Hopf Galois extensions, smash products, and Morita equivalence
type: Item
instance of: scholarly article
title: Hopf Galois extensions, smash products, and Morita equivalence
zbMATH Open document ID: 0706.16023
DOI: 10.1016/0021-8693(90)90274-R
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let \(H\) be a finite-dimensional Hopf algebra over a field, \(A\) an \(H\)-module algebra, \(A^ H\) its ring of invariants. The authors discuss various relations between \(A^ H\), \(A\) and the smash-product \(A{\#}H\), in particular concerning \(H\) (or \(H^*\))-Galois extensions, Morita contexts and normal bases. We only mention a few which are of more general interest. \(A/A^ H\) is Galois if \(A{\#}H\) is simple (it is also assumed here that \(A\) is left irreducible over \(A{\#}H\) and has finite left Goldie rank). They show that \(A^ H\) and \(A{\#}H\) are connected via a Morita context. They also show that if \(A{\#}H\) is simple artinian, then \(A/A^ H\) has a normal basis. A major application is for \(A=D\), a division ring. They show that \((D:D^ H)=\dim D\) is equivalent to each of an extensive list of conditions, e.g. \(D/D^ H\) is an \(H^*\)-Galois extension or \(D/D^ H\) has the normal bases property. In this case, \(D{\#}H\) is isomorphic to a full matrix algebra over \(D^ H\).
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S40
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 4158888
zbMATH Keywords: finite-dimensional Hopf algebras
zbMATH Keywords: \(H\)-module algebras
zbMATH Keywords: rings of invariants
zbMATH Keywords: smash products
zbMATH Keywords: Galois extensions
zbMATH Keywords: Morita contexts
zbMATH Keywords: normal bases
author: Miriam Cohen
author: Davida Fischman
author: Susan Montgomery
reviewed by: Earl J. Taft
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4158888