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
- for all subobjects B and each family of subobjects {Aα} 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,
- is finite, and the multiplicity[4]
- 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
[edit]- 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 1.
Notes
[edit]- ^ In the sense that it admits arbitrary direct limits of subobjects and every object is a union of its subobjects of finite length.
- ^ Cline, Parshall & Scott 1988, §3
- ^ Here, a composition factor of an object A in C is, by definition, a composition factor of one of its finite length subobjects.
- ^ 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
[edit]- Cline, E.; Parshall, B.; Scott, L. (January 1988). "Finite-dimensional algebras and highest-weight categories" (PDF). Journal für die reine und angewandte Mathematik. 1988 (391). Berlin, Germany: Walter de Gruyter: 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.