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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