Equivalence of \(\chi\)-Hopf algebras
type: Item
instance of: scholarly article
title: Equivalence of \(\chi\)-Hopf algebras
zbMATH Open document ID: 1029.16025
DOI: 10.1016/S0252-9602(17)30227-8
published in: Acta Mathematica Scientia. Series B. (English Edition)
publication date: 2004-02-04T00:00:00Z
review text: Let \(\mathbb{Z}\) and \(\mathbb{N}_0\) be respectively the set of integers and the set of nonnegative integers. Denote by \(\mathbb{Z} I\) the free Abelian group with \(I\) as basis, and denote by \(\mathbb{N}_0 I\) the submonoid with \(I\) as basis. For a bilinear form \(\chi\colon\mathbb{Z} I\times\mathbb{Z} I\to\mathbb{Z}\), denote by \(\chi^t\) the bilinear form given by \(\chi^t(x,y)=\chi(y,x)\). Let \(K\) be a field and \(0\neq c\in K\). For an \(\mathbb{N}_0 I\)-graded algebra \(A=(A,m,e)\) over \(K\), let \(A_\chi=(A,m_\chi,e)\) be the twisted algebra given by \(m_\chi(a\otimes b)=c^{\chi(|a|,|b|)}m(a\otimes b)\), and let \(_\chi A=(A,{_\chi m},e)\) with \(_{-\chi}m=m_{-\chi}T\), where \(T\) is the usual twist map. Dually, for an \(\mathbb{N}_0 I\)-graded coalgebra \(A=(A,\delta,\varepsilon)\), one can form \(A_\chi=(A,\delta_\chi,\varepsilon)\) and \(_\chi A=(A,{_\chi\delta},\varepsilon)\) with \(_\chi\delta=T\delta_{-\chi}\), where \(\delta_\chi(a)=\sum_{|a|=|a_1|+|a_2|}c^{\chi(|a_1|,|a_2|)}a_1\otimes a_2\). This paper discusses some equivalences of \(\chi\)-Hopf algebras. It is proved that \((A,m,e,\delta,\varepsilon)\) is a \(\chi\)-bialgebra if and only if \((A,m,e,{_{\chi^t}\delta},\varepsilon)\) is a \((-\chi^t)\)-bialgebra if and only if \((A,{_\chi m},e,\delta,\varepsilon)\) is a \((-\chi^t)\)-bialgebra if and only if \((A,{_\chi m},e,{_{-\chi}\delta},\varepsilon)\) is a \(\chi\)-bialgebra if and only if \((A,{_{-\chi^t}m},e,{_{\chi^t}\delta},\varepsilon)\) is a \(\chi\)-bialgebra. It is also shown that \((A,m,e,\delta,\varepsilon,s)\) is a \(\chi\)-Hopf algebra if and only if \((A,m,e,{_{\chi^t}\delta},\varepsilon,s^{-1})\) is a \((-\chi^t)\)-Hopf algebra if and only if \((A,{_\chi m},e,\delta,\varepsilon,s^{-1})\) is a \((-\chi^t)\)-Hopf algebra if and only if \((A,{_\chi m},e,{_{-\chi}\delta},\varepsilon,s)\) is a \(\chi\)-Hopf algebra if and only if \((A,{_{-\chi^t}m},e,{_{\chi^t}\delta},\varepsilon,s)\) is a \(\chi\)-Hopf algebra, where \(s^{-…
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 1931041
zbMATH Keywords: graded coalgebras
zbMATH Keywords: graded algebras
zbMATH Keywords: free Abelian groups
zbMATH Keywords: bilinear forms
zbMATH Keywords: twisted algebras
zbMATH Keywords: Hopf algebras
zbMATH Keywords: bialgebras
zbMATH Keywords: antipodes
author: Jianhua Sun
MaRDI profile type: Publication
full work available at URL: s0252-9602(17)30227-8
OpenAlex ID: W2783995918
description: scientific article; zbMATH DE number 1931041