Tannaka–Krein duality
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Tannaka–Krein duality theory concerns the interaction of a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
and its
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of
linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
s. It is a natural extension of
Pontryagin duality In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
, between compact and discrete
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
topological groups, to groups that are compact but
noncommutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
. The theory is named after
Tadao Tannaka was a Japanese mathematician who worked in algebraic number theory. Biography Tannaka was born in Matsuyama, Ehime Prefecture on December 27, 1908. After receiving a Bachelor of Science in mathematics from Tohoku Imperial University in 1932, he ...
and Mark Grigorievich Krein. In contrast to the case of commutative groups considered by
Lev Pontryagin Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely d ...
, the notion dual to a noncommutative
compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
is not a group, but a
category of representations In representation theory, the category of representations of some algebraic structure has the representations of as objects and equivariant maps as morphisms between them. One of the basic thrusts of representation theory is to understand the con ...
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 group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
s via Tannakian formalism. Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by
mathematical physicists Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. A generalization of Tannaka–Krein theory provides the natural framework for studying representations of
quantum groups In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebr ...
, and is currently being extended to quantum supergroups, quantum groupoids and their dual
Hopf algebroid In mathematics, in the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf ''k''-algebroids. If ''k'' is a field, a commutative ''k''-algebroid is a cogroupoid object i ...
s.


The idea of Tannaka–Krein duality: category of representations of a group

In Pontryagin duality theory for
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
commutative groups, the dual object to a group ''G'' is its character group \hat, which consists of its one-dimensional
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
s. If we allow the group ''G'' to be noncommutative, the most direct analogue of the character group is the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of irreducible
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
s of ''G''. The analogue of the product of characters is the
tensor product of representations In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be ...
. 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 \Pi(G) of all finite-dimensional representations, and treat it as a
monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...
, 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 \Pi(G) is a monoidal
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
from the identity
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
\operatorname_ to itself. In other words, it is a non-zero function \varphi that associates with any T \in \operatorname\Pi(G) an endomorphism of the space of ''T'' and satisfies the conditions of compatibility with tensor products, \varphi(T\otimes U)=\varphi(T)\otimes\varphi(U), and with arbitrary
intertwining operator In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
s f\colon T\to U, namely, f\circ \varphi(T) = \varphi(U) \circ f. The collection \Gamma(\Pi(G)) of all representations of the category \Pi(G) can be endowed with multiplication \varphi\psi(T)=\varphi(T)\psi(T) and
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, in which convergence is defined
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
, i.e., a sequence \ converges to some \varphi if and only if \ converges to \varphi(T) for all T \in \operatorname\Pi(G). It can be shown that the set \Gamma(\Pi(G)) thus becomes a compact (topological) group.


Theorems of Tannaka and Krein

Tannaka's theorem provides a way to reconstruct the
compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
''G'' from its category of representations Π(''G''). Let ''G'' be a compact group and let ''F:'' Π(''G'') → VectC be the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
from finite-dimensional
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
representations of ''G'' to complex
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s. One puts a topology on the
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
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 \tau \mapsto \tau_V (taking a natural transformation \tau to its component \tau_V at V \in \Pi(G)) is a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
. 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 \tau_ = \tau_V \otimes \tau_W. We also say that ''τ'' is self-conjugate if \overline = \tau where the bar denotes complex conjugation. Then the set \mathcal(G) 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 G \to \mathcal(G). 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. The following conditions are necessary and sufficient in order for Π to be a dual object to a compact group ''G''. : 1. There exists an object I with the property that I\otimes A \approx A for all objects ''A'' of Π (which will necessarily be unique up to isomorphism). : 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.


Generalization

Interest in Tannaka–Krein duality theory was reawakened in the 1980s with the discovery of
quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebr ...
s in the work of
Drinfeld Vladimir Gershonovich Drinfeld ( uk, Володи́мир Ге́ршонович Дрінфельд; russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowne ...
and
Jimbo Jimbo is a diminutive form of the given name James. It is also a Japanese surname, and it means state or province in Swahili. It may refer to: Given name or nickname * Jimbo (drag queen), Canadian drag queen * Jimbo Aquino (born 1985), Filipi ...
. 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 In mathematics, a ''commutativity constraint'' \gamma on a monoidal category ''\mathcal'' is a choice of isomorphism \gamma_ : A\otimes B \rightarrow B\otimes A for each pair of objects ''A'' and ''B'' which form a "natural family." In partic ...
. 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 In theoretical physics, a rational conformal field theory is a special type of two-dimensional conformal field theory with a finite number of conformal primaries. In these theories, all dimensions (and the central charge) are rational numbers that ...
. 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

The Doplicher–Roberts theorem (due to
Sergio Doplicher Sergio Doplicher (born 30 December 1940) is an Italian mathematical physicist, who mainly dealt with the mathematical foundations of quantum field theory and quantum gravity. Biography Sergio Doplicher graduated in Physics at the Sapienza Uni ...
and John E. Roberts) characterises Rep(''G'') in terms of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, as a type of
subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
of the category of
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s. Such subcategories of compact group unitary representations on Hilbert spaces are: # a strict symmetric monoidal C*-category with conjugates # a subcategory having
subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory ...
s and
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
s, such that the C*-algebra of endomorphisms of the monoidal unit contains only scalars.


See also

*
Gelfand–Naimark theorem In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra ''A'' is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 ...


Notes


External links

* * * {{DEFAULTSORT:Tannaka-Krein duality Monoidal categories Unitary representation theory Harmonic analysis Topological groups Duality theories