Details for SatakeDiagrams
Description
Examples
For each family of non-compact simple matrix algebras (see SimpleLieAlgebraData, SimpleLieAlgebraProperties), we list the simple roots and draw the corresponding Satake diagram.
withDifferentialGeometry:withLieAlgebras:
Algebra
Type
Simple Roots
Sataka Diagram
1.
sl9
A
1−10000000,01−1000000,001−100000,0001−10000,00001−1000,000001−100,0000001−10,00000001−1,111111112
2.
su7,3
1−10−II0000,01−10−II000,001−I−I−2I−I00,000IIII−I0,0000000I−I,000−I−I−III2I,001II2I−I−I−I,01−10I−I000,1−10I−I0000
3.
su5,5
1−1000I−I00,01−1000I−I0,001−1000I−I,0001−1III2I,000020000,0001−1−I−I−I−2I,001−1000−II,01−1000−II0,1−1000−II00
4.
su∗10
000002I000,1−2100−I−I00,0000002I00,01−2100−I−I0,00000002I0,001−2000−I−I,000000002I,1001−I000−I,00002I0000
5.
su10
I−I0000000,0I−I000000,00I−I00000,000I−I0000,0000I−I000,00000I−I00,000000I−I0,0000000I−I,IIIIIIII2I
6.
so6,3
B
1−100,01−10,001−I,000I
7.
so9
I−I00,0I−I0,00I−I,000I
8.
sp10, ℝ
C
1−1000,01−100,001−10,0001−1,00002
9.
sp6, 4
002I00,1−1−I−I0,0002I0,010−I−I,00002I
10.
sp6, 6
0002I00,1−10−I−I0,00002I0,01−10−I−I,000002I,00200−2I
11.
sp10
I−I000,0I−I00,00I−I0,000I−I,00002I
12.
so5, 5
D
1−1000,01−100,001−10,0001−1,00011
13.
so8, 2
1−1000,01−I00,00I−I0,000I−I,000II
14.
so6, 4
1−1000,01−100,001−10,0001−I,0001I
15.
so12
I−I0000,0I−I000,00I−I00,000I−I0,0000I−I,0000II
16.
so∗10
002I00,1−1−I−I0,0002I0,010−I−I,010−II
17.
so*(12)
0002I00,1−10−I−I0,00002I0,01−10−I−I,000002I,002000
Calculations
Example 1.
LD1≔SimpleLieAlgebraDatasl(10),alg1:
DGsetupLD1
Lie algebra: alg1
T1≔SimpleLieAlgebraPropertiesalg1:
SR1≔T1SimpleRoots
Example 2.
LD2≔SimpleLieAlgebraDatasu(7,3),alg:
DGsetupLD2
Lie algebra: alg
T2≔SimpleLieAlgebraPropertiesalg:
SR2≔T2SimpleRoots
Example 3.
LD2≔SimpleLieAlgebraDatasu(5, 5),alg2:
Lie algebra: alg2
T2≔SimpleLieAlgebraPropertiesalg2:
Example 4.
LD4≔SimpleLieAlgebraDatasu*(10),alg4:
DGsetupLD4
Lie algebra: alg3
T3≔SimpleLieAlgebraPropertiesalg4:
SR4≔T4SimpleRoots
Example 5.
LD5≔SimpleLieAlgebraDatasu(10),alg5:
DGsetupLD5
Lie algebra: alg5
T5≔SimpleLieAlgebraPropertiesalg5:
SR5≔T5SimpleRoots
Example 6.
LD6≔SimpleLieAlgebraDataso(6, 3),alg6:
DGsetupLD6
Lie algebra: alg6
T6≔SimpleLieAlgebraPropertiesalg6:
SR6≔T6SimpleRoots
LD7≔SimpleLieAlgebraDataso(9),alg7:
DGsetupLD7
Lie algebra: alg7
T7≔SimpleLieAlgebraPropertiesalg7:
SR7≔T7SimpleRoots
Example 8.
LD8≔SimpleLieAlgebraDatasp(10, R),alg8:
DGsetupLD8
Lie algebra: alg8
T8≔SimpleLieAlgebraPropertiesalg8:
SR8≔T8SimpleRoots
Example 9.
LD9≔SimpleLieAlgebraDatasp(6, 4),alg9:
DGsetupLD9
Lie algebra: alg9
T9≔SimpleLieAlgebraPropertiesalg9:
SR9≔T9SimpleRoots
Example 10.
LD10≔SimpleLieAlgebraDatasp(6, 6),alg10:
DGsetupLD10
Lie algebra: alg10
T10≔SimpleLieAlgebraPropertiesalg10:
SR10≔T10SimpleRoots
Example 11.
LD11≔SimpleLieAlgebraDatasp(10),alg11:
DGsetupLD11
Lie algebra: alg11
T11≔SimpleLieAlgebraPropertiesalg11:
SR11≔T11SimpleRoots
Example 12.
LD12≔SimpleLieAlgebraDataso(5, 5),alg12:
DGsetupLD12
Lie algebra: alg12
T12≔SimpleLieAlgebraPropertiesalg12:
SR12≔T12SimpleRoots
Example 13.
LD13≔SimpleLieAlgebraDataso(8, 2),alg13:
DGsetupLD13
Lie algebra: alg13
T13≔SimpleLieAlgebraPropertiesalg13:
SR12≔T13SimpleRoots
Example 14.
LD14≔SimpleLieAlgebraDataso(6, 4),alg14:
DGsetupLD14
Lie algebra: alg14
T14≔SimpleLieAlgebraPropertiesalg14:
SR14≔T14SimpleRoots
Example 15.
LD15≔SimpleLieAlgebraDataso(12),alg15:
DGsetupLD15
Lie algebra: alg15
T15≔SimpleLieAlgebraPropertiesalg15:
SR15≔T15SimpleRoots
Example 16.
LD16≔SimpleLieAlgebraDataso*(10),alg16:
DGsetupLD16
Lie algebra: alg16
T16≔SimpleLieAlgebraPropertiesalg16:
SR16≔T16SimpleRoots
Example 17.
LD17≔SimpleLieAlgebraDataso*(12),alg17:
DGsetupLD17
Lie algebra: alg17
T17≔SimpleLieAlgebraPropertiesalg17:
SR17≔T17SimpleRoots
See Also
DifferentialGeometry
LieAlgebras
DynkinDiagram
CartanMatrix
SimpleLieAlgebraData
SimpleLieAlgebraProperties
SatakeDiagram
SimpleRoots
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