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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 {*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,

- 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*

- 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.

**^**In the sense that it admits arbitrary direct limits of subobjects and every object is a union of its subobjects of finite length.**^**Cline & 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*.

- 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.

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