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Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of Etingof-...

The finite-dimensional representations over $\mathbb C(q)$ of $U_q(\mathfrak{sl}(2))$ are all highest weight. There are two irreducible modules of each dimension. In one, the highest weight vector $v$ ...

I was told that, if I understand correctly, that the enveloping algebra of semisimple Lie algebra admits one family of quantum deformation as Hopf algebra, which was proved by Drinfeld. Anyone can ...