Highest-weight category
# 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.
