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)&nbsp;→&nbsp;A(λ) such that all composition factors S(μ) of A(λ)/S(λ) satisfy μ&nbsp;<&nbsp;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

#

# 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&nbsp;1.

Notes

References

See also

• Category O