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[1]
• has enough injectives
• satisfies

${\displaystyle B\cap \left(\bigcup _{\alpha }A_{\alpha }\right)=\bigcup _{\alpha }\left(B\cap A_{\alpha }\right)}$

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,

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

is finite, and the multiplicity[4]

${\displaystyle [A(\lambda ):S(\mu )]}$

is also finite.

• Each S(λ) has an injective envelope I(λ) in C equipped with an increasing filtration

${\displaystyle 0=F_{0}(\lambda )\subseteq F_{1}(\lambda )\subseteq \dots \subseteq I(\lambda )}$

such that

1. ${\displaystyle F_{1}(\lambda )=A(\lambda )}$
2. for n > 1, ${\displaystyle F_{n}(\lambda )/F_{n-1}(\lambda )\cong A(\mu )}$ for some μ = μ(n) > λ
3. for each μ in Λ, μ(n) = μ for only finitely many n
4. ${\displaystyle \bigcup _{i}F_{i}(\lambda )=I(\lambda ).}$