Dual aspects of the quasitriangular bialgebras and the braided bialgebras
The concept of a braided bialgebra was introduced by \textit{R. G. Larson} and \textit{J. Towber} [Commun. Algebra 19, No. 12, 3295-3345 (1991; Zbl 0751.16014)] as a bialgebra \(A\) such that the category of (left and right) \(A\)-comodules is a braided monoidal category. They proved that this is equivalent to the existence of a bilinear form on \(A\) with certain compatibility conditions. Since the category of (left and right) modules over a bialgebra \(A\) is a braided monoidal category if and only if \(A\) is quasitriangular, it is not surprising that the dual of a quasitriangular bialgebra is braided and that the dual of a braided bialgebra (with a certain finiteness condition) is quasitriangular which is one of the main results of the paper under review. This was also observed in the finite-dimensional case by \textit{T. Hayashi} [J. Algebra 152, No. 1, 146-165 (1992)] and in a topological context by \textit{R. G. Larson} and \textit{J. Towber} [in the paper cited above]. The author uses this duality to translate the well-known formula for the square of the antipode of a quasitriangular Hopf algebra to the dual formula for a braided Hopf algebra which implies that the antipode is bijective. These results were already obtained before by \textit{P. Schauenburg} [On coquasitriangular Hopf algebras and the quantum Yang-Baxter equation (Algebra-Berichte 67, Verlag Reinhard Fischer, München) (1992; Zbl 0841.16039)] and also by \textit{Y. Doi} [Commun. Algebra 21, No. 5, 1731-1749 (1993; Zbl 0779.16015)]. Finally, the author shows that a finite-dimensional braided bialgebra with a non-degenerate defining bilinear form has an antipode and then translates this to the dual situation.
scientific article; zbMATH DE number 1053810