Subalgebra \(A^{36}_1\) ↪ \(E^{1}_6\)
17 out of 119
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: \(\displaystyle E^{1}_6\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{36}_1\): (8, 10, 14, 20, 14, 8): 72
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-6}+g_{-8}+g_{-9}+g_{-10}+g_{-19}\)
Positive simple generators: \(\displaystyle 9g_{19}+5g_{10}+5g_{9}+g_{8}+8g_{6}+8g_{1}\)
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: \(\displaystyle 2V_{10\omega_{1}}\oplus 2V_{8\omega_{1}}\oplus 2V_{6\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\)
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 12) ; the vectors are over the primal subalgebra.\(-g_{15}-1/5g_{14}-1/5g_{13}-8/5g_{11}-8/5g_{7}+1/3g_{4}\)\(g_{19}+5/9g_{10}+5/9g_{9}+8/9g_{6}+8/9g_{1}\)\(g_{8}\)\(-g_{25}-g_{22}+5/9g_{16}+5/9g_{12}\)\(-g_{21}+1/5g_{20}-g_{18}+1/5g_{17}\)\(g_{23}-8/5g_{20}+8/5g_{17}\)\(g_{28}+g_{26}\)\(g_{31}+g_{29}+8/5g_{27}\)\(g_{34}+1/9g_{30}\)\(g_{33}+g_{32}\)\(g_{35}\)\(g_{36}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(4\omega_{1}\)\(4\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(6\omega_{1}\)\(8\omega_{1}\)\(8\omega_{1}\)\(10\omega_{1}\)\(10\omega_{1}\)
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.
\(-9/8g_{19}-5/8g_{10}-5/8g_{9}-1/8g_{8}-g_{6}-g_{1}\)
\(h_{6}+7/4h_{5}+5/2h_{4}+7/4h_{3}+5/4h_{2}+h_{1}\)
\(1/4g_{-1}+1/4g_{-6}+1/4g_{-8}+1/4g_{-9}+1/4g_{-10}+1/4g_{-19}\)
\(g_{19}+5/9g_{10}+5/9g_{9}+8/9g_{6}+8/9g_{1}\)
\(-8/9h_{6}-14/9h_{5}-19/9h_{4}-14/9h_{3}-h_{2}-8/9h_{1}\)
\(-2/9g_{-1}-2/9g_{-6}-2/9g_{-9}-2/9g_{-10}-2/9g_{-19}\)
\(-g_{15}-1/5g_{14}-1/5g_{13}-8/5g_{11}-8/5g_{7}+1/3g_{4}\)
\(2/5g_{5}-2/5g_{3}+2/5g_{2}+2/3g_{-2}-2/15g_{-3}+2/15g_{-5}\)
\(-2/5g_{-4}+2/15g_{-7}+2/15g_{-11}+2/15g_{-13}+2/15g_{-14}+2/15g_{-15}\)
\(-g_{25}-g_{22}+5/9g_{16}+5/9g_{12}\)
\(-5/9g_{10}+5/9g_{9}-4/9g_{6}+4/9g_{1}\)
\(4/9h_{6}+5/9h_{5}-5/9h_{3}-4/9h_{1}\)
\(-1/3g_{-1}+1/3g_{-6}-2/3g_{-9}+2/3g_{-10}\)
\(1/3g_{-12}+1/3g_{-16}-1/3g_{-22}-1/3g_{-25}\)
\(-g_{21}+1/5g_{20}-g_{18}+1/5g_{17}\)
\(-1/5g_{14}+1/5g_{13}-4/5g_{11}+4/5g_{7}\)
\(3/5g_{5}+3/5g_{3}+1/5g_{-3}+1/5g_{-5}\)
\(-1/5g_{-7}+1/5g_{-11}-2/5g_{-13}+2/5g_{-14}\)
\(1/5g_{-17}-1/5g_{-18}+1/5g_{-20}-1/5g_{-21}\)
\(g_{23}-8/5g_{20}+8/5g_{17}\)
\(-g_{15}+3/5g_{14}+3/5g_{13}-8/5g_{11}-8/5g_{7}-g_{4}\)
