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| {{Lie groups}}
| | I am Christa from Kropfing studying Political Science. I did my schooling, secured 89% and hope to find someone with same interests in Model Aircraft Hobbies.<br><br>Here is my weblog: Www.Hostgator1Centcoupon.info, [http://iaco.ajou.ac.kr/?document_srl=514632 iaco.ajou.ac.kr], |
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| This article gives a table of some common [[Lie group]]s and their associated [[Lie algebra]]s.
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| The following are noted: the [[Topology|topological]] properties of the group ([[dimension]]; [[Connected space|connectedness]]; [[Compact space|compactness]]; the nature of the [[fundamental group]]; and whether or not they are [[simply connected]]) as well as on their algebraic properties ([[Abelian group|abelian]]; [[Simple group|simple]]; [[Semisimple group|semisimple]]).
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| 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]].
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| == Real Lie groups and their algebras ==
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| Column legend
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| * '''CM''': Is this group ''G'' [[Compact space|compact]]? (Yes or No)
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| * '''<math>\pi_0</math>''': Gives the [[group of components]] of ''G''. The order of the component group gives the number of [[connected space|connected components]]. The group is [[connected space|connected]] if and only if the component group is [[trivial group|trivial]] (denoted by 0).
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| * '''<math>\pi_1</math>''': Gives the [[fundamental group]] of ''G'' whenever ''G'' is connected. The group is [[simply connected]] if and only if the fundamental group is [[trivial group|trivial]] (denoted by 0).
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| * '''UC''': If ''G'' is not simply connected, gives the [[universal cover]] of ''G''.
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| {{clr}}
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| {| border="1" cellpadding="2" cellspacing="0"
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| |- style="background-color:#eee"
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| ! Lie group
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| ! Description
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| ! CM
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| ! <math>\pi_0</math>
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| ! <math>\pi_1</math>
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| ! UC
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| ! Remarks
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| ! Lie algebra
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| ! dim/'''R'''
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| |-
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| | align=center | '''R'''<sup>''n''</sup>
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| | [[Euclidean space]] with addition
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| | N
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| | 0
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| | 0
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| | abelian
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| | align=center | '''R'''<sup>''n''</sup>
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| | align=center | ''n''
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| |-
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| | align=center | '''R'''<sup>×</sup>
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| | nonzero [[real number]]s with multiplication
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| | N
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| | '''Z'''<sub>2</sub>
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| | –
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| | abelian
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| | align=center | '''R'''
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| | align=center | 1
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| |-
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| | align=center | '''R'''<sup>+</sup>
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| | positive real numbers with multiplication
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| | N
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| | 0
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| | 0
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| | abelian
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| | align=center | '''R'''
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| | align=center | 1
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| |-
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| | align=center | ''S''<sup>1</sup> = U(1)
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| | the [[circle group]]: [[complex number]]s of absolute value 1, with multiplication;
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| | Y
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| | 0
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| | '''Z'''
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| | '''R'''
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| | abelian, isomorphic to SO(2), Spin(2), and '''R'''/'''Z'''
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| | align=center | '''R'''
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| | align=center | 1
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| |-
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| | align=center | [[Affine group|Aff(1)]]
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| | invertible [[affine transformation]]s from '''R''' to '''R'''.
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| | N
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| | '''Z'''<sub>2</sub>
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| | 0
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| | [[solvable group|solvable]], [[semidirect product]] of '''R'''<sup>+</sup> and '''R'''<sup>×</sup>
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| | align=center | <math>\left\{\left[\begin{smallmatrix}a & b \\ 0 & 0\end{smallmatrix}\right] : a,b \in \mathbb{R}\right\}</math>
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| | align=center | 2
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| |-
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| | align=center | '''H'''<sup>×</sup>
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| | non-zero [[quaternions]] with multiplication
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| | N
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| | 0
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| | 0
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| |
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| | align=center | '''H'''
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| | align=center | 4
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| |-
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| | align=center | ''S''<sup>3</sup> = Sp(1)
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| | [[quaternions]] of [[absolute value]] 1, with multiplication; topologically a [[3-sphere]]
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| | Y
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| | 0
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| | 0
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| | isomorphic to [[SU(2)]] and to [[Spin(3)]]; [[Double covering group|double cover]] of [[SO(3)]]
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| | align=center | Im('''H''')
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| | align=center | 3
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| |-
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| | align=center | GL(''n'','''R''')
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| | [[general linear group]]: [[invertible matrix|invertible]] ''n''×''n'' real [[matrix (mathematics)|matrices]]
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| | N
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| | '''Z'''<sub>2</sub>
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| | –
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| | align=center | M(''n'','''R''')
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| | align=center | ''n''<sup>2</sup>
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| |-
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| | align=center | GL<sup>+</sup>(''n'','''R''')
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| | ''n''×''n'' real matrices with positive [[determinant]]
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| | N
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| | 0
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| | '''Z''' ''n''=2<br>'''Z'''<sub>2</sub> ''n''>2
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| | GL<sup>+</sup>(1,'''R''') is isomorphic to '''R'''<sup>+</sup> and is simply connected
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| | align=center | M(''n'','''R''')
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| | align=center | ''n''<sup>2</sup>
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| |-
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| | align=center | SL(''n'','''R''')
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| | [[special linear group]]: real matrices with [[determinant]] 1
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| | N
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| | 0
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| | '''Z''' ''n''=2<br>'''Z'''<sub>2</sub> ''n''>2
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| | SL(1,'''R''') is a single point and therefore compact and simply connected
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| | align=center | sl(''n'','''R''')
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| | align=center | ''n''<sup>2</sup>−1
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| |-
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| | align=center | [[SL2(R)|SL(2,'''R''')]]
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| | Orientation-preserving isometries of the [[Poincaré half-plane]], isomorphic to SU(1,1), isomorphic to Sp(2,'''R''').
