On smash products of transitive module algebras.
type: Item
instance of: scholarly article
title: On smash products of transitive module algebras.
zbMATH Open document ID: 1215.16022
author: Shenglin Zhu
author: Caihong Wang
published in: Chinese Annals of Mathematics. Series B
publication date: 2010-11-17T00:00:00Z
review text: Let \(k\) be a field of characteristic zero. Let \(H\) be a Hopf algebra over \(k\). An \(H\)-module algebra is called transitive if \(A^H=k1_A\) and \(A\) has no proper \(H\)-ideal. The main result of the paper says that if \(H\) is a finite-dimensional semisimple Hopf algebra over \(k\) and \(A\) is an \(s\)-dimensional transitive \(H\)-module algebra with a \(1\)-dimensional ideal \(k\lambda\), then \(A\#H\) is isomorphic to \(M_s(N)\), where \(N=\{h\in H\mid\sum h_{(1)}\cdot\lambda\otimes h_{(2)}=\lambda\otimes h\}\). This result is applied to the case of \(A=k(X)\), the function algebra on a finite set \(X\). The authors present counterexamples of finite-dimensional transitive \(H\)-module algebras over a non-semisimple Hopf algebra for which \(A\#H\) is non-isomorphic to \(M_{\dim A}(N)\).
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16S40
zbMATH DE Number: 5816618
zbMATH Keywords: semisimple Hopf algebras
zbMATH Keywords: smash products
zbMATH Keywords: transitive module algebras
MaRDI profile type: Publication
full work available at URL: s11401-010-0586-3
OpenAlex ID: W1975803218
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DOI: 10.1007/S11401-010-0586-3
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description: scientific article; zbMATH DE number 5816618