For a bialgebra \(A\), the category of left \(A\)-modules is a monoidal category. If \(A\) is quasitriangular, then \(_ A\text{Mod}\) is a braided monoidal category. In this case, the matrix \(R=\sum_ iR_ i\otimes R_ i'\) in \(A\otimes A\) involved in the definition is used to construct a map \(\sigma(E,F)\) of \(E\otimes F\) to \(F\otimes E\), for \(E\), \(F\) in \(_ A\text{Mod}\), namely \(\sigma(E,F)\) \((e\otimes f)=\sum_ i(R_ i,f)\otimes (R_ i',e)\), thus replacing the usual twist map which is not a map of left \(A\)-modules. If \(A\) is not finite-dimensional, some applications require \(R\) to be taken in a suitable topological completion of \(A\otimes A\). Majid has tried to circumvent this by considering pairings of bialgebras to the base field. In the paper under review, the authors propose a different approach.
They first consider the category of left \(A\)-comodules, which is a monoidal category in an obviuos way, and ask when it is a braided monoidal category. The answer involves a pairing of \(A\otimes A\) to the base field, and in this case, \(A\) is called a braided bialgebra, and suitable equivalent sets of axioms are developed. If \(R: V\otimes V\to V\otimes V\) for a finite dimensional vector space \(V\), then a somewhat universal construction of a bialgebra \(A(R)\) is given, such that \(V\) is a left \(A(R)\)-comodule and \(R\) is an \(A(R)\)-comodule map. If \(R\) is Yang-Baxter, then \(A(R)\) is a braided bialgebra. The authors feel that working with comodules, rather than modules, avoids topological questions involving completions. The final section of the paper extends the concept of a quasitriangular bialgebra to what the authors call a “completed-triangular” bialgebra. Here the bialgebras are cofinitary topological modules over the base ring \(K\) (a Dedekind domain), the \(R\) involved is in the completed tensor product, and the category of continuous right modules is a braided monoidal category. Also there is a continuous dual which is a braided bialgebra, and over a field there is a suitable converse (i.e. every braided bialgebra gives rise to a completed-triangular bialgebra).