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 as well as on their algebraic properties.
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 legendCpt: Is this group G compact? ': 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.

': Gives the fundamental group of G whenever G is connected. The group is simply connected if and only if the fundamental group is trivial.UC: If G is not simply connected, gives the universal cover of G.

Lie group	Description	Cpt			UC	Remarks	Lie algebra	dim/R
Rⁿ	Euclidean space with addition	N	0	0		abelian	Rⁿ	n
R^×	nonzero real numbers with multiplication	N	Z₂	-		abelian	R	1
R⁺	positive real numbers with multiplication	N	0	0		abelian	R	1
S¹ = U	the circle group: complex numbers of absolute value 1 with multiplication;	Y	0	Z	R	abelian, isomorphic to SO, Spin, and R/'Z	R'	1
Aff(1)	invertible affine transformations from R to R.	N	Z₂	-		solvable, semidirect product of R⁺ and R^×		2
H^×	non-zero quaternions with multiplication	N	0	0			H	4
S³ = Sp	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)	Im	3
GL	general linear group: invertible n×n real matrices	N	Z₂	-			M	n²
GL⁺	n×n real matrices with positive determinant	N	0	Z n=2 Z₂ n>2		GL⁺ is isomorphic to R⁺ and is simply connected	M	n²
SL	special linear group: real matrices with determinant 1	N	0	Z n=2 Z₂ n>2		SL is a single point and therefore compact and simply connected	sl	n²−1
SL(2,R)	Orientation-preserving isometries of the Poincaré half-plane, isomorphic to SU, isomorphic to Sp.	N	0	Z		The universal cover has no finite-dimensional faithful representations.	sl	3
O	orthogonal group: real orthogonal matrices	Y	Z₂	-		The symmetry group of the sphere or hypersphere.	so	n/2
SO	special orthogonal group: real orthogonal matrices with determinant 1	Y	0	Z n=2 Z₂ n>2	Spin n>2	SO is a single point and SO is isomorphic to the circle group, SO is the rotation group of the sphere.	so	n/2
SE	special euclidean group: group of rigid body motions in n-dimensional space.	N	0				se	n + n/2
Spin	spin group: double cover of SO	Y	0 n>1	0 n>2		Spin is isomorphic to Z₂ and not connected; Spin is isomorphic to the circle group and not simply connected	so	n/2
Sp	symplectic group: real symplectic matrices	N	0	Z			sp	n
Sp	compact symplectic group: quaternionic n×n unitary matrices	Y	0	0			sp	n
Mp	metaplectic group: double cover of real symplectic group Sp	Y	0	Z		Mp is a Lie group that is not algebraic	sp	n
U	unitary group: complex n×n unitary matrices	Y	0	Z	R×SU	For n=1: isomorphic to S¹. Note: this is not a complex Lie group/algebra	u	n²
SU	special unitary group: complex n×n unitary matrices with determinant 1	Y	0	0		Note: this is not a complex Lie group/algebra	su	n²−1

Real Lie algebras

Lie algebra	Description	Simple?	Semi-simple?	Remarks	dim/R
R	the real numbers, the Lie bracket is zero				1
Rⁿ	the Lie bracket is zero				n
R³	the Lie bracket is the cross product				3
H	quaternions, with Lie bracket the commutator				4
Im	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 and to so				3
M	n×n matrices, with Lie bracket the commutator				n²
sl	square matrices with trace 0, with Lie bracket the commutator				n²−1
so	skew-symmetric square real matrices, with Lie bracket the commutator.	, except n=4		Exception: so is semi-simple, but not simple.	n/2
sp	real matrices that satisfy JA + A^TJ = 0 where J is the standard skew-symmetric matrix				n
sp	square quaternionic matrices A satisfying A = −A^∗, with Lie bracket the commutator				n
u	square complex matrices A satisfying A = −A^∗, with Lie bracket the commutator			Note: this is not a complex Lie algebra	n²
su n≥2	square complex matrices A with trace 0 satisfying A = −A^∗, with Lie bracket the commutator			Note: this is not a complex Lie algebra	n²−1

Complex Lie groups and their algebras

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.

Lie group	Description	Cpt			UC	Remarks	Lie algebra	dim/C
Cⁿ	group operation is addition	N	0	0		abelian	Cⁿ	n
C^×	nonzero complex numbers with multiplication	N	0	Z		abelian	C	1
GL	general linear group: invertible n×n complex matrices	N	0	Z		For n=1: isomorphic to C^×	M	n²
SL	special linear group: complex matrices with determinant 1	N	0	0		for n=1 this is a single point and thus compact.	sl	n²−1
SL	Special case of SL for n=2	N	0	0		Isomorphic to Spin, isomorphic to Sp	sl	3
PSL	Projective special linear group	N	0	Z₂	SL	Isomorphic to the Möbius group, isomorphic to the restricted Lorentz group SO⁺, isomorphic to SO.	sl	3
O	orthogonal group: complex orthogonal matrices	N	Z₂	-		finite for n=1	so	n/2
SO	special orthogonal group: complex orthogonal matrices with determinant 1	N	0	Z n=2 Z₂ n>2		SO is abelian and isomorphic to C^×; nonabelian for n>2. SO is a single point and thus compact and simply connected	so	n/2
Sp	symplectic group: complex symplectic matrices	N	0	0			sp	n

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.

Lie algebra	Description	Simple?	Semi-simple?	Remarks	dim/C
C	the complex numbers				1
Cⁿ	the Lie bracket is zero				n
M	n×n matrices with Lie bracket the commutator				n²
sl	square matrices with trace 0 with Lie bracket the commutator				n²−1
sl	Special case of sl with n=2			isomorphic to su C	3
so	skew-symmetric square complex matrices with Lie bracket the commutator	, except n=4		Exception: so is semi-simple, but not simple.	n/2
sp	complex matrices that satisfy JA + A^TJ = 0 where J is the standard skew-symmetric matrix				n

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.