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 AH ⊗ 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, AH 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.
|Number of pages||6|
|Journal||Bulletin of the London Mathematical Society|
|State||Published - 1 Jan 1986|
ASJC Scopus subject areas
- Mathematics (all)