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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

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)
Full Text: DOI
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