SatakeDiagram Details - Maple Help

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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

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

Calculations

Example 1.

 > $\mathrm{LD1}≔\mathrm{SimpleLieAlgebraData}\left("sl\left(10\right)",\mathrm{alg1}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD1}\right)$
 ${\mathrm{Lie algebra: alg1}}$ (2.1.1)
 > $\mathrm{T1}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg1}\right):$
 > $\mathrm{SR1}≔\mathrm{T1}\left["SimpleRoots"\right]$
 ${\mathrm{SR1}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\\ {1}\\ {1}\\ {1}\\ {1}\\ {2}\end{array}\right]\right]$ (2.1.2)

Example 2.

 > $\mathrm{LD2}≔\mathrm{SimpleLieAlgebraData}\left("su\left(7,3\right)",\mathrm{alg}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: alg}}$ (2.1.3)
 > $\mathrm{T2}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg}\right):$
 > $\mathrm{SR2}≔\mathrm{T2}\left["SimpleRoots"\right]$
 ${\mathrm{SR1}}{:=}\left[\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {-}{I}\\ {I}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{1}\\ {0}\\ {-}{I}\\ {I}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {-}{I}\\ {-}{I}\\ {-}{2}{}{I}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {I}\\ {I}\\ {I}\\ {I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {-}{I}\\ {-}{I}\\ {-}{I}\\ {I}\\ {I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {I}\\ {I}\\ {2}{}{I}\\ {-}{I}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{1}\\ {0}\\ {I}\\ {-}{I}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {I}\\ {-}{I}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]\right]$ (2.1.4)

Example 3.

 > $\mathrm{LD2}≔\mathrm{SimpleLieAlgebraData}\left("su\left(5, 5\right)",\mathrm{alg2}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: alg2}}$ (2.1.5)
 > $\mathrm{T2}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg2}\right):$
 > $\mathrm{SR2}≔\mathrm{T2}\left["SimpleRoots"\right]$
 ${\mathrm{SR2}}{:=}\left[\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {1}\\ {-}{1}\\ {I}\\ {I}\\ {I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {0}\\ {2}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {1}\\ {-}{1}\\ {-}{I}\\ {-}{I}\\ {-}{I}\\ {-}{2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {-}{I}\\ {I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\\ {-}{I}\\ {I}\\ {0}\\ {0}\end{array}\right]\right]$ (2.1.6)

Example 4.

 > $\mathrm{LD4}≔\mathrm{SimpleLieAlgebraData}\left("su*\left(10\right)",\mathrm{alg4}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD4}\right)$
 ${\mathrm{Lie algebra: alg3}}$ (2.1.7)
 > $\mathrm{T3}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg4}\right):$
 > $\mathrm{SR4}≔\mathrm{T4}\left["SimpleRoots"\right]$
 ${\mathrm{SR3}}{:=}\left[\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{2}\\ {1}\\ {0}\\ {0}\\ {-}{I}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{2}\\ {1}\\ {0}\\ {0}\\ {-}{I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {-}{2}\\ {0}\\ {0}\\ {0}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {0}\\ {1}\\ {-}{I}\\ {0}\\ {0}\\ {0}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]\right]$ (2.1.8)

Example 5.

 > $\mathrm{LD5}≔\mathrm{SimpleLieAlgebraData}\left("su\left(10\right)",\mathrm{alg5}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD5}\right)$
 ${\mathrm{Lie algebra: alg5}}$ (2.1.9)
 > $\mathrm{T5}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg5}\right):$
 > $\mathrm{SR5}≔\mathrm{T5}\left["SimpleRoots"\right]$
 ${\mathrm{SR5}}{:=}\left[\left[\begin{array}{c}{I}\\ {-}{I}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {I}\\ {-}{I}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {I}\\ {-}{I}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {I}\\ {I}\\ {I}\\ {I}\\ {I}\\ {I}\\ {I}\\ {2}{}{I}\end{array}\right]\right]$ (2.1.10)



Example 6.