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| | N
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| | 0
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| | '''Z'''
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| | The [[universal cover]] has no finite-dimensional faithful representations.
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| | align=center | sl(2,'''R''')
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| | align=center | 3
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| |-
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| | align=center | O(''n'')
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| | [[orthogonal group]]: real [[orthogonal matrix|orthogonal matrices]]
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| | Y
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| | '''Z'''<sub>2</sub>
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| | –
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| | The symmetry group of the [[sphere]] (n=3) or [[hypersphere]].
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| | align=center | so(''n'')
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| | align=center | ''n''(''n''−1)/2
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| |-
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| | align=center | SO(''n'')
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| | [[special orthogonal group]]: real orthogonal matrices with determinant 1
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| | Y
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| | 0
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| | '''Z''' ''n''=2<br>'''Z'''<sub>2</sub> ''n''>2
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| | Spin(''n'')<br>''n''>2
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| | SO(1) is a single point and SO(2) is isomorphic to the [[circle group]], SO(3) is the rotation group of the sphere.
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| | align=center | so(''n'')
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| | align=center | ''n''(''n''−1)/2
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| |-
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| | align=center | Spin(''n'')
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| | [[spin group]]: [[Double covering group|double cover]] of SO(''n'')
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| | Y
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| | 0 ''n''>1
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| | 0 ''n''>2
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| | Spin(1) is isomorphic to '''Z'''<sub>2</sub> and not connected; Spin(2) is isomorphic to the circle group and not simply connected
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| | align=center | so(''n'')
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| | align=center | ''n''(''n''−1)/2
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| |-
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| | align=center | Sp(2''n'','''R''')
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| | [[symplectic group]]: real [[symplectic matrix|symplectic matrices]]
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| | N
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| | 0
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| | '''Z'''
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| | align=center | sp(2''n'','''R''')
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| | align=center | ''n''(2''n''+1)
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| |-
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| | align=center | Sp(''n'')
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| | [[compact symplectic group]]: quaternionic ''n''×''n'' [[unitary matrix|unitary matrices]]
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| | Y
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| | 0
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| | 0
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| | align=center | sp(''n'')
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| | align=center | ''n''(2''n''+1)
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| |-
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| | align=center | U(''n'')
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| | [[unitary group]]: [[complex number|complex]] ''n''×''n'' [[unitary matrix|unitary matrices]]
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| | Y
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| | 0
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| | '''Z'''
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| | '''R'''×SU(''n'')
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| | For ''n''=1: isomorphic to S<sup>1</sup>. Note: this is ''not'' a complex Lie group/algebra
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| | align=center | u(''n'')
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| | align=center | ''n''<sup>2</sup>
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| |-
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| | align=center | SU(''n'')
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| | [[special unitary group]]: [[complex number|complex]] ''n''×''n'' [[unitary matrix|unitary matrices]] with determinant 1
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| | Y
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| | 0
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| | 0
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| | Note: this is ''not'' a complex Lie group/algebra
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| | align=center | su(''n'')
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| | align=center | ''n''<sup>2</sup>−1
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| |-
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| |}
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| ==Real Lie algebras==
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| Table legend:
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| * '''S''': Is this algebra simple? (Yes or No)
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| * '''SS''': Is this algebra [[Semisimple Lie algebra|semi-simple]]? (Yes or No)
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| {| border="1" cellpadding="2" cellspacing="0"
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| |- style="background-color:#eee"
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| ! Lie algebra
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| ! Description
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| ! S
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| ! SS
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| ! Remarks
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| ! dim/'''R'''
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| |-
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| | align=center | '''R'''
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| | the [[real number]]s, the Lie bracket is zero
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| | align=center | 1
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| |-
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| | align=center | '''R'''<sup>''n''</sup>
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| | the Lie bracket is zero
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| | align=center | ''n''
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| |-
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| | align=center | '''R'''<sup>''3''</sup>
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| | the Lie bracket is the [[cross product]]
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| | align=center | ''3''
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| |-
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| | align=center | '''H'''
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| | [[quaternions]], with Lie bracket the commutator
