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6.8 Summary – Tits–Satake diagrams for non-compact real forms

To summarize the analysis, we provide the Tits–Satake diagrams for all noncompact real forms of all simple Lie algebras [593Jump To The Next Citation Point]. We do not give explicitly the Tits–Satake diagrams of the compact real forms as these are simply obtained by painting in black all the roots of the standard Dynkin diagrams.

Theorem: The simple real Lie algebras are:


Table 26: All classical real Lie algebras of š”°š”², š”°š”¬, š”°š”­ and š”°š”© type.
Algebra
Real rank Restricted root lattice
š”°š”²(p,q) p ≥ q > 0p + q ≥ 2 q (BC )q if p > q, Cq if p = q
       
š”°š”¬(p,q) p > q > 0 p + q = 2n + 1 ≥ 5 q Bq
p ≥ q > 0p + q = 2n ≥ 8 q Bq if p > q, Dq if p = q
       
š”°š”­(p,q) p ≥ q > 0p + q ≥ 3 q (BC ) q if p > q, C q if p = q
       
š”°š”­(n,ā„ ) n ≥ 3 n Cn
       
š”°š”¬∗(2n ) n ≥ 5 [nāˆ•2 ] C n 2 if n even, (BC )n−1 2 if n odd
       
š”°š”©(n, ā„) n ≥ 3 n − 1 An− 1
       
š”°š”©(n, ā„) n ≥ 2 n − 1 A n− 1


Table 27: All exceptional real Lie algebras.
Algebra Real rank Restricted root lattice
G 2 G2
F I 4 F4
F II 1 (BC )1
E I 6 E 6
E II 4 F4
E III 2 (BC ) 2
E IV 2 A2
E V 7 E7
E V I 4 F4
E V II 3 C3
E V III 8 E8
E IX 4 F4


Table 28: Tits–Satake diagrams (An series)
An series n ≥ 1 Tits–Satake diagram Restricted root system
š”°š”©(n,ā„ ), n ≥ 3

A I
PIC PIC
An
š”°š”² ∗(n + 1 ), n = 2k + 1

A II
PIC
(k+1) black and k white roots alternate.
PIC
A2k
š”°š”² (p, n + 1 − p)

A III
PIC
The p(> 0) first and p last roots
are white and connected.
PIC
BCp
š”°š”² (n+21, n+21), n = 2k + 1

A III
PIC PIC
C (k+1)
š”°š”² (1, n − 1), n ≥ 3

A IV
PIC
Only the first and last roots
are white and connected.
PIC
A1


Table 29: Tits–Satake diagrams (Bn series)
Bn series n ≥ 4 Tits–Satake diagram Restricted root system
š”°š”¬ (p, 2n − p + 1), p ≥ 1

B I
PIC
The p(≥ 2) first roots are white.
PIC
Bp
š”°š”¬ (1,2n )

B II
PIC
Only the first root is white.
PIC
A1


Table 30: Tits–Satake diagrams (Cn series)
Cn series n ≥ 3 Tits–Satake diagram Restricted root system
š”°š”­ (n, ā„)

C I
PIC PIC
Cn
š”°š”­ (p, n − p)

C II
PIC
The 2p first roots are alternatively
white and black, the n − 2p remaining are black
PIC
Bp
n n š”°š”­ (2,2), n = 2k

C II
PIC PIC
C n 2


Table 31: Tits–Satake diagrams (Dn series)
Dn series n ≥ 4 Tits–Satake diagram Restricted root system
š”°š”¬ (p, 2n − p), p ≤ n − 2

D I
PIC
The p ≤n − 2 first roots are white.
PIC
Bp
š”°š”¬ (n − 1,n + 1)

D I
PIC PIC
B(n−1)
š”°š”¬ (n, n)

D I
PIC PIC
Dn
š”°š”¬ (1,2n − 1)

D II
PIC PIC
A1
š”°š”¬ (∗(2 n)), n = 2k

D III
PIC PIC
C2k−1
∗ š”°š”¬ ((2n )), n = 2k + 1

D III
PIC PIC
BC2k


Table 32: Tits–Satake diagrams (G2 series)
G2 series Tits–Satake diagram Restricted root system
G2 (2)

G
PIC PIC


Table 33: Tits–Satake diagrams (F4 series)
F4 series Tits–Satake diagram Restricted root system
F4 (4)

F I
PIC PIC
F4 (− 20)

F II
PIC PIC


Table 34: Tits–Satake diagrams (E6 series)
E6 series Tits–Satake diagram Restricted root system
E6 (6)

E I
PIC PIC
E6 (2)

E II
PIC PIC
E6 (− 14)

E III
PIC PIC
E6 (− 26)

E IV
PIC PIC


Table 35: Tits–Satake diagrams (E7 series)
E7 series Tits–Satake diagram Restricted root system
E7 (7)

E V
PIC PIC
E7 (− 5)

E V I
PIC PIC
E7 (− 25)

E V II
PIC PIC


Table 36: Tits–Satake diagrams (E8 series)
E8 series Tits–Satake diagram Restricted root system
E8 (8)

E V III
PIC PIC
E 8(− 24)

E IX
PIC PIC


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