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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 Spin group ： ウィキペディア英語版 Spin group In mathematics the spin group Spin(''n'') 〔 page 14〕〔 page 15〕 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups :$1\; \backslash to\; \backslash mathrm\_2\; \backslash to\; \backslash operatorname(n)\; \backslash to\; \backslash operatorname(n)\; \backslash to\; 1.$ As a Lie group, Spin(''n'') therefore shares its dimension, , and its Lie algebra with the special orthogonal group. For ''n'' > 2, Spin(''n'') is simply connected and so coincides with the universal cover of SO(''n''). The non-trivial element of the kernel is denoted −1 , which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −''I'' . Spin(''n'') can be constructed as a subgroup of the invertible elements in the Clifford algebra ''C''ℓ(''n''). ==Accidental isomorphisms== In low dimensions, there are isomorphisms among the classical Lie groups called ''accidental isomorphisms''. For instance, there are isomorphisms between low-dimensional spin groups and certain classical Lie groups, owing to low-dimensional isomorphisms between the root systems (and corresponding isomorphisms of Dynkin diagrams) of the different families of simple Lie algebras. Specifically, we have :Spin(1) = O(1).     dim=0 :Spin(2) = U(1) = SO(2), which acts on in by double phase rotation .     dim=1 :Spin(3) = Sp(1) = SU(2), corresponding to $B\_1\; \backslash cong\; A\_1.$     dim=3 :Spin(4) = SU(2) × SU(2), corresponding to $D\_2\; \backslash cong\; A\_1\; \backslash times\; A\_1.$     dim=6 :Spin(5) = Sp(2), corresponding to $B\_2\; \backslash cong\; C\_2.$     dim=10 :Spin(6) = SU(4), corresponding to $D\_3\; \backslash cong\; A\_3.$     dim=15 There are certain vestiges of these isomorphisms left over for  = 7, 8 (see Spin(8) for more details). For higher , these isomorphisms disappear entirely. 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「Spin group」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.