Root subsystems of A^{1}

sl(9), type $A^{1}_8$
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.

Page generated by the calculator project.

g: A^{1}_8. There are 30 table entries (= 28 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).

Type k_{ss}: A^{1}_8 (Full subalgebra) Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_7 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_6+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_5+A^{1}_2 Type C(k_{ss})_{ss}: 0
Type k_{ss}: A^{1}_4+A^{1}_3 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_6 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_5+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_4+A^{1}_2 Type C(k_{ss})_{ss}: 0
Type k_{ss}: A^{1}_4+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 2A^{1}_3 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_3+A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 3A^{1}_2 Type C(k_{ss})_{ss}: 0
Type k_{ss}: A^{1}_5 Type C(k_{ss})_{ss}: A^{1}_2	Type k_{ss}: A^{1}_4+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_3+A^{1}_2 Type C(k_{ss})_{ss}: A^{1}_1	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}: A^{1}_2+3A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_4 Type C(k_{ss})_{ss}: A^{1}_3	Type k_{ss}: A^{1}_3+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_2
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}: A^{1}_1	Type k_{ss}: 4A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_3 Type C(k_{ss})_{ss}: A^{1}_4
Type k_{ss}: A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_3	Type k_{ss}: 3A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_2	Type k_{ss}: A^{1}_2 Type C(k_{ss})_{ss}: A^{1}_5	Type k_{ss}: 2A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_4
Type k_{ss}: A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_6	Type k_{ss}: 0 (Cartan subalgebra) Type C(k_{ss})_{ss}: A^{1}_8

There are 30 parabolic, 0 pseudo-parabolic but not parabolic and 0 non pseudo-parabolic root subsystems.

The roots needed to generate the root subsystems are listed below.
["parabolic","0", []],
["parabolic","A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1]]],
["parabolic","2A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 1, 1, 1, 1, 1, 1, 0]]],
["parabolic","A^{1}_2", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1]]],
["parabolic","3A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 1, 1, 1, 1, 1, 1, 0], [0, 0, 1, 1, 1, 1, 0, 0]]],
["parabolic","A^{1}_2+A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 1, 1, 1, 1, 1, 0, 0]]],
["parabolic","A^{1}_3", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0]]],
["parabolic","4A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 1, 1, 1, 1, 1, 1, 0], [0, 0, 1, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0]]],
["parabolic","A^{1}_2+2A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 1, 1, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0, 0]]],
["parabolic","2A^{1}_2", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0]]],
["parabolic","A^{1}_3+A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 1, 1, 1, 1, 0, 0, 0]]],
["parabolic","A^{1}_4", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0]]],
["parabolic","A^{1}_2+3A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 1, 1, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0]]],
["parabolic","2A^{1}_2+A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0]]],
["parabolic","A^{1}_3+2A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 1, 1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0]]],
["parabolic","A^{1}_3+A^{1}_2", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 1, 1, 1, 1, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0]]],
["parabolic","A^{1}_4+A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 1, 1, 1, 0, 0, 0, 0]]],
["parabolic","A^{1}_5", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0]]],
["parabolic","3A^{1}_2", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["parabolic","A^{1}_3+A^{1}_2+A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 1, 1, 1, 1, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0]]],
["parabolic","2A^{1}_3", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 1, 1, 1, 1, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["parabolic","A^{1}_4+2A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_4+A^{1}_2", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 1, 1, 1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["parabolic","A^{1}_5+A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_6", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["parabolic","A^{1}_4+A^{1}_3", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 1, 1, 1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_5+A^{1}_2", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_6+A^{1}_1", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_7", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_8", [[1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0]]]

LaTeX table with root subalgebra details.
\documentclass{article}
\usepackage{longtable, amssymb, lscape}
\begin{document}
Lie algebra type: $A^{1}_8$. There are 30 table entries (= 28 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
$A^{1}_8$&$0$&$36$&$0$&$0$&$8$&$0$&\\
$A^{1}_7$&$0$&$28$&$0$&$0$&$7$&$0$&\\
$A^{1}_6+A^{1}_1$&$0$&$22$&$0$&$1$&$7$&$0$&\\
$A^{1}_5+A^{1}_2$&$0$&$18$&$0$&$0$&$7$&$0$&\\
$A^{1}_4+A^{1}_3$&$0$&$16$&$0$&$0$&$7$&$0$&\\
$A^{1}_6$&$A^{1}_1$&$21$&$1$&$0$&$6$&$1$&\\
$A^{1}_5+A^{1}_1$&$0$&$16$&$0$&$1$&$6$&$0$&\\
$A^{1}_4+A^{1}_2$&$0$&$13$&$0$&$0$&$6$&$0$&\\
$A^{1}_4+2A^{1}_1$&$0$&$12$&$0$&$2$&$6$&$0$&\\
$2A^{1}_3$&$0$&$12$&$0$&$0$&$6$&$0$&\\
$A^{1}_3+A^{1}_2+A^{1}_1$&$0$&$10$&$0$&$1$&$6$&$0$&\\
$3A^{1}_2$&$0$&$9$&$0$&$0$&$6$&$0$&\\
$A^{1}_5$&$A^{1}_2$&$15$&$3$&$0$&$5$&$2$&\\
$A^{1}_4+A^{1}_1$&$A^{1}_1$&$11$&$1$&$1$&$5$&$1$&\\
$A^{1}_3+A^{1}_2$&$A^{1}_1$&$9$&$1$&$0$&$5$&$1$&\\
$A^{1}_3+2A^{1}_1$&$0$&$8$&$0$&$2$&$5$&$0$&\\
$2A^{1}_2+A^{1}_1$&$0$&$7$&$0$&$1$&$5$&$0$&\\
$A^{1}_2+3A^{1}_1$&$0$&$6$&$0$&$3$&$5$&$0$&\\
$A^{1}_4$&$A^{1}_3$&$10$&$6$&$0$&$4$&$3$&\\
$A^{1}_3+A^{1}_1$&$A^{1}_2$&$7$&$3$&$1$&$4$&$2$&\\
$2A^{1}_2$&$A^{1}_2$&$6$&$3$&$0$&$4$&$2$&\\
$A^{1}_2+2A^{1}_1$&$A^{1}_1$&$5$&$1$&$2$&$4$&$1$&\\
$4A^{1}_1$&$0$&$4$&$0$&$4$&$4$&$0$&\\
$A^{1}_3$&$A^{1}_4$&$6$&$10$&$0$&$3$&$4$&\\
$A^{1}_2+A^{1}_1$&$A^{1}_3$&$4$&$6$&$1$&$3$&$3$&\\
$3A^{1}_1$&$A^{1}_2$&$3$&$3$&$3$&$3$&$2$&\\
$A^{1}_2$&$A^{1}_5$&$3$&$15$&$0$&$2$&$5$&\\
$2A^{1}_1$&$A^{1}_4$&$2$&$10$&$2$&$2$&$4$&\\
$A^{1}_1$&$A^{1}_6$&$1$&$21$&$1$&$1$&$6$&\\
$0$&$A^{1}_8$&$0$&$36$&$0$&$0$&$8$&\\
\end{longtable}
\end{document}