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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 adjoint representation ： ウィキペディア英語版 adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, in the case where ''G'' is the Lie group of invertible matrices of size ''n'', ''GL(n)'', the Lie algebra is the vector space of all (not necessarily invertible) ''n''-by-''n'' matrices. So in this case the adjoint representation is the vector space of ''n''-by-''n'' matrices $x$, and any element ''g'' in ''GL(n)'' acts as a linear transformation of this vector space given by conjugation: $x\; \backslash mapsto\; g\; x\; g^$. For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. ==Definition== Let ''G'' be a Lie group and let $\backslash mathfrak\; g$ be its Lie algebra (which we identify with ''TeG'', the tangent space to the identity element in ''G''). Define the map :$\backslash Psi\; :\; G\; \backslash to\; \backslash mathrm(G),\; \backslash ,\; g\; \backslash mapsto\; \backslash Psi\_g$ where Aut(''G'') is the automorphism group of ''G'' and the automorphism Ψ''g'' is defined by :$\backslash Psi\_g(h)\; =\; ghg^\backslash ,$ for all ''h'' in ''G''. The differential of Ψ''g'' at the identity is an automorphism of the Lie algebra $\backslash mathfrak\; g$. We denote this map by Ad''g'': :$d(\backslash Psi\_g)\_e=\backslash mathrm\_g\backslash colon\; \backslash mathfrak\; g\; \backslash to\; \backslash mathfrak\; g.$ To say that Ad''g'' is a Lie algebra automorphism is to say that Ad''g'' is a linear transformation of $\backslash mathfrak\; g$ that preserves the Lie bracket. The map :$\backslash mathrm\backslash colon\; G\; \backslash to\; \backslash mathrm(\backslash mathfrak\; g),\; \backslash ,\; g\; \backslash mapsto\; \backslash mathrm\_g$ is called the adjoint representation of ''G''. This is indeed a representation of ''G'' since $\backslash mathrm(\backslash mathfrak\; g)$ is a closed〔The condition that a linear map is a Lie algebra homomorphism is a closed condition.〕 Lie subgroup of $\backslash mathrm(\backslash mathfrak\; g)$ and the above adjoint map is a Lie group homomorphism. Note Ad is a trivial map if ''G'' is abelian. If ''G'' is an (immersed) Lie subgroup of the general linear group $GL\_n(\backslash mathbb)$, then, since the exponential map is the matrix exponential: $\backslash operatorname(X)\; =\; e^X$, taking the derivative of $\backslash Psi\_g(\backslash operatorname(tX))\; =\; ge^g^$ at ''t'' = 0, one gets: for ''g'' in ''G'' and ''X'' in $\backslash mathfrak$, :$\backslash operatorname\_g(X)\; =\; gX\; g^$ where on the right we have the products of matrices. 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「adjoint 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.