\(6/5g_{5}-6/5g_{3}-6/5g_{2}+2g_{-2}+2/5g_{-3}-2/5g_{-5}\)
\(-18/5g_{-4}-2/5g_{-7}-2/5g_{-11}+6/5g_{-13}+6/5g_{-14}-2/5g_{-15}\)
\(-8/5g_{-17}+8/5g_{-20}-8/5g_{-23}\)
\(g_{28}+g_{26}\)
\(-g_{21}-g_{20}-g_{18}-g_{17}\)
\(g_{14}-g_{13}-2g_{11}+2g_{7}\)
\(3g_{5}+3g_{3}-g_{-3}-g_{-5}\)
\(g_{-7}-g_{-11}-4g_{-13}+4g_{-14}\)
\(5g_{-17}+g_{-18}+5g_{-20}+g_{-21}\)
\(6g_{-26}+6g_{-28}\)
\(g_{31}+g_{29}+8/5g_{27}\)
\(3/5g_{25}-3/5g_{22}-g_{16}+g_{12}\)
\(-6/5g_{19}+g_{10}+g_{9}-2/5g_{6}-2/5g_{1}\)
\(2/5h_{6}+1/5h_{5}-4/5h_{4}+1/5h_{3}+6/5h_{2}+2/5h_{1}\)
\(3/5g_{-1}+3/5g_{-6}-12/5g_{-9}-12/5g_{-10}+8/5g_{-19}\)
\(3g_{-12}-3g_{-16}-g_{-22}+g_{-25}\)
\(-2g_{-27}-2g_{-29}-2g_{-31}\)
\(g_{34}+1/9g_{30}\)
\(1/9g_{28}-1/9g_{26}+8/9g_{24}\)
\(-2/9g_{23}+7/9g_{21}-1/9g_{20}-7/9g_{18}+1/9g_{17}\)
\(-4/3g_{15}+1/3g_{14}+1/3g_{13}+2/3g_{11}+2/3g_{7}+2/9g_{4}\)
\(-5/3g_{5}+5/3g_{3}-2/3g_{2}+10/9g_{-2}-5/9g_{-3}+5/9g_{-5}\)
\(8/3g_{-4}+5/9g_{-7}+5/9g_{-11}+20/9g_{-13}+20/9g_{-14}-16/9g_{-15}\)
\(-5/3g_{-17}+7/3g_{-18}+5/3g_{-20}-7/3g_{-21}+16/3g_{-23}\)
\(14/3g_{-24}-14/3g_{-26}+14/3g_{-28}\)
\(-14/3g_{-30}-14/3g_{-34}\)
\(g_{33}+g_{32}\)
\(g_{31}-g_{29}\)
\(-g_{25}-g_{22}-g_{16}-g_{12}\)
\(g_{10}-g_{9}-2g_{6}+2g_{1}\)
\(2h_{6}-h_{5}+h_{3}-2h_{1}\)
\(-5g_{-1}+5g_{-6}+4g_{-9}-4g_{-10}\)
\(-9g_{-12}-9g_{-16}-5g_{-22}-5g_{-25}\)
\(14g_{-29}-14g_{-31}\)
\(14g_{-32}+14g_{-33}\)
\(g_{35}\)
\(-g_{33}+g_{32}\)
\(-g_{31}-g_{29}+2g_{27}\)
\(3g_{25}-3g_{22}+g_{16}-g_{12}\)
\(-6g_{19}-g_{10}-g_{9}+4g_{6}+4g_{1}\)
\(-4h_{6}+7h_{5}+8h_{4}+7h_{3}+6h_{2}-4h_{1}\)
\(-15g_{-1}-15g_{-6}+6g_{-9}+6g_{-10}+20g_{-19}\)
\(-21g_{-12}+21g_{-16}-35g_{-22}+35g_{-25}\)
\(-70g_{-27}+56g_{-29}+56g_{-31}\)
\(126g_{-32}-126g_{-33}\)
\(-252g_{-35}\)
\(g_{36}\)
\(g_{34}-g_{30}\)
\(-g_{28}+g_{26}+2g_{24}\)
\(2g_{23}+3g_{21}+g_{20}-3g_{18}-g_{17}\)
\(-8g_{15}-3g_{14}-3g_{13}+4g_{11}+4g_{7}-2g_{4}\)
\(-15g_{5}+15g_{3}+6g_{2}+10g_{-2}+5g_{-3}-5g_{-5}\)
\(36g_{-4}-5g_{-7}-5g_{-11}+30g_{-13}+30g_{-14}+16g_{-15}\)
\(-35g_{-17}-21g_{-18}+35g_{-20}+21g_{-21}+112g_{-23}\)
\(-42g_{-24}-168g_{-26}+168g_{-28}\)
\(-378g_{-30}+42g_{-34}\)
\(-420g_{-36}\)
