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.
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.
is a monoidal natural transformation from the identity functor
an endomorphism of the space of T and satisfies the conditions of compatibility with tensor products,
and topology, in which convergence is defined pointwise, i.e., a sequence
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 let F: Π(G) → VectC be the forgetful functor from finite-dimensional complex representations of G to complex finite-dimensional vector spaces.
One puts a topology on the natural transformations τ: F → F by setting it to be the coarsest topology possible such that each of the projections End(F) → End(V) given by
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
of all tensor-preserving, self-conjugate natural transformations of F is a closed subset 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
Tannaka's theorem then says that this map is an isomorphism.
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.
Interest in Tannaka–Krein duality theory was reawakened in the 1980s with the discovery of quantum groups in the work of Drinfeld 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 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.
The Doplicher–Roberts theorem (due to Sergio Doplicher and John E. Roberts) characterises Rep(G) in terms of category theory, as a type of subcategory of the category of Hilbert spaces.
[1] Such subcategories of compact group unitary representations on Hilbert spaces are: