Coactions, smash products, and Hopf modules
type: Item
instance of: scholarly article
title: Coactions, smash products, and Hopf modules
zbMATH Open document ID: 0814.16035
author: Chuanren Cai
author: Huixiang Chen
published in: Journal of Algebra
publication date: 1994-08-16T00:00:00Z
review text: Blattner and Montgomery (1985) have shown that the smash product \(H \# H^*\) of a finite dimensional Hopf algebra \(H\) with its dual \(H^*\) is naturally isomorphic to the algebra of linear transformations on the underlying vector space of \(H\). In the present paper the following generalization of this result for infinite dimensional Hopf algebras is proven: if \(H^{* \text{rat}}\neq 0\), then \(H\# H^{* \text{rat}}\) is a central simple algebra and also a dense ring of linear transformations with finite ranks on the vector space \(H\). Here \(H^{* \text{rat}}\) is the maximal rational submodule on the left (or, equivalently, right) regular module \(H^*\), and \(H^{* \text{rat}}\neq 0\) iff there exists a left (right) integral in \(H^*\). The proof uses correspondences between \(H\)-comodules and rational modules over a dense subalgebra of \(H^*\), and between relative Hopf modules and modules over certain smash products.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S40
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 614199
zbMATH Keywords: left integral
zbMATH Keywords: smash product
zbMATH Keywords: finite dimensional Hopf algebra
zbMATH Keywords: algebra of linear transformations
zbMATH Keywords: infinite dimensional Hopf algebras
zbMATH Keywords: central simple algebra
zbMATH Keywords: dense ring of linear transformations
zbMATH Keywords: maximal rational submodule
zbMATH Keywords: regular module
zbMATH Keywords: \(H\)-comodules
zbMATH Keywords: rational modules
zbMATH Keywords: relative Hopf modules
reviewed by: Yuri Bespalov
MaRDI profile type: Publication
full work available at URL: jabr.1994.1176
OpenAlex ID: W2016145206
DOI: 10.1006/JABR.1994.1176
description: scientific article; zbMATH DE number 614199