Tannaka-Krein duality
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In mathematics, Tannaka-Krein duality theory concerns the interaction of a compact topological group and its category of linear representations.
It extends an important mathematical duality between compact and discrete commutative topological groups, known as Pontryagin duality, to groups that are compact, but noncommutative. The theory is named for two men, the Ukrainian mathematician Mark Grigorievich Krein, and the Japanese 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 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 of algebraic groups: see tannakian category. Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by mathematical physisists. A generalization of Tannaka-Krein theory provides natural framework for studying the representations of quantum groups.
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[edit] 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 G is its character group 
which consists of its one-dimensional unitary representations. 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 representations of G. The analogue of the product of characters is the tensor product of representations. However, irreducible representations of G 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 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 non-zero function φ that associates with any 
an endomorphism of the space of T and satisfies the conditions of compatibility with tensor products, 
and with arbitrary intertwining operators 
namely,
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 
if it's true pointwise, i.e. 
for all 
It can be shown that the set Γ(Π(G)) thus becomes a compact topological group.
[edit] Theorems of Tannaka and Krein
Tannaka's theorem provides a way to reconstruct the compact group G from its category of representations Π(G).
Let G be a compact group and φg be the representation of the category Π(G) given by the formula
- φg(T) = T(g),
where T is an object of the category Π(G), i.e. a representation of the group G. Then the map 
is an isomorphism of topological groups G and Γ(Π(G)).
Krein's theorem answers the following question: which categories can arise as a dual object to a compact group?
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.
- 1.There exists a unique up to isomorphism object with the property
for all objects A of Π. - 2.Every object A of Π can be decomposed into a sum of minimal objects.
- 3.If A and B are two minimal objects then the space of homomorphisms HomΠ(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.
[edit] Generalization
Interest to Tannaka-Krein duality theory was reawakened in the 1980s with the discovery of quantum groups in the the work of Drinfel'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 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. Tanaka-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
[edit] Doplicher-Roberts theorem
This result[1] characterises Rep(G) in terms of category theory, as a type of subcategory of the category of Hilbert spaces. 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
- having subobjects and direct sums, such that
- the C*-algebra of endomorphisms of the monoidal unit is of scalars.
[edit] Notes
- ^ S. Doplicher and J. Roberts. A new duality theory for compact groups. Inventiones Mathematicae, 98:157--218, 1989.
