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&alpha;} 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.


 * Each S(λ) has an injective envelope I(λ) in C equipped with an increasing filtration
 * $$0=F_0(\lambda)\subseteq F_1(\lambda)\subseteq\dots\subseteq I(\lambda)$$
 * 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
 * $$\bigcup_iF_i(\lambda)=I(\lambda).$$

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.