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| In [[mathematics]], '''Tannaka–Krein duality''' theory concerns the interaction of a [[compact group|compact]] [[topological group]] and its category of [[linear representation]]s. It is a natural extension of [[Pontryagin duality]], between compact and discrete [[commutative]] topological groups, to groups that are compact but [[noncommutative]]. The theory is named for two men, the Soviet mathematician [[Mark Grigorievich Krein]], and the Japanese [[Tadao Tannaka]]. In contrast to the case of [[commutative]] groups considered by [[Lev Pontryagin]], the notion dual to a noncommutative [[compact group]] is not a group, but a [[category (mathematics)|category]] Π(''G'') with some additional structures, formed by the finite-dimensional representations of ''G''.
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| Duality theorems of Tannaka and Krein describe the converse passage from the category Π(''G'') back to the group ''G'', 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. [[Alexander Grothendieck]] later showed that by a similar process, Tannaka duality can be extended to the case of [[algebraic group]]s: see [[tannakian category]]. Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by [[mathematical physics|mathematical physicists]]. A generalization of Tannaka–Krein theory provides the natural framework for studying representations of [[quantum groups]], and is currently being extended to quantum [[Supergroup (physics)|supergroups]], quantum [[groupoids]] and their dual quantum [[Lie algebroid|algebroids]].
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| == The idea of Tannaka–Krein duality: category of representations of a group ==
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| In Pontryagin duality theory for [[locally compact]] commutative groups, the dual object to a group ''G'' is its [[character group]] <math>\hat{G},</math> which consists of its one-dimensional [[unitary representation]]s. If we allow the group ''G'' to be noncommutative, the most direct analogue of the character group is the set of equivalence classes of irreducible [[unitary representation]]s of ''G''. The analogue of the product of characters is the tensor product of representations. However, [[irreducible representation]]s of ''G'' in general fail to form a group, because a tensor product of irreducible representations is not necessarily irreducible. It turns out that one needs to consider the set Π(''G'') of all finite-dimensional representations, and treat it as [[monoidal category]], where the product is the usual tensor product of representations, and the dual object is given by the operation of the [[contragredient representation]]. A '''representation''' of the category Π(''G'') is a monoidal [[natural transformation]] from the identity functor <math>id_{\Pi(G)}</math> to itself. In other words, it is a non-zero function φ that associates with any <math>T\in Ob \Pi(G)</math> an endomorphism of the space of ''T'' and satisfies the conditions of compatibility with tensor products, <math>\phi(T\otimes U)=\phi(T)\otimes\phi(U)</math>, and with arbitrary [[intertwining operator]]s ''f:'' ''T'' → ''U'', namely, <math>f\circ \phi(T) = \phi(U) \circ f</math>. The collection Γ(Π(''G'')) of all representations of the category Π(''G'') can be endowed with multiplication φψ(''T'') = φ(''T'') ψ(''T'') and topology, in which <math>\phi_a\to\phi</math> if it's true pointwise, i.e. <math>\phi_a(T)\to\phi(T)</math> for all <math>T\in Ob\Pi(G)</math>. It can be shown that the set Γ(Π(''G'')) thus becomes a compact (topological) group.
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| == Theorems of Tannaka and Krein ==
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| '''Tannaka's theorem''' provides a way to reconstruct the [[compact group]] ''G'' from its category of representations Π(''G'').
