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Since you mention classification results for $R$-matrices: For finite abelian groups, there is a bijection between the set of universal $R$-matrices of the group hopf algebra $\mathbb C[G]$, the set o …

There have been some classic papers, on the development of a Morita theory for equivalent Categories of comodules over coalgebras. I believe one of the oldest and most complete works, which develops t …

I will try to provide an answer for a particular case of your last question: Let us consider (following my comment above) the case of $H=\mathbb{CZ}_2$ i.e. the group hopf algebra equipped with its no …

The OP asks more than one different things: the classification of fin dim, semisimple (or not), braided Hopf algebras is still a wide open area (up to my knowledge of course). The classification of …

I think that -apart from the applications in CFT and TFT already mentioned in previous answers- one of the most fundamental applications of braided Hopf algebras (with both non-trivial and "calculable …

I am not sure whether the following point of view is what you are asking for, in the sense that it is not some further development of your observations, but a seemingly different categorical descripti …