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

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 [5 , 93

]. 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:

The Lie algebras $𝔤ℝ$ where $𝔤$ is one of the complex simple Lie algebras $An (n ≥ 1), Bn (n ≥ 2), Cn (n ≥ 3), Dn (n ≥ 4), G2, F4, E6, E7, or E8$ , and the compact real forms of these.
The classical real Lie algebras of types $𝔰𝔲$ , $𝔰𝔬$ , $𝔰𝔭$ and $𝔰𝔩$ . These are listed in Table 26.
The twelve exceptional real Lie algebras, listed in Table 27 (our conventions are due to Cartan).

Table 26:

All classical real Lie algebras of $𝔰𝔲$ , $𝔰𝔬$ , $𝔰𝔭$ and $𝔰𝔩$ type.

Algebra		Real rank	Restricted root lattice
$𝔰𝔲(p,q)$	$p ≥ q > 0p + q ≥ 2$	$q$	$(BC )q$ if $p > q$ , $Cq$ if $p = q$

$𝔰𝔬(p,q)$	$p > q > 0 p + q = 2n + 1 ≥ 5$	$q$	$Bq$
	$p ≥ q > 0p + q = 2n ≥ 8$	$q$	$Bq$ if $p > q$ , $Dq$ if $p = q$

$𝔰𝔭(p,q)$	$p ≥ q > 0p + q ≥ 3$	$q$	$(BC ) q$ if $p > q$ , $C q$ if $p = q$

$𝔰𝔭(n,ℝ )$	$n ≥ 3$	$n$	$Cn$

$𝔰𝔬∗(2n )$	$n ≥ 5$	$[n∕2 ]$	$C n 2$ if $n$ even, $(BC )n−1 2$ if $n$ odd

$𝔰𝔩(n, ℝ)$	$n ≥ 3$	$n − 1$	$An− 1$

$𝔰𝔩(n, ℍ)$	$n ≥ 2$	$n − 1$	$A n− 1$

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
$𝔰𝔩(n,ℝ ), n ≥ 3$ $A I$		$An$
$𝔰𝔲 ∗(n + 1 ), n = 2k + 1$ $A II$	(k+1) black and $k$ white roots alternate.	$A2k$
$𝔰𝔲 (p, n + 1 − p)$ $A III$	The $p(> 0)$ first and $p$ last roots are white and connected.	$BCp$
$𝔰𝔲 (n+21, n+21), n = 2k + 1$ $A III$		$C (k+1)$
$𝔰𝔲 (1, n − 1), n ≥ 3$ $A IV$	Only the first and last roots are white and connected.	$A1$

Table 29:

Tits–Satake diagrams ( $Bn$ series)

$Bn$ series $n ≥ 4$	Tits–Satake diagram	Restricted root system
$𝔰𝔬 (p, 2n − p + 1), p ≥ 1$ $B I$	The $p(≥ 2)$ first roots are white.	$Bp$
$𝔰𝔬 (1,2n )$ $B II$	Only the first root is white.	$A1$

Table 30:

Tits–Satake diagrams ( $Cn$ series)

$Cn$ series $n ≥ 3$	Tits–Satake diagram	Restricted root system
$𝔰𝔭 (n, ℝ)$ $C I$		$Cn$
$𝔰𝔭 (p, n − p)$ $C II$	The $2p$ first roots are alternatively white and black, the $n − 2p$ remaining are black	$Bp$
$n n 𝔰𝔭 (2,2), n = 2k$ $C II$		$C n 2$

Table 31:

Tits–Satake diagrams ( $Dn$ series)

$Dn$ series $n ≥ 4$	Tits–Satake diagram	Restricted root system
$𝔰𝔬 (p, 2n − p), p ≤ n − 2$ $D I$	The $p ≤n − 2$ first roots are white.	$Bp$
$𝔰𝔬 (n − 1,n + 1)$ $D I$		$B(n−1)$
$𝔰𝔬 (n, n)$ $D I$		$Dn$
$𝔰𝔬 (1,2n − 1)$ $D II$		$A1$
$𝔰𝔬 (∗(2 n)), n = 2k$ $D III$		$C2k−1$
$∗ 𝔰𝔬 ((2n )), n = 2k + 1$ $D III$		$BC2k$

Table 32:

Tits–Satake diagrams ( $G2$ series)

$G2$ series	Tits–Satake diagram	Restricted root system
$G2 (2)$ $G$

Table 33:

Tits–Satake diagrams ( $F4$ series)

$F4$ series	Tits–Satake diagram	Restricted root system
$F4 (4)$ $F I$
$F4 (− 20)$ $F II$

Table 34:

Tits–Satake diagrams ( $E6$ series)

$E6$ series	Tits–Satake diagram	Restricted root system
$E6 (6)$ $E I$
$E6 (2)$ $E II$
$E6 (− 14)$ $E III$
$E6 (− 26)$ $E IV$

Table 35:

Tits–Satake diagrams ( $E7$ series)

$E7$ series	Tits–Satake diagram	Restricted root system
$E7 (7)$ $E V$
$E7 (− 5)$ $E V I$
$E7 (− 25)$ $E V II$

Table 36:

Tits–Satake diagrams ( $E8$ series)

$E8$ series	Tits–Satake diagram	Restricted root system
$E8 (8)$ $E V III$
$E 8(− 24)$ $E IX$