title: Highest-weight category
text: # 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
  
         $B\cap \left(\bigcup _{\alpha }A_{\alpha }\right)=\bigcup _{\alpha }\left(B\cap A_{\alpha }\right)$
      for all subobjects B and each family of subobjects ${A_α}$ 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,
  
         $\dim _{k}\operatorname {Hom} _{k}(A(\lambda ),A(\mu ))$
      is finite, and the multiplicity
         $[A(\lambda ):S(\mu )]$
      is also finite.
         $0=F_{0}(\lambda )\subseteq F_{1}(\lambda )\subseteq \dots \subseteq I(\lambda )$
      such that
     * $F_{1}(\lambda )=A(\lambda )$
  
     * for n > 1, $F_{n}(\lambda )/F_{n-1}(\lambda )\cong A(\mu )$ for some μ = λ(n) > λ
  
     * for each μ in Λ, λ(n) = μ for only finitely many n
  
     * $\bigcup _{i}F_{i}(\lambda )=I(\lambda ).$
  
  ## Examples
  
  * The module category of the $k$-algebra of upper triangular $n\times n$ matrices over $k$.
  * This concept is named after the category of highest-weight modules of Lie-algebras.
  * A finite-dimensional $k$-algebra $A$ is quasi-hereditary iff its module category is a highest-weight category. In particular all module-categories over semisimple and hereditary algebras are highest-weight categories.
  * A cellular algebra over a field is quasi-hereditary (and hence its module category a highest-weight category) iff its Cartan-determinant is 1.
  