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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 Jordan triple system ： ウィキペディア英語版 Triple system In algebra, a triple system (or ternar) is a vector space ''V'' over a field F together with a F-trilinear map :$(\backslash cdot,\backslash cdot,\backslash cdot)\; \backslash colon\; V\backslash times\; V\; \backslash times\; V\backslash to\; V.$ The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators , ''w''] and triple anticommutators {''u'', {''v'', ''w''}}. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations ((R-space )]s and their noncompact duals). ==Lie triple systems== A triple system is said to be a Lie triple system if the trilinear form, denoted (), satisfies the following identities: :$()\; =\; -()$ :$()\; +\; ()\; +\; ()\; =\; 0$ :$,x,y]\; +\; +\; +\; +\; +\; +\; ,\; is\; a\; derivation\; (algebra)">derivation\; of\; the\; triple\; product.\; The\; identity\; also\; shows\; that\; the\; space\; k\; =\; span\; is\; closed\; under\; commutator\; bracket,\; hence\; a\; Lie\; algebra.$ Writing m in place of ''V'', it follows that :$\backslash mathfrak\; g\; :=\; \backslash mathfrak\; k\; \backslash oplus\backslash mathfrak\; m$ can be made into a Lie algebra with bracket :$()\; =\; (()+L\_,\; L(v)\; -\; M(u)).$ The decomposition of g is clearly a symmetric decomposition for this Lie bracket, and hence if ''G'' is a connected Lie group with Lie algebra g and ''K'' is a subgroup with Lie algebra k, then ''G''/''K'' is a symmetric space. Conversely, given a Lie algebra g with such a symmetric decomposition (i.e., it is the Lie algebra of a symmetric space), the triple bracket (''v'' ), ''w''] makes m into a Lie triple system. 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「Triple system」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.