Highest-weight category

• is locally artinian
• has enough injectives
• satisfies

• The poset Λ indexes an exhaustive set of non-isomorphic simple objects in C.
• Λ also indexes a collection of objects of objects of C such that there exist embeddings S → A such that all composition factors S of A/''S satisfy μ'' < λ.
• For all μ, λ in Λ,
• Each S has an injective envelope I in C equipped with an increasing filtration

  Examples

• The module category of the -algebra of upper triangular matrices over.
• This concept is named after the category of highest-weight modules of Lie-algebras.
• A finite-dimensional -algebra 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 iff its Cartan-determinant is 1.