- Tannaka-Krein duality
In
mathematics , Tannaka-Krein duality theory concerns the interaction of acompact topological group and its category oflinear representation s.It extends an important
mathematical duality between compact and discretecommutative topological groups, known asPontryagin duality , to groups that are compact, butnoncommutative . The theory is named for two men, the Soviet mathematicianMark Grigorievich Krein , and the JapaneseShiro Tannaka . In contrast to the case ofcommutative groups considered byLev Pontryagin , the notion dual to a noncommutativecompact group is not a group, but a category Π("G") with some additional structures, formed by the finite-dimensional representations of "G".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 ofalgebraic group s: seetannakian category . Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by mathematical physicists. A generalization of Tannaka-Krein theory provides natural framework for studying the representations ofquantum groups .The idea of Tannaka-Krein duality: category of representations of a group
In Pontryagin duality theory for
locally compact commutative groups, the dual object to a group is itscharacter group which consists of its one-dimensionalunitary representation s. If we allow the group to be noncommutative, the most direct analogue of the character group is the set of equivalence classes of irreducibleunitary representation s of The analogue of the product of characters is the tensor product of representations. However,irreducible representation s of in general fail to form a group, because tensor product of irreducible representations is not necessarily irreducible. It turns out that one needs to consider the set 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 thecontragredient representation . A representation of the category is a monoidalnatural transformation from the identity functor to itself. In other words, it is a non-zero function that associates with any an endomorphism of the space of and satisfies the conditions of compatibility with tensor products, and with arbitraryintertwining operator s namely, The collection of all representations of the category can be endowed with multiplication and topology, in which if it's true pointwise, i.e. for all It can be shown that the set thus becomes a compact topological group.Theorems of Tannaka and Krein
Tannaka's theorem provides a way to reconstruct the compact group from its category of representations
Let be a compact group and be the representation of the category given by the formula: where is an object of the category i.e. a representation of the group Then the map is an
isomorphism of topological groups andKrein's theorem answers the following question: which categories can arise as a dual object to a compact group?
Let 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 to be a dual object to a compact group : 1.There exists a unique up to isomorphism object with the property for all objects of : 2.Every object of can be decomposed into a sum of minimal objects.: 3.If and are two minimal objects then the space of homomorphisms is either one-dimensional (when they are isomorphic) or is equal to zero. If all these conditions are satisfied then the category where is the group of the representations of
Generalization
Interest to Tannaka-Krein duality theory was reawakened in the 1980s with the discovery of
quantum group s in the work ofDrinfel'd and 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 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 inrational 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.Doplicher-Roberts theorem
This result [S. Doplicher and J. Roberts. "A new duality theory for compact groups". Inventiones Mathematicae, 98:157--218, 1989.] characterises Rep(G) in terms of
category theory , as a type ofsubcategory of the category ofHilbert space s. Such categories of unitary representation of a compact group are the same as certain subcategories, the required properties being:# a strict symmetric monoidal
C*-category with conjugates
# havingsubobject s anddirect sum s, such that
#the C*-algebra of endomorphisms of themonoidal unit is of scalars.Notes
External links
* [http://front.math.ucdavis.edu/author/Amini-M* UC Davis site with three articles on Tannaka Krein Duality]
* [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]
* [http://intl.pnas.org/cgi/content/full/97/2/541 Quantum groups with invariant integrals by Alfons Van Daele(involves Tannaka-Klein)]
*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.
Wikimedia Foundation. 2010.