In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. 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 after Tadao Tannaka and Mark Grigorievich Krein. 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 of representations Π(G) with some additional structure, 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 via Tannakian formalism. Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by 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 supergroups, quantum groupoids and their dual Hopf algebroids.
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 G ^ , {\displaystyle {\hat {G}},} 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, 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 Π ( G ) {\displaystyle \Pi (G)} of all finite-dimensional representations, and treat it as a 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 ) {\displaystyle \Pi (G)} is a monoidal natural transformation from the identity functor id Π ( G ) {\displaystyle \operatorname {id} _{\Pi (G)}} to itself. In other words, it is a non-zero function φ {\displaystyle \varphi } that associates with any T ∈ Ob Π ( G ) {\displaystyle T\in \operatorname {Ob} \Pi (G)} an endomorphism of the space of T and satisfies the conditions of compatibility with tensor products, φ ( T ⊗ U ) = φ ( T ) ⊗ φ ( U ) {\displaystyle \varphi (T\otimes U)=\varphi (T)\otimes \varphi (U)} , and with arbitrary intertwining operators f : T → U {\displaystyle f\colon T\to U} , namely, f ∘ φ ( T ) = φ ( U ) ∘ f {\displaystyle f\circ \varphi (T)=\varphi (U)\circ f} . The collection Γ ( Π ( G ) ) {\displaystyle \Gamma (\Pi (G))} of all representations of the category Π ( G ) {\displaystyle \Pi (G)} can be endowed with multiplication φ ψ ( T ) = φ ( T ) ψ ( T ) {\displaystyle \varphi \psi (T)=\varphi (T)\psi (T)} and topology, in which convergence is defined pointwise, i.e., a sequence { φ a } {\displaystyle \{\varphi _{a}\}} converges to some φ {\displaystyle \varphi } if and only if { φ a ( T ) } {\displaystyle \{\varphi _{a}(T)\}} converges to φ ( T ) {\displaystyle \varphi (T)} for all T ∈ Ob Π ( G ) {\displaystyle T\in \operatorname {Ob} \Pi (G)} . It can be shown that the set Γ ( Π ( G ) ) {\displaystyle \Gamma (\Pi (G))} thus becomes a compact (topological) group.
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