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

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:^[2]

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.^[3]
For all μ, λ in Λ,

\dim _{k}\operatorname {Hom} _{k}(A(\lambda ),A(\mu ))

is finite, and the multiplicity^[4]

[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
$\bigcup _{i}F_{i}(\lambda )=I(\lambda ).$

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.

↑ In the sense that it admits arbitrary direct limits of subobjects and every object is a union of its subobjects of finite length.
↑ Cline, Parshall & Scott 1988, §3
↑ Here, a composition factor of an object A in C is, by definition, a composition factor of one of its finite length subobjects.
↑ Here, 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.