Highest-weight category

: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*) → *A*(*λ*) such that all composition factors *S*(*μ*) of *A*(*λ*)/*S*(*λ*) satisfy *μ* - For all *μ*, *λ* in Λ, ::\dim_k\operatorname{Hom}_k(A(\lambda),A(\mu)) :is finite, and the multiplicity ::[A(\lambda):S(\mu)] :is also finite. - Each *S*(*λ*) has an injective envelope *I*(*λ*) in **C** equipped with an increasing filtration ::0=F_0(\lambda)\subseteq F_1(\lambda)\subseteq\dots\subseteq I(\lambda) :such that :# F_1(\lambda)=A(\lambda) :# for *n* 1, F_n(\lambda)/F_{n-1}(\lambda)\cong A(\mu) for some *μ* = *λ*(*n*) *λ* :# for each *μ* in Λ, *λ*(*n*) = *μ* for only finitely many *n* :# \bigcup_iF_i(\lambda)=I(\lambda). ## Examples - The module category of the k-algebra of upper triangular n\times n matrices over k. - This concept is named after the category of highest-weight modules of Lie-algebras. - A finite-dimensional k-algebra 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 ## References - {{cite journal | access-date=2012-07-17 ## References 1. In the sense that it admits arbitrary [[direct limit]]s of [[subobject]]s and every object is a union of its subobjects of [[finite length object. finite length]]. 2. {{harvnb. 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. 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''. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Highest-weight_category) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Highest-weight_category?action=history). ::