SatakeDiagram Details

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Details for SatakeDiagrams

Description

•	For each family of non-compact simple matrix algebras (see SimpleLieAlgebraData, SimpleLieAlgebraProperties), we list the simple roots and draw the corresponding Satake diagram.

Examples

>	$with (DifferentialGeometry) &colon;$ $with (LieAlgebras) &colon;$

	Algebra	Type	Simple Roots	Sataka Diagram
1.	$sl (9)$	$A$	$[[\begin{array}{r} 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ - 1 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ - 1 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ - 1 \end{array}], [\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 2 \end{array}]]$
2.	$su (7, 3)$	$A$	$[[\begin{array}{c} 1 \\ - 1 \\ 0 \\ - I \\ I \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 1 \\ - 1 \\ 0 \\ - I \\ I \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 1 \\ - I \\ - I \\ - 2 I \\ - I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ I \\ I \\ I \\ I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ - I \\ - I \\ - I \\ I \\ I \\ 2 I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 1 \\ I \\ I \\ 2 I \\ - I \\ - I \\ - I \end{array}], [\begin{array}{c} 0 \\ 1 \\ - 1 \\ 0 \\ I \\ - I \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 1 \\ - 1 \\ 0 \\ I \\ - I \\ 0 \\ 0 \\ 0 \\ 0 \end{array}]]$
3.	$su (5, 5)$	$A$	$[\begin{array}{c} 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ I \\ - I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\ - 1 \\ I \\ I \\ I \\ 2 I \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 0 \\ 2 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\ - 1 \\ - I \\ - I \\ - I \\ - 2 I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ - I \\ I \end{array}], [\begin{array}{c} 0 \\ 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ - I \\ I \\ 0 \end{array}], [\begin{array}{c} 1 \\ - 1 \\ 0 \\ 0 \\ 0 \\ - I \\ I \\ 0 \\ 0 \end{array}]$
4.	$su * (10)$	$A$	$[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 2 I \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 1 \\ - 2 \\ 1 \\ 0 \\ 0 \\ - I \\ - I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 2 I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 1 \\ - 2 \\ 1 \\ 0 \\ 0 \\ - I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 2 I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 1 \\ - 2 \\ 0 \\ 0 \\ 0 \\ - I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 2 I \end{array}], [\begin{array}{c} 1 \\ 0 \\ 0 \\ 1 \\ - I \\ 0 \\ 0 \\ 0 \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 2 I \\ 0 \\ 0 \\ 0 \\ 0 \end{array}]$
5.	$su (10)$	$A$	$[[\begin{array}{c} I \\ - I \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ I \\ - I \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ I \\ - I \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ I \\ - I \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ I \\ - I \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ I \\ - I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ I \\ - I \end{array}], [\begin{array}{c} I \\ I \\ I \\ I \\ I \\ I \\ I \\ I \\ 2 I \end{array}]]$
6.	$so (6, 3)$	$B$	$[[\begin{array}{r} 1 \\ - 1 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 1 \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ I \end{array}]]$
7.	$so (9)$	$B$	$[[\begin{array}{c} I \\ - I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ I \end{array}]]$
8.	$sp (10, &reals;)$	$C$	$[[\begin{array}{r} 1 \\ - 1 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 1 \\ - 1 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 0 \\ 2 \end{array}]]$
9.	$sp (6, 4)$	$C$	$[[\begin{array}{c} 0 \\ 0 \\ 2 I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 1 \\ - 1 \\ - I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 2 I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 1 \\ 0 \\ - I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 2 I \end{array}]]$
10.	$sp (6, 6)$	$C$	$[[\begin{array}{c} 0 \\ 0 \\ 0 \\ 2 I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 1 \\ - 1 \\ 0 \\ - I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 2 I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 1 \\ - 1 \\ 0 \\ - I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 2 I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 2 \\ 0 \\ 0 \\ - 2 I \end{array}]]$
11.	$sp (10)$	$C$	$[[\begin{array}{c} I \\ - I \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ I \\ - I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 2 I \end{array}]]$
12.	$so (5, 5)$	$D$	$[[\begin{array}{r} 1 \\ - 1 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 1 \\ - 1 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 0 \\ 1 \\ 1 \end{array}]]$
13.	$so (8, 2)$	$D$	$[[\begin{array}{r} 1 \\ - 1 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 1 \\ - I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ I \\ I \end{array}]]$
14.	$so (6, 4)$	D	$[[\begin{array}{r} 1 \\ - 1 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \\ 0 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\ I \end{array}]]$
15.	$so (12)$	D	$[[\begin{array}{c} I \\ - I \\ 0 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ I \\ - I \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ I \\ - I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ I \\ I \end{array}]]$
16.	$so * (10)$	D	$[[\begin{array}{c} 0 \\ 0 \\ 2 I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 1 \\ - 1 \\ - I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 2 I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 1 \\ 0 \\ - I \\ - I \end{array}], [\begin{array}{c} 0 \\ 1 \\ 0 \\ - I \\ I \end{array}]]$
17.	$so *(12)$	D	$[[\begin{array}{c} 0 \\ 0 \\ 0 \\ 2 I \\ 0 \\ 0 \end{array}], [\begin{array}{c} 1 \\ - 1 \\ 0 \\ - I \\ - I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 2 I \\ 0 \end{array}], [\begin{array}{c} 0 \\ 1 \\ - 1 \\ 0 \\ - I \\ - I \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 2 I \end{array}], [\begin{array}{r} 0 \\ 0 \\ 2 \\ 0 \\ 0 \\ 0 \end{array}]]$

