Atlas of Lie Groups and Representations

Galois Cohomology

The tables gives |H1(Γ,G)| for every real form of a simple connected complex group, except for some intermediate covers in type A. Also the component groups of the adjoint groups are given. Here is some detail about the mathematics, the realex code used to produce the tables, and the output of the script (H1(Γ,G) for all simple, simply connected groups up to rank 8).

 Simply Connected Classical groups Group |H1(Γ,G)| SL(n,R), GL(n,R), Sp(2n,R) 1 SL(n,H), Spin*(2n) 2 SU(p,q) ⌊p/2⌋ + ⌊q/2⌋+1 Sp(p,q) p+q+1 Spin(p,q) ⌊(p+q)/4⌋+δ(p,q)

In the last row δ(p,q) depends on p,q mod(4), according to the following table:

 p\q 0 1 2 3 0 3 2 2 2 1 2 1 1 0 2 2 1 1 0 3 2 0 0 0

 Simply Connected Exceptional groups inner class group K real rank name |H1(Γ,G)| compact E6 A5A1 4 quasisplitquaternionic 3 E6 D5T 2 Hermitian 3 E6 E6 0 compact 3 split E6 C4 6 split 2 E6 F4 2 quasicompact 2 compact E7 A7 7 split 2 E7 D6A1 4 quaternionic 4 E7 E6T 3 Hermitian 2 E7 E7 2 compact 4 compact E8 D8 8 split 3 E8 A7A1 4 quaternionic 3 E8 E8 0 compact 3 compact F4 C3A1 4 split 3 F4 B4 1 3 F4 F4 0 compact 3 compact G2 A1A1 2 split 2 G2 G2 0 compact 2

 Special orthgonal groups group |H1k(Γ,G)| SO(p,q) ⌊p/2⌋+⌊q/2⌋+1 SO*(2n) 2

group0(G(R))||H1(Γ,G)|
PSL(n,R)
 2 n even 1 n odd
 2 n even 1 n odd
PSL(n,H)12
PSU(p,q)
 2 p=q 1 otherwise
⌊(p+q)/2⌋+1
PSO(p,q)
 1 pq=0 1 p,q odd and p≠q 4 p=q even 2 otherwise
 ⌊(p+q+2)/4⌋ p,q odd ⌊(p+q)/4⌋+3 p,q even, p+q=0(4) ⌊(p+q)/4⌋+2 p,q even, p+q=2(4) (p+q+1)/2 p,q opposite parity
PSO*(2n)
 2 n even 1 n odd
 n/2+3 n even (n-1)/2+2 n odd
PSp(2n,R)2⌊n/2⌋+2
PSp(p,q)
 2 p=q 1 otherwise
⌊(p+q)/2⌋+2

In types E8, F4 and G2 the adjoint group is simply connected. In simply connected type E6 the center has order 3, and the adjoint and simply connected groups have the same cohomology (and the real points are connected). The only essentially new adjoint case is E7.

 Adjoint exceptional groups group K real rank name |π0(G(R))| |H1(Γ,G)| E7 A7 7 split 2 4 E7 D6A1 4 quaternionic 1 4 E7 E6T 3 Hermitian 2 4 E7 E7 0 compact 1 4