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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 Filtered algebra ： ウィキペディア英語版 Filtered algebra In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory. A filtered algebra over the field $k$ is an algebra $(A,\backslash cdot)$ over $k$ which has an increasing sequence $\backslash \; \backslash subset\; F\_0\; \backslash subset\; F\_1\; \backslash subset\; \backslash cdots\; \backslash subset\; F\_i\; \backslash subset\; \backslash cdots\; \backslash subset\; A$ of subspaces of $A$ such that :$A=\backslash cup\_,\backslash qquad\; F\_m\backslash cdot\; F\_n\backslash subset\; F\_.$ ==Associated graded algebra== In general there is the following construction that produces a graded algebra out of a filtered algebra. If $A$ is a filtered algebra then the ''associated graded algebra'' $\backslash mathcal(A)$ is defined as follows: G_n\,, where, :$G\_0=F\_0,$ and :$\backslash forall\; n>0,\; \backslash quad\; G\_n=F\_n/F\_\backslash ,,$ |2= the multiplication is defined by :$(x+F\_)(y+F\_)=x\backslash cdot\; y+F\_$ for all $x\backslash in\; F\_n$ and $y\backslash in\; F\_m$. (More precisely, the multiplication map $\backslash mathcal(A)\backslash times\; \backslash mathcal(A)\; \backslash to\; \backslash mathcal(A)$ is combined from the maps :$(F\_n\; /\; F\_)\; \backslash times\; (F\_m\; /\; F\_)\; \backslash to\; F\_/F\_,\; \backslash \; \backslash \; \backslash \; \backslash \; \backslash \; \backslash left(x+F\_,y+F\_\backslash right)\; \backslash mapsto\; x\backslash cdot\; y+F\_$ for all $n\backslash geq\; 0$ and $m\backslash geq\; 0$.) }} The multiplication is well defined and endows $\backslash mathcal(A)$ with the structure of a graded algebra, with gradation $\backslash \_(A)$. Also if $A$ is unital, such that the unit lies in $F\_0$, then $\backslash mathcal(A)$ will be unital as well. As algebras $A$ and $\backslash mathcal(A)$ are distinct (with the exception of the trivial case that $A$ is graded) but as vector spaces they are isomorphic. 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「Filtered algebra」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.