Subalgebra \(A^{12}_1\) ↪ \(F^{1}_4\)
10 out of 59
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{12}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{12}_1\): (6, 12, 16, 8): 24
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-2}-2g_{-9}+g_{-11}+g_{-18}\)
Positive simple generators: \(\displaystyle 4g_{18}+g_{16}+2g_{11}-g_{9}+4g_{5}+4g_{2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/6\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}24\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 2V_{6\omega_{1}}\oplus 4V_{4\omega_{1}}\oplus 6V_{2\omega_{1}}\)
In the table below we indicate the highest weight vectors of the decomposition of the ambient Lie algebra as a module over the semisimple part. The second row indicates weights of the highest weight vectors relative to the Cartan of the semisimple subalgebra.
Highest vectors of representations (total 12) ; the vectors are over the primal subalgebra.\(g_{11}+1/4g_{9}+2g_{5}+1/2g_{2}\)\(-g_{12}+1/2g_{10}\)\(g_{15}+g_{13}\)\(-g_{16}+1/2g_{9}-8/3g_{5}+1/3g_{2}\)\(g_{18}-1/4g_{9}-2/3g_{5}+5/6g_{2}\)\(g_{8}+1/4g_{6}\)\(g_{21}\)\(g_{22}-2g_{20}+2g_{14}\)\(g_{17}\)\(g_{19}\)\(g_{23}\)\(g_{24}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(4\omega_{1}\)\(4\omega_{1}\)\(4\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(6\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 4 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)\(\displaystyle V_{4\omega_{1}} \) → (4)\(\displaystyle V_{6\omega_{1}} \) → (6)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{18}-1/4g_{16}-1/2g_{11}+1/4g_{9}-g_{5}-g_{2}\)
\(2h_{4}+4h_{3}+3h_{2}+3/2h_{1}\)
\(1/2g_{-2}-g_{-9}+1/2g_{-11}+1/2g_{-18}\)
\(g_{8}+1/4g_{6}\)
\(1/4g_{3}-1/2g_{-3}\)
\(1/4g_{-8}\)
\(g_{11}+1/4g_{9}+2g_{5}+1/2g_{2}\)
\(-h_{3}-h_{2}-h_{1}-1/4g_{-1}\)
\(-1/4g_{-5}-1/2g_{-11}\)
\(-g_{12}+1/2g_{10}\)
\(1/2g_{7}-g_{-7}\)
\(g_{-10}-1/2g_{-12}\)
\(g_{15}+g_{13}\)
\(-g_{4}-g_{-4}\)
\(-2g_{-13}\)
\(-g_{16}+1/2g_{9}-8/3g_{5}+1/3g_{2}\)
\(8/3g_{1}+2h_{3}+2/3h_{2}+1/2g_{-1}\)
\(-2/3g_{-2}+1/2g_{-5}-4/3g_{-9}+1/3g_{-11}+8/3g_{-16}-2/3g_{-18}\)
\(g_{21}\)
\(g_{6}\)
\(g_{3}+2g_{-3}\)
\(-4g_{-6}+g_{-8}\)
\(4g_{-21}\)
\(g_{22}-2g_{20}+2g_{14}\)
\(g_{16}+2g_{11}+g_{9}-4g_{5}\)
\(8g_{1}-2h_{1}-2g_{-1}\)
\(2g_{-2}-2g_{-5}+4g_{-9}+2g_{-11}+8g_{-16}-2g_{-18}\)
\(-4g_{-14}+4g_{-20}-8g_{-22}\)
\(g_{17}\)
\(g_{15}\)
\(g_{4}-g_{-4}\)
\(-2g_{-13}+2g_{-15}\)
\(-2g_{-17}\)
\(g_{19}\)
\(-2g_{12}-g_{10}\)
\(-g_{7}-2g_{-7}\)
\(2g_{-10}+g_{-12}\)
\(4g_{-19}\)
\(g_{23}\)
\(g_{22}+2g_{20}\)
\(4g_{18}+g_{16}+g_{9}-2g_{2}\)
\(-8h_{4}-4h_{3}-4h_{1}-2g_{-1}\)
\(8g_{-2}-2g_{-5}-16g_{-9}+4g_{-11}-12g_{-18}\)
\(20g_{-20}+40g_{-22}\)
\(-80g_{-23}+20g_{-24}\)
\(g_{24}\)
\(-g_{20}-g_{14}\)
\(-g_{18}-g_{11}+2g_{5}+2g_{2}\)
\(-4g_{1}+2h_{4}+4h_{3}+2h_{1}\)
\(-6g_{-2}-4g_{-9}+4g_{-11}-4g_{-16}+4g_{-18}\)
\(-10g_{-14}-10g_{-20}\)
\(20g_{-23}-20g_{-24}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 5M_{2\omega_{1}}\oplus 5M_{0}\oplus 5M_{-2\omega_{1}}\)\(\displaystyle 4M_{4\omega_{1}}\oplus 4M_{2\omega_{1}}\oplus 4M_{0}\oplus 4M_{-2\omega_{1}}\oplus 4M_{-4\omega_{1}}\)\(\displaystyle 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 12399874 arithmetic operations while solving the Serre relations polynomial system.
