Equivalence of \(\chi\)-Hopf algebras (Q1809885)

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 \(s^{-1}\) is the inverse of the antipode \(s\).