 > $\mathrm{LD6}≔\mathrm{SimpleLieAlgebraData}\left("so\left(6, 3\right)",\mathrm{alg6}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD6}\right)$
 ${\mathrm{Lie algebra: alg6}}$ (2.1.11)
 > $\mathrm{T6}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg6}\right):$
 > $\mathrm{SR6}≔\mathrm{T6}\left["SimpleRoots"\right]$
 ${\mathrm{SR6}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {I}\end{array}\right]\right]$ (2.1.12)



Example 6.

 > $\mathrm{LD7}≔\mathrm{SimpleLieAlgebraData}\left("so\left(9\right)",\mathrm{alg7}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD7}\right)$
 ${\mathrm{Lie algebra: alg7}}$ (2.1.13)
 > $\mathrm{T7}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg7}\right):$
 > $\mathrm{SR7}≔\mathrm{T7}\left["SimpleRoots"\right]$
 ${\mathrm{SR7}}{:=}\left[\left[\begin{array}{c}{I}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {I}\end{array}\right]\right]$ (2.1.14)



Example 8.

 > $\mathrm{LD8}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(10, R\right)",\mathrm{alg8}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD8}\right)$
 ${\mathrm{Lie algebra: alg8}}$ (2.1.15)
 > $\mathrm{T8}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg8}\right):$
 > $\mathrm{SR8}≔\mathrm{T8}\left["SimpleRoots"\right]$
 ${\mathrm{SR8}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {0}\\ {2}\end{array}\right]\right]$ (2.1.16)



Example 9.

 > $\mathrm{LD9}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(6, 4\right)",\mathrm{alg9}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD9}\right)$
 ${\mathrm{Lie algebra: alg9}}$ (2.1.17)
 > $\mathrm{T9}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg9}\right):$
 > $\mathrm{SR9}≔\mathrm{T9}\left["SimpleRoots"\right]$
 ${\mathrm{SR9}}{:=}\left[\left[\begin{array}{c}{0}\\ {0}\\ {2}{}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {-}{I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {0}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\end{array}\right]\right]$ (2.1.18)



Example 10.

 > $\mathrm{LD10}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(6, 6\right)",\mathrm{alg10}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD10}\right)$
 ${\mathrm{Lie algebra: alg10}}$ (2.1.19)
 > $\mathrm{T10}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg10}\right):$
 > $\mathrm{SR10}≔\mathrm{T10}\left["SimpleRoots"\right]$
 ${\mathrm{SR10}}{:=}\left[\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {-}{I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{1}\\ {0}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {2}\\ {0}\\ {0}\\ {-}{2}{}{I}\end{array}\right]\right]$ (2.1.20)



Example 11.

 > $\mathrm{LD11}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(10\right)",\mathrm{alg11}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD11}\right)$
 ${\mathrm{Lie algebra: alg11}}$ (2.1.21)
 > $\mathrm{T11}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg11}\right):$
 > $\mathrm{SR11}≔\mathrm{T11}\left["SimpleRoots"\right]$
 ${\mathrm{SR11}}{:=}\left[\left[\begin{array}{c}{I}\\ {-}{I}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {I}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\end{array}\right]\right]$ (2.1.22)



Example 12.

 > $\mathrm{LD12}≔\mathrm{SimpleLieAlgebraData}\left("so\left(5, 5\right)",\mathrm{alg12}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD12}\right)$
 ${\mathrm{Lie algebra: alg12}}$ (2.1.23)
 > $\mathrm{T12}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg12}\right):$
 > $\mathrm{SR12}≔\mathrm{T12}\left["SimpleRoots"\right]$
 ${\mathrm{SR12}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\\ {1}\end{array}\right]\right]$ (2.1.24)



Example 13.

 > $\mathrm{LD13}≔\mathrm{SimpleLieAlgebraData}\left("so\left(8, 2\right)",\mathrm{alg13}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD13}\right)$
 ${\mathrm{Lie algebra: alg13}}$ (2.1.25)
 > $\mathrm{T13}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{alg13}\right):$
 > $\mathrm{SR12}≔\mathrm{T13}\left["SimpleRoots"\right]$
 ${\mathrm{SR12}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]{,}\right]$