Subalgebra \(A^{36}_1\) ↪ \(C^{1}_4\)
12 out of 46
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{36}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{36}_1\): (10, 16, 18, 10): 72
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-4}+g_{-10}\)
Positive simple generators: \(\displaystyle 9g_{10}+g_{4}+8g_{2}+5g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/18\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}72\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus 2V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}}\)
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.\(g_{10}+8/9g_{2}+5/9g_{1}\)\(g_{4}\)\(g_{9}+5g_{8}\)\(-g_{14}+5/8g_{13}\)\(g_{11}\)\(g_{16}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(6\omega_{1}\)\(10\omega_{1}\)
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)
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.
\(-9/5g_{10}-1/5g_{4}-8/5g_{2}-g_{1}\)
\(2h_{4}+18/5h_{3}+16/5h_{2}+2h_{1}\)
\(2/5g_{-1}+2/5g_{-2}+2/5g_{-4}+2/5g_{-10}\)
\(g_{10}+8/9g_{2}+5/9g_{1}\)
\(-h_{4}-2h_{3}-16/9h_{2}-10/9h_{1}\)
\(-2/9g_{-1}-2/9g_{-2}-2/9g_{-10}\)
\(g_{9}+5g_{8}\)
\(g_{7}+4g_{6}\)
\(3g_{3}-g_{-3}\)
\(g_{-6}+2g_{-7}\)
\(-g_{-8}-g_{-9}\)
\(-g_{14}+5/8g_{13}\)
\(-3/8g_{12}-5/8g_{5}\)
\(-3/4g_{10}-1/4g_{2}+5/8g_{1}\)
\(3/4h_{4}+3/2h_{3}+1/2h_{2}-5/4h_{1}\)
\(-3/2g_{-1}+3/8g_{-2}+g_{-10}\)
\(-15/8g_{-5}-5/8g_{-12}\)
\(-5/4g_{-13}+5/4g_{-14}\)
\(g_{11}\)
\(g_{9}-g_{8}\)
\(g_{7}-2g_{6}\)
\(-3g_{3}-g_{-3}\)
\(g_{-6}-4g_{-7}\)
\(-g_{-8}+5g_{-9}\)
\(-6g_{-11}\)
\(g_{16}\)
\(g_{15}\)
\(2g_{14}+g_{13}\)
\(3g_{12}-g_{5}\)
\(6g_{10}-4g_{2}+g_{1}\)
\(-6h_{4}-12h_{3}+8h_{2}-2h_{1}\)
\(-6g_{-1}+15g_{-2}-20g_{-10}\)
\(-21g_{-5}+35g_{-12}\)
\(-56g_{-13}-70g_{-14}\)
\(126g_{-15}\)
\(-252g_{-16}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 1620703 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_{-10}, x_{12} g_{10}+x_{11} g_{7}+x_{10} g_{6}+x_{9} g_{4}+x_{8} g_{2}+x_{7} g_{1})
h: (10, 16, 18, 10), e = combination of g_{1} g_{2} g_{4} g_{6} g_{7} g_{10} , f= combination of g_{-1} g_{-2} g_{-4} g_{-6} g_{-7} g_{-10} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: V_{10\omega_{1}}+V_{8\omega_{1}}+3V_{6\omega_{1}}+4V_{4\omega_{1}}+6V_{2\omega_{1}}+6V_{0}+6V_{-2\omega_{1}}+4V_{-4\omega_{1}}+3V_{-6\omega_{1}}+V_{-8\omega_{1}}+V_{-10\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{7} -5= 0
x_{4} x_{10} +x_{2} x_{8} -8= 0
x_{6} x_{11} +x_{5} x_{9} -x_{4} x_{8} = 0
x_{6} x_{12} +2x_{5} x_{11} +x_{3} x_{9} -10= 0
x_{5} x_{12} +x_{3} x_{11} -x_{2} x_{10} = 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -9= 0
The above system after transformation.
x_{1} x_{7} -5= 0
x_{4} x_{10} +x_{2} x_{8} -8= 0
x_{6} x_{11} +x_{5} x_{9} -x_{4} x_{8} = 0
x_{6} x_{12} +2x_{5} x_{11} +x_{3} x_{9} -10= 0
x_{5} x_{12} +x_{3} x_{11} -x_{2} x_{10} = 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -9= 0
For the calculator:
(DynkinType =A^{36}_1; ElementsCartan =((10, 16, 18, 10)); generators =(x_{1} g_{-1}+x_{2} g_{-2}+x_{3} g_{-4}+x_{4} g_{-6}+x_{5} g_{-7}+x_{6} g_{-10}, x_{12} g_{10}+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} -5, x_{4} x_{10} +x_{2} x_{8} -8, x_{6} x_{11} +x_{5} x_{9} -x_{4} x_{8} , x_{6} x_{12} +2x_{5} x_{11} +x_{3} x_{9} -10, x_{5} x_{12} +x_{3} x_{11} -x_{2} x_{10} , x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -9 )