# Table of Lie groups > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Table_of_Lie_groups > Markdown URL: https://mediated.wiki/source/Table_of_Lie_groups.md > Source: https://en.wikipedia.org/wiki/Table_of_Lie_groups > Source revision: 1281237084 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Lie groups and their associated Lie algebras}} {{Lie groups}} This article gives a table of some common [Lie group](/source/Lie_group)s and their associated [Lie algebra](/source/Lie_algebra)s. The following are noted: the [topological](/source/Topology) properties of the group ([dimension](/source/dimension); [connectedness](/source/Connected_space); [compactness](/source/Compact_space); the nature of the [fundamental group](/source/fundamental_group); and whether or not they are [simply connected](/source/simply_connected)) as well as on their algebraic properties ([abelian](/source/Abelian_group); [simple](/source/Simple_Lie_group); [semisimple](/source/Semisimple_Lie_group)). For more examples of Lie groups and other related topics see the [list of simple Lie groups](/source/list_of_simple_Lie_groups); the [Bianchi classification](/source/Bianchi_classification) of groups of up to three dimensions; see [classification of low-dimensional real Lie algebras](/source/classification_of_low-dimensional_real_Lie_algebras) for up to four dimensions; and the [list of Lie group topics](/source/list_of_Lie_group_topics). == Real Lie groups and their algebras == Column legend * '''Cpt''': Is this group ''G'' [compact](/source/Compact_space)? (Yes or No) * '''\pi_0''': Gives the [group of components](/source/group_of_components) of ''G''. The order of the component group gives the number of [connected components](/source/connected_space). The group is [connected](/source/connected_space) if and only if the component group is [trivial](/source/trivial_group) (denoted by 0). * '''\pi_1''': Gives the [fundamental group](/source/fundamental_group) of ''G'' whenever ''G'' is connected. The group is [simply connected](/source/simply_connected) if and only if the fundamental group is [trivial](/source/trivial_group) (denoted by 0). * '''UC''': If ''G'' is not simply connected, gives the [universal cover](/source/universal_cover) of ''G''. {{clr}} {| class="wikitable" |- style="background-color:#eee" ! Lie group ! Description ! Cpt ! \pi_0 ! \pi_1 ! UC ! Remarks ! Lie algebra ! dim/'''R''' |- | align=center | '''R'''''n'' | [Euclidean space](/source/Euclidean_space) with addition | N | 0 | 0 | | abelian | align=center | '''R'''''n'' | align=center | ''n'' |- | align=center | '''R'''× | nonzero [real number](/source/real_number)s with multiplication | N | '''Z'''2 | – | | abelian | align=center | '''R''' | align=center | 1 |- | align=center | '''R'''+ | [positive real numbers](/source/positive_real_numbers) with multiplication | N | 0 | 0 | | abelian | align=center | '''R''' | align=center | 1 |- | align=center | ''S''1 = U(1) | the [circle group](/source/circle_group): [complex number](/source/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 | [Aff(1)](/source/Affine_group) | invertible [affine transformation](/source/affine_transformation)s from '''R''' to '''R'''. | N | '''Z'''2 | – | | [solvable](/source/solvable_group), [semidirect product](/source/semidirect_product) of '''R'''+ and '''R'''× | align=center | \left\{\left[\begin{smallmatrix}a & b \\ 0 & 1\end{smallmatrix}\right] : a\in \R^*,b \in \mathbb{R}\right\} | align=center | 2 |- | align=center | '''H'''× | non-zero [quaternions](/source/quaternions) with multiplication | N | 0 | 0 | | | align=center | '''H''' | align=center | 4 |- | align=center | ''S''3 = Sp(1) | [quaternions](/source/quaternions) of [absolute value](/source/absolute_value) 1 with multiplication; topologically a [3-sphere](/source/3-sphere) | Y | 0 | 0 | | isomorphic to [SU(2)](/source/SU(2)) and to [Spin(3)](/source/Spin(3)); [double cover](/source/Double_covering_group) of [SO(3)](/source/SO(3)) | align=center | Im('''H''') | align=center | 3 |- | align=center | GL(''n'','''R''') | [general linear group](/source/general_linear_group): [invertible](/source/invertible_matrix) ''n''×''n'' real [matrices](/source/matrix_(mathematics)) | N | '''Z'''2 | – | | | align=center | M(''n'','''R''') | align=center | ''n''2 |- | align=center | GL+(''n'','''R''') | ''n''×''n'' real matrices with positive [determinant](/source/determinant) | N | 0 | '''Z'''  ''n''=2
'''Z'''2 ''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](/source/special_linear_group): real matrices with [determinant](/source/determinant) 1 | N | 0 | '''Z'''  ''n''=2
