Highest-weight category
In the mathematical field of representation theory, a highest-weight category is a k-linear category C that
- 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.