The total number of arithmetic operations I needed to solve the Serre relations polynomial system was larger than 1 000 000. I am printing out the Serre relations system for you: maybe that can help improve the polynomial system algorithms.
Subalgebra realized.
1*2 (unknown) gens:
(
x_{1} g_{-2}+x_{2} g_{-5}+x_{3} g_{-6}+x_{4} g_{-8}+x_{5} g_{-9}+x_{6} g_{-10}+x_{7} g_{-11}+x_{8} g_{-12} \\ +x_{9} g_{-13}+x_{10} g_{-15}+x_{11} g_{-16}+x_{12} g_{-18}, x_{24} g_{18}+x_{23} g_{16}+x_{22} g_{15}+x_{21} g_{13}+x_{20} g_{12}+x_{19} g_{11}+x_{18} g_{10}+x_{17} g_{9} \\ +x_{16} g_{8}+x_{15} g_{6}+x_{14} g_{5}+x_{13} g_{2})
h: (6, 12, 16, 8), e = combination of g_{2} g_{5} g_{6} g_{8} g_{9} g_{10} g_{11} g_{12} g_{13} g_{15} g_{16} g_{18} , f= combination of g_{-2} g_{-5} g_{-6} g_{-8} g_{-9} g_{-10} g_{-11} g_{-12} g_{-13} g_{-15} g_{-16} g_{-18} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: 2V_{6\omega_{1}}+6V_{4\omega_{1}}+12V_{2\omega_{1}}+12V_{0}+12V_{-2\omega_{1}}+6V_{-4\omega_{1}}+2V_{-6\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{12} x_{24} +x_{11} x_{23} +2x_{10} x_{22} +2x_{9} x_{21} +2x_{8} x_{20} +x_{7} x_{19} +2x_{6} x_{18}
+x_{5} x_{17} +2x_{4} x_{16} +2x_{3} x_{15} +x_{2} x_{14} +x_{1} x_{13} -12= 0
x_{12} x_{23} +2x_{10} x_{21} +2x_{8} x_{18} +x_{7} x_{17} +2x_{4} x_{15} +x_{2} x_{13} = 0
x_{10} x_{20} +x_{9} x_{18} +x_{7} x_{16} +x_{5} x_{15} -x_{4} x_{14} -x_{3} x_{13} = 0
x_{12} x_{20} +x_{11} x_{18} -x_{10} x_{16} -x_{9} x_{15} +x_{8} x_{14} +x_{6} x_{13} = 0
x_{11} x_{24} +2x_{9} x_{22} +2x_{6} x_{20} +x_{5} x_{19} +2x_{3} x_{16} +x_{1} x_{14} = 0
x_{12} x_{24} +2x_{10} x_{22} +2x_{8} x_{20} +x_{7} x_{19} +2x_{4} x_{16} +x_{2} x_{14} -6= 0
x_{8} x_{22} +x_{6} x_{21} +x_{4} x_{19} +x_{3} x_{17} -x_{2} x_{16} -x_{1} x_{15} = 0
x_{12} x_{24} +x_{11} x_{23} +2x_{10} x_{22} +2x_{9} x_{21} +x_{8} x_{20} +x_{7} x_{19} +x_{6} x_{18}
+x_{5} x_{17} +x_{4} x_{16} +x_{3} x_{15} -8= 0
x_{12} x_{22} +x_{11} x_{21} -x_{10} x_{19} -x_{9} x_{17} -x_{8} x_{16} -x_{6} x_{15} = 0
x_{8} x_{24} +x_{6} x_{23} -x_{4} x_{22} -x_{3} x_{21} +x_{2} x_{20} +x_{1} x_{18} = 0
x_{10} x_{24} +x_{9} x_{23} -x_{7} x_{22} -x_{5} x_{21} -x_{4} x_{20} -x_{3} x_{18} = 0
x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{8} x_{20} +x_{6} x_{18} -4= 0
The above system after transformation.
