After the technical work of the preceding three chapters, we now take a break and return to more conceptual issues, i.e. to the general theory of Hopf algebras and quasitriangular structures. We have already seen early on, in Exercise 1.6.8, that if a Hopf algebra is represented in vector spaces V, W then it is also naturally represented in V ⊗ W. So the representations of the Hopf algebra have among themselves a tensor product operation ⊗. This is a key property of group representations, and is just as important for Hopf algebras. We have already used it several times in previous chapters in the course of our Hopf algebra constructions. In this chapter, we want to study this phenomenon quite systematically and more conceptually using the language of category theory. It should be possible to come to this chapter directly after Chapters 1 and 2, viewing the intermediate ones as providing examples and applications. For readers who want to skim this chapter, the key calculation, which also contains the essence of the entire chapter, is Theorem 9.2.4.

Category theory itself is often considered to be a hard subject by physicists, and for this reason we will try to be informal and nontechnical as much as possible. In truth, a category C just means a collection of objects (in our case they will be representations), and a specification of what it is to be an allowed map or morphism between any two objects.