Editing Table of Lie groups

{{Short description|Lie groups and their associated Lie algebras}}
{{Lie groups}}

This article gives a table of some common [[Lie group]]s and their associated [[Lie algebra]]s.

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 Lie group|simple]]; [[Semisimple Lie group|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; see [[classification of low-dimensional real Lie algebras]] for up to four dimensions; and the [[list of Lie group topics]].

== Real Lie groups and their algebras ==

Column legend
* '''Cpt''': Is this group ''G'' [[Compact space|compact]]? (Yes or No)
* '''<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).
* '''<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).
* '''UC''': If ''G'' is not simply connected, gives the [[universal cover]] of ''G''.
{{clr}}
{| class="wikitable"
|- style="background-color:#eee"
! Lie group
! Description
! Cpt
! <math>\pi_0</math>
! <math>\pi_1</math>
! UC
! Remarks
! Lie algebra
! dim/'''R'''
|- 
| align=center | '''R'''<sup>''n''</sup>
| [[Euclidean space]] with addition
| N
| 0
| 0
|
| abelian
| align=center | '''R'''<sup>''n''</sup>
| align=center | ''n''
|- 
| align=center | '''R'''<sup>&times;</sup>
| nonzero [[real number]]s with multiplication
| N
| '''Z'''<sub>2</sub>
| &ndash;
|
| abelian
| align=center | '''R'''
| align=center | 1
|- 
| align=center | '''R'''<sup>+</sup>
| [[positive real numbers]] with multiplication
| N
| 0
| 0
|
| abelian
| align=center | '''R'''
| align=center | 1
|- 
| align=center | ''S''<sup>1</sup>&nbsp;=&nbsp;U(1)
| the [[circle group]]: [[complex number]]s of absolute value 1 with multiplication;
| Y
| 0
| '''Z'''
| '''R'''
| abelian,  isomorphic to SO(2), Spin(2), and '''R'''/'''Z'''
| align=center | '''R'''
| align=center | 1
|- 
| align=center | [[Affine group|Aff(1)]]
| invertible [[affine transformation]]s from '''R''' to '''R'''.
| N
| '''Z'''<sub>2</sub>
| &ndash;
| 
| [[solvable group|solvable]], [[semidirect product]] of '''R'''<sup>+</sup> and '''R'''<sup>&times;</sup>
| align=center | <math>\left\{\left[\begin{smallmatrix}a & b \\ 0 & 1\end{smallmatrix}\right] : a\in \R^*,b \in \mathbb{R}\right\}</math>
| align=center | 2
|- 
| align=center | '''H'''<sup>&times;</sup>
| non-zero [[quaternions]] with multiplication
| N
| 0
| 0
| 
|
| align=center | '''H'''
| align=center | 4
|- 
| align=center | ''S''<sup>3</sup>&nbsp;=&nbsp;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 covering group|double cover]] of [[SO(3)]]
| align=center | Im('''H''')
| align=center | 3
|- 
| align=center | GL(''n'','''R''')
| [[general linear group]]: [[invertible matrix|invertible]] ''n''&times;''n'' real [[matrix (mathematics)|matrices]]
| N
| '''Z'''<sub>2</sub>
| &ndash;
|
| 
| align=center | M(''n'','''R''')
| align=center | ''n''<sup>2</sup>
|- 
| align=center | GL<sup>+</sup>(''n'','''R''')
| ''n''&times;''n'' real matrices with positive [[determinant]]
| N
| 0
| '''Z'''&nbsp;&nbsp;''n''=2<br>'''Z'''<sub>2</sub>&nbsp;''n''&gt;2
|
| GL<sup>+</sup>(1,'''R''') is isomorphic to '''R'''<sup>+</sup> and is simply connected
| align=center | M(''n'','''R''')
| align=center | ''n''<sup>2</sup>
|- 
| align=center | SL(''n'','''R''')
| [[special linear group]]: real matrices with [[determinant]] 1
| N
| 0
| '''Z'''&nbsp;&nbsp;''n''=2<br>'''Z'''<sub>2</sub>&nbsp;''n''&gt;2
| 
| SL(1,'''R''') is a single point and therefore compact and simply connected
| align=center | sl(''n'','''R''')
| align=center | ''n''<sup>2</sup>&minus;1
|- 
| align=center | [[SL2(R)|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 matrix|orthogonal matrices]]
| Y
| '''Z'''<sub>2</sub>
| &ndash;
|
| The symmetry group of the [[sphere]] (''n''=3) or [[hypersphere]].
| align=center | so(''n'')
| align=center | ''n''(''n''&minus;1)/2
|- 
| align=center | SO(''n'')
| [[special orthogonal group]]: real orthogonal matrices with determinant 1
| Y
| 0
| '''Z'''&nbsp;&nbsp;''n''=2<br>'''Z'''<sub>2</sub>&nbsp;''n''&gt;2
| Spin(''n'')<br>''n''&gt;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''&minus;1)/2
|- 
| align=center | SE(''n'')
| special [[euclidean group]]: group of rigid body motions in n-dimensional space.
| N
| 0
| 
| 
|
| align=center | se(''n'')
| align=center | ''n'' + ''n''(''n''&minus;1)/2
|- 
| align=center | Spin(''n'')
| [[spin group]]: [[Double covering group|double cover]] of SO(''n'')
| Y
| 0&nbsp;''n''&gt;1
| 0&nbsp;''n''&gt;2
|
| Spin(1) is isomorphic to '''Z'''<sub>2</sub> and not connected; Spin(2) is isomorphic to the circle group and not simply connected
| align=center | so(''n'')
| align=center | ''n''(''n''&minus;1)/2
|- 
| align=center | Sp(2''n'','''R''')
| [[symplectic group]]: real [[symplectic matrix|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''&times;''n'' [[unitary matrix|unitary matrices]]
| Y
| 0
| 0
|
| 
| align=center | sp(''n'')
| align=center | ''n''(2''n''+1)
|- 
| align=center | Mp(''2n'','''R''')
| [[metaplectic group]]: double cover of [[symplectic group|real symplectic group]] Sp(''2n'','''R''')
| Y
| 0
| '''Z'''
|
| Mp(2,'''R''') is a Lie group that is not [[algebraic group|algebraic]]
| align=center | sp(''2n'','''R''')
| align=center | ''n''(2''n''+1)
|- 
| align=center | U(''n'')
| [[unitary group]]: [[complex number|complex]] ''n''&times;''n'' [[unitary matrix|unitary matrices]]
| Y
| 0
| '''Z'''
| '''R'''&times;SU(''n'')
| For ''n''=1: isomorphic to S<sup>1</sup>. Note: this is ''not'' a complex Lie group/algebra
| align=center | u(''n'')
| align=center | ''n''<sup>2</sup>
|- 
| align=center | SU(''n'')
| [[special unitary group]]: [[complex number|complex]] ''n''&times;''n'' [[unitary matrix|unitary matrices]] with determinant 1
| Y
| 0
| 0
|
| Note: this is ''not'' a complex Lie group/algebra
| align=center | su(''n'')
| align=center | ''n''<sup>2</sup>&minus;1
|- 
|}