Calculations

Example 1.

>	$LD1 ≔ SimpleLieAlgebraData (sl(10), alg1) &colon;$

>	$DGsetup (LD1)$

$Lie algebra: alg1$

(2.1.1)

>	$T1 ≔ SimpleLieAlgebraProperties (alg1) &colon;$

>	$SR1 ≔ T1 [SimpleRoots]$

Example 2.

>	$LD2 ≔ SimpleLieAlgebraData (su(7,3), alg) &colon;$

>	$DGsetup (LD2)$

$Lie algebra: alg$

(2.1.2)

>	$T2 ≔ SimpleLieAlgebraProperties (alg) &colon;$

>	$SR2 ≔ T2 [SimpleRoots]$

Example 3.

>	$LD2 ≔ SimpleLieAlgebraData (su(5, 5), alg2) &colon;$

>	$DGsetup (LD2)$

$Lie algebra: alg2$

(2.1.3)

>	$T2 ≔ SimpleLieAlgebraProperties (alg2) &colon;$

>	$SR2 ≔ T2 [SimpleRoots]$

Example 4.

>	$LD4 ≔ SimpleLieAlgebraData (su*(10), alg4) &colon;$

>	$DGsetup (LD4)$

$Lie algebra: alg3$

(2.1.4)

>	$T3 ≔ SimpleLieAlgebraProperties (alg4) &colon;$

>	$SR4 ≔ T4 [SimpleRoots]$

Example 5.

>	$LD5 ≔ SimpleLieAlgebraData (su(10), alg5) &colon;$

>	$DGsetup (LD5)$

$Lie algebra: alg5$

(2.1.5)

>	$T5 ≔ SimpleLieAlgebraProperties (alg5) &colon;$

>	$SR5 ≔ T5 [SimpleRoots]$

Example 6.

>	$LD6 ≔ SimpleLieAlgebraData (so(6, 3), alg6) &colon;$

>	$DGsetup (LD6)$

$Lie algebra: alg6$

(2.1.6)

>	$T6 ≔ SimpleLieAlgebraProperties (alg6) &colon;$

>	$SR6 ≔ T6 [SimpleRoots]$

Example 6.

