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= Category of representations =
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Introduction
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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 conditions under which
this category is semisimple; i.e., whether an object decomposes into
simple objects (see Maschke's theorem for the case of finite groups).
The Tannakian formalism gives conditions under which a group 'G' may
be recovered from the category of representations of it together with
the forgetful functor to the category of vector spaces.
The Grothendieck ring of the category of finite-dimensional
representations of a group 'G' is called the representation ring of
'G'.
Definitions
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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 spaces over and representations of
on that vector space
* morphisms: equivariant maps
* composition: the composition of equivariant maps
* identities: the identity function (which is indeed an equivariant
map).
The category is denoted by \operatorname{Rep}_F(G) or
\operatorname{Rep}(G).
For a Lie group, one typically requires the representations to be
smooth or admissible. For the case of a Lie algebra, see Lie algebra
representation. See also: category O.
The category of modules over the group ring
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There is an isomorphism of categories between the category of
representations of a group over a field (described above) and the
category of modules over the group ring [], denoted []-Mod.
Category-theoretic definition
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Every group can be viewed as a category with a single object, where
morphisms in this category are the elements of and composition is
given by the group operation; so is the automorphism group of the
unique object. Given an arbitrary category , a 'representation' of in
is a functor from to . Such a functor sends the unique object to an
object say ' in and induces a group homomorphism G \to
\operatorname{Aut}(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, and a linear representation is equivalent to a
functor to Vect, the category of vector spaces over a field.
In this setting, the category of linear representations of over is
the functor category � Vect, which has natural transformations as its
morphisms.
Properties
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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 states that when the characteristic of doesn't
divide the order of , the category of representations of over is
semisimple.
Restriction and induction
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Given a group with a subgroup , there are two fundamental functors
between the categories of representations of and (over a fixed
field): one is a forgetful functor called the restriction functor
:
\begin{align}
\operatorname{Res}_H^G : \operatorname{Rep}(G) &\longrightarrow
\operatorname{Rep}(H) \\
\pi &\longmapsto \pi|_H
\end{align}
and the other, the induction functor
:\operatorname{Ind}_H^G : \operatorname{Rep}(H) \to
\operatorname{Rep}(G).
When and are finite groups, they are adjoint to each other
:\operatorname{Hom}_G(\operatorname{Ind}_H^G W, U) \cong
\operatorname{Hom}_H(W, \operatorname{Res}_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
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Tannaka-Krein duality concerns the interaction of a compact
topological group 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.
External links
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*
https://ncatlab.org/nlab/show/category+of+representations
License
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License URL:
http://creativecommons.org/licenses/by-sa/3.0/
Original Article:
http://en.wikipedia.org/wiki/Category_of_representations