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\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\)\(\displaystyle 3M_{4\omega_{1}}\oplus 3M_{2\omega_{1}}\oplus 3M_{0}\oplus 3M_{-2\omega_{1}}\oplus 3M_{-4\omega_{1}}\)\(\displaystyle 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\)\(\displaystyle 2M_{8\omega_{1}}\oplus 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\oplus 2M_{-4\omega_{1}}
\oplus 2M_{-6\omega_{1}}\oplus 2M_{-8\omega_{1}}\)		\(\displaystyle 2M_{10\omega_{1}}\oplus 2M_{8\omega_{1}}\oplus 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}
\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\oplus 2M_{-8\omega_{1}}\oplus 2M_{-10\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 14218918 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_{-4}+x_{3} g_{-6}+x_{4} g_{-7}+x_{5} g_{-8}+x_{6} g_{-9}+x_{7} g_{-10}+x_{8} g_{-11} \\ +x_{9} g_{-13}+x_{10} g_{-14}+x_{11} g_{-15}+x_{12} g_{-19}, x_{24} g_{19}+x_{23} g_{15}+x_{22} g_{14}+x_{21} g_{13}+x_{20} g_{11}+x_{19} g_{10}+x_{18} g_{9}+x_{17} g_{8} \\ +x_{16} g_{7}+x_{15} g_{6}+x_{14} g_{4}+x_{13} g_{1})
h: (8, 10, 14, 20, 14, 8), e = combination of g_{1} g_{4} g_{6} g_{7} g_{8} g_{9} g_{10} g_{11} g_{13} g_{14} g_{15} g_{19} , f= combination of g_{-1} g_{-4} g_{-6} g_{-7} g_{-8} g_{-9} g_{-10} g_{-11} g_{-13} g_{-14} g_{-15} g_{-19} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: 2V_{10\omega_{1}}+4V_{8\omega_{1}}+6V_{6\omega_{1}}+9V_{4\omega_{1}}+12V_{2\omega_{1}}+12V_{0}+12V_{-2\omega_{1}}+9V_{-4\omega_{1}}+6V_{-6\omega_{1}}+4V_{-8\omega_{1}}+2V_{-10\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{4} x_{16} +x_{1} x_{13} -8= 0
x_{12} x_{22} +x_{11} x_{19} +x_{9} x_{17} +x_{6} x_{14} -x_{4} x_{13} = 0
x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{7} x_{19} +x_{6} x_{18} +x_{5} x_{17} +x_{2} x_{14}
-20= 0
x_{12} x_{23} +x_{10} x_{19} +x_{9} x_{18} +x_{5} x_{14} = 0
x_{12} x_{21} +x_{11} x_{18} +x_{10} x_{17} -x_{8} x_{15} +x_{7} x_{14} = 0
x_{8} x_{20} +x_{3} x_{15} -8= 0
x_{10} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{2} x_{18} -x_{1} x_{16} = 0
x_{12} x_{24} +x_{11} x_{23} +x_{9} x_{21} +x_{6} x_{18} +x_{4} x_{16} -14= 0
x_{11} x_{24} +x_{7} x_{22} +x_{6} x_{21} +x_{2} x_{17} = 0
x_{12} x_{24} +x_{10} x_{22} +x_{9} x_{21} +x_{5} x_{17} -10= 0
x_{9} x_{24} +x_{6} x_{23} +x_{5} x_{22} -x_{3} x_{20} +x_{2} x_{19} = 0
x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{8} x_{20} +x_{7} x_{19} -14= 0
The above system after transformation.
