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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 Dual representation ： ウィキペディア英語版 Dual representation In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:〔Lecture 1 of 〕 : is the transpose of , that is, = for all . The dual representation is also known as the contragredient representation. If is a Lie algebra and is a representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:〔Lecture 8 of 〕 : = for all . In both cases, the dual representation is a representation in the usual sense. ==Motivation== In representation theory, both vectors in and linear functionals in are considered as ''column vectors'' so that the representation can act (by matrix multiplication) from the ''left''. Given a basis for and the dual basis for , the action of a linear functional on , can be expressed by matrix multiplication, :$\backslash langle\backslash varphi,\; v\backslash rangle\; \backslash equiv\; \backslash varphi(v)\; =\; \backslash varphi^Tv$, where the superscript is matrix transpose. Consistency requires :$\backslash langle^$ *(g)\varphi, \rho(g)v\rangle = \langle\varphi, v\rangle.〔Lecture 1, page 4 of 〕 With the definition given, :$\backslash langle^$ *(g)\varphi, \rho(g)v\rangle = \langle\rho(g^)^T\varphi, \rho(g)v\rangle = (\rho(g^)^T\varphi)^T \rho(g)v = \varphi^T\rho(g^)\rho(g)v = \varphi^Tv = \langle\varphi, v\rangle. For the Lie algebra representation one chooses consistency with a possible group representation. Generally, if is a representation of a Lie group, then given by :$\backslash pi(X)\; =\; \backslash frac\backslash Pi(e^)|\_.$ is a representation of its Lie algebra. If is dual to , then its corresponding Lie algebra representation is given by :$\backslash pi^$ *(X) = \frac\Pi^ *(e^)|_ = \frac\Pi(e^)^T|_ = -\pi(X)^T..〔Lecture 8, page 111 of 〕 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「Dual 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.