Radford's biproduct for quasi-Hopf algebras and bosonization
type: Item
instance of: scholarly article
title: Radford's biproduct for quasi-Hopf algebras and bosonization
zbMATH Open document ID: 1014.16036
DOI: 10.1016/S0022-4049(02)00014-2
published in: Journal of Pure and Applied Algebra
publication date: 2002-12-02T00:00:00Z
review text: Let \(H\) be a Hopf algebra and \(B\) a linear space such that \(B\) is an algebra in the category of left \(H\)-modules, and a coalgebra in the category of left \(H\)-comodules. Suppose that \(B\) is a Hopf algebra in the braided category of Yetter-Drinfeld modules over \(H\). Then the smash product algebra structure and the smash coproduct coalgebra structure make \(B\otimes H\) a Hopf algebra. This is in fact a necessary and sufficient condition for the smash product algebra structure and the smash coproduct coalgebra structure to afford a Hopf algebra structure on \(B\otimes H\) and is due to Radford. Moreover, Hopf algebras in the category of Yetter-Drinfeld modules over \(H\) are in correspondence with Hopf algebras \(A\) equipped with projections \(\pi\colon A\to H\) which admit sections \(i\colon H\to A\). This correspondence when formulated in the language of braided categories is known as bosonization and is due to Majid. In the present paper, the authors generalize the above result for quasi-Hopf algebras and describe the Majid bosonization for quasi-Hopf algebras.
reviewed by: Goutam Mukherjee
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 57T05
Mathematics Subject Classification ID: 81R05
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 1839047
zbMATH Keywords: Hopf algebras
zbMATH Keywords: categories of left modules
zbMATH Keywords: coalgebras
zbMATH Keywords: categories of left comodules
zbMATH Keywords: braided categories
zbMATH Keywords: Yetter-Drinfeld modules
zbMATH Keywords: smash products
zbMATH Keywords: bosonization
zbMATH Keywords: quasi-Hopf algebras
author: Daniel Bulacu
author: Erna Nauwelaerts
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1839047