Lie algebra \(G^{1}_2\)
Semisimple complex Lie subalgebras
Up to linear equivalence, there are total 7 semisimple subalgebras (including the full subalgebra). The subalgebras are ordered by rank, Dynkin indices of simple constituents and dimensions of simple constituents.
The upper index indicates the Dynkin index, the lower index indicates the rank of the subalgebra.
Computation time in seconds: 1.868.
419467 total arithmetic operations performed = 369610 additions and 49857 multiplications.
The base field over which the subalgebras were realized is: \(\displaystyle \mathbb Q\)
Number of root subalgebras other than the Cartan and full subalgebra: 4
Number of sl(2)'s: 4
Subalgebra \(A^{1}_1\) ↪ \(G^{1}_2\)
1 out of 7
Subalgebra type: \(\displaystyle A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: \(\displaystyle A^{3}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_1\)
Basis of Cartan of centralizer: 1 vectors:
(1, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (3, 2): 2
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-6}\)
Positive simple generators: \(\displaystyle g_{6}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus 4V_{\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+2/3\psi}\oplus V_{4/3\psi}\oplus V_{\omega_{1}-2/3\psi}\oplus V_{0}\oplus V_{\omega_{1}-2\psi}
\oplus V_{-4/3\psi}\)
Made total 274 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1\) ↪ \(G^{1}_2\)
2 out of 7
Subalgebra type: \(\displaystyle A^{3}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{3}_1\)
Basis of Cartan of centralizer: 1 vectors:
(0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{3}_1+A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (6, 3): 6
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-4}\)
Positive simple generators: \(\displaystyle g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 2V_{3\omega_{1}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi}\oplus V_{0}\oplus V_{-4\psi}\)
Made total 274 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1\) ↪ \(G^{1}_2\)
3 out of 7
Subalgebra type: \(\displaystyle A^{4}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (6, 4): 8
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-2}+g_{-5}\)
Positive simple generators: \(\displaystyle 2g_{5}+2g_{2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\)
Made total 338565 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{28}_1\) ↪ \(G^{1}_2\)
4 out of 7
Subalgebra type: \(\displaystyle A^{28}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{28}_1\): (18, 10): 56
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}\)
Positive simple generators: \(\displaystyle 10g_{2}+6g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/14\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}56\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{2\omega_{1}}\)
Made total 1774 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1+A^{1}_1\) ↪ \(G^{1}_2\)
5 out of 7
Subalgebra type: \(\displaystyle A^{3}_1+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (6, 3): 6, \(\displaystyle A^{1}_1\): (0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-4}\), \(\displaystyle g_{-2}\)
Positive simple generators: \(\displaystyle g_{4}\), \(\displaystyle g_{2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & 0\\
0 & 2\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & 0\\
0 & 2\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 365 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2\) ↪ \(G^{1}_2\)
6 out of 7
Subalgebra type: \(\displaystyle A^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (3, 2): 2, (0, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \(\displaystyle g_{-6}\), \(\displaystyle g_{2}\)
Positive simple generators: \(\displaystyle g_{6}\), \(\displaystyle g_{-2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\)
Made total 357 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(G^{1}_2\) ↪ \(G^{1}_2\)
7 out of 7
Subalgebra type: \(\displaystyle G^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle G^{1}_2\): (6, 3): 6, (-3, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: \(\displaystyle g_{-4}\), \(\displaystyle g_{5}\)
Positive simple generators: \(\displaystyle g_{4}\), \(\displaystyle g_{-5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1\\
-1 & 2\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\
-3 & 2\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}}\)
Made total 359 arithmetic operations while solving the Serre relations polynomial system.
Of the 4 h element conjugacy classes 1 had their Weyl group orbits computed and buffered. The h elements and their computed orbit sizes follow. | h element | orbit size |
| (18, 10) | size not computed |
| (6, 4) | size not computed |
| (6, 3) | size not computed |
| (3, 2) | 6 |
Number of sl(2) subalgebras: 4.
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.
