SatakeDiagram Details

All Products Maple MapleSim

Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : SatakeDiagram Details

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)

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

>	$SR17 ≔ {T17}_{SimpleRoots}$

What kind of issue would you like to report? (Optional)
Please add your Comment (Optional)
E-mail Address (Optional)
What is ?	This question helps us to combat spam

Maple

Erweiterungen für Maple

MapleSim

Erweiterungen für MapleSim

Systementwicklung

Beratende Dienstleistungen

Produkte zur Online-Ausbildung

Ausbildung

Industrie

Automobil und Luftfahrt

Robotik

Maschinendesign & industrielle Automation

Andere

Lösungen für die Industrie

Kaufen: Einzelheiten und Preise

Kaufen

Institutionelle Studentenlizenzierung

Elite-Wartungsprogramm

Support

Produktschulung

Online Hilfe zu Produkten

Webinare und Veranstaltungen

Veröffentlichungen

Content Hubs

Anwendungsbeispiele

Community

Über Maplesoft

Pressezentrum

User Community

Kontakt

Online Help

All Products Maple MapleSim

Was this information helpful?

Tell us what we can do better:

Thanks for your Comment

Maple

Leistungsfähige intuitive Mathematiksoftware

Erweiterungen für Maple

MapleSim

Modellierung auf Systemebene

Erweiterungen für MapleSim

Systementwicklung

Beratende Dienstleistungen

Produkte zur Online-Ausbildung

Ausbildung

Industrie

Automobil und Luftfahrt

Robotik

Maschinendesign & industrielle Automation

Andere

Lösungen für die Industrie

Kaufen: Einzelheiten und Preise

Kaufen

Institutionelle Studentenlizenzierung

Elite-Wartungsprogramm

Support

Produktschulung

Online Hilfe zu Produkten

Webinare und Veranstaltungen

Veröffentlichungen

Content Hubs

Anwendungsbeispiele

Community

Über Maplesoft

Pressezentrum

User Community

Kontakt

Online Help

All Products Maple MapleSim

Was this information helpful?

Tell us what we can do better:

Thanks for your Comment