  Representations of a group $G$ in vector spaces over a field $K$ form a
category. One can reconstruct the given group $G$ from its representations to
vector spaces as the full group of monoidal automorphisms of the underlying
functor. This is a special example of Tannaka-Krein theory. This theory was
used in recent years to reconstruct quantum groups (quasitriangular Hopf
algebras) in the study of algebraic quantum field theory and other
applications.
  We show that a similar study of representations in spaces with additional
structure (super vector spaces, graded vector spaces, comodules, braided
monoidal categories) produces additional symmetries, called ``hidden
symmetries''. More generally, reconstructed quantum groups tend to decompose
into a smash product of the given quantum group and a quantum group of
``hidden'' symmetries of the base category.