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 G^{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: 56
Simple basis ambient algebra w.r.t defining h: 2 vectors:
(1, 0), (0, 1)
Containing regular semisimple subalgebra number 1: G^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\(
h = 10h_{2}+18h_{1}\)
\(
e = 5g_{2}+6g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}3x_{1} x_{3} -18~\\x_{2} x_{4} -10~\\\end{array}\)
h-characteristic: (0, 2)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 2 vectors:
(1, 0), (0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: G^{1}_2
Containing regular semisimple subalgebra number 2: A^{1}_2
Containing regular semisimple subalgebra number 3: A^{3}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{4\psi}+3V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\(
h = 4h_{2}+6h_{1}\)
\(
e = -2/5g_{5}+g_{4}-2g_{3}+5g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{3} x_{8} +2x_{2} x_{7} +x_{1} x_{6} ~\\x_{4} x_{7} +2x_{3} x_{6} +x_{2} x_{5} ~\\3x_{4} x_{8} +6x_{3} x_{7} +3x_{2} x_{6} -6~\\x_{4} x_{8} +3x_{3} x_{7} +3x_{2} x_{6} +x_{1} x_{5} -4~\\\end{array}\)
h-characteristic: (1, 0)
Length of the weight dual to h: 6
Simple basis ambient algebra w.r.t defining h: 2 vectors:
(1, 0), (0, 1)
Containing regular semisimple subalgebra number 1: A^{3}_1
sl(2)-module decomposition of the ambient Lie algebra: \(2V_{3\psi}+V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\(
h = 3h_{2}+6h_{1}\)
\(
e = g_{4}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}6x_{1} x_{2} -6~\\3x_{1} x_{2} -3~\\\end{array}\)
h-characteristic: (0, 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: \(V_{2\psi}+4V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\(
h = 2h_{2}+3h_{1}\)
\(
e = g_{6}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}3x_{1} x_{2} -3~\\2x_{1} x_{2} -2~\\\end{array}\)
Calculator input for loading subalgebras directly without recomputation. Subalgebras found so far: 7
Orbit sizes: A^28_1: n/a; A^4_1: n/a; A^3_1: n/a; A^1_1: 6;
Current subalgebra chain length: 0
SetOutputFile("subalgebras_G^{1}_2");
LoadSemisimpleSubalgebras {}(AmbientDynkinType=G^{1}{}\left(2\right);CurrentChain=\left(\right);NumExploredTypes=\left(\right);NumExploredHs=\left(\right);Subalgebras=\left((DynkinType=A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}3 & 2 \end{pmatrix};generators=\left(g{}\left(-6\right), g{}\left(6\right)\right)), (DynkinType=A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 3 \end{pmatrix};generators=\left(g{}\left(-4\right), g{}\left(4\right)\right)), (DynkinType=A^{4}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 4 \end{pmatrix};generators=\left(g{}\left(-2\right)+g{}\left(-5\right), 2 g{}\left(2\right)+2 g{}\left(5\right)\right)), (DynkinType=A^{28}{}\left(1\right);ElementsCartan=\begin{pmatrix}18 & 10 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right), 6 g{}\left(1\right)+10 g{}\left(2\right)\right)), (DynkinType=A^{3}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 3\\
0 & 1 \end{pmatrix};generators=\left(g{}\left(-4\right), g{}\left(4\right), g{}\left(-2\right), g{}\left(2\right)\right)), (DynkinType=A^{1}{}\left(2\right);ElementsCartan=\begin{pmatrix}3 & 2\\
0 & -1 \end{pmatrix};generators=\left(g{}\left(-6\right), g{}\left(6\right), g{}\left(2\right), g{}\left(-2\right)\right)), (DynkinType=G^{1}{}\left(2\right);ElementsCartan=\begin{pmatrix}6 & 3\\
-3 & -1 \end{pmatrix};generators=\left(g{}\left(-4\right), g{}\left(4\right), g{}\left(5\right), g{}\left(-5\right)\right))\right))