Highest-weight category

In the

• 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

• The module category of the ${\displaystyle k}$-algebra of upper triangular ${\displaystyle n\times n}$ matrices over ${\displaystyle k}$.
• This concept is named after the category of
  highest-weight modules
  of Lie-algebras.
• A finite-dimensional ${\displaystyle k}$-algebra ${\displaystyle A}$ is
  semisimple and hereditary
  algebras are highest-weight categories.
• A
  iff
  its Cartan-determinant is 1.