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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Semisimplicity criteria for irreducible Hopf algebras in positive characteristic


Author: Akira Masuoka
Journal: Proc. Amer. Math. Soc. 137 (2009), 1925-1932
MSC (2000): Primary 16W30
DOI: https://doi.org/10.1090/S0002-9939-09-09863-3
Published electronically: February 9, 2009
MathSciNet review: 2480272
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Abstract: We prove that a finite-dimensional irreducible Hopf algebra $ H$ in positive characteristic is semisimple if and only if it is commutative and semisimple if and only if the restricted Lie algebra $ P(H)$ of the primitives is a torus. This generalizes Hochschild's theorem on restricted Lie algebras, and also generalizes Demazure and Gabriel's and Sweedler's results on group schemes in the special but essential situation with a finiteness assumption added.


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Additional Information

Akira Masuoka
Affiliation: Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan
Email: akira@math.tsukuba.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-09-09863-3
Keywords: Hopf algebra in positive characteristic, restricted Lie algebra
Received by editor(s): July 14, 2008
Published electronically: February 9, 2009
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.