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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Highest-weight category ： ウィキペディア英語版 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''〔In the sense that it admits arbitrary direct limits of subobjects and every object is a union of its subobjects of finite length.〕 *has enough injectives *satisfies ::$B\backslash cap\backslash left(\backslash bigcup\_\backslash alpha\; A\_\backslash alpha\backslash right)=\backslash bigcup\_\backslash alpha\backslash left(B\backslash cap\; A\_\backslash alpha\backslash right)$ :for all subobjects ''B'' and each family of subobjects 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 in C. * Λ also indexes a collection of objects 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''.〔Here, a composition factor of an object ''A'' in C is, by definition, a composition factor of one of its finite length subobjects.〕 * For all ''μ'', ''λ'' in Λ, ::$\backslash dim\_k\backslash operatorname\_k(A(\backslash lambda),A(\backslash mu))$ :is finite, and the multiplicity〔Here, if ''A'' is an object in C and ''S'' is a simple object in C, the multiplicity () is, by definition, the supremum of the multiplicity of ''S'' in all finite length subobjects of ''A''.〕 ::$()$ :is also finite. *Each ''S''(''λ'') has an injective envelope ''I''(''λ'') in C equipped with an increasing filtration ::$0=F\_0(\backslash lambda)\backslash subseteq\; F\_1(\backslash lambda)\backslash subseteq\backslash dots\backslash subseteq\; I(\backslash lambda)$ :such that :# $F\_1(\backslash lambda)=A(\backslash lambda)$ :# for ''n'' > 1, $F\_n(\backslash lambda)/F\_(\backslash lambda)\backslash cong\; A(\backslash mu)$ for some ''μ'' = ''μ''(''n'') > ''λ'' :# for each ''μ'' in Λ, ''μ''(''n'') = ''μ'' for only finitely many ''n'' :# $\backslash bigcup\_iF\_i(\backslash lambda)=I(\backslash lambda).$ == Examples == * The module category of the $k$-algebra of upper triangular $n\backslash times\; n$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Highest-weight category」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.