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.Comment: 42 pages, amslatex, figures generated with bezier.sty, replaced to
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