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Crossed products of semisimple cocommutative Hopf algebras


Author: William Chin
Journal: Proc. Amer. Math. Soc. 116 (1992), 321-327
MSC: Primary 16W30; Secondary 16S30, 16S35
DOI: https://doi.org/10.1090/S0002-9939-1992-1100646-7
MathSciNet review: 1100646
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Abstract: We provide a short proof of an analog of Nagata's theorem for finite-dimensional Hopf algebras. The result, proved Hopf-algebraically by Sweedler and using group schemes by Demazure and Gabriel, says that a finite-dimensional cocommutative semisimple irreducible Hopf algebra is commutative. With mild base field assumptions such a Hopf algebra is just the dual of a $ p$-group algebra. We give en route an easy proof of a version of Hochschild's theorem on semisimple restricted enveloping algebras.

Let $ R{\char93 _t}H$ denote a crossed product with an invertible cocycle $ t$, where $ H$ is a semisimple cocommutative Hopf algebra $ H$ over a perfect field. The result above is applied to show that $ R{\char93 _t}H$ is semiprime if and only if $ R$ is $ H$-semiprime. The approach relies on results on ideals of the crossed product that are stable under the action of the dual of $ H$ and the Fisher-Montgomery theorem for crossed products of finite groups.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1100646-7
Article copyright: © Copyright 1992 American Mathematical Society

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