SatakeDiagram Details - Maple Help

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Details for SatakeDiagrams 

     

 

Description

Examples

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

withDifferentialGeometry:withLieAlgebras:

 

 

 

Algebra

Type

Simple Roots

Sataka Diagram

1.

sl9

A

110000000,011000000,001100000,000110000,000011000,000001100,000000110,000000011,111111112

2.

su7,3

A

110II0000,0110II000,001II2II00,000IIIII0,0000000II,000IIIII2I,001II2IIII,0110II000,110II0000

3.

su5,5

A

11000II00,011000II0,0011000II,00011III2I,000020000,00011III2I,0011000II,011000II0,11000II00

4.

su10

A

000002I000,12100II00,0000002I00,012100II0,00000002I0,0012000II,000000002I,1001I000I,00002I0000

5.

su10

A

II0000000,0II000000,00II00000,000II0000,0000II000,00000II00,000000II0,0000000II,IIIIIIII2I

6.

so6,3

B

1100,0110,001I,000I

7.

so9

B

II00,0II0,00II,000I

8.

sp10, ℝ

C

11000,01100,00110,00011,00002

9.

sp6, 4

C

002I00,11II0,0002I0,010II,00002I

10.

sp6, 6

C

0002I00,110II0,00002I0,0110II,000002I,002002I

11.

sp10

 

C

II000,0II00,00II0,000II,00002I

12.

so5, 5

D

11000,01100,00110,00011,00011

13.

so8, 2

D

11000,01I00,00II0,000II,000II

14.

so6, 4

   D

11000,01100,00110,0001I,0001I

15.

so12

D

II0000,0II000,00II00,000II0,0000II,0000II

16.

so10

D

002I00,11II0,0002I0,010II,010II

17.

so*(12)

D

0002I00,110II0,00002I0,0110II,000002I,002000

 

 

Calculations

 

Example 1.

LD1SimpleLieAlgebraDatasl(10),alg1:

DGsetupLD1

Lie algebra: alg1

(2.1.1)

T1SimpleLieAlgebraPropertiesalg1:

SR1T1SimpleRoots

SR1:=110000000,011000000,001100000,000110000,000011000,000001100,000000110,000000011,111111112

(2.1.2)

 

Example 2.

LD2SimpleLieAlgebraDatasu(7,3),alg:

DGsetupLD2

Lie algebra: alg

(2.1.3)

T2SimpleLieAlgebraPropertiesalg:

SR2T2SimpleRoots

SR1:=110II0000,0110II000,001II2II00,000IIIII0,0000000II,000IIIII2I,001II2IIII,0110II000,110II0000

(2.1.4)

 

Example 3.

LD2SimpleLieAlgebraDatasu(5, 5),alg2:

DGsetupLD2

Lie algebra: alg2

(2.1.5)

T2SimpleLieAlgebraPropertiesalg2:

SR2T2SimpleRoots

SR2:=11000II00,011000II0,0011000II,00011III2I,000020000,00011III2I,0011000II,011000II0,11000II00

(2.1.6)

 

Example 4.

LD4SimpleLieAlgebraDatasu*(10),alg4:

DGsetupLD4

Lie algebra: alg3

(2.1.7)

T3SimpleLieAlgebraPropertiesalg4:

SR4T4SimpleRoots

SR3:=000002I000,12100II00,0000002I00,012100II0,00000002I0,0012000II,000000002I,1001I000I,00002I0000

(2.1.8)

 

Example 5.

LD5SimpleLieAlgebraDatasu(10),alg5:

DGsetupLD5

Lie algebra: alg5

(2.1.9)

T5SimpleLieAlgebraPropertiesalg5:

SR5T5SimpleRoots

SR5:=II0000000,0II000000,00II00000,000II0000,0000II000,00000II00,000000II0,0000000II,IIIIIIII2I

(2.1.10)

Example 6.

LD6SimpleLieAlgebraDataso(6, 3),alg6:

DGsetupLD6

Lie algebra: alg6

(2.1.11)

T6SimpleLieAlgebraPropertiesalg6:

SR6T6SimpleRoots

SR6:=1100,0110,001I,000I

(2.1.12)

Example 6.

LD7SimpleLieAlgebraDataso(9),alg7:

DGsetupLD7

Lie algebra: alg7

(2.1.13)

T7SimpleLieAlgebraPropertiesalg7:

SR7T7SimpleRoots

SR7:=II00,0II0,00II,000I

(2.1.14)

Example 8.

LD8SimpleLieAlgebraDatasp(10, R),alg8:

DGsetupLD8

Lie algebra: alg8

(2.1.15)

T8SimpleLieAlgebraPropertiesalg8:

SR8T8SimpleRoots

SR8:=11000,01100,00110,00011,00002

(2.1.16)

Example 9.

LD9SimpleLieAlgebraDatasp(6, 4),alg9:

DGsetupLD9

Lie algebra: alg9

(2.1.17)

T9SimpleLieAlgebraPropertiesalg9:

SR9T9SimpleRoots

SR9:=002I00,11II0,0002I0,010II,00002I

(2.1.18)

Example 10.

LD10SimpleLieAlgebraDatasp(6, 6),alg10:

DGsetupLD10

Lie algebra: alg10

(2.1.19)

T10SimpleLieAlgebraPropertiesalg10:

SR10T10SimpleRoots

SR10:=0002I00,110II0,00002I0,0110II,000002I,002002I

(2.1.20)

Example 11.

LD11SimpleLieAlgebraDatasp(10),alg11:

DGsetupLD11

Lie algebra: alg11

(2.1.21)

T11SimpleLieAlgebraPropertiesalg11:

SR11T11SimpleRoots

SR11:=II000,0II00,00II0,000II,00002I

(2.1.22)

Example 12.

LD12SimpleLieAlgebraDataso(5, 5),alg12:

DGsetupLD12

Lie algebra: alg12

(2.1.23)

T12SimpleLieAlgebraPropertiesalg12:

SR12T12SimpleRoots

SR12:=11000,01100,00110,00011,00011

(2.1.24)

Example 13.

LD13SimpleLieAlgebraDataso(8, 2),alg13:

DGsetupLD13

Lie algebra: alg13

(2.1.25)