Semisimplicity criteria for irreducible Hopf algebras in positive characteristic

2008/12/22 by Akira Masuoka, Masuoka, Akira
Mathematics · #16W30 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:16W30

paper · pdf · doi:10.48550/arxiv.0812.4115

8 pages; to appear in the Proc. Amer. Math. Soc

arxiv created 2008/12/22 · arxiv updated 2009/12/01

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 finiteness assumption added.

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