A class of twisted braided groups
type: Item
instance of: scholarly article
title: A class of twisted braided groups
zbMATH Open document ID: 0961.16024
author: Zhengming Jiao
author: Shuanhong Wang
published in: Kodai Mathematical Journal
publication date: 2000-12-10T00:00:00Z
review text: Let \(k\) be a field. In this paper the concept of a \(T\)-bialgebra over \(k\) is introduced; this is a vector space \(H\) endowed with algebra and coalgebra structures such that the comultiplication \(\Delta_H\colon H\to H\otimes H\) is an algebra map, when considering the algebra structure on \(H\otimes H\) obtained by ``twisting'' by some linear map \(T\colon H\otimes H\to H\otimes H\) the usual tensor product algebra structure. If there exists an antipode for \(H\), then \(H\) is called a \(T\)-Hopf algebra. It is shown that if \(A\) is an involutory Hopf algebra and \(\sigma\colon A\otimes A\to k\) is a convolution invertible normal \(2\)-cocycle satisfying an extra condition, then there exists a \(T_\sigma\)-Hopf algebra structure, denoted \(A_\sigma\), on the vector space \(A\): this is obtained by deforming both the multiplication and comultiplication on \(A\) using \(\sigma\) and \(\sigma^{-1}\); here \(T_\sigma\colon A\otimes A\to A\otimes A\) is given by the rule \[ T_\sigma(a\otimes b)=\sigma^{-1}(b_2{\mathcal S}(b_4),a_2{\mathcal S}(a_4))(b_3\otimes a_3)\sigma(a_1{\mathcal S}(a_5),b_1{\mathcal S}(b_5)). \] This construction applies, in particular, if \(A\) is a commutative Hopf algebra with an invertible normal \(2\)-cocycle \(\sigma\). In this case, one can also consider the coquasitriangular Hopf algebra \(A^\sigma\) obtained by twisting the algebra structure on \(A\) through \(\sigma\). Then the braided group \(\underline{A^\sigma}\), as introduced by \textit{S. Majid} [in J. Pure Appl. Algebra 86, No. 2, 187-221 (1993; Zbl 0797.17004)], which is a Hopf algebra in the braided category of left \(A^\sigma\)-comodules, is a \(T\)-Hopf algebra with respect to the braiding in this category. The main result of the paper states that \(A_\sigma\) is isomorphic to \(\underline{A^\sigma}\) as \(T\)-Hopf algebras.
reviewed by: Sonia Natale
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1496085
zbMATH Keywords: Hopf algebras
zbMATH Keywords: braided groups
zbMATH Keywords: bialgebras
zbMATH Keywords: antipodes
zbMATH Keywords: cocycles
zbMATH Keywords: braided categories
zbMATH Keywords: comodules
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138044157
OpenAlex ID: W1979175851
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DOI: 10.2996/KMJ/1138044157
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description: scientific article; zbMATH DE number 1496085