tannaka_krein_duality
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| ====== Tannaka–Krein duality ====== | ====== Tannaka–Krein duality ====== | ||
| - | **Tannaka–Krein duality** for compact quantum groups is a theorem that generalizes 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). | + | **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 ===== | ===== 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. | 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 ===== | ===== 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 | ||
| + | )] | ||
| + | |||
| + | ~~REFNOTES ref ~~ | ||
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