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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 Lie algebra homomorphism ： ウィキペディア英語版 Lie algebra In mathematics, a Lie algebra (, not ) is a vector space together with a non-associative multiplication called "Lie bracket" $(y)$. It was introduced to study the concept of infinitesimal transformations. Hermann Weyl introduced the term "Lie algebra" (after Sophus Lie) in the 1930s. In older texts, the name "infinitesimal group" is used. Lie algebras are closely related to Lie groups which are groups that are also smooth manifolds, with the property that the group operations of multiplication and inversion are smooth maps. Any Lie group gives rise to a Lie algebra. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering (Lie's third theorem). This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras. == Definitions == A Lie algebra is a vector space $\backslash ,\backslash mathfrak$ over some field ''F'' together with a binary operation $():\; \backslash mathfrak\backslash times\backslash mathfrak\backslash to\backslash mathfrak$ called the Lie bracket that satisfies the following axioms: * Bilinearity, ::$(x\; +\; b\; y,\; z)\; =\; a\; (z)\; +\; b\; (z),\; \backslash quad\; (a\; x\; +\; b\; y)\; =\; a(x)\; +\; b\; (y)$ :for all scalars ''a'', ''b'' in ''F'' and all elements ''x'', ''y'', ''z'' in $\backslash mathfrak$. * Alternativity, ::$()=0\backslash$ :for all ''x'' in $\backslash mathfrak$. * The Jacobi identity, :: $]\; +$ and using alternativity shows that $()\; +\; ()=0\backslash$ for all elements ''x'', ''y'' in $\backslash mathfrak$, showing that bilinearity and alternativity together imply *Anticommutativity, ::, :for all elements ''x'', ''y'' in $\backslash mathfrak$. Anticommutativity only implies the alternating property if the field's characteristic is not 2.〔Humphreys p. 1〕 It is customary to express a Lie algebra in lower-case fraktur, like $\backslash mathfrak$. If a Lie algebra is associated with a Lie group, then the spelling of the Lie algebra is the same as that Lie group. For example, the Lie algebra of SU(''n'') is written as $\backslash mathfrak(n)$. 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」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.