A monoidal structure on the category of relative Hopf modules.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1277.16022
DOI: 10.1142/S0219498811005506
publication date: 2012-05-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.4802
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 16T10
zbMATH DE Number: 6037674
zbMATH Keywords: braided monoidal categories
zbMATH Keywords: comodule algebras
zbMATH Keywords: actions
zbMATH Keywords: coactions
zbMATH Keywords: relative Hopf modules
zbMATH Keywords: Yetter-Drinfeld modules
zbMATH Keywords: braided bialgebras
zbMATH Keywords: coalgebras
author: Daniel Bulacu
author: Stefaan Caenepeel
MaRDI profile type: Publication
arXiv ID: 1011.4802
cites work: Q3153375
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OpenAlex ID: W2963733562
published in: Journal of Algebra and its Applications
title: A monoidal structure on the category of relative Hopf modules.
review text: Let \(B\) be a bialgebra in a braided monoidal category \(\mathcal C\) and let \(A\) be a left \(B\)-comodule algebra. The objects of the category \(^B\mathcal C_A\) of relative \((B,A)\)-Hopf modules have a left \(B\)-coaction and a right \(A\)-action that satisfy the appropriate variant of the Hopf module compatibility condition.NEWLINENEWLINE If \(A\) is also a coalgebra and there is a given morphism \(\varphi\) from \(B\otimes A\) to \(A\), it is possible to use these structure elements to define a potential right \(A\)-action on the tensor product of two relative \((B,A)\)-Hopf modules. This tensor product has in addition the standard left \(B\)-coaction.NEWLINENEWLINE The main result of the paper under review says that the category \(^B\mathcal C_A\) of relative \((B,A)\)-Hopf modules becomes a monoidal category in this way if, and only if, the coproduct and the additional morphism \(\varphi\) turn \(A\) into a braided bialgebra, i.e., a bialgebra in the category of left Yetter-Drinfeld modules over \(B\).
reviewed by: Jörg Feldvoss
description: scientific article; zbMATH DE number 6037674