Category Of Representations
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In
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, the category of representations of some
algebraic structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
has the representations of as
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an a ...
s and equivariant maps as
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s between them. One of the basic thrusts of representation theory is to understand the conditions under which this
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * 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 group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s). The Tannakian formalism gives conditions under which a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
''G'' may be recovered from the category of representations of it together with the
forgetful functor In mathematics, more specifically 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 mapping to the output. For an algebraic structure of ...
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 rin ...
. The Grothendieck ring 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 represent ...
of ''G''.


Definitions

Depending on the types of 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s over and representations of on that vector space * Morphisms: Equivariant maps * Composition: The
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...
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 (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
, 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 ident ...
, 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 i ...
. See also: category O.


The category of modules over the group ring

There is an
isomorphism of categories In category theory, two categories ''C'' and ''D'' are isomorphic if there exist functors ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'' that are mutually inverse to each other, i.e. ''FG'' = 1''D'' (the identity functor on ''D'') and ''GF'' ...
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 ...
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, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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 the g ...
of the unique object. Given an arbitrary category , a ''representation'' of in is a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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) whe ...
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 by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
, 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 rin ...
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 natur ...
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: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...
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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
, there are two fundamental functors between the categories of representations of and (over a fixed field): one is a
forgetful functor In mathematics, more specifically 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 mapping to the output. For an algebraic structure of ...
called the restriction functor : \begin \operatorname_H^G : \operatorname(G) &\longrightarrow \operatorname(H) \\ \pi &\longmapsto \pi, _H \end and the other, the induction functor :\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. 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
topological group In mathematics, topological groups are 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 structures ...
and its category 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 essen ...
s. 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

* {{refend


External links

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