翻訳と辞書 Words near each other ・ Lidé na kře ・ Lidérc ・ Lidó Rico ・ Lidón ・ Lie ・ Lie (disambiguation) ・ Lie (song) ・ Lie (surname) ・ Lie (T-ara song) ・ Lie a Little Better ・ Lie About Us ・ Lie algebra ・ Lie algebra bundle ・ Lie algebra cohomology ・ Lie algebra extension ・ Lie algebra representation ・ Lie algebra-valued differential form ・ Lie algebroid ・ Lie Back and Enjoy It ・ Lie Back and Think of England ・ Lie bialgebra ・ Lie bialgebroid ・ Lie bracket of vector fields ・ Lie Cliff ・ Lie coalgebra ・ Lie conformal algebra ・ Lie derivative ・ Lie detection ・ Lie Detector (disambiguation) ・ Lie Detector (TV series) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lie algebra representation ： ウィキペディア英語版 Lie algebra representation In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays a decisive role. The universality of this construction of this ring says that the category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra. ==Formal definition== A representation of a Lie algebra $\backslash mathfrak\; g$ is a Lie algebra homomorphism :$\backslash rho\backslash colon\; \backslash mathfrak\; g\; \backslash to\; \backslash mathfrak(V)$ from $\backslash mathfrak\; g$ to the Lie algebra of endomorphisms on a vector space ''V'' (with the commutator as the Lie bracket), sending an element ''x'' of $\backslash mathfrak\; g$ to an element ''ρ''''x'' of $\backslash mathfrak(V)$. Explicitly, this means that :$\backslash rho\_\; =\; ()\; =\; \backslash rho\_x\backslash rho\_y\; -\; \backslash rho\_y\backslash rho\_x\backslash ,$ for all ''x,y'' in $\backslash mathfrak\; g$. The vector space ''V'', together with the representation ρ, is called a $\backslash mathfrak\; g$-module. (Many authors abuse terminology and refer to ''V'' itself as the representation). The representation $\backslash rho$ is said to be faithful if it is injective. One can equivalently define a $\backslash mathfrak\; g$-module as a vector space ''V'' together with a bilinear map $\backslash mathfrak\; g\; \backslash times\; V\backslash to\; V$ such that :$()\backslash cdot\; v\; =\; x\backslash cdot(y\backslash cdot\; v)\; -\; y\backslash cdot(x\backslash cdot\; v)$ for all ''x,y'' in $\backslash mathfrak\; g$ and ''v'' in ''V''. This is related to the previous definition by setting ''x'' ⋅ ''v'' = ρ''x'' (v). 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「Lie algebra 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.