Reconstruction of Hidden Symmetries

Abstract

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.