'''Z'''2 ''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''')](/source/SL2(R)) | Orientation-preserving isometries of the [Poincaré half-plane](/source/Poincar%C3%A9_half-plane), isomorphic to SU(1,1), isomorphic to Sp(2,'''R'''). | N | 0 | '''Z''' | | The [universal cover](/source/universal_cover) has no finite-dimensional faithful representations. | align=center | sl(2,'''R''') | align=center | 3 |- | align=center | O(''n'') | [orthogonal group](/source/orthogonal_group): real [orthogonal matrices](/source/orthogonal_matrix) | Y | '''Z'''2 | – | | The symmetry group of the [sphere](/source/sphere) (''n''=3) or [hypersphere](/source/hypersphere). | align=center | so(''n'') | align=center | ''n''(''n''−1)/2 |- | align=center | SO(''n'') | [special orthogonal group](/source/special_orthogonal_group): real orthogonal matrices with determinant 1 | Y | 0 | '''Z'''  ''n''=2
'''Z'''2 ''n''>2 | Spin(''n'')
''n''>2 | SO(1) is a single point and SO(2) is isomorphic to the [circle group](/source/circle_group), SO(3) is the rotation group of the sphere. | align=center | so(''n'') | align=center | ''n''(''n''−1)/2 |- | align=center | SE(''n'') | special [euclidean group](/source/euclidean_group): group of rigid body motions in n-dimensional space. | N | 0 | | | | align=center | se(''n'') | align=center | ''n'' + ''n''(''n''−1)/2 |- | align=center | Spin(''n'') | [spin group](/source/spin_group): [double cover](/source/Double_covering_group) of SO(''n'') | Y | 0 ''n''>1 | 0 ''n''>2 | | Spin(1) is isomorphic to '''Z'''2 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](/source/symplectic_group): real [symplectic matrices](/source/symplectic_matrix) | N | 0 | '''Z''' | | | align=center | sp(2''n'','''R''') | align=center | ''n''(2''n''+1) |- | align=center | Sp(''n'') | [compact symplectic group](/source/compact_symplectic_group): quaternionic ''n''×''n'' [unitary matrices](/source/unitary_matrix) | Y | 0 | 0 | | | align=center | sp(''n'') | align=center | ''n''(2''n''+1) |- | align=center | Mp(''2n'','''R''') | [metaplectic group](/source/metaplectic_group): double cover of [real symplectic group](/source/symplectic_group) Sp(''2n'','''R''') | Y | 0 | '''Z''' | | Mp(2,'''R''') is a Lie group that is not [algebraic](/source/algebraic_group) | align=center | sp(''2n'','''R''') | align=center | ''n''(2''n''+1) |- | align=center | U(''n'') | [unitary group](/source/unitary_group): [complex](/source/complex_number) ''n''×''n'' [unitary matrices](/source/unitary_matrix) | 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](/source/special_unitary_group): [complex](/source/complex_number) ''n''×''n'' [unitary matrices](/source/unitary_matrix) with determinant 1 | Y | 0 | 0 | | Note: this is ''not'' a complex Lie group/algebra | align=center | su(''n'') | align=center | ''n''2−1 |- |} ==Real Lie algebras== {{Main|Classification of low-dimensional real Lie algebras}} {| class="wikitable" |- style="background-color:#eee" ! Lie algebra ! Description ! Simple? ! [Semi-simple](/source/Semisimple_Lie_algebra)? ! Remarks ! dim/'''R''' |- | align=center | '''R''' | the [real number](/source/real_number)s, the Lie bracket is zero | | | | align=center | 1 |- | align=center | '''R'''''n'' | the Lie bracket is zero | | | | align=center | ''n'' |- | align=center | '''R'''''3'' | the Lie bracket is the [cross product](/source/cross_product) | {{Yes}} | {{Yes}} | | align=center | ''3'' |- | align=center | '''H''' | [quaternions](/source/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](/source/cross_product); also isomorphic to su(2) and to so(3,'''R''') | {{Yes}} | {{Yes}} | | align=center | 3 |- | align=center | M(''n'','''R''') | ''n''×''n'' matrices, with Lie bracket the commutator | | | | align=center | ''n''2 |- | align=center | sl(''n'','''R''') | square matrices with [trace](/source/trace_of_a_matrix) 0, with Lie bracket the commutator | {{Yes}} | {{Yes}} | | align=center | ''n''2−1 |- | align=center | so(''n'') | [skew-symmetric](/source/skew-symmetric_matrix) 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''−1)/2 |- | align=center | sp(2''n'','''R''') | real matrices that satisfy ''JA'' + ''A''T''J'' = 0 where ''J'' is the standard [skew-symmetric matrix](/source/skew-symmetric_matrix) | {{Yes}} | {{Yes}} | | align=center | ''n''(2''n''+1) |- | align=center | sp(''n'') | square quaternionic matrices ''A'' satisfying ''A'' = −''A''∗, with Lie bracket the commutator | {{Yes}} | {{Yes}} | | align=center | ''n''(2''n''+1) |- | align=center | u(''n'') | square complex matrices ''A'' satisfying ''A'' = −''A''∗, with Lie bracket the commutator | | | Note: this is ''not'' a complex Lie algebra | align=center | ''n''2 |- | align=center | su(''n'')
