Representations of algebraic quantum groups and reconstruction theorems for tensor categories

We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka - Krein reconstruction problem. We show that every concrete semisimple tensor *-category with conjugates is equivalent to the category of finite-dimensional nondegenerate *-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka - Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well-known general result concerning discrete multiplier Hopf *-algebras.