====== Tannaka–Krein duality ======

**Tannaka–Krein duality** was formulated for compact quantum groups by Woronowicz [(ref:Wor88)] as a generalization of the classical [[wp>Tannaka–Krein duality|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 [(ref: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 =====

  * [[nlab>Tannaka+duality|Tannaka duality]]
  * [[nlab>rigid monoidal category|Rigid monoidal category]]

===== References =====

[(ref:Wor88>>
author    : Stanisław L. Woronowicz
title     : Tannaka–Krein duality for compact matrix pseudogroups. Twisted SU(N) groups
journal   : Inventiones mathematicae
year      : 1988
volume    : 93
number    : 1
pages     : 35--76
url       : http://dx.doi.org/10.1007/BF01393687
)]

[(ref:NT13>>
author    : Sergey Neshveyev and Lars Tuset
title     : Compact Quantum Groups and Their Representation Categories
publisher : Société Mathématique de France
address   : Paris
year      : 2013
isbn      : 978-2-85629-777-3
url       : https://www.sciencesmaths-paris.fr/en/comptact-quantum-groups-and-their-representation-categories-629.htm
)]

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