A class of twisted braided groups (Q1577707)

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.