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Hopf algebra actions. (English) Zbl 0591.16005

Let H be a finite dimensional Hopf algebra over the field k and let A be an H-module algebra. Then the smash product A#H is a ring extension of A and the ring A H of H-invariants defined by A H ={aA| h·a=ϵ(h)a} is a subring of A. This paper studies the connection between these rings using the key fact that if H is semisimple, then it has a left integral, namely an element x with hx=ϵ(h)x for all hH, satisfying ϵ (x)0·

Section 1 obtains a Maschke-type theorem for smash products. Indeed it is shown that if H is semisimple and if WV are A#H-modules, then W is an A-direct summand of V if and only if it is an A#H-direct summand.

Section 2 introduces a Morita context [B, B A A#H , A#H A B ,A#H] where B=A H and with [, ] nondegenerate. Then criteria are given for (, ) to be nondegenerate and the context is used to relate A H to A#H. The paper also proposes an interesting question. Namely if H is a finite dimensional semisimple Hopf algebra and if A is semiprime, must A#H also be semiprime? Some partial results are given.

Reviewer: D.S.Passman

MSC:
16W30Hopf algebras (assoc. rings and algebras) (MSC2000)
16D70Structure and classification of associative ring and algebras