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| | align=center | 4
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| |-
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| | align=center | Im('''H''')
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| | quaternions with zero real part, with Lie bracket the commutator; isomorphic to real 3-vectors,
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| with Lie bracket the [[cross product]]; also isomorphic to su(2) and to so(3,'''R''')
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| | Y
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| | Y
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| | align=center | 3
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| |-
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| | align=center | M(''n'','''R''')
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| | ''n''×''n'' matrices, with Lie bracket the commutator
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| | align=center | ''n''<sup>2</sup>
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| |-
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| | align=center | sl(''n'','''R''')
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| | square matrices with [[trace of a matrix|trace]] 0, with Lie bracket the commutator
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| | Y
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| | Y
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| | align=center | ''n''<sup>2</sup>−1
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| |-
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| | align=center | so(''n'')
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| | [[skew-symmetric]] square real matrices, with Lie bracket the commutator.
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| | Y
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| | Y
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| | Exception: so(4) is semi-simple, but ''not'' simple.
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| | align=center | ''n''(''n''−1)/2
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| |-
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| | align=center | sp(2''n'','''R''')
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| | real matrices that satisfy ''JA'' + ''A''<sup>''T''</sup>''J'' = 0 where ''J'' is the standard [[skew-symmetric matrix]]
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| | Y
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| | Y
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| | align=center | ''n''(2''n''+1)
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| |-
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| | align=center | sp(''n'')
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| | square quaternionic matrices ''A'' satisfying ''A'' = −''A''<sup>*</sup>, with Lie bracket the commutator
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| | Y
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| | Y
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| | align=center | ''n''(2''n''+1)
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| |-
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| | align=center | u(''n'')
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| | square complex matrices ''A'' satisfying ''A'' = −''A''<sup>*</sup>, with Lie bracket the commutator
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| | align=center | ''n''<sup>2</sup>
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| |-
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| | align=center | su(''n'') <br>''n''≥2
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| | square complex matrices ''A'' with trace 0 satisfying ''A'' = −''A''<sup>*</sup>, with Lie bracket the commutator
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| | Y
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| | Y
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| | align=center | ''n''<sup>2</sup>−1
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| |-
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| |}
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| == [[Complex Lie group]]s and their algebras ==
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| The dimensions given are dimensions over '''C'''. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.
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| {| border="1" cellpadding="2" cellspacing="0"
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| |- style="background-color:#eee"
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| ! Lie group
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| ! Description
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| ! CM
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| ! <math>\pi_0</math>
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| ! <math>\pi_1</math>
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| ! UC
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| ! Remarks
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| ! Lie algebra
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| ! dim/'''C'''
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| |-
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| | align="center" | '''C'''<sup>''n''</sup>
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| | group operation is addition
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| | N
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| | 0
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| | 0
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| | abelian
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| | align="center" | '''C'''<sup>''n''</sup>
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| | align="center" | ''n''
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| |-
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| | align="center" | '''C'''<sup>×</sup>
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| | nonzero [[complex number]]s with multiplication
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| | N
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| | 0
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| | '''Z'''
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| |
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| | abelian
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| | align="center" | '''C'''
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| | align="center" | 1
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| |-
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| | align="center" | GL(''n'','''C''')
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| | [[general linear group]]: [[invertible matrix|invertible]] ''n''×''n'' complex [[Matrix (mathematics)|matrices]]
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| | N
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| | 0
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| | '''Z'''
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| | For ''n''=1: isomorphic to '''C'''<sup>×</sup>
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| | align="center" | M(''n'','''C''')
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| | align="center" | ''n''<sup>2</sup>
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| |-
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| | align="center" | SL(''n'','''C''')
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| | [[special linear group]]: complex matrices with [[determinant]]
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| 1
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| | N
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| | 0
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| | 0
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| |
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| | for n=1 this is a single point and thus compact.
