Simple Lie Groups of Small Dimension
The following table lists some Lie groups with simple Lie algebras of small dimension. The groups on a given line all have the same Lie algebra. In the dimension 1 case, the groups are abelian and not simple.
| Dim | Groups | Symmetric space | Compact dual | Rank | Dim | |
|---|---|---|---|---|---|---|
| 1 | R, S1=U(1)=SO2(R)=Spin(2) | Abelian | Real line | 0 | 1 | |
| 3 | S3=Sp(1)=SU(2)=Spin(3), SO3(R)=PSU(2) | Compact | ||||
| 3 | SL2(R)=Sp2(R), SO2,1(R) | Split, Hermitian, hyperbolic | Hyperbolic plane H2 | Sphere S2 | 1 | 2 |
| 6 | SL2(C)=Sp2(C), SO3,1(R), SO3(C) | Complex | Hyperbolic space H3 | Sphere S3 | 1 | 3 |
| 8 | SL3(R) | Split | Euclidean structures on R3 | Real structures on C3 | 2 | 5 |
| 8 | SU(3) | Compact | ||||
| 8 | SU(1,2) | Hermitian, quasi-split, quaternionic | Complex hyperbolic plane | Complex projective plane | 1 | 4 |
| 10 | Sp(2)=Spin(5), SO5(R) | Compact | ||||
| 10 | SO4,1(R), Sp2,2(R) | Hyperbolic, quaternionic | Hyperbolic space H4 | Sphere S4 | 1 | 4 |
| 10 | SO3,2(R),Sp4(R) | Split, Hermitian | Siegel upper half space | Complex structures on H2 | 2 | 6 |
| 14 | G2 | Compact | ||||
| 14 | G2 | Split, quaternionic | Non-division quaternionic subalgebras of non-division octonions | Quaternionic subalgebras of octonions | 2 | 8 |
| 15 | SU(4)=Spin(6), SO6(R) | Compact | ||||
| 15 | SL4(R), SO3,3(R) | Split | R3 in R3,3 | Grassmannian G(3,3) | 3 | 9 |
| 15 | SU(3,1) | Hermitian | Complex hyperbolic space | Complex projective space | 1 | 6 |
| 15 | SU(2,2), SO4,2(R) | Hermitian, quasi-split, quaternionic | R2 in R2,4 | Grassmannian G(2,4) | 2 | 8 |
| 15 | SL2(H), SO5,1(R) | Hyperbolic | Hyperbolic space H5 | Sphere S5 | 1 | 5 |
| 16 | SL3(C) | Complex | SU(3) | 2 | 8 | |
| 20 | SO5(C), Sp4(C) | Complex | Spin5(R) | 2 | 10 | |
| 21 | SO7(R) | Compact | ||||
| 21 | SO6,1(R) | Hyperbolic | Hyperbolic space H6 | Sphere S6 | ||
| 21 | SO5,2(R) | Hermitian | ||||
| 21 | SO4,3(R) | Split, quaternionic | ||||
| 21 | Sp(3) | Compact | ||||
| 21 | Sp6(R) | Split, hermitian | ||||
| 21 | Sp4,2(R) | Quaternionic | ||||
| 24 | SU(5) | Compact | ||||
| 24 | SL5(R) | Split | ||||
| 24 | SU4,1 | Hermitian | ||||
| 24 | SU3,2 | Hermitian, quaternionic | ||||
| 28 | SO8(R) | Compact | ||||
| 28 | SO7,1(R) | Hyperbolic | Hyperbolic space H7 | Sphere S7 | ||
| 28 | SO6,2(R) | Hermitian | ||||
| 28 | SO5,3(R) | Quasi-split | ||||
| 28 | SO4,4(R) | Split, quaternionic | ||||
| 28 | SO*8(R) | Hermitian | ||||
| 28 | G2(C) | Complex | ||||
| 30 | SL4(C) | Complex |
Read more about this topic: List Of Simple Lie Groups
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