Hopf-Galois extensions and smash products
type: Item
instance of: scholarly article
title: Hopf-Galois extensions and smash products
zbMATH Open document ID: 0611.16006
DOI: 10.1016/0021-8693(87)90080-9
author: Kenji Yokogawa
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let K be a field, let H be a commutative and cocommutative Hopf algebra which is finite dimensional over K, and let \(H^*\) denote the dual Hopf algebra. If A and B are algebras over K, such that A is H-Galois and B is \(H^*\)-Galois; one may form a smash product A{\#}B, and A{\#}B is an \(H\otimes H^*\)-Galois algebra over K. Every \(H\otimes H^*\)-Galois algebra over K can be constructed from a unit in \(H\otimes H^*\otimes H\otimes H^*\), which is a 2-cocycle in the Harrison cohomology for \(H\otimes H^*\). The author describes a normalization for such 2- cocycles; and he associates with a normal 2-cocycle, a homomorphism \(\sigma\) of the bialgebra H into itself. The \(H\otimes H^*\)-Galois algebra corresponding to the 2-cocycle is a smash product if, and only if, this map \(\sigma\) is an isomorphism. The author has made effective use of Harrison and Amitsur cohomology to prove and generalize results which were previously derivable in the language of group schemes.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16E40
zbMATH DE Number: 3987491
zbMATH Keywords: commutative and cocommutative Hopf algebra
zbMATH Keywords: dual Hopf algebra
zbMATH Keywords: smash product
zbMATH Keywords: Galois algebra
zbMATH Keywords: Harrison cohomology
zbMATH Keywords: 2-cocycles
zbMATH Keywords: bialgebra
reviewed by: H. F. Kreimer
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90080-9
OpenAlex ID: W2041145893
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description: scientific article; zbMATH DE number 3987491