List of Simple Lie Groups - Simple Lie Groups of Small Dimension

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

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