Hopf algebra actions.

*(English)*Zbl 0591.16005Let 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.

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) |

##### Keywords:

finite dimensional Hopf algebra; Maschke-type theorem; smash products; direct summand; Morita context
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\textit{M. Cohen} and \textit{D. Fishman}, J. Algebra 100, 363--379 (1986; Zbl 0591.16005)

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