''n''≥2 | square complex matrices ''A'' with trace 0 satisfying ''A'' = −''A''∗, with Lie bracket the commutator | {{Yes}} | {{Yes}} | Note: this is ''not'' a complex Lie algebra | align=center | ''n''2−1 |- |} == 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 ! \pi_0 ! \pi_1 ! UC ! Remarks ! Lie algebra ! dim/'''C''' |- | align="center" | '''C'''''n'' | group operation is addition | N | 0 | 0 | | abelian | align="center" | '''C'''''n'' | align="center" | ''n'' |- | align="center" | '''C'''× | nonzero [complex number](/source/complex_number)s with multiplication | N | 0 | '''Z''' | | abelian | align="center" | '''C''' | align="center" | 1 |- | align="center" | GL(''n'','''C''') | [general linear group](/source/general_linear_group): [invertible](/source/invertible_matrix) ''n''×''n'' complex [matrices](/source/Matrix_(mathematics)) | N | 0 | '''Z''' | | For ''n''=1: isomorphic to '''C'''× | align="center" | M(''n'','''C''') | align="center" | ''n''2 |- | align="center" | SL(''n'','''C''') | [special linear group](/source/special_linear_group): complex matrices with [determinant](/source/determinant) 1 | N | 0 | 0 | | for ''n''=1 this is a single point and thus compact. | align="center" | sl(''n'','''C''') | align="center" | ''n''2−1 |- | 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'''2 | SL(2,'''C''') | Isomorphic to the [Möbius group](/source/M%C3%B6bius_group), isomorphic to the restricted [Lorentz group](/source/Lorentz_group) SO+(3,1,'''R'''), isomorphic to SO(3,'''C'''). | align="center" | sl(2,'''C''') | align="center" | 3 |- | align="center" | O(''n'','''C''') | [orthogonal group](/source/orthogonal_group): complex [orthogonal matrices](/source/orthogonal_matrix) | N | '''Z'''2 | – | | finite for ''n''=1 | align="center" | so(''n'','''C''') | align="center" | ''n''(''n''−1)/2 |- | align="center" | SO(''n'','''C''') | [special orthogonal group](/source/special_orthogonal_group): complex orthogonal matrices with determinant 1 | N | 0 | '''Z'''  ''n''=2
'''Z'''2 ''n''>2 | | SO(2,'''C''') is abelian and isomorphic to '''C'''×; nonabelian for ''n''>2. SO(1,'''C''') is a single point and thus compact and simply connected | align="center" | so(''n'','''C''') | align="center" | ''n''(''n''−1)/2 |- | align="center" | Sp(2''n'','''C''') | [symplectic group](/source/symplectic_group): complex [symplectic matrices](/source/symplectic_matrix) | 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](/source/complex_number)s | | | | align="center" | 1 |- | align="center" | '''C'''''n'' | the Lie bracket is zero | | | | align="center" | ''n'' |- | align="center" | M(''n'','''C''') | ''n''×''n'' matrices with Lie bracket the commutator | | | | align="center" | ''n''2 |- | align="center" | sl(''n'','''C''') | square matrices with [trace](/source/trace_of_a_matrix) 0 with Lie bracket the commutator | {{Yes}} | {{Yes}} | | align="center" | ''n''2−1 |- | align="center" | sl(2,'''C''') | Special case of sl(''n'','''C''') with ''n''=2 | {{Yes}} | {{Yes}} | isomorphic to su(2) \otimes '''C''' | align="center" | 3 |- | align="center" | so(''n'','''C''') | [skew-symmetric](/source/skew-symmetric_matrix) 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''−1)/2 |- | align="center" | sp(2''n'','''C''') | complex matrices that satisfy ''JA'' + ''A''T''J'' = 0 where ''J'' is the standard [skew-symmetric matrix](/source/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](/source/Classification_of_low-dimensional_real_Lie_algebras) * [Simple Lie group#Full classification](/source/Simple_Lie_group) ==References== * {{Fulton-Harris}} Category:Lie groups Category:Lie algebras Lie groups --- Adapted from the Wikipedia article [Table of Lie groups](https://en.wikipedia.org/wiki/Table_of_Lie_groups) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Table_of_Lie_groups?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.