Subalgebra \(A^{36}_1\) ↪ \(F^{1}_4\)
13 out of 59
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: \(\displaystyle F^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{36}_1\): (10, 20, 28, 16): 72
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-4}+g_{-5}+g_{-6}+g_{-11}\)
Positive simple generators: \(\displaystyle 9g_{11}+5g_{6}+g_{5}+8g_{4}\)
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: \(\displaystyle 2V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\)
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 8) ; the vectors are over the primal subalgebra.\(-g_{9}+1/5g_{8}+8/5g_{7}+1/3g_{2}\)\(g_{11}+5/9g_{6}+8/9g_{4}\)\(g_{5}\)\(g_{14}+8/5g_{12}\)\(g_{19}+8/5g_{18}\)\(g_{22}+1/9g_{20}\)\(g_{23}\)\(g_{24}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(8\omega_{1}\)\(10\omega_{1}\)\(10\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 6 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)\(\displaystyle V_{4\omega_{1}} \) → (4)\(\displaystyle V_{6\omega_{1}} \) → (6)\(\displaystyle V_{8\omega_{1}} \) → (8)\(\displaystyle V_{10\omega_{1}} \) → (10)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-9/8g_{11}-5/8g_{6}-1/8g_{5}-g_{4}\)
\(2h_{4}+7/2h_{3}+5/2h_{2}+5/4h_{1}\)
\(1/4g_{-4}+1/4g_{-5}+1/4g_{-6}+1/4g_{-11}\)
\(g_{11}+5/9g_{6}+8/9g_{4}\)
\(-16/9h_{4}-28/9h_{3}-19/9h_{2}-h_{1}\)
\(-2/9g_{-4}-2/9g_{-6}-2/9g_{-11}\)
\(-g_{9}+1/5g_{8}+8/5g_{7}+1/3g_{2}\)
\(-2/5g_{3}-2/5g_{1}+2/3g_{-1}+2/15g_{-3}\)
\(2/5g_{-2}+2/15g_{-7}+2/15g_{-8}-2/15g_{-9}\)
\(g_{14}+8/5g_{12}\)
\(-g_{9}-3/5g_{8}+8/5g_{7}-g_{2}\)
\(-6/5g_{3}+6/5g_{1}+2g_{-1}-2/5g_{-3}\)
\(18/5g_{-2}-2/5g_{-7}+6/5g_{-8}+2/5g_{-9}\)
\(8/5g_{-12}+8/5g_{-14}\)
\(g_{19}+8/5g_{18}\)
\(3/5g_{15}-g_{10}\)
\(-6/5g_{11}+g_{6}-2/5g_{4}\)
\(4/5h_{4}+2/5h_{3}-4/5h_{2}+6/5h_{1}\)
\(3/5g_{-4}-12/5g_{-6}+8/5g_{-11}\)
\(-3g_{-10}+g_{-15}\)
\(-2g_{-18}-2g_{-19}\)
\(g_{22}+1/9g_{20}\)
\(1/9g_{17}+8/9g_{16}\)
\(-2/9g_{14}+7/9g_{13}+1/9g_{12}\)
\(-4/3g_{9}-1/3g_{8}-2/3g_{7}+2/9g_{2}\)
\(5/3g_{3}+2/3g_{1}+10/9g_{-1}+5/9g_{-3}\)
\(-8/3g_{-2}+5/9g_{-7}+20/9g_{-8}+16/9g_{-9}\)
\(5/3g_{-12}+7/3g_{-13}-16/3g_{-14}\)
\(-14/3g_{-16}-14/3g_{-17}\)
\(14/3g_{-20}+14/3g_{-22}\)
\(g_{23}\)
\(g_{21}\)
\(g_{19}-2g_{18}\)
\(-3g_{15}-g_{10}\)
\(6g_{11}+g_{6}-4g_{4}\)
\(8h_{4}-14h_{3}-8h_{2}-6h_{1}\)
\(15g_{-4}-6g_{-6}-20g_{-11}\)
\(-21g_{-10}-35g_{-15}\)
\(70g_{-18}-56g_{-19}\)
\(126g_{-21}\)
\(-252g_{-23}\)
\(g_{24}\)
\(g_{22}-g_{20}\)
\(-g_{17}+2g_{16}\)
\(2g_{14}+3g_{13}-g_{12}\)
\(-8g_{9}+3g_{8}-4g_{7}-2g_{2}\)
\(15g_{3}-6g_{1}+10g_{-1}-5g_{-3}\)
\(-36g_{-2}-5g_{-7}+30g_{-8}-16g_{-9}\)
\(35g_{-12}-21g_{-13}-112g_{-14}\)
\(42g_{-16}-168g_{-17}\)
\(378g_{-20}-42g_{-22}\)
