Semisimplicity criteria for irreducible Hopf algebras in positive characteristic

TL;DR

This paper establishes that finite-dimensional irreducible Hopf algebras in positive characteristic are semisimple precisely when their primitive Lie algebra is a torus, extending classical theorems to this setting.

Contribution

It generalizes Hochschild's theorem and classical results on group schemes to finite-dimensional irreducible Hopf algebras in positive characteristic.

Findings

Semisimplicity characterized by primitive Lie algebra being a torus

Extension of Hochschild's theorem to Hopf algebras

Generalization of classical group scheme results

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.