  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 non-degenerate
*-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.
