Reconstruction in braided categories and a notion of commutative bialgebra (Q1380017)

Super-bialgebras and super-commutativity were the first steps towards defining bialgebras and commutativity in a symmetric monoidal category. A unified approach for bialgebras in general symmetric categories was indicated by Manin in his Montréal Lecture Notes [\textit{Yu. I. Manin}, ``Quantum groups and non-commutative geometry'' (1988; Zbl 0724.17006)]. This generalization works naturally with no problems. When it comes now to define ``commutative'' bialgebras in a braided category things are different. \textit{S. Majid} was the first to realize the difficulty and he proposed to replace the usual definition of commutativity by a weaker condition which has to be formulated with respect to a specified class of comodules such that a compatibility condition is satisfied for every module in that class. The authors of this paper remark that there is a very natural class of coalgebras that comes equipped with a specified class of comodules, these are the coalgebras obtained by reconstruction theory from a category \(\omega:{\mathcal C}\to{\mathcal A}\) where \(({\mathcal A},\tau)\) is the base braided category, and the class of comodules is \(\{\omega(X)\mid X\) in \({\mathcal C}\}\). In this paper, the authors propose a different commutativity condition for bialgebras. They prove that a coalgebra obtained by reconstruction from a category \(\omega:{\mathcal C}\to{\mathcal A}\) over the base braided category \(({\mathcal A},\tau)\) has the additional structure of being an object of the center \({\mathcal Z}({\mathcal A}\)-\({\mathcal C}oalg)\) of the category of coalgebras. They proved then (theorem 10) that braided groups (Hopf algebras in a category) which are obtained by reconstruction from a braided category over \({\mathcal A}\) are commutative algebras over the center of \({\mathcal A}\)-\({\mathcal C}oalg\). In fact if \(B\) is reconstructed from a braided category \(({\mathcal A},\tau)\) then the paper proves that there is a new flip \(\sigma: B\otimes B\to B\otimes B\) which is a solution of the QYBE and such that the multiplication of \(B\) satisfies: \(\nabla= \nabla\circ\sigma\). Now one can see that the commutativity is expressed directly with no reference to a class of comodules which is indeed nice. Beware, while \(B\) is a bialgebra with respect to the original braiding \(\tau\) in \({\mathcal A}\), it is commutative with respect to a different braiding namely that of \({\mathcal Z}({\mathcal A}\)-\({\mathcal C}oalg)\). This new flip has other nice features and it can be used to define a structure of bialgebra in \({\mathcal Z}({\mathcal A}\)-\({\mathcal C}oalg)\) on the tensor product \(B\otimes H\), when \(B\) and \(H\) are coalgebras in \({\mathcal Z}({\mathcal A}\)-\({\mathcal C}oalg)\). In addition the tensor product becomes a Hopf algebra if the factors \(B\) and \(H\) are Hopf algebras.