Category of representations

In representation theory, the category of representations of some algebraic structure Template:Var has the representations of Template:Var 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 representations one wants to consider, it is typical to use slightly different definitions.

For a finite group Template:Var and a field Template:Var, the category of representations of Template:Var over Template:Var has

• Objects: Pairs (Template:Var, Template:Var) of vector spaces Template:Var over Template:Var and representations Template:Var of Template:Var 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 ${\mathrm{Rep}}_F(G)$ or $\mathrm{Rep}(G)$.

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

There is an isomorphism of categories between the category of representations of a group Template:Var over a field Template:Var (described above) and the category of modules over the group ring Template:Var[[[:Template:Var]]], denoted Template:Var[[[:Template:Var]]]-Mod.

Category-theoretic definition

Every group Template:Var can be viewed as a category with a single object, where morphisms in this category are the elements of Template:Var and composition is given by the group operation; so Template:Var is the automorphism group of the unique object. Given an arbitrary category Template:Var, a representation of Template:Var in Template:Var is a functor from Template:Var to Template:Var. Such a functor sends the unique object to an object say Template:Var in Template:Var and induces a group homomorphism $G\to \mathrm{Aut}(X)$; see Automorphism group#In category theory for more. For example, a [[Group action (mathematics)|Template:Var-set]] is equivalent to a functor from Template:Var to Set, the category of sets, and a linear representation is equivalent to a functor to Vect_Template:Var, the category of vector spaces over a field Template:Var.

In this setting, the category of linear representations of Template:Var over Template:Var is the functor category Template:Var → Vect_Template:Var, which has natural transformations as its morphisms.

Properties

Template:Expand section 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 Template:Var doesn't divide the order of Template:Var, the category of representations of Template:Var over Template:Var is semisimple.

Restriction and induction

Given a group Template:Var with a subgroup Template:Var, there are two fundamental functors between the categories of representations of Template:Var and Template:Var (over a fixed field): one is a forgetful functor called the restriction functor

$\begin{array}{cc}{\mathrm{Res}}_H^G:\mathrm{Rep}(G)& ⟶\mathrm{Rep}(H)\\ \pi & ⟼\pi {|}_H\end{array}$

and the other, the induction functor

${\mathrm{Ind}}_H^G:\mathrm{Rep}(H)\to \mathrm{Rep}(G)$.

When Template:Var and Template:Var are finite groups, they are adjoint to each other

${\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,U)\cong {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}U)$,

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

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 Template:Var back to the group Template:Var, 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.

References

• Template:Citation

External links

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