Root subsystems of E^{1}

$E^{1}_6$
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.

Page generated by the calculator project.

g: E^{1}_6. There are 21 table entries (= 19 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).

Type k_{ss}: E^{1}_6 (Full subalgebra) Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_5+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 3A^{1}_2 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_5 Type C(k_{ss})_{ss}: 0
Type k_{ss}: A^{1}_5 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_4+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_3+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 2A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: D^{1}_4 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_4 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_3+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: 2A^{1}_2 Type C(k_{ss})_{ss}: A^{1}_2
Type k_{ss}: A^{1}_2+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 4A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_3 Type C(k_{ss})_{ss}: 2A^{1}_1	Type k_{ss}: A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_2
Type k_{ss}: 3A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_2 Type C(k_{ss})_{ss}: 2A^{1}_2	Type k_{ss}: 2A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_3	Type k_{ss}: A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_5
Type k_{ss}: 0 (Cartan subalgebra) Type C(k_{ss})_{ss}: E^{1}_6

There are 17 parabolic, 4 pseudo-parabolic but not parabolic and 0 non pseudo-parabolic root subsystems. The roots needed to generate the root subsystems are listed below using the root indices in GAP.
["parabolic","0", []],
["parabolic","A^{1}_1", [36]],
["parabolic","2A^{1}_1", [36, 23]],
["parabolic","A^{1}_2", [36, 2]],
["parabolic","3A^{1}_1", [36, 23, 15]],
["parabolic","A^{1}_2+A^{1}_1", [36, 2, 7]],
["parabolic","A^{1}_3", [36, 2, 4]],
["parabolic","A^{1}_2+2A^{1}_1", [36, 2, 7, 11]],
["parabolic","2A^{1}_2", [36, 2, 7, 3]],
["parabolic","A^{1}_3+A^{1}_1", [36, 2, 4, 1]],
["parabolic","A^{1}_4", [36, 2, 4, 5]],
["parabolic","D^{1}_4", [36, 2, 4, 15]],
["parabolic","2A^{1}_2+A^{1}_1", [36, 2, 7, 3, 11]],
["parabolic","A^{1}_4+A^{1}_1", [36, 2, 4, 5, 1]],
["parabolic","A^{1}_5", [36, 2, 4, 5, 6]],
["parabolic","D^{1}_5", [36, 2, 4, 5, 3]],
["parabolic","E^{1}_6", [36, 3, 2, 4, 5, 6]],
["pseudoParabolicNonParabolic","4A^{1}_1", [36, 23, 15, 4]],
["pseudoParabolicNonParabolic","A^{1}_3+2A^{1}_1", [36, 2, 4, 1, 6]],
["pseudoParabolicNonParabolic","3A^{1}_2", [36, 2, 7, 3, 11, 6]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_1", [36, 2, 4, 5, 6, 1]]

The roots needed to generate the root subsystems are listed below.
["parabolic","0", []],
["parabolic","A^{1}_1", [[1, 2, 2, 3, 2, 1]]],
["parabolic","2A^{1}_1", [[1, 2, 2, 3, 2, 1], [1, 0, 1, 1, 1, 1]]],
["parabolic","A^{1}_2", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0]]],
["parabolic","3A^{1}_1", [[1, 2, 2, 3, 2, 1], [1, 0, 1, 1, 1, 1], [0, 0, 1, 1, 1, 0]]],
["parabolic","A^{1}_2+A^{1}_1", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0]]],
["parabolic","A^{1}_3", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0]]],
["parabolic","A^{1}_2+2A^{1}_1", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 1]]],
["parabolic","2A^{1}_2", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0]]],
["parabolic","A^{1}_3+A^{1}_1", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_4", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0]]],
["parabolic","D^{1}_4", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, -1, -1, -1, 0]]],
["parabolic","2A^{1}_2+A^{1}_1", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 1]]],
["parabolic","A^{1}_4+A^{1}_1", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_5", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1]]],
["parabolic","D^{1}_5", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0], [0, 0, -1, 0, 0, 0]]],
["parabolic","E^{1}_6", [[1, 2, 2, 3, 2, 1], [0, 0, -1, 0, 0, 0], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1]]],
["pseudoParabolicNonParabolic","4A^{1}_1", [[1, 2, 2, 3, 2, 1], [1, 0, 1, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_3+2A^{1}_1", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]]],
["pseudoParabolicNonParabolic","3A^{1}_2", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, -1]]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_1", [[1, 2, 2, 3, 2, 1], [0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1], [1, 0, 0, 0, 0, 0]]]

LaTeX table with root subalgebra details.
\documentclass{article}
\usepackage{longtable, amssymb, lscape}
\begin{document}
Lie algebra type: $E^{1}_6$. There are 21 table entries (= 19 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).
Let $\mathfrak g$ stand for the type of the regular subalgebra and $C(\mathfrak g)$ for the type of the centralizer. Let $r$ stand for the rank of $\mathfrak g$, let $r_c$ stand for the rank of the semisimple part of the centralizer, let $p$ stand for the number of positive roots of $\mathfrak g$, let $q$ stand for the number of positive roots of the centralizer, and let $m$ stand for the number of $A_1$ components (of all root lengths) of $\mathfrak g$. \begin{longtable}{cccccccc}
$\mathfrak g$ & $C(\mathfrak g)$& $p$ & $q$& $m$& $r$ & $c_r$ \\\endhead
$E^{1}_6$&$0$&$36$&$0$&$0$&$6$&$0$&\\
$A^{1}_5+A^{1}_1$&$0$&$16$&$0$&$1$&$6$&$0$&\\
$3A^{1}_2$&$0$&$9$&$0$&$0$&$6$&$0$&\\
$D^{1}_5$&$0$&$20$&$0$&$0$&$5$&$0$&\\
$A^{1}_5$&$A^{1}_1$&$15$&$1$&$0$&$5$&$1$&\\
$A^{1}_4+A^{1}_1$&$0$&$11$&$0$&$1$&$5$&$0$&\\
$A^{1}_3+2A^{1}_1$&$0$&$8$&$0$&$2$&$5$&$0$&\\
$2A^{1}_2+A^{1}_1$&$0$&$7$&$0$&$1$&$5$&$0$&\\
$D^{1}_4$&$0$&$12$&$0$&$0$&$4$&$0$&\\
$A^{1}_4$&$A^{1}_1$&$10$&$1$&$0$&$4$&$1$&\\
$A^{1}_3+A^{1}_1$&$A^{1}_1$&$7$&$1$&$1$&$4$&$1$&\\
$2A^{1}_2$&$A^{1}_2$&$6$&$3$&$0$&$4$&$2$&\\
$A^{1}_2+2A^{1}_1$&$0$&$5$&$0$&$2$&$4$&$0$&\\
$4A^{1}_1$&$0$&$4$&$0$&$4$&$4$&$0$&\\
$A^{1}_3$&$2A^{1}_1$&$6$&$2$&$0$&$3$&$2$&\\
$A^{1}_2+A^{1}_1$&$A^{1}_2$&$4$&$3$&$1$&$3$&$2$&\\
$3A^{1}_1$&$A^{1}_1$&$3$&$1$&$3$&$3$&$1$&\\
$A^{1}_2$&$2A^{1}_2$&$3$&$6$&$0$&$2$&$4$&\\
$2A^{1}_1$&$A^{1}_3$&$2$&$6$&$2$&$2$&$3$&\\
$A^{1}_1$&$A^{1}_5$&$1$&$15$&$1$&$1$&$5$&\\
$0$&$E^{1}_6$&$0$&$36$&$0$&$0$&$6$&\\
\end{longtable}
\end{document}