Table of Contents

Tannaka–Krein duality

Tannaka–Krein duality was formulated for compact quantum groups by Woronowicz [Wor88] as a generalization of the classical Tannaka–Krein duality for compact groups. Basic idea of the theorem is that any compact quantum group can be recovered from its representation theory (i.e. from the structure of the category of its representations).

The statement

There are several possibilities, how to formulate this result, which differ in their generality and in the amount of categorical formulations involved. Here, we present some of them starting with the most abstract one and going more concrete.

Category with a fiber functor

The following abstract formulation was taken from [NT13].

Theorem. Let $\Cscr$ be a rigid monoidal $*$-category, $F\colon\Cscr\to\FinHilb$ be a unitary monoidal functor. Then there exist a compact quantum group $G$ and a unitary monoidal equivalence $E\colon\Cscr\to\Rep_G$ such that $F$ is naturally unitarily monoidally isomorphic to the composition of the canonical fiber functor $\Rep_G\to\FinHilb$ with $E$. Furthermore, the Hopf $*$-algebra $(C[G];\Delta)$ for such a $G$ is uniquely determined up to isomorphism.

Such a monoidal functor $\Cscr\to\FinHilb$ is called a fiber functor.

Concrete categories

Applications

Further reading

References

[Wor88] Stanisław L. Woronowicz, 1988. Tannaka–Krein duality for compact matrix pseudogroups. Twisted SU(N) groups. Inventiones mathematicae, 93(1), pp.35–76.
[NT13] Sergey Neshveyev and Lars Tuset, 2013. Compact Quantum Groups and Their Representation Categories. Paris: Société Mathématique de France, ISBN 978-2-85629-777-3.