x_{12} x_{24} +x_{11} x_{23} +2x_{10} x_{22} +2x_{9} x_{21} +2x_{8} x_{20} +x_{7} x_{19} +2x_{6} x_{18}
+x_{5} x_{17} +2x_{4} x_{16} +2x_{3} x_{15} +x_{2} x_{14} +x_{1} x_{13} -12= 0
x_{12} x_{23} +2x_{10} x_{21} +2x_{8} x_{18} +x_{7} x_{17} +2x_{4} x_{15} +x_{2} x_{13} = 0
x_{10} x_{20} +x_{9} x_{18} +x_{7} x_{16} +x_{5} x_{15} -x_{4} x_{14} -x_{3} x_{13} = 0
x_{12} x_{20} +x_{11} x_{18} -x_{10} x_{16} -x_{9} x_{15} +x_{8} x_{14} +x_{6} x_{13} = 0
x_{11} x_{24} +2x_{9} x_{22} +2x_{6} x_{20} +x_{5} x_{19} +2x_{3} x_{16} +x_{1} x_{14} = 0
x_{12} x_{24} +2x_{10} x_{22} +2x_{8} x_{20} +x_{7} x_{19} +2x_{4} x_{16} +x_{2} x_{14} -6= 0
x_{8} x_{22} +x_{6} x_{21} +x_{4} x_{19} +x_{3} x_{17} -x_{2} x_{16} -x_{1} x_{15} = 0
x_{12} x_{24} +x_{11} x_{23} +2x_{10} x_{22} +2x_{9} x_{21} +x_{8} x_{20} +x_{7} x_{19} +x_{6} x_{18}
+x_{5} x_{17} +x_{4} x_{16} +x_{3} x_{15} -8= 0
x_{12} x_{22} +x_{11} x_{21} -x_{10} x_{19} -x_{9} x_{17} -x_{8} x_{16} -x_{6} x_{15} = 0
x_{8} x_{24} +x_{6} x_{23} -x_{4} x_{22} -x_{3} x_{21} +x_{2} x_{20} +x_{1} x_{18} = 0
x_{10} x_{24} +x_{9} x_{23} -x_{7} x_{22} -x_{5} x_{21} -x_{4} x_{20} -x_{3} x_{18} = 0
x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{8} x_{20} +x_{6} x_{18} -4= 0
For the calculator:
(DynkinType =A^{12}_1; ElementsCartan =((6, 12, 16, 8)); generators =(x_{1} g_{-2}+x_{2} g_{-5}+x_{3} g_{-6}+x_{4} g_{-8}+x_{5} g_{-9}+x_{6} g_{-10}+x_{7} g_{-11}+x_{8} g_{-12}+x_{9} g_{-13}+x_{10} g_{-15}+x_{11} g_{-16}+x_{12} g_{-18}, x_{24} g_{18}+x_{23} g_{16}+x_{22} g_{15}+x_{21} g_{13}+x_{20} g_{12}+x_{19} g_{11}+x_{18} g_{10}+x_{17} g_{9}+x_{16} g_{8}+x_{15} g_{6}+x_{14} g_{5}+x_{13} g_{2}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{12} x_{24} +x_{11} x_{23} +2x_{10} x_{22} +2x_{9} x_{21} +2x_{8} x_{20} +x_{7} x_{19} +2x_{6} x_{18} +x_{5} x_{17} +2x_{4} x_{16} +2x_{3} x_{15} +x_{2} x_{14} +x_{1} x_{13} -12, x_{12} x_{23} +2x_{10} x_{21} +2x_{8} x_{18} +x_{7} x_{17} +2x_{4} x_{15} +x_{2} x_{13} , x_{10} x_{20} +x_{9} x_{18} +x_{7} x_{16} +x_{5} x_{15} -x_{4} x_{14} -x_{3} x_{13} , x_{12} x_{20} +x_{11} x_{18} -x_{10} x_{16} -x_{9} x_{15} +x_{8} x_{14} +x_{6} x_{13} , x_{11} x_{24} +2x_{9} x_{22} +2x_{6} x_{20} +x_{5} x_{19} +2x_{3} x_{16} +x_{1} x_{14} , x_{12} x_{24} +2x_{10} x_{22} +2x_{8} x_{20} +x_{7} x_{19} +2x_{4} x_{16} +x_{2} x_{14} -6, x_{8} x_{22} +x_{6} x_{21} +x_{4} x_{19} +x_{3} x_{17} -x_{2} x_{16} -x_{1} x_{15} , x_{12} x_{24} +x_{11} x_{23} +2x_{10} x_{22} +2x_{9} x_{21} +x_{8} x_{20} +x_{7} x_{19} +x_{6} x_{18} +x_{5} x_{17} +x_{4} x_{16} +x_{3} x_{15} -8, x_{12} x_{22} +x_{11} x_{21} -x_{10} x_{19} -x_{9} x_{17} -x_{8} x_{16} -x_{6} x_{15} , x_{8} x_{24} +x_{6} x_{23} -x_{4} x_{22} -x_{3} x_{21} +x_{2} x_{20} +x_{1} x_{18} , x_{10} x_{24} +x_{9} x_{23} -x_{7} x_{22} -x_{5} x_{21} -x_{4} x_{20} -x_{3} x_{18} , x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{8} x_{20} +x_{6} x_{18} -4 )