25:
1553:. Tannaka–Krein philosophy suggests that braided monoidal categories arising from conformal field theory can also be obtained from quantum groups, and in a series of papers, Kazhdan and Lusztig proved that it was indeed so. On the other hand, braided monoidal categories arising from certain quantum groups were applied by Reshetikhin and Turaev to construction of new invariants of knots.
1549:. It turned out that a good duality theory of Tannaka–Krein type also exists in this case and plays an important role in the theory of quantum groups by providing a natural setting in which both the quantum groups and their representations can be studied. Shortly afterwards different examples of braided monoidal categories were found in
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Let Π be a category of finite-dimensional vector spaces, endowed with operations of tensor product and involution. The following conditions are necessary and sufficient in order for Π to be a dual object to a compact group
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in general fail to form a group, or even a monoid, because a tensor product of irreducible representations is not necessarily irreducible. It turns out that one needs to consider the set
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195:, allowing one to recover the group from its category of representations. Moreover, they in effect completely characterize all categories that can arise from a group in this fashion.
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1537:. One of the main approaches to the study of a quantum group proceeds through its finite-dimensional representations, which form a category akin to the
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answers the following question: which categories can arise as a dual object to a compact group?
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Durkdević, Mićok (December 1996). "Quantum principal bundles and
Tannaka-Krein duality theory".
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gives rise to a tensor-preserving self-conjugate natural transformation via multiplication by
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Interest in
Tannaka–Krein duality theory was reawakened in the 1980s with the discovery of
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Duality theorems of
Tannaka and Krein describe the converse passage from the category Π(
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later showed that by a similar process, Tannaka duality can be extended to the case of
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1589:. Such subcategories of compact group unitary representations on Hilbert spaces are:
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Doplicher, S.; Roberts, J. (1989). "A new duality theory for compact groups".
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to be noncommutative, the most direct analogue of the character group is the
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The idea of
Tannaka–Krein duality: category of representations of a group
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of all tensor-preserving, self-conjugate natural transformations of
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and satisfies the conditions of compatibility with tensor products,
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552:{\displaystyle \varphi (T\otimes U)=\varphi (T)\otimes \varphi (U)}
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1401:. Tannaka's theorem then says that this map is an isomorphism.
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If all these conditions are satisfied then the category Π = Π(
164:. In contrast to the case of commutative groups considered by
1257:{\displaystyle \tau _{V\otimes W}=\tau _{V}\otimes \tau _{W}}
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of all finite-dimensional representations, and treat it as a
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are two minimal objects then the space of homomorphisms Hom
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if it is the identity map on the trivial representation of
1803:, Lecture Notes in Mathematics, vol. 1488, Springer,
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of Π (which will necessarily be unique up to isomorphism).
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where the bar denotes complex conjugation. Then the set
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topology possible such that each of the projections End(
1794:"An introduction to Tannaka duality and quantum groups"
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of Π can be decomposed into a sum of minimal objects.
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to itself. In other words, it is a non-zero function
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769:{\displaystyle \varphi \psi (T)=\varphi (T)\psi (T)}
1608:, such that the C*-algebra of endomorphisms of the
305:. The analogue of the product of characters is the
152:topological groups, to groups that are compact but
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1737:Proceedings of the National Academy of Sciences
1346:), which is in fact a (compact) group whenever
942:{\displaystyle T\in \operatorname {Ob} \Pi (G)}
483:{\displaystyle T\in \operatorname {Ob} \Pi (G)}
239:commutative groups, the dual object to a group
16:Duality between a group and its representations
422:{\displaystyle \operatorname {id} _{\Pi (G)}}
215:, and is currently being extended to quantum
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987:thus becomes a compact (topological) group.
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1197:. We say that a natural transformation is
309:. However, irreducible representations of
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1733:"Quantum groups with invariant integrals"
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1298:{\displaystyle {\overline {\tau }}=\tau }
109:Learn how and when to remove this message
1005:from its category of representations Π(
688:of all representations of the category
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1394:{\displaystyle G\to {\mathcal {T}}(G)}
1104:{\displaystyle \tau \mapsto \tau _{V}}
278:which consists of its one-dimensional
168:, the notion dual to a noncommutative
129:theory concerns the interaction of a
1350:is a (compact) group. Every element
47:adding citations to reliable sources
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1464:{\displaystyle I\otimes A\approx A}
872:{\displaystyle \{\varphi _{a}(T)\}}
717:can be endowed with multiplication
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235:In Pontryagin duality theory for
980:{\displaystyle \Gamma (\Pi (G))}
810:{\displaystyle \{\varphi _{a}\}}
681:{\displaystyle \Gamma (\Pi (G))}
490:an endomorphism of the space of
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1684:Reports on Mathematical Physics
1551:rational conformal field theory
949:. It can be shown that the set
148:, between compact and discrete
144:. It is a natural extension of
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1724:10.1016/S0034-4877(97)84884-7
1545:), but of more general type,
1539:symmetric monoidal categories
1052:. One puts a topology on the
991:Theorems of Tannaka and Krein
348:contragredient representation
1593:a strict symmetric monoidal
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1186:{\displaystyle V\in \Pi (G)}
156:. The theory is named after
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1016:be a compact group and let
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271:{\displaystyle {\hat {G}},}
174:category of representations
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1731:Van Daele, Alfons (2000).
1419:1. There exists an object
1342:is a closed subset of End(
1563:Doplicher–Roberts theorem
1557:Doplicher–Roberts theorem
1547:braided monoidal category
1151:{\displaystyle \tau _{V}}
449:that associates with any
1644:Inventiones Mathematicae
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1066:by setting it to be the
1037:from finite-dimensional
830:{\displaystyle \varphi }
442:{\displaystyle \varphi }
282:. If we allow the group
1623:Gelfand–Naimark theorem
1439:with the property that
1054:natural transformations
710:{\displaystyle \Pi (G)}
379:{\displaystyle \Pi (G)}
335:{\displaystyle \Pi (G)}
299:unitary representations
280:unitary representations
209:mathematical physicists
162:Mark Grigorievich Krein
58:"Tannaka–Krein duality"
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1836:Monoidal categories
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1657:1989InMat..98..157D
1585:of the category of
1264:. We also say that
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1041:representations of
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205:Tannakian formalism
1851:Topological groups
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1846:Harmonic analysis
1818:978-3-540-46435-8
1432:{\displaystyle I}
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1199:tensor-preserving
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1035:forgetful functor
996:Tannaka's theorem
650:. The collection
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134:topological group
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1597:with conjugates
1581:, as a type of
1579:category theory
1571:John E. Roberts
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41:Please help
36:verification
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