On the braiding on a Hopf algebra in a braided category
type: Item
instance of: scholarly article
title: On the braiding on a Hopf algebra in a braided category
zbMATH Open document ID: 0914.16017
author: Peter Schauenburg
published in: The New York Journal of Mathematics
publication date: 1999-02-09T00:00:00Z
full work available at URL: https://eudml.org/doc/119774
full work available at URL: Vol4.html
review text: The paper is based upon the following simple observation: If \(H\) is a Hopf algebra with an antipode, the compatibility condition of a bialgebra can be solved for the braiding. In particular, the braiding \(\tau_{HH}\colon H\otimes H\to H\otimes H\) is uniquely determined by the algebra and coalgebra structure \(\nabla\) and \(\Delta\) if an antipode \(S\) exists, that is, \(\tau_{HH}=(\nabla\otimes\nabla)(S\otimes\Delta\nabla\otimes S)(\Delta\otimes\Delta)\). This result implies that a monoidal functor between prebraided monoidal categories that maps a Hopf algebra \(H\) to a Hopf algebra, necessarily preserves the braiding on \(H\otimes H\). Several applications are listed. Notably, the observation rules out that any nontrivial examples of commutative (or, cocommutative) Hopf algebras in nonsymmetric braided categories exist. This is a rigorous proof of a version of Majid's observation that commutativity is too restrictive a condition for Hopf algebras in braided categories.
reviewed by: Fang Li
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18D35
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1249605
zbMATH Keywords: quantum groups
zbMATH Keywords: quantum commutativity
zbMATH Keywords: Hopf algebras
zbMATH Keywords: antipodes
zbMATH Keywords: braidings
zbMATH Keywords: prebraided monoidal categories
zbMATH Keywords: nonsymmetric braided categories
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1249605