## Abstract

We show that actions of finite-dimensional semisimple commutative Hopf algebras H on H-module algebras A are essentially group-gradings. Moreover we show that the centralizer of H in the smash product A # H equals A^{H} ⊗ H. Using these we invoke results about group graded algebras and results about centralizers of separable subalgebras to give connections between the ideal structure of A, A^{H} and A # H. Examples of the above occur naturally when one considers: (1) finite abelian groups G of automorphisms of an algebra A with G ^{−1} ɛ A; (2) G-graded algebras, for finite groups G; (3) finite-dimensional restricted Lie algebras L, with semisimple restricted enveloping algebra u(L), acting as derivations on an algebra A.

Original language | English |
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Pages (from-to) | 159-164 |

Number of pages | 6 |

Journal | Bulletin of the London Mathematical Society |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1986 |

## ASJC Scopus subject areas

- General Mathematics