Table of Lie groups

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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[edit]

Column legend


Lie group Description CM \pi_0 \pi_1 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(1) the circle group: complex numbers of absolute value 1, with multiplication; Y 0 Z R abelian, isomorphic to SO(2), Spin(2), and R/Z R 1
Aff(1) invertible affine transformations from R to R. N Z2 0 solvable, semidirect product of R+ and R× \left\{\left[\begin{smallmatrix}a & b \\ 0 & 0\end{smallmatrix}\right] : a,b \in \mathbb{R}\right\} 2
H× non-zero quaternions with multiplication N 0 0 H 4
S3 = 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) Im(H) 3
GL(n,R) general linear group: invertible n×n real matrices N Z2 M(n,R) n2
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 M(n,R) n2
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 sl(n,R) n2−1
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. sl(2,R) 3
O(n) orthogonal group: real orthogonal matrices Y Z2 The symmetry group of the sphere (n=3) or hypersphere. so(n) n(n−1)/2
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. so(n) n(n−1)/2
Spin(n) spin group: double cover of SO(n) Y n>1 n>2 Spin(1) is isomorphic to Z2 and not connected; Spin(2) is isomorphic to the circle group and not simply connected so(n) n(n−1)/2
Sp(2n,R) symplectic group: real symplectic matrices N 0 Z sp(2n,R) n(2n+1)
Sp(n) compact symplectic group: quaternionic n×n unitary matrices Y 0 0 sp(n) n(2n+1)
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 u(n) n2
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 su(n) n2−1

Real Lie algebras[edit]

Table legend:

  • S: Is this algebra simple? (Yes or No)
  • SS: Is this algebra semi-simple? (Yes or No)
Lie algebra Description S SS 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(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)

Y Y 3
M(n,R) n×n matrices, with Lie bracket the commutator n2
sl(n,R) square matrices with trace 0, with Lie bracket the commutator Y Y n2−1
so(n) skew-symmetric square real matrices, with Lie bracket the commutator. Y Y Exception: so(4) is semi-simple, but not simple. n(n−1)/2
sp(2n,R) real matrices that satisfy JA + ATJ = 0 where J is the standard skew-symmetric matrix Y Y n(2n+1)
sp(n) square quaternionic matrices A satisfying A = −A*, with Lie bracket the commutator Y Y n(2n+1)
u(n) square complex matrices A satisfying A = −A*, with Lie bracket the commutator n2
su(n)
n≥2
square complex matrices A with trace 0 satisfying A = −A*, with Lie bracket the commutator Y Y n2−1

Complex Lie groups and their algebras[edit]

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.

Lie group Description CM \pi_0 \pi_1 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(n,C) general linear group: invertible n×n complex matrices N 0 Z For n=1: isomorphic to C× M(n,C) n2
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. sl(n,C) n2−1
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) sl(2,C) 3
PSL(2,C) Projective special linear group N 0 Z2 SL(2,C) Isomorphic to the Möbius group, isomorphic to the restricted Lorentz group SO+(3,1,R), isomorphic to SO(3,C). sl(2,C) 3
O(n,C) orthogonal group: complex orthogonal matrices N Z2 compact for n=1 so(n,C) n(n−1)/2
SO(n,C) special orthogonal group: complex orthogonal matrices with determinant 1 N 0 Z  n=2
Z2 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 so(n,C) n(n−1)/2
Sp(2n,C) symplectic group: complex symplectic matrices N 0 0 sp(2n,C) n(2n+1)

Complex Lie algebras[edit]

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 S SS Remarks dim/C
C the complex numbers 1
Cn the Lie bracket is zero n
M(n,C) n×n matrices, with Lie bracket the commutator n2
sl(n,C) square matrices with trace 0, with Lie bracket

the commutator

Y Y n2−1
sl(2,C) Special case of sl(n,C) with n=2 Y Y isomorphic to su(2) \otimes C 3
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. n(n−1)/2
sp(2n,C) complex matrices that satisfy JA + ATJ = 0

where J is the standard skew-symmetric matrix

Y Y n(2n+1)

References[edit]