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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 $$A^ H=\{a\in A|$$ $$h\cdot a=\epsilon (h)a\}$$ 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 $$hx=\epsilon (h)x$$ for all $$h\in H$$, satisfying $$\epsilon$$ (x)$$\neq 0.$$
Section 1 obtains a Maschke-type theorem for smash products. Indeed it is shown that if H is semisimple and if $$W\subseteq V$$ 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 $$[B,_ BA_{A\#H},_{A\#H}A_ B,A\#H]$$ 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:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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