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 
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 an equivariant map).

The category is denoted by  or .

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 

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 

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 ; 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 

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 
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

and the other, the induction functor
.

When  and  are finite groups, they are adjoint to each other
,
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 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.

Notes

References

External links 
 https://ncatlab.org/nlab/show/category+of+representations

Representation theory
Category theory