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In representation theory, the category of representations of some algebraic structure A has the representations of A 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 Grothendieck ring of the category of finite-dimensional representations of a group G is called the representation ring of G.

Attributes	Values
rdf:type	Thing
rdfs:label	Category of representations (en)
rdfs:comment	In representation theory, the category of representations of some algebraic structure A has the representations of A 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 Grothendieck ring of the category of finite-dimensional representations of a group G is called the representation ring of G. (en)
rdfs:seeAlso	Equivariant map Representation theory of finite groups Modules dbr:The_convolution_algebra
dcterms:subject	Category theory Representation theory
Wikipage page ID	55872661 (xsd:integer)
Wikipage revision ID	1068615965 (xsd:integer)
Link from a Wikipage to another Wikipage	Monoidal category Representation theory Algebraic structure Vector space Lie algebra representation Lie group Mackey theory Maschke's theorem Natural transformation Function composition Morphism Lie algebra Compact group Frobenius reciprocity Functor Subgroup Admissible representation Category of sets Topological group Tuple Adjoint functor Field (mathematics) Forgetful functor Isomorphism of categories Group action (mathematics) Group homomorphism Finite groups Category theory Representation theory Characteristic (algebra) Tensor product of representations Automorphism group Group ring Identity function Order (group theory) Category (mathematics) Category O Category of modules Category of vector spaces Restricted representation Representation ring Tannakian formalism Linear representation Grothendieck ring Induction functor Simple object Smooth representation Object (category theory) Semisimple category
Link from a Wikipage to an external page	https://ncatlab.org/nlab/show/category+of+representations
sameAs	Category of representations Category of representations
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Expand_section dbt:Main dbt:Refbegin dbt:Refend dbt:Reflist dbt:See_also dbt:Var
has abstract	In representation theory, the category of representations of some algebraic structure A has the representations of A 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. (en)
prov:wasDerivedFrom	wikipedia-en:Category_of_representations?oldid=1068615965&ns=0
page length (characters) of wiki page	6798 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Category_of_representations
is Link from a Wikipage to another Wikipage of	Semi-simplicity Tannaka–Krein duality Maschke's theorem Glossary of representation theory Representation theory of finite groups Pontryagin duality Category of modules Polynomial functor
is foaf:primaryTopic of	wikipedia-en:Category_of_representations

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