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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 Table of Lie groups ： ウィキペディア英語版 Table of Lie groups This article gives a table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple). For more examples of Lie groups and other related topics see the list of simple Lie groups; the Bianchi classification of groups of up to three dimensions; and the list of Lie group topics. == Real Lie groups and their algebras == Column legend * Cpt: Is this group ''G'' compact? (Yes or No) * $\backslash pi\_0$: Gives the group of components of ''G''. The order of the component group gives the number of connected components. The group is connected if and only if the component group is trivial (denoted by 0). * $\backslash pi\_1$: Gives the fundamental group of ''G'' whenever ''G'' is connected. The group is simply connected if and only if the fundamental group is trivial (denoted by 0). * UC: If ''G'' is not simply connected, gives the universal cover of ''G''. a">& b \\ 0 & 0\end\right ) : a,b \in \mathbb\right\} | align=center | 2 |- | align=center | H× | non-zero quaternions with multiplication | N | 0 | 0 | | | align=center | H | align=center | 4 |- | align=center | ''S''3 = Sp(1) | quaternions of absolute value 1, with multiplication; topologically a 3-sphere | Y | 0 | 0 | | isomorphic to SU(2) and to Spin(3); double cover of SO(3) | align=center | Im(H) | align=center | 3 |- | align=center | GL(''n'',R) | general linear group: invertible ''n''×''n'' real matrices | N | Z2 | – | | | align=center | M(''n'',R) | align=center | ''n''2 |- | align=center | GL+(''n'',R) | ''n''×''n'' real matrices with positive determinant | N | 0 | Z  ''n''=2 Z2 ''n''>2 | | GL+(1,R) is isomorphic to R+ and is simply connected | align=center | M(''n'',R) | align=center | ''n''2 |- | align=center | SL(''n'',R) | special linear group: real matrices with determinant 1 | N | 0 | Z  ''n''=2 Z2 ''n''>2 | | SL(1,R) is a single point and therefore compact and simply connected | align=center | sl(''n'',R) | align=center | ''n''2−1 |- | align=center | SL(2,R) | Orientation-preserving isometries of the Poincaré half-plane, isomorphic to SU(1,1), isomorphic to Sp(2,R). | N | 0 | Z | | The universal cover has no finite-dimensional faithful representations. | align=center | sl(2,R) | align=center | 3 |- | align=center | O(''n'') | orthogonal group: real orthogonal matrices | Y | Z2 | – | | The symmetry group of the sphere (n=3) or hypersphere. | align=center | so(''n'') | align=center | ''n''(''n''−1)/2 |- | align=center | SO(''n'') | special orthogonal group: real orthogonal matrices with determinant 1 | Y | 0 | Z  ''n''=2 Z2 ''n''>2 | Spin(''n'') ''n''>2 | SO(1) is a single point and SO(2) is isomorphic to the circle group, SO(3) is the rotation group of the sphere. | align=center | so(''n'') | align=center | ''n''(''n''−1)/2 |- | align=center | Spin(''n'') | spin group: double cover of SO(''n'') | Y | 0 ''n''>1 | 0 ''n''>2 | | Spin(1) is isomorphic to Z2 and not connected; Spin(2) is isomorphic to the circle group and not simply connected | align=center | so(''n'') | align=center | ''n''(''n''−1)/2 |- | align=center | Sp(2''n'',R) | symplectic group: real symplectic matrices | N | 0 | Z | | | align=center | sp(2''n'',R) | align=center | ''n''(2''n''+1) |- | align=center | Sp(''n'') | compact symplectic group: quaternionic ''n''×''n'' unitary matrices | Y | 0 | 0 | | | align=center | sp(''n'') | align=center | ''n''(2''n''+1) |- | align=center | U(''n'') | unitary group: complex ''n''×''n'' unitary matrices | Y | 0 | Z | R×SU(''n'') | For ''n''=1: isomorphic to S1. Note: this is ''not'' a complex Lie group/algebra | align=center | u(''n'') | align=center | ''n''2 |- | align=center | SU(''n'') | special unitary group: complex ''n''×''n'' unitary matrices with determinant 1 | Y | 0 | 0 | | Note: this is ''not'' a complex Lie group/algebra | align=center | su(''n'') | align=center | ''n''2−1 |- |} 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「Table of Lie groups」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.