DifferentialGeometry/LieAlgebras/SatakeDiagramDetails

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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

	Algebra	Type	Simple Roots	Sataka Diagram
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.		D
15.		D
16.		D
17.		D

Calculations

Example 1.

(2.1.1)

(2.1.2)

Example 2.

(2.1.3)

(2.1.4)

Example 3.

(2.1.5)

(2.1.6)

Example 4.

(2.1.7)

(2.1.8)

Example 5.

(2.1.9)

(2.1.10)

Example 6.

(2.1.11)

SR6 := [Vector(4, {(1) = 1, (2) = -1, (3) = 0, (4) = 0}), Vector(4, {(1) = 0, (2) = 1, (3) = -1, (4) = 0}), Vector(4, {(1) = 0, (2) = 0, (3) = 1, (4) = -I}), Vector(4, {(1) = 0, (2) = 0, (3) = 0, (4) = I})]

(2.1.12)

Example 6.

(2.1.13)

SR7 := [Vector(4, {(1) = I, (2) = -I, (3) = 0, (4) = 0}), Vector(4, {(1) = 0, (2) = I, (3) = -I, (4) = 0}), Vector(4, {(1) = 0, (2) = 0, (3) = I, (4) = -I}), Vector(4, {(1) = 0, (2) = 0, (3) = 0, (4) = I})]

(2.1.14)

Example 8.

(2.1.15)

SR8 := [Vector(5, {(1) = 1, (2) = -1, (3) = 0, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = 1, (3) = -1, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 1, (4) = -1, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 1, (5) = -1}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 2})]

(2.1.16)

Example 9.

(2.1.17)

SR9 := [Vector(5, {(1) = 0, (2) = 0, (3) = 2*I, (4) = 0, (5) = 0}), Vector(5, {(1) = 1, (2) = -1, (3) = -I, (4) = -I, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 2*I, (5) = 0}), Vector(5, {(1) = 0, (2) = 1, (3) = 0, (4) = -I, (5) = -I}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 2*I})]

(2.1.18)

Example 10.

(2.1.19)

(2.1.20)

Example 11.

(2.1.21)

SR11 := [Vector(5, {(1) = I, (2) = -I, (3) = 0, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = I, (3) = -I, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = I, (4) = -I, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = I, (5) = -I}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 2*I})]

(2.1.22)

Example 12.

(2.1.23)

SR12 := [Vector(5, {(1) = 1, (2) = -1, (3) = 0, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = 1, (3) = -1, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 1, (4) = -1, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 1, (5) = -1}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 1, (5) = 1})]

(2.1.24)

Example 13.

(2.1.25)

SR12 := [Vector(5, {(1) = 1, (2) = -1, (3) = 0, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = 1, (3) = -I, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = I, (4) = -I, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = I, (5) = -I}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = I, (5) = I})]

(2.1.26)

Example 14.

(2.1.27)

SR14 := [Vector(5, {(1) = 1, (2) = -1, (3) = 0, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = 1, (3) = -1, (4) = 0, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 1, (4) = -1, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 1, (5) = -I}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 1, (5) = I})]

(2.1.28)

Example 15.

(2.1.29)

(2.1.30)

Example 16.

(2.1.31)

SR16 := [Vector(5, {(1) = 0, (2) = 0, (3) = 2*I, (4) = 0, (5) = 0}), Vector(5, {(1) = 1, (2) = -1, (3) = -I, (4) = -I, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = 0, (4) = 2*I, (5) = 0}), Vector(5, {(1) = 0, (2) = 1, (3) = 0, (4) = -I, (5) = -I}), Vector(5, {(1) = 0, (2) = 1, (3) = 0, (4) = -I, (5) = I})]