The Drinfeld double D(H) is a very useful quasi-triangular Hopf algebra. S Majid generalized the Drinfeld double and constructed a double crossproduct A H(1.2). Double bicrossproduct A???????-ev H, which was constructed by W Zhao, S Wang and Z Jiao, is a more generalized product. The double bicrossproduct generalizes double cross products (coproducts), bicrossproducts, biproducts, Drinfeld double, and smash products (coproducts). Braided tensor categories were introduced by A Joyal and R Street. Algebraic structures within them, especially Hopf algebras were introduced by S Majid. The author Shouchuan Zhang and H Chen constructed the double bicrossproduct D = A????????? H in braided tensor categories and gave the necessary and sufficient conditions for D to be a bialgebra. Y Bespalove and B Drabant stripped off some conditions of double bicrossproduct and defined the cross product bialgebras in braided tensor categories [6.7]. We show that cross product bialgebra D = A??1.2 ?? ??2.1 H is a Hopf algebra when both A and H have antipodes.
Han, Y. ying, Zhang, S. chuan, & Tang, L. (2002). Cross product Hopf algebras. Acta Scientiarum Naturalium Universitatis Normalis Hunanensis, 25(3), 14–19.