\(420g_{-24}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle 2M_{10\omega_{1}}\oplus 2M_{8\omega_{1}}\oplus 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}
\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\oplus 2M_{-8\omega_{1}}\oplus 2M_{-10\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 4345721 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_{-4}+x_{3} g_{-5}+x_{4} g_{-6}+x_{5} g_{-7}+x_{6} g_{-8}+x_{7} g_{-9}+x_{8} g_{-11}, x_{16} g_{11}+x_{15} g_{9}+x_{14} g_{8}+x_{13} g_{7}+x_{12} g_{6}+x_{11} g_{5}+x_{10} g_{4}+x_{9} g_{2})
h: (10, 20, 28, 16), e = combination of g_{2} g_{4} g_{5} g_{6} g_{7} g_{8} g_{9} g_{11} , f= combination of g_{-2} g_{-4} g_{-5} g_{-6} g_{-7} g_{-8} g_{-9} g_{-11} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: 2V_{10\omega_{1}}+3V_{8\omega_{1}}+4V_{6\omega_{1}}+5V_{4\omega_{1}}+8V_{2\omega_{1}}+8V_{0}+8V_{-2\omega_{1}}+5V_{-4\omega_{1}}+4V_{-6\omega_{1}}+3V_{-8\omega_{1}}+2V_{-10\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{8} x_{16} +x_{7} x_{15} +2x_{6} x_{14} +2x_{4} x_{12} +x_{3} x_{11} +x_{1} x_{9} -20= 0
x_{8} x_{15} +2x_{6} x_{12} +x_{3} x_{9} = 0
x_{8} x_{14} +x_{7} x_{12} -x_{6} x_{11} +x_{5} x_{10} -x_{4} x_{9} = 0
x_{5} x_{13} +x_{2} x_{10} -8= 0
x_{7} x_{16} +2x_{4} x_{14} +x_{1} x_{11} = 0
x_{8} x_{16} +2x_{6} x_{14} +x_{3} x_{11} -10= 0
x_{6} x_{16} +x_{4} x_{15} -x_{3} x_{14} +x_{2} x_{13} -x_{1} x_{12} = 0
x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} -14= 0
The above system after transformation.
x_{8} x_{16} +x_{7} x_{15} +2x_{6} x_{14} +2x_{4} x_{12} +x_{3} x_{11} +x_{1} x_{9} -20= 0
x_{8} x_{15} +2x_{6} x_{12} +x_{3} x_{9} = 0
x_{8} x_{14} +x_{7} x_{12} -x_{6} x_{11} +x_{5} x_{10} -x_{4} x_{9} = 0
x_{5} x_{13} +x_{2} x_{10} -8= 0
x_{7} x_{16} +2x_{4} x_{14} +x_{1} x_{11} = 0
x_{8} x_{16} +2x_{6} x_{14} +x_{3} x_{11} -10= 0
x_{6} x_{16} +x_{4} x_{15} -x_{3} x_{14} +x_{2} x_{13} -x_{1} x_{12} = 0
x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} -14= 0
For the calculator:
(DynkinType =A^{36}_1; ElementsCartan =((10, 20, 28, 16)); generators =(x_{1} g_{-2}+x_{2} g_{-4}+x_{3} g_{-5}+x_{4} g_{-6}+x_{5} g_{-7}+x_{6} g_{-8}+x_{7} g_{-9}+x_{8} g_{-11}, x_{16} g_{11}+x_{15} g_{9}+x_{14} g_{8}+x_{13} g_{7}+x_{12} g_{6}+x_{11} g_{5}+x_{10} g_{4}+x_{9} g_{2}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{8} x_{16} +x_{7} x_{15} +2x_{6} x_{14} +2x_{4} x_{12} +x_{3} x_{11} +x_{1} x_{9} -20, x_{8} x_{15} +2x_{6} x_{12} +x_{3} x_{9} , x_{8} x_{14} +x_{7} x_{12} -x_{6} x_{11} +x_{5} x_{10} -x_{4} x_{9} , x_{5} x_{13} +x_{2} x_{10} -8, x_{7} x_{16} +2x_{4} x_{14} +x_{1} x_{11} , x_{8} x_{16} +2x_{6} x_{14} +x_{3} x_{11} -10, x_{6} x_{16} +x_{4} x_{15} -x_{3} x_{14} +x_{2} x_{13} -x_{1} x_{12} , x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} +x_{4} x_{12} -14 )