Semisimplicity criteria for irreducible Hopf algebras in positive characteristic

Masuoka, Akira

doi:10.1090/S0002-9939-09-09863-3

Semisimplicity criteria for irreducible Hopf algebras in positive characteristic
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Proc. Amer. Math. Soc. 137 (2009), 1925-1932 Request permission

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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