Semisimplicity criteria for irreducible Hopf algebras in positive characteristic

概要

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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.