Semisimplicity criteria for irreducible Hopf algebras in positive characteristic

Details Semisimplicity criteria for irreducible Hopf algebras in positive characteristic Semisimplicity criteria for irreducible Hopf algebras in positive characteristic

2008-12-22

arxiv.org/abs/0812.4115v1

Mathematics

Rings and Algebras

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

Scientific paper

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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