>	$LD7 ≔ SimpleLieAlgebraData (so(9), alg7) &colon;$

>	$DGsetup (LD7)$

$Lie algebra: alg7$

(2.1.7)

>	$T7 ≔ SimpleLieAlgebraProperties (alg7) &colon;$

>	$SR7 ≔ T7 [SimpleRoots]$

Example 8.

>	$LD8 ≔ SimpleLieAlgebraData (sp(10, R), alg8) &colon;$

>	$DGsetup (LD8)$

$Lie algebra: alg8$

(2.1.8)

>	$T8 ≔ SimpleLieAlgebraProperties (alg8) &colon;$

>	$SR8 ≔ T8 [SimpleRoots]$

Example 9.

>	$LD9 ≔ SimpleLieAlgebraData (sp(6, 4), alg9) &colon;$

>	$DGsetup (LD9)$

$Lie algebra: alg9$

(2.1.9)

>	$T9 ≔ SimpleLieAlgebraProperties (alg9) &colon;$

>	$SR9 ≔ T9 [SimpleRoots]$

Example 10.

>	$LD10 ≔ SimpleLieAlgebraData (sp(6, 6), alg10) &colon;$

>	$DGsetup (LD10)$

$Lie algebra: alg10$

(2.1.10)

>	$T10 ≔ SimpleLieAlgebraProperties (alg10) &colon;$

>	$SR10 ≔ T10 [SimpleRoots]$

Example 11.

>	$LD11 ≔ SimpleLieAlgebraData (sp(10), alg11) &colon;$

>	$DGsetup (LD11)$

$Lie algebra: alg11$

(2.1.11)

>	$T11 ≔ SimpleLieAlgebraProperties (alg11) &colon;$

>	$SR11 ≔ T11 [SimpleRoots]$

Example 12.

>	$LD12 ≔ SimpleLieAlgebraData (so(5, 5), alg12) &colon;$

>	$DGsetup (LD12)$

$Lie algebra: alg12$

(2.1.12)

>	$T12 ≔ SimpleLieAlgebraProperties (alg12) &colon;$

>	$SR12 ≔ T12 [SimpleRoots]$

Example 13.

>	$LD13 ≔ SimpleLieAlgebraData (so(8, 2), alg13) &colon;$

>	$DGsetup (LD13)$

$Lie algebra: alg13$

(2.1.13)

>	$T13 ≔ SimpleLieAlgebraProperties (alg13) &colon;$

>	$SR12 ≔ T13 [SimpleRoots]$

Example 14.

>	$LD14 ≔ SimpleLieAlgebraData (so(6, 4), alg14) &colon;$

>	$DGsetup (LD14)$

$Lie algebra: alg14$

(2.1.14)

>	$T14 ≔ SimpleLieAlgebraProperties (alg14) &colon;$

>	$SR14 ≔ T14 [SimpleRoots]$

Example 15.

>	$LD15 ≔ SimpleLieAlgebraData (so(12), alg15) &colon;$

>	$DGsetup (LD15)$

$Lie algebra: alg15$

(2.1.15)

>	$T15 ≔ SimpleLieAlgebraProperties (alg15) &colon;$

>	$SR15 ≔ T15 [SimpleRoots]$

Example 16.

>	$LD16 ≔ SimpleLieAlgebraData (so*(10), alg16) &colon;$

>	$DGsetup (LD16)$

$Lie algebra: alg16$

(2.1.16)

>	$T16 ≔ SimpleLieAlgebraProperties (alg16) &colon;$

>	$SR16 ≔ T16 [SimpleRoots]$

Example 17.

>	$LD17 ≔ SimpleLieAlgebraData (so*(12), alg17) &colon;$

>	$DGsetup (LD17)$

$Lie algebra: alg17$

(2.1.17)

>	$T17 ≔ SimpleLieAlgebraProperties (alg17) &colon;$

>	$SR17 ≔ T17 [SimpleRoots]$

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