Cross Product Bialgebras, I

The subject of this article is bialgebra factorizations or cross product bialgebras without cocycles. We establish a theory characterizing cross product bialgebras universally in terms of projections and injections. Especially all known types of biproduct, double cross product, and bicross product bialgebras can be described by this theory. Furthermore the theory provides new families of (cocycle-free) cross product bialgebras. Besides the universal characterization we find an equivalent (co)modular description of certain types of cross product bialgebras in terms of so-called Hopf data. With the help of Hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras. We are working in the general setting of braided monoidal categories, which allows us to apply our results in particular to the braided category of Hopf bimodules over a Hopf algebra. Majidʹs double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category. This resembles the case of the Drinfelʹd double which can be constructed as a twist of a specific cross product.