The generalized smash product and coproduct
type: Item
instance of: scholarly article
title: The generalized smash product and coproduct
zbMATH Open document ID: 0963.16036
DOI: 10.1142/S0252959900000406
author: Haibin Kan
published in: Chinese Annals of Mathematics. Series B
publication date: 2001-04-01T00:00:00Z
review text: It is shown in this article that some well known constructions of algebras made up from two algebras with certain additional structure can be understood in terms of generalized smash products. Explicitly, for two bialgebras \(A\) and \(H\) with an invertible skew pairing \(\tau\colon A\otimes H\to k\), a bialgebra \(A\bowtie_\tau H\) can be constructed by twisting the multiplication of \(A\otimes H\) by \(\tau\) (the Drinfeld double is an example of this construction). Another construction is \(A*H\), for \(H\) a Hopf algebra and \(A\) an \(H\)-bimodule algebra. On the other hand, let \(H\) be a bialgebra, \(A\) a left \(H\)-module algebra and \(X\) a left \(H\)-comodule algebra. The generalized smash product of \(A\) and \(X\) is the tensor product \(A\#X\) with the multiplication twisted by \(xa=\sum(x_{(-1)}a)x_{(0)}\). A first result explains when \(A\#X\) is a bialgebra and when it is a Hopf algebra. Then, it is proved that \(A\bowtie_\tau H\) as well as \(A*H\) can be constructed as generalized smash coproducts. A partial reciprocal statement establishes that if \(A\#X\) is a coquasitriangular bialgebra then it can be constructed as \(A\bowtie_\tau X\) for a given \(\tau\) (and \(A\) and \(X\) are coquasitriangular). At the end of the article, these results are dualized.
reviewed by: Matías Graña
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S40
zbMATH DE Number: 1530108
zbMATH Keywords: generalized smash products
zbMATH Keywords: smash coproducts
zbMATH Keywords: bialgebras
zbMATH Keywords: Drinfeld doubles
zbMATH Keywords: Hopf algebras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1530108