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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 root datum ： ウィキペディア英語版 root datum In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970. ==Definition== A root datum consists of a quadruple :$(X^\backslash ast,\; \backslash Phi,\; X\_\backslash ast,\; \backslash Phi^\backslash vee)$, where * $X^\backslash ast$ and $X\_\backslash ast$ are free abelian groups of finite rank together with a perfect pairing between them with values in $\backslash mathbb$ which we denote by ( , ) (in other words, each is identified with the dual lattice of the other). * $\backslash Phi$ is a finite subset of $X^\backslash ast$ and $\backslash Phi^\backslash vee$ is a finite subset of $X\_\backslash ast$ and there is a bijection from $\backslash Phi$ onto $\backslash Phi^\backslash vee$, denoted by $\backslash alpha\backslash mapsto\backslash alpha^\backslash vee$. * For each $\backslash alpha$, $(\backslash alpha,\; \backslash alpha^\backslash vee)=2$. * For each $\backslash alpha$, the map $x\backslash mapsto\; x-(x,\backslash alpha^\backslash vee)\backslash alpha$ induces an automorphism of the root datum (in other words it maps $\backslash Phi$ to $\backslash Phi$ and the induced action on $X\_\backslash ast$ maps $\backslash Phi^\backslash vee$ to $\backslash Phi^\backslash vee$) The elements of $\backslash Phi$ are called the roots of the root datum, and the elements of $\backslash Phi^\backslash vee$ are called the coroots. The elements of $X^\backslash ast$ are sometimes called weights and those of $X\_\backslash ast$ accordingly coweights. If $\backslash Phi$ does not contain $2\backslash alpha$ for any $\backslash alpha\backslash in\backslash Phi$, then the root datum is called reduced. 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「root datum」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.