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 artinianIn the sense that it admits arbitrary direct limits of subobjects and every object is a union of its subobjects of finite length. has enough injectives satisfies
for all subobjects B and each family of subobjects {Aα} 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.Here, a composition factor of an object A in C is, by definition, a composition factor of one of its finite length subobjects. For all μ, λ in Λ,
is finite, and the multiplicityHere, if A is an object in C and S is a simple object in C, the multiplicity [A:S] is, by definition, the supremum of the multiplicity of S in all finite length subobjects of A.
is also finite. Each S(λ) has an injective envelope I(λ) in C equipped with an increasing filtration
such that
for n > 1, for some μ = λ(n) > λ for each μ in Λ, λ(n) = μ for only finitely many n
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 (and hence its module category a highest-weight category) iff its Cartan-determinant is 1.
Notes
References
See also Category O