Reconstruction of Hidden Symmetries

Details Reconstruction of Hidden Symmetries Reconstruction of Hidden Symmetries

1994-12-09

arxiv.org/abs/hep-th/9412085v2

Physics

High Energy Physics

High Energy Physics - Theory

42 pages, amslatex, figures generated with bezier.sty, replaced to facilitate mailing

Scientific paper

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.

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