Lie groups and Lie algebras |
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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; 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※
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- Cpt: Is this group G compact? (Yes/No)
- : 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 (denoted by, 0).
- : Gives the fundamental group of G whenever G is connected. The group is simply connected if and only if the fundamental group is trivial (denoted by 0).
- UC: If G is not simply connected, gives the universal cover of G.
Z2 n>2
Z2 n>2
Z2 n>2
n>2
Real Lie algebras※
Lie algebra | Description | Simple? | Semi-simple? | Remarks | dim/R |
---|---|---|---|---|---|
R | the real numbers, the Lie bracket is zero | 1 | |||
R | the Lie bracket is zero | n | |||
R | the Lie bracket is the cross product | Yes | Yes | 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) |
Yes | Yes | 3 | |
M(n,R) | n×n matrices, with Lie bracket the commutator | n | |||
sl(n,R) | square matrices with trace 0, with Lie bracket the commutator | Yes | Yes | n−1 | |
so(n) | skew-symmetric square real matrices, "with Lie bracket the commutator." | Yes, except n=4 | Yes | Exception: so(4) is semi-simple,
but not simple. |
n(n−1)/2 |
sp(2n,R) | real matrices that satisfy JA + AJ = 0 where J is the standard skew-symmetric matrix | Yes | Yes | n(2n+1) | |
sp(n) | square quaternionic matrices A satisfying A = −A, with Lie bracket the commutator | Yes | Yes | n(2n+1) | |
u(n) | square complex matrices A satisfying A = −A, with Lie bracket the commutator | Note: this is not a complex Lie algebra | n | ||
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 | n−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 | ||
---|---|---|---|---|---|---|---|---|
C | group operation is addition | N | 0 | 0 | abelian | C | 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) | n | |
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) | n−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 | – | finite 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※
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 | |||
C | the Lie bracket is zero | n | |||
M(n,C) | n×n matrices with Lie bracket the commutator | n | |||
sl(n,C) | square matrices with trace 0 with Lie bracket
the commutator |
Yes | Yes | n−1 | |
sl(2,C) | Special case of sl(n,C) with n=2 | Yes | Yes | isomorphic to su(2) C | 3 |
so(n,C) | 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. |
n(n−1)/2 |
sp(2n,C) | complex matrices that satisfy JA + AJ = 0
where J is the standard skew-symmetric matrix |
Yes | Yes | n(2n+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※
References※
- Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.