 119991, Russia
Email: sonya.spr@gmail.com

DOI: https://doi.org/10.1090/S1061-0022-2014-01319-2
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