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| Let ''G'' be a compact group and let ''F:'' Π(''G'') → Vect<sub>'''C'''</sub> be the forgetful functor from finite-dimensional complex representations of ''G'' to complex finite-dimensional vector spaces. One puts a topology on the [[natural transformation]]s ''τ:'' ''F'' → ''F'' by setting it to be the coarsest topology possible such that each of the projections End(''F'') → End(''V'') given by <math>\tau \mapsto \tau_V</math> is a continuous function. We say that a natural transformation is '''tensor-preserving''' if it is the identity map on the trivial representation of ''G'', and if it preserves tensor products in the sense that <math>\tau_{V \otimes W} = \tau_V \otimes \tau_W</math>. We also say that τ is '''self-conjugate''' if <math>\overline{\tau} = \tau</math> where the bar denotes complex conjugation. Then the set <math>\mathcal{T}(G)</math> of all tensor-preserving, self-conjugate natural transformations of ''F'' is a closed set of End(''F''), which is in fact a (compact) group whenever ''G'' is a (compact) group. Every element ''x'' of ''G'' gives rise to a tensor-preserving self-conjugate natural transformation via multiplication by ''x'' on each representation, and hence one has a map <math>G \to \mathcal{T}(G)</math>. Tannaka's theorem then says that this map is an isomorphism.
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| '''Krein's theorem''' answers the following question: which categories can arise as a dual object to a compact group?
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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 ''G''.
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| : 1. There exists a unique up to isomorphism object with the property <math>I\otimes A \approx A</math> for all objects ''A'' of Π.
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| : 2. Every object ''A'' of Π can be decomposed into a sum of minimal objects.
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| : 3. If ''A'' and ''B'' are two minimal objects then the space of homomorphisms Hom<sub>Π</sub>(''A'', ''B'') is either one-dimensional (when they are isomorphic) or is equal to zero. If all these conditions are satisfied then the category Π = Π(''G''), where ''G'' is the group of the representations of Π.
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| == Generalization ==
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| Interest to Tannaka–Krein duality theory was reawakened in the 1980s with the discovery of [[quantum group]]s in the work of [[Drinfeld]] and [[Michio Jimbo|Jimbo]]. 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 [[symmetric monoidal category|symmetric monoidal categories]] Π(''G''), but of more general type, [[braided monoidal category]]. 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 [[rational conformal field theory]]. 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.
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| ==Doplicher–Roberts theorem==
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| This result (due to Sergio Doplicher and John E. Roberts)<ref>S. Doplicher and J. Roberts. ''A new duality theory for compact groups''. Inventiones Mathematicae, 98:157–218, 1989.</ref> characterises Rep(''G'') in terms of [[category theory]], as a type of [[subcategory]] of the category of [[Hilbert space]]s. Such subcategories of compact group unitary representations on Hilbert spaces are:
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| # a strict symmetric monoidal [[C*-category]] with conjugates
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| # a subcategory having [[subobject]]s and [[direct sum]]s, such that the C*-algebra of endomorphisms of the [[monoidal unit]] contains only scalars.
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| ==Notes==
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| <references/>
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| ==External links==
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| *[http://scholar.google.com/scholar?hl=en&lr=&safe=active&q=cache:Ct8t2Xa4-nIJ:arxiv.org/pdf/q-alg/9507018+tannaka-krein+duality Quantum Principal Bundles and Tannaka–Krein Duality by Mico Durdevic]
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| *[http://intl.pnas.org/cgi/content/full/97/2/541 Quantum groups with invariant integrals by Alfons Van Daele (involves Tanaka–Krein)]
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| *André Joyal and Ross Street, [http://www.maths.mq.edu.au/~street/CT90Como.pdf An introduction to Tannaka duality and quantum groups], in Part II of ''Category Theory, Proceedings, Como 1990'', eds. A. Carboni, M. C. Pedicchio and G. Rosolini, Lectures Notes in Mathematics '''1488''', Springer, Berlin, 1991, 411–492.
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| {{DEFAULTSORT:Tannaka-Krein duality}}
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| [[Category:Monoidal categories]]
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| [[Category:Unitary representation theory]]
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| [[Category:Harmonic analysis]]
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| [[Category:Topological groups]]
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| [[Category:Duality theories]]
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The title of the author is Numbers but it's not the most masucline title out there. To collect cash is a factor that I'm totally addicted to. Hiring is my occupation. Minnesota has usually been his home but his wife wants them to transfer.
Feel free to surf to my homepage; over the counter std test, you could check here,