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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 Coadjoint representation ： ウィキペディア英語版 Coadjoint representation In mathematics, the coadjoint representation $K$ of a Lie group $G$ is the dual of the adjoint representation. If $\backslash mathfrak$ denotes the Lie algebra of $G$, the corresponding action of $G$ on $\backslash mathfrak^$ *, the dual space to $\backslash mathfrak$, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on $G$. The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups $G$ a basic role in their representation theory is played by coadjoint orbit. In the Kirillov method of orbits, representations of $G$ are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of $G$, which again may be complicated, while the orbits are relatively tractable. ==Formal definition== Let $G$ be a Lie group and $\backslash mathfrak$ be its Lie algebra. Let $\backslash mathrm\; :\; G\; \backslash rightarrow\; \backslash mathrm(\backslash mathfrak)$ denote the adjoint representation of $G$. Then the coadjoint representation $K\; :\; G\; \backslash rightarrow\; \backslash mathrm(\backslash mathfrak^$ *) is defined as $\backslash mathrm^$ *(g^) := \mathrm(g^)^ *. More explicitly, :$\backslash langle\; K(g)F,\; Y\; \backslash rangle\; =\; \backslash langle\; F,\; \backslash mathrm(g^)Y\; \backslash rangle$ for $g\; \backslash in\; G,\; Y\; \backslash in\; \backslash mathfrak,\; F\; \backslash in\; \backslash mathfrak^$ *, where $\backslash langle\; F,\; Y\; \backslash rangle$ denotes the value of a linear functional $F$ on a vector $Y$. Let $K\_$ denote the representation of the Lie algebra $\backslash mathfrak$ on $\backslash mathfrak^$ * induced by the coadjoint representation of the Lie group $G$. Then for $X\; \backslash in\; \backslash mathfrak,\; K\_(X)\; =\; -\backslash mathrm(X)^$ * where $\backslash mathrm$ is the adjoint representation of the Lie algebra $\backslash mathfrak$. One may make this observation from the infinitesimal version of the defining equation for $K$ above, which is as follows : :$\backslash langle\; K\_(X)F,\; Y\; \backslash rangle\; =\; \backslash langle\; F,\; -\; \backslash mathrm(X)Y\; \backslash rangle$ for $X,\; Y\; \backslash in\; \backslash mathfrak,\; F\; \backslash in\; \backslash mathfrak^$ *. . 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「Coadjoint representation」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.