A duality theorem for Hopf module algebras
type: Item
instance of: scholarly article
title: A duality theorem for Hopf module algebras
zbMATH Open document ID: 0589.16010
DOI: 10.1016/0021-8693(85)90099-7
author: Robert J. Blattner
author: Susan Montgomery
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let \(H\) be a Hopf algebra over a field \(F\), \(A\) an \(H\)-module algebra. The smash product algebra \(A\#H\) is an \(H^ o\)-module algebra for the continuous dual \(H^ o\) of \(H\). Under certain technical conditions, the authors prove that \((A\#H)\#U\cong A\otimes (H\#U)\) for \(U\) a Hopf subalgebra of \(H^ o\). This had been proved previously by \textit{M. Cohen} and \textit{S. Montgomery} [Trans. Am. Math. Soc. 282, 237-258 (1984; Zbl 0533.16001)] for \(H=FG\), the group algebra of a finite group \(G\), or \(H=(FG)^*\), \(U=H^*\). In both of these cases, \((A\#H)\#H^*\) and \(A\otimes (H\#H^*)\) are isomorphic to the matrix algebra \(M_ n(A)\), \(n=|G|\), and the authors' theorem yields \((A\#H)\#H^*\cong M_ n(A)\) for \(H\) finite-dimensional. The technical conditions in the authors' theorem require \(H\) and \(U\) to have bijective antipodes, that the action of \(H\) on \(A\) be \(U\)-locally finite and that \(U\) satisfy a certain symmetry condition with respect to \(H\): An example is given to show that the local finiteness condition is needed. The generality of the theorem allows the authors to treat universal enveloping algebras of Lie algebras and group algebras. A sharpened version of the theorem for residually linear \(H\), and \(U\) dense in \(H^*\), yields a dense embedding of \(H\#U\) in \(\text{End\,}H\), and has some interesting applications to \(U(L)\) for \(L\) a nilpotent finite-dimensional Lie algebra over \(F\) of characteristic zero, and to algebras \(A\) graded by a group \(G\) which is residually \(F\)-linear with finite conjugacy classes.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 3946379
zbMATH Keywords: Hopf algebras
zbMATH Keywords: smash products
zbMATH Keywords: continuous dual
zbMATH Keywords: matrix algebras
zbMATH Keywords: antipodes
zbMATH Keywords: universal enveloping algebras
zbMATH Keywords: Lie algebras
zbMATH Keywords: group algebras
zbMATH Keywords: dense embeddings
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90099-7
OpenAlex ID: W2050236291
cites work: Q3933122
cites work: Group-Graded Rings, Smash Products, and Group Actions
cites work: On representations of Lie algebras
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cites work: Simplicity of Smash Products
cites work: Duality for crossed products of von Neumann algebras
cites work: Q4406395
cites work: Complete Sets of Representations of Algebras
cites work: Q5590089
cites work: Q3329592
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description: scientific article; zbMATH DE number 3946379