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| | align="center" | sl(''n'','''C''')
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| | align="center" | ''n''<sup>2</sup>−1
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| |-
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| | align="center" | SL(2,'''C''')
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| | Special case of SL(''n'','''C''') for ''n''=2
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| | N
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| | 0
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| | 0
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| |
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| | Isomorphic to Spin(3,'''C'''), isomorphic to Sp(2,'''C''')
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| | align="center" | sl(2,'''C''')
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| | align="center" | 3
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| |-
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| | align="center" | PSL(2,'''C''')
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| | Projective special linear group
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| | N
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| | 0
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| | '''Z'''<sub>2</sub>
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| | SL(2,'''C''')
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| | Isomorphic to the [[Möbius group]], isomorphic to the restricted [[Lorentz group]] SO<sup>+</sup>(3,1,'''R'''), isomorphic to SO(3,'''C''').
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| | align="center" | sl(2,'''C''')
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| | align="center" | 3
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| |-
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| | align="center" | O(''n'','''C''')
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| | [[orthogonal group]]: complex [[orthogonal matrix|orthogonal matrices]]
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| | N
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| | '''Z'''<sub>2</sub>
| |
| | –
| |
| |
| |
| | compact for n=1
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| | align="center" | so(''n'','''C''')
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| | align="center" | ''n''(''n''−1)/2
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| |-
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| | align="center" | SO(''n'','''C''')
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| | [[special orthogonal group]]: complex orthogonal matrices with determinant 1
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| | N
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| | 0
| |
| | '''Z''' ''n''=2<br>'''Z'''<sub>2</sub> ''n''>2
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| |
| |
| | SO(2,'''C''') is abelian and isomorphic to '''C'''<sup>×</sup>; nonabelian for ''n''>2. SO(1,'''C''') is a single point and thus compact and simply connected
| |
| | align="center" | so(''n'','''C''')
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| | align="center" | ''n''(''n''−1)/2
| |
| |-
| |
| | align="center" | Sp(2''n'','''C''')
| |
| | [[symplectic group]]: complex [[symplectic matrix|symplectic matrices]]
| |
| | N
| |
| | 0
| |
| | 0
| |
| |
| |
| |
| |
| | align="center" | sp(2''n'','''C''')
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| | align="center" | ''n''(2''n''+1)
| |
| |-
| |
| |}
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| | |
| == Complex Lie algebras ==
| |
| | |
| The dimensions given are dimensions over '''C'''. Note that every complex Lie algebra can also be viewed as a real Lie algebra of twice the dimension.
| |
| | |
| {| border="1" cellpadding="2" cellspacing="0"
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| |- style="background-color:#eee"
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| ! Lie algebra
| |
| ! Description
| |
| ! S
| |
| ! SS
| |
| ! Remarks
| |
| ! dim/'''C'''
| |
| |-
| |
| | align="center" | '''C'''
| |
| | the [[complex number]]s
| |
| |
| |
| |
| |
| |
| |
| | align="center" | 1
| |
| |-
| |
| | align="center" | '''C'''<sup>''n''</sup>
| |
| | the Lie bracket is zero
| |
| |
| |
| |
| |
| |
| |
| | align="center" | ''n''
| |
| |-
| |
| | align="center" | M(''n'','''C''')
| |
| | ''n''×''n'' matrices, with Lie bracket the commutator
| |
| |
| |
| |
| |
| |
| |
| | align="center" | ''n''<sup>2</sup>
| |
| |-
| |
| | align="center" | sl(''n'','''C''')
| |
| | square matrices with [[trace of a matrix|trace]] 0, with Lie bracket
| |
| the commutator
| |
| | Y
| |
| | Y
| |
| |
| |
| | align="center" | ''n''<sup>2</sup>−1
| |
| |-
| |
| | align="center" | sl(2,'''C''')
| |
| | Special case of sl(''n'','''C''') with ''n''=2
| |
| | Y
| |
| | Y
| |
| | isomorphic to su(2) <math>\otimes</math> '''C'''
| |
| | align="center" | 3
| |
| |-
| |
| | align="center" | so(''n'','''C''')
| |
| | [[skew-symmetric]] square complex matrices, with Lie bracket
| |
| the commutator
| |
| | Y
| |
| | Y
| |
| | Exception: so(4,'''C''') is semi-simple, but not simple.
| |
| | align="center" | ''n''(''n''−1)/2
| |
| |-
| |
| | align="center" | sp(2''n'','''C''')
| |
| | complex matrices that satisfy ''JA'' + ''A''<sup>''T''</sup>''J'' = 0
| |
| where ''J'' is the standard [[skew-symmetric matrix]]
| |
| | Y
| |
| | Y
| |
| |
| |
| | align="center" | ''n''(2''n''+1)
| |
| |-
| |
| |}
| |
| | |
| ==References==
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| * {{Fulton-Harris}}
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| [[Category:Lie groups]]
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| [[Category:Lie algebras]]
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| [[Category:Mathematics-related lists|Lie groups]]
| |