Highest-weight category
In the mathematical field of representation theory, a highest-weight category is a k-linear category C (here k is a field) that is locally artinian has enough injectives satisfies B ∩ ( ⋃ α A α ) = ⋃ α ( B ∩ A α ) {\displaystyle B\cap \left(\bigcup _{\alpha }A_{\alpha }\right)=\bigcup _{\alpha }\left(B\cap A_{\alpha }\right)} for all subobjects B and each family of subobjects {Aα} of each object X and such that there is a locally finite poset Λ (whose elements are called the weights of C) that satisfies the following conditions: The poset Λ indexes an exhaustive set of non-isomorphic simple objects {S(λ)} in C. Λ also indexes a collection of objects {A(λ)} of objects of C such that there exist embeddings S(λ) → A(λ) such that all composition factors S(μ) of A(λ)/S(λ) satisfy μ < λ. For all μ, λ in Λ, dim k Hom k ( A ( λ ) , A ( μ ) ) {\displaystyle \dim _{k}\operatorname {Hom} _{k}(A(\lambda ),A(\mu ))} is finite, and the multiplicity [ A ( λ ) : S ( μ ) ] {\displaystyle [A(\lambda ):S(\mu )]} is also finite.