==Real Lie algebras==
{{Main|Classification of low-dimensional real Lie algebras}}

{| class="wikitable"
|- style="background-color:#eee"
! Lie algebra
! Description
! Simple?
! [[Semisimple Lie algebra|Semi-simple]]?
! Remarks
! dim/'''R'''
|- 
| align=center | '''R'''
| the [[real number]]s, the Lie bracket is zero
|
|
|
| align=center | 1
|- 
| align=center | '''R'''<sup>''n''</sup>
| the Lie bracket is zero
|
|
|
| align=center | ''n''
|-
| align=center | '''R'''<sup>''3''</sup>
| the Lie bracket is the [[cross product]]
| {{Yes}}
| {{Yes}}
|
| align=center | ''3''
|-
| align=center | '''H'''
| [[quaternions]], with Lie bracket the commutator
|
|
|
| align=center | 4
|- 
| align=center | Im('''H''')
| quaternions with zero real part, with Lie bracket the commutator; isomorphic to real 3-vectors,
with Lie bracket the [[cross product]]; also isomorphic to su(2) and to so(3,'''R''')
| {{Yes}}
| {{Yes}}
|
| align=center | 3
|- 
| align=center | M(''n'','''R''')
| ''n''&times;''n'' matrices, with Lie bracket the commutator
|
|
|
| align=center | ''n''<sup>2</sup>
|- 
| align=center | sl(''n'','''R''')
| square matrices with [[trace of a matrix|trace]] 0, with Lie bracket the commutator
| {{Yes}}
| {{Yes}}
| 
| align=center | ''n''<sup>2</sup>&minus;1
|- 
| align=center | so(''n'')
| [[skew-symmetric matrix|skew-symmetric]] square real matrices, with Lie bracket the commutator.
| {{Yes}}, except ''n''=4
| {{Yes}}
| Exception: so(4) is semi-simple,
but ''not'' simple.
| align=center | ''n''(''n''&minus;1)/2
|- 
| align=center | sp(2''n'','''R''')
| real matrices that satisfy ''JA'' + ''A''<sup>T</sup>''J'' = 0 where ''J'' is the standard [[skew-symmetric matrix]]
| {{Yes}}
| {{Yes}}
|
| align=center | ''n''(2''n''+1)
|- 
| align=center | sp(''n'')
| square quaternionic matrices ''A'' satisfying ''A'' = &minus;''A''<sup>∗</sup>, with Lie bracket the commutator
| {{Yes}}
| {{Yes}}
|
| align=center | ''n''(2''n''+1)
|- 
| align=center | u(''n'')
| square complex matrices ''A'' satisfying ''A'' = &minus;''A''<sup>∗</sup>, with Lie bracket the commutator
|
|
| Note: this is ''not'' a complex Lie algebra
| align=center | ''n''<sup>2</sup>
|- 
| align=center | su(''n'') <br>''n''≥2
| square complex matrices ''A'' with trace 0 satisfying ''A'' = &minus;''A''<sup>∗</sup>, with Lie bracket the commutator
| {{Yes}}
| {{Yes}}
| Note: this is ''not'' a complex Lie algebra
| align=center | ''n''<sup>2</sup>&minus;1
|- 
|}

== Complex Lie groups and their algebras ==
{{main|Complex Lie group}}
Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below 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.

