sl(4), type \(A^{1}_3\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.
Page generated by the calculator project.
g: A^{1}_3. There are 5 table entries (= 3 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).
Type k_{ss}: A^{1}_3
(Full subalgebra)
Type C(k_{ss})_{ss}: 0		 Type k_{ss}: A^{1}_2

Type C(k_{ss})_{ss}: 0		 Type k_{ss}: 2A^{1}_1

Type C(k_{ss})_{ss}: 0		 Type k_{ss}: A^{1}_1

Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: 0
(Cartan subalgebra)
Type C(k_{ss})_{ss}: A^{1}_3
There are 5 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]]],
["parabolic","2A^{1}_1", [[1, 1, 1], [0, 1, 0]]],
["parabolic","A^{1}_2", [[1, 1, 1], [0, 0, -1]]],
["parabolic","A^{1}_3", [[1, 1, 1], [0, 0, -1], [0, -1, 0]]]
LaTeX table with root subalgebra details.
\documentclass{article}
\usepackage{longtable, amssymb, lscape}
\begin{document}
Lie algebra type: $A^{1}_3$. There are 5 table entries (= 3 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
$A^{1}_3$&$0$&$6$&$0$&$0$&$3$&$0$&\\
$A^{1}_2$&$0$&$3$&$0$&$0$&$2$&$0$&\\
$2A^{1}_1$&$0$&$2$&$0$&$2$&$2$&$0$&\\
$A^{1}_1$&$A^{1}_1$&$1$&$1$&$1$&$1$&$1$&\\
$0$&$A^{1}_3$&$0$&$6$&$0$&$0$&$3$&\\
\end{longtable}
\end{document}