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

has enough injectives

satisfies

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:

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.

For all μ, λ in Λ,

is finite, and the multiplicity is also finite.

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

such that

1. F 1 ( λ ) = A ( λ )
2. for n > 1, F n ( λ ) / F n − 1 ( λ ) ≅ A ( μ ) for some μ = μ(n) > λ
3. for each μ in Λ, μ(n) = μ for only finitely many n
4. ⋃ i F i ( λ ) = I ( λ ) .