{| class="wikitable"
|- style="background-color:#eee"
! Lie group
! Description
! Cpt
! <math>\pi_0</math>
! <math>\pi_1</math>
! UC
! Remarks
! Lie algebra
! dim/'''C'''
|-
| align="center" | '''C'''<sup>''n''</sup>
| group operation is addition
| N
| 0
| 0
| 
| abelian
| align="center" | '''C'''<sup>''n''</sup>
| align="center" | ''n''
|-
| align="center" | '''C'''<sup>&times;</sup>
| nonzero [[complex number]]s with multiplication
| N
| 0
| '''Z'''
| 
| abelian
| align="center" | '''C'''
| align="center" | 1
|-
| align="center" | GL(''n'','''C''')
| [[general linear group]]: [[invertible matrix|invertible]] ''n''&times;''n'' complex [[Matrix (mathematics)|matrices]]
| N
| 0 
| '''Z'''
| 
| For ''n''=1: isomorphic to '''C'''<sup>&times;</sup>
| align="center" | M(''n'','''C''')
| align="center" | ''n''<sup>2</sup>
|-
| align="center" | SL(''n'','''C''') 
| [[special linear group]]: complex matrices with [[determinant]]
1
| N
| 0
| 0
| 
| for ''n''=1 this is a single point and thus compact.
| align="center" | sl(''n'','''C''')
| align="center" | ''n''<sup>2</sup>&minus;1
|-
| align="center" | SL(2,'''C''')
| Special case of SL(''n'','''C''') for ''n''=2
| N
| 0
| 0
|
| Isomorphic to Spin(3,'''C'''), isomorphic to Sp(2,'''C''')
| align="center" | sl(2,'''C''')
| align="center" | 3
|- 
| align="center" | PSL(2,'''C''')
| Projective special linear group
| N
| 0
| '''Z'''<sub>2</sub>
| SL(2,'''C''')
| Isomorphic to the [[Möbius group]], isomorphic to the restricted [[Lorentz group]] SO<sup>+</sup>(3,1,'''R'''), isomorphic to SO(3,'''C'''). 
| align="center" | sl(2,'''C''')
| align="center" | 3
|-
| align="center" | O(''n'','''C''')
| [[orthogonal group]]: complex [[orthogonal matrix|orthogonal matrices]]
| N
| '''Z'''<sub>2</sub>
| &ndash;
| 
| finite for ''n''=1
| align="center" | so(''n'','''C''')
| align="center" | ''n''(''n''&minus;1)/2
|-
| align="center" | SO(''n'','''C''')
| [[special orthogonal group]]: complex orthogonal matrices with determinant 1
| N
| 0
| '''Z'''&nbsp;&nbsp;''n''=2<br>'''Z'''<sub>2</sub>&nbsp;''n''&gt;2
|
| SO(2,'''C''') is abelian and isomorphic to '''C'''<sup>&times;</sup>; nonabelian for ''n''&gt;2. SO(1,'''C''') is a single point and thus compact and simply connected
| align="center" | so(''n'','''C''')
| align="center" | ''n''(''n''&minus;1)/2
|-
| align="center" | Sp(2''n'','''C''')
| [[symplectic group]]: complex [[symplectic matrix|symplectic matrices]]
| N
| 0
| 0
|
|
| align="center" | sp(2''n'','''C''')
| align="center" | ''n''(2''n''+1)
|-
|}

== Complex Lie algebras ==
{{main|Complex Lie algebra}}
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.

{| class="wikitable"
|- style="background-color:#eee"
! Lie algebra
! Description
! Simple?
! Semi-simple?
! 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''&times;''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
| {{Yes}}
| {{Yes}}
|
| align="center" | ''n''<sup>2</sup>&minus;1
|-
| align="center" | sl(2,'''C''')
| Special case of sl(''n'','''C''') with ''n''=2
| {{Yes}}
| {{Yes}}
| isomorphic to su(2) <math>\otimes</math> '''C'''
| align="center" | 3
|- 
| align="center" | so(''n'','''C''')
| [[skew-symmetric matrix|skew-symmetric]] square complex matrices with Lie bracket
the commutator
| {{Yes}}, except ''n''=4
| {{Yes}}
| Exception: so(4,'''C''') is semi-simple,
but ''not'' simple.
| align="center" | ''n''(''n''&minus;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]]
| {{Yes}}
| {{Yes}}
|
| align="center" | ''n''(2''n''+1)
|-
|}

The Lie algebra of affine transformations of dimension two, in fact, exist for any field. An instance has already been listed in the first table for real Lie algebras.

== See also ==

* [[Classification of low-dimensional real Lie algebras]]

* [[Simple Lie group#Full classification]]

==References==
* {{Fulton-Harris}}

[[Category:Lie groups]]
[[Category:Lie algebras]]
[[Category:Mathematics-related lists|Lie groups]]