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In
representation theory 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 essen ...
, the
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 representations of some
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
has the representations of as objects and equivariant maps as
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
s between them. One of the basic thrusts of representation theory is to understand the conditions under which this category is
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
; i.e., whether an object decomposes into simple objects (see
Maschke's theorem In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
for the case of
finite groups Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...
). The
Tannakian formalism In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear ...
gives conditions under which a group ''G'' may be recovered from the category of representations of it together with 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 sig ...
to the
category of vector spaces In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ri ...
. The
Grothendieck ring In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined below). Almost all Noetherian rings that occur naturally in algebraic geometry or number t ...
of the category of finite-dimensional representations of a group ''G'' is called the
representation ring In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear represen ...
of ''G''.


Definitions

Depending on the types of the representations one wants to consider, it is typical to use slightly different definitions. For a finite group and a field , the category of representations of over has * objects: pairs (, ) of
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 ...
s over and representations of on that vector space * morphisms: equivariant maps * composition: the composition of equivariant maps * identities: the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
(which is an equivariant map). The category is denoted by \operatorname_F(G) or \operatorname(G). For a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
, one typically requires the representations to be smooth or admissible. For the case of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
, see
Lie algebra representation In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is ...
. See also: category O.


The category of modules over the group ring

There is an isomorphism of categories between the category of representations of a group over a field (described above) and the
category of modules In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ri ...
over the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
[], denoted []-Mod.


Category-theoretic definition

Every group can be viewed as a category with a single object, where
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
s in this category are the elements of and composition is given by the group operation; so is the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is th ...
of the unique object. Given an arbitrary category , a ''representation'' of in is a
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, an ...
from to . Such a functor sends the unique object to an object say ' in and induces a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) ...
G \to \operatorname(X); see Automorphism group#In category theory for more. For example, a -set is equivalent to a functor from to Set, the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...
, and a linear representation is equivalent to a functor to Vect, the
category of vector spaces In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ri ...
over a field . In this setting, the category of linear representations of over is the functor category → Vect, which has
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 na ...
s as its morphisms.


Properties

The category of linear representations of a group has a monoidal structure given by the tensor product of representations, which is an important ingredient in Tannaka-Krein duality (see below).
Maschke's theorem In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
states that when the characteristic of doesn't divide the
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of , the category of representations of over is
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
.


Restriction and induction

Given a group with a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
, there are two fundamental functors between the categories of representations of and (over a fixed field): one is a
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 sig ...
called the restriction functor : \begin \operatorname_H^G : \operatorname(G) &\longrightarrow \operatorname(H) \\ \pi &\longmapsto \pi, _H \end and the other, the
induction functor In group theory, the induced representation is a group representation, representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "m ...
:\operatorname_H^G : \operatorname(H) \to \operatorname(G). When and are finite groups, they are adjoint to each other :\operatorname_G(\operatorname_H^G W, U) \cong \operatorname_H(W, \operatorname_H^G U), a theorem called
Frobenius reciprocity In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting. It can be used to leverage knowledge about representations of a subgroup to find a ...
. The basic question is whether the decomposition into irreducible representations (simple objects of the category) behaves under restriction or induction. The question may be attacked for instance by the Mackey theory.


Tannaka-Krein duality

Tannaka–Krein duality 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 British ...
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 ...
and its category of linear representations. Tannaka's theorem describes the converse passage from the category of finite-dimensional representations of a group back to the group , allowing one to recover the group from its category of representations. Krein's theorem in effect completely characterizes all categories that can arise from a group in this fashion. These concepts can be applied to representations of several different structures, see the main article for details.


Notes


References

*{{Citation , last1=André , first1=Yves , title=Une introduction aux motifs (motifs purs, motifs mixtes, périodes) , publisher=Société Mathématique de France , location=Paris , series=Panoramas et Synthèses , isbn=978-2-85629-164-1 , mr=2115000 , year=2004 , volume=17


External links

* https://ncatlab.org/nlab/show/category+of+representations Representation theory Category theory