\(sl(2)\)-subalgebras of sl(3), type \(A^{1}_2\)

sl(3), type \(A^{1}_2\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.

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Number of sl(2) subalgebras: 2.

Let h be in the Cartan subalgebra. Let \(\alpha_1, ..., \alpha_n\) be simple roots with respect to h. Then the h-characteristic, as defined by E. Dynkin, is the n-tuple \((\alpha_1(h), ..., \alpha_n(h))\).

The actual realization of h. The coordinates of h are given with respect to the fixed original simple basis. Note that the h-characteristic is computed using a possibly different simple basis, more precisely, with respect to any h-positive simple basis.

A regular semisimple subalgebra might contain an sl(2) such that the sl(2) has no centralizer in the regular semisimple subalgebra, but the regular semisimple subalgebra might fail to be minimal containing. This happens when another minimal containing regular semisimple subalgebra of equal rank nests as a root subalgebra in the containing SA. See Dynkin, Semisimple Lie subalgebras of semisimple Lie algebras, remark before Theorem 10.4.

The \(sl(2)\) submodules of the ambient Lie algebra are parametrized by their highest weight with respect to the Cartan element h of \(sl(2)\). In turn, the highest weight is a positive integer multiple of the fundamental highest weight \(\psi\). \(V_{l\psi}\) is \(l + 1\)-dimensional.

Type + realization link	h-Characteristic	Realization of h	sl(2)-module decomposition of the ambient Lie algebra \(\psi=\) the fundamental \(sl(2)\)-weight.	Centralizer dimension	Type of semisimple part of centralizer, if known	The square of the length of the weight dual to h.	Dynkin index	Minimal containing regular semisimple SAs	Containing regular semisimple SAs in which the sl(2) has no centralizer
\(A^{4}_1\)	(2, 2)	(2, 2)	\(V_{4\psi}+V_{2\psi}\)	0	\(\displaystyle 0\)	8	4	A^{1}_2;	A^{1}_2;
\(A^{1}_1\)	(1, 1)	(1, 1)	\(V_{2\psi}+2V_{\psi}+V_{0}\)	1	\(\displaystyle 0\)	2	1	A^{1}_1;	A^{1}_1;

Length longest root ambient algebra squared/4= 1/2

Given a root subsystem P, and a root subsubsystem P_0, in (10.2) of Semisimple subalgebras of semisimple Lie algebras, E. Dynkin defines a numerical constant e(P, P_0) (which we call Dynkin epsilon).
In Theorem 10.3, Dynkin proves that if an sl(2) is an S-subalgebra in the root subalgebra generated by P, such that it has characteristic 2 for all simple roots of P lying in P_0, then e(P, P_0)= 0. It turns out by direct computation that, in the current case of A^{1}_2, e(P,P_0)= 0 implies that an S-sl(2) subalgebra of the root subalgebra generated by P with characteristic with 2's in the simple roots of P_0 always exists. Note that Theorem 10.3 is stated in one direction only.

h-characteristic: (2, 2)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 2 vectors: (1, 0), (0, 1)
Containing regular semisimple subalgebra number 1: A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{4\psi}+V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 2h_{2}+2h_{1}\)
\( e = g_{2}+2g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{1} x_{3} -2~\\x_{2} x_{4} -2~\\\end{array}\)

h-characteristic: (1, 1)
Length of the weight dual to h: 2
Simple basis ambient algebra w.r.t defining h: 2 vectors: (1, 0), (0, 1)
Containing regular semisimple subalgebra number 1: A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{2\psi}+2V_{\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = h_{2}+h_{1}\)
\( e = g_{3}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{1} x_{2} -1~\\x_{1} x_{2} -1~\\\end{array}\)