Subalgebra \(A^{60}_1\) ↪ \(F^{1}_4\)
14 out of 59
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{60}_1\) (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: \(\displaystyle A^{60}_1\): (14, 26, 36, 20): 120
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-7}+g_{-9}\)
Positive simple generators: \(\displaystyle 8g_{9}+10g_{7}+18g_{2}+14g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/30\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}120\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{14\omega_{1}}\oplus 2V_{10\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}\)
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 6) ; the vectors are over the primal subalgebra.\(g_{9}+5/4g_{7}+9/4g_{2}+7/4g_{1}\)\(-g_{13}+7/5g_{8}\)\(g_{16}+14/25g_{14}+7/10g_{12}\)\(g_{22}+9/4g_{20}\)\(g_{21}\)\(g_{24}\)
weight\(2\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(10\omega_{1}\)\(10\omega_{1}\)\(14\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 5 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)\(\displaystyle V_{4\omega_{1}} \) → (4)\(\displaystyle V_{6\omega_{1}} \) → (6)\(\displaystyle V_{10\omega_{1}} \) → (10)\(\displaystyle V_{14\omega_{1}} \) → (14)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-4/7g_{9}-5/7g_{7}-9/7g_{2}-g_{1}\)
\(10/7h_{4}+18/7h_{3}+13/7h_{2}+h_{1}\)
\(1/7g_{-1}+1/7g_{-2}+1/7g_{-7}+1/7g_{-9}\)
\(-g_{13}+7/5g_{8}\)
\(2/5g_{6}-g_{4}\)
\(2/5g_{3}+3/5g_{-3}\)
\(2/5g_{-4}-1/5g_{-6}\)
\(1/5g_{-8}-1/5g_{-13}\)
\(g_{16}+14/25g_{14}+7/10g_{12}\)
\(14/25g_{11}-3/10g_{10}-21/25g_{5}\)
\(14/25g_{9}-3/10g_{7}-6/25g_{2}+7/25g_{1}\)
\(3/5h_{4}-13/25h_{3}-8/25h_{2}-7/25h_{1}\)
\(-6/25g_{-1}+4/25g_{-2}+9/25g_{-7}-21/25g_{-9}\)
\(-2/5g_{-5}-1/5g_{-10}+3/5g_{-11}\)
\(-1/5g_{-12}-1/5g_{-14}-2/5g_{-16}\)
\(g_{22}+9/4g_{20}\)
\(g_{19}+5/4g_{18}\)
\(5/4g_{16}-2g_{14}-1/4g_{12}\)
\(-2g_{11}-3/2g_{10}-3/2g_{5}\)
\(-2g_{9}-3/2g_{7}+3/2g_{2}+7/2g_{1}\)
\(3h_{4}+7h_{3}+1/2h_{2}-7/2h_{1}\)
\(-15/2g_{-1}-5/2g_{-2}+9/2g_{-7}+15/2g_{-9}\)
\(-5g_{-5}-7g_{-10}-15g_{-11}\)
\(2g_{-12}+20g_{-14}-14g_{-16}\)
\(18g_{-18}+18g_{-19}\)
\(-18g_{-20}-18g_{-22}\)
\(g_{21}\)
\(g_{17}\)
\(g_{15}\)
\(g_{13}+g_{8}\)
\(2g_{6}+g_{4}\)
\(2g_{3}-3g_{-3}\)
\(2g_{-4}+5g_{-6}\)
\(-5g_{-8}-7g_{-13}\)
\(12g_{-15}\)
\(-12g_{-17}\)
\(12g_{-21}\)
\(g_{24}\)
\(g_{23}\)
\(g_{22}-g_{20}\)
\(g_{19}-2g_{18}\)
\(-2g_{16}-2g_{14}+3g_{12}\)
\(-2g_{11}+5g_{10}-8g_{5}\)
\(-2g_{9}+5g_{7}-18g_{2}+10g_{1}\)
\(-10h_{4}-6h_{3}+20h_{2}-10h_{1}\)
\(-40g_{-1}+56g_{-2}-28g_{-7}+14g_{-9}\)
\(-96g_{-5}+84g_{-10}-54g_{-11}\)
\(-180g_{-12}+150g_{-14}+168g_{-16}\)
\(-528g_{-18}+330g_{-19}\)
\(528g_{-20}-1188g_{-22}\)
\(1716g_{-23}\)
\(-1716g_{-24}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle 2M_{10\omega_{1}}\oplus 2M_{8\omega_{1}}\oplus 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}
\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\oplus 2M_{-8\omega_{1}}\oplus 2M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 1189835 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_{-1}+x_{2} g_{-2}+x_{3} g_{-4}+x_{4} g_{-6}+x_{5} g_{-7}+x_{6} g_{-9}, x_{12} g_{9}+x_{11} g_{7}+x_{10} g_{6}+x_{9} g_{4}+x_{8} g_{2}+x_{7} g_{1})
h: (14, 26, 36, 20), e = combination of g_{1} g_{2} g_{4} g_{6} g_{7} g_{9} , f= combination of g_{-1} g_{-2} g_{-4} g_{-6} g_{-7} g_{-9} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: V_{14\omega_{1}}+V_{12\omega_{1}}+3V_{10\omega_{1}}+3V_{8\omega_{1}}+4V_{6\omega_{1}}+5V_{4\omega_{1}}+6V_{2\omega_{1}}+6V_{0}+6V_{-2\omega_{1}}+5V_{-4\omega_{1}}+4V_{-6\omega_{1}}+3V_{-8\omega_{1}}+3V_{-10\omega_{1}}+V_{-12\omega_{1}}+V_{-14\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{7} -14= 0
x_{6} x_{12} +2x_{4} x_{10} +x_{2} x_{8} -26= 0
x_{6} x_{10} +x_{5} x_{9} -x_{4} x_{8} = 0
x_{5} x_{11} +x_{3} x_{9} -10= 0
x_{4} x_{12} +x_{3} x_{11} -x_{2} x_{10} = 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -18= 0
The above system after transformation.
x_{1} x_{7} -14= 0
x_{6} x_{12} +2x_{4} x_{10} +x_{2} x_{8} -26= 0
x_{6} x_{10} +x_{5} x_{9} -x_{4} x_{8} = 0
x_{5} x_{11} +x_{3} x_{9} -10= 0
x_{4} x_{12} +x_{3} x_{11} -x_{2} x_{10} = 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -18= 0
For the calculator:
(DynkinType =A^{60}_1; ElementsCartan =((14, 26, 36, 20)); generators =(x_{1} g_{-1}+x_{2} g_{-2}+x_{3} g_{-4}+x_{4} g_{-6}+x_{5} g_{-7}+x_{6} g_{-9}, x_{12} g_{9}+x_{11} g_{7}+x_{10} g_{6}+x_{9} g_{4}+x_{8} g_{2}+x_{7} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{1} x_{7} -14, x_{6} x_{12} +2x_{4} x_{10} +x_{2} x_{8} -26, x_{6} x_{10} +x_{5} x_{9} -x_{4} x_{8} , x_{5} x_{11} +x_{3} x_{9} -10, x_{4} x_{12} +x_{3} x_{11} -x_{2} x_{10} , x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -18 )