x_{4} x_{16} +x_{1} x_{13} -8= 0
x_{12} x_{22} +x_{11} x_{19} +x_{9} x_{17} +x_{6} x_{14} -x_{4} x_{13} = 0
x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{7} x_{19} +x_{6} x_{18} +x_{5} x_{17} +x_{2} x_{14}
-20= 0
x_{12} x_{23} +x_{10} x_{19} +x_{9} x_{18} +x_{5} x_{14} = 0
x_{12} x_{21} +x_{11} x_{18} +x_{10} x_{17} -x_{8} x_{15} +x_{7} x_{14} = 0
x_{8} x_{20} +x_{3} x_{15} -8= 0
x_{10} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{2} x_{18} -x_{1} x_{16} = 0
x_{12} x_{24} +x_{11} x_{23} +x_{9} x_{21} +x_{6} x_{18} +x_{4} x_{16} -14= 0
x_{11} x_{24} +x_{7} x_{22} +x_{6} x_{21} +x_{2} x_{17} = 0
x_{12} x_{24} +x_{10} x_{22} +x_{9} x_{21} +x_{5} x_{17} -10= 0
x_{9} x_{24} +x_{6} x_{23} +x_{5} x_{22} -x_{3} x_{20} +x_{2} x_{19} = 0
x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{8} x_{20} +x_{7} x_{19} -14= 0
For the calculator:
(DynkinType =A^{36}_1; ElementsCartan =((8, 10, 14, 20, 14, 8)); generators =(x_{1} g_{-1}+x_{2} g_{-4}+x_{3} g_{-6}+x_{4} g_{-7}+x_{5} g_{-8}+x_{6} g_{-9}+x_{7} g_{-10}+x_{8} g_{-11}+x_{9} g_{-13}+x_{10} g_{-14}+x_{11} g_{-15}+x_{12} g_{-19}, x_{24} g_{19}+x_{23} g_{15}+x_{22} g_{14}+x_{21} g_{13}+x_{20} g_{11}+x_{19} g_{10}+x_{18} g_{9}+x_{17} g_{8}+x_{16} g_{7}+x_{15} g_{6}+x_{14} g_{4}+x_{13} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{4} x_{16} +x_{1} x_{13} -8, x_{12} x_{22} +x_{11} x_{19} +x_{9} x_{17} +x_{6} x_{14} -x_{4} x_{13} , x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{9} x_{21} +x_{7} x_{19} +x_{6} x_{18} +x_{5} x_{17} +x_{2} x_{14} -20, x_{12} x_{23} +x_{10} x_{19} +x_{9} x_{18} +x_{5} x_{14} , x_{12} x_{21} +x_{11} x_{18} +x_{10} x_{17} -x_{8} x_{15} +x_{7} x_{14} , x_{8} x_{20} +x_{3} x_{15} -8, x_{10} x_{24} +x_{7} x_{23} +x_{5} x_{21} +x_{2} x_{18} -x_{1} x_{16} , x_{12} x_{24} +x_{11} x_{23} +x_{9} x_{21} +x_{6} x_{18} +x_{4} x_{16} -14, x_{11} x_{24} +x_{7} x_{22} +x_{6} x_{21} +x_{2} x_{17} , x_{12} x_{24} +x_{10} x_{22} +x_{9} x_{21} +x_{5} x_{17} -10, x_{9} x_{24} +x_{6} x_{23} +x_{5} x_{22} -x_{3} x_{20} +x_{2} x_{19} , x_{12} x_{24} +x_{11} x_{23} +x_{10} x_{22} +x_{8} x_{20} +x_{7} x_{19} -14 )