Generalized cocommutativity of some Hopf algebras and their relationship with finite fields
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S. Yu. Spiridonova
Translated by: N. B. Lebedinskaya - St. Petersburg Math. J. 25 (2014), 855-868
- DOI: https://doi.org/10.1090/S1061-0022-2014-01319-2
- Published electronically: July 18, 2014
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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.References
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Bibliographic 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
- Received by editor(s): July 7, 2012
- Published electronically: July 18, 2014
- Additional Notes: Partially supported by RFBR (grant no. 12-01-00070)
- © Copyright 2014 American Mathematical Society
- Journal: St. Petersburg Math. J. 25 (2014), 855-868
- MSC (2010): Primary 16T05
- DOI: https://doi.org/10.1090/S1061-0022-2014-01319-2
- MathSciNet review: 3184611