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 = λ(n) > λ
3. for each μ in Λ, λ(n) = μ for only finitely many n
4. ${\displaystyle \bigcup _{i}F_{i}(\lambda )=I(\lambda ).}$

Examples

• 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 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.

Notes

1. In the sense that it admits arbitrary direct limits of subobjects and every object is a union of its subobjects of finite length.
2. Cline, Parshall & Scott 1988, §3
3. Here, a composition factor of an object A in C is, by definition, a composition factor of one of its finite length subobjects.
4. 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.

References

• Cline, E.; Parshall, B.; Scott, L. (January 1988). "Finite-dimensional algebras and highest-weight categories" (PDF). Journal für die reine und angewandte Mathematik. Berlin, Germany: Walter de Gruyter. 1988 (391): 85–99. CiteSeerX 10.1.1.112.6181. doi:10.1515/crll.1988.391.85. ISSN 0075-4102. OCLC 1782270. S2CID 118202731. Retrieved 2012-07-17.

See also

• Category O