Subalgebra \(A^{85}_1\) ↪ \(C^{1}_5\)
22 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{85}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{85}_1\): (14, 24, 30, 32, 17): 170
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-5}+g_{-13}\)
Positive simple generators: \(\displaystyle 16g_{13}+g_{5}+15g_{3}+12g_{2}+7g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/85\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}170\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus 2V_{6\omega_{1}}\oplus 2V_{2\omega_{1}}\)
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 7) ; the vectors are over the primal subalgebra.\(g_{13}+15/16g_{3}+3/4g_{2}+7/16g_{1}\)\(g_{5}\)\(g_{15}+7g_{14}\)\(-g_{19}+4/5g_{18}+7/20g_{10}\)\(g_{17}\)\(-g_{23}+7/12g_{22}\)\(g_{25}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(6\omega_{1}\)\(6\omega_{1}\)\(8\omega_{1}\)\(10\omega_{1}\)\(14\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 6 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)\(\displaystyle V_{6\omega_{1}} \) → (6)\(\displaystyle V_{8\omega_{1}} \) → (8)\(\displaystyle V_{10\omega_{1}} \) → (10)\(\displaystyle V_{14\omega_{1}} \) → (14)
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.
\(-16/7g_{13}-1/7g_{5}-15/7g_{3}-12/7g_{2}-g_{1}\)
\(17/7h_{5}+32/7h_{4}+30/7h_{3}+24/7h_{2}+2h_{1}\)
\(2/7g_{-1}+2/7g_{-2}+2/7g_{-3}+2/7g_{-5}+2/7g_{-13}\)
\(g_{13}+15/16g_{3}+3/4g_{2}+7/16g_{1}\)
\(-h_{5}-2h_{4}-15/8h_{3}-3/2h_{2}-7/8h_{1}\)
\(-1/8g_{-1}-1/8g_{-2}-1/8g_{-3}-1/8g_{-13}\)
\(g_{15}+7g_{14}\)
\(g_{12}+6g_{11}\)
\(g_{9}+5g_{8}\)
\(4g_{4}-g_{-4}\)
\(g_{-8}+3g_{-9}\)
\(-g_{-11}-2g_{-12}\)
\(g_{-14}+g_{-15}\)
\(-g_{19}+4/5g_{18}+7/20g_{10}\)
\(-1/5g_{16}-9/20g_{7}-7/20g_{6}\)
\(-2/5g_{13}-1/4g_{3}+1/10g_{2}+7/20g_{1}\)
\(2/5h_{5}+4/5h_{4}+1/2h_{3}-1/5h_{2}-7/10h_{1}\)
\(-3/5g_{-1}-1/10g_{-2}+1/5g_{-3}+3/10g_{-13}\)
\(-1/2g_{-6}-3/10g_{-7}-1/10g_{-16}\)
\(-1/5g_{-10}-1/5g_{-18}+1/5g_{-19}\)
\(g_{17}\)
\(g_{15}-g_{14}\)
\(g_{12}-2g_{11}\)
\(g_{9}-3g_{8}\)
\(-4g_{4}-g_{-4}\)
\(g_{-8}-5g_{-9}\)
\(-g_{-11}+6g_{-12}\)
\(g_{-14}-7g_{-15}\)
\(8g_{-17}\)
\(-g_{23}+7/12g_{22}\)
\(-5/12g_{21}+7/12g_{20}\)
\(-5/6g_{19}+1/6g_{18}-7/12g_{10}\)
\(-2/3g_{16}-3/4g_{7}+7/12g_{6}\)
\(-4/3g_{13}-1/12g_{3}+4/3g_{2}-7/12g_{1}\)
\(4/3h_{5}+8/3h_{4}+1/6h_{3}-8/3h_{2}+7/6h_{1}\)
\(5/2g_{-1}-10/3g_{-2}+1/6g_{-3}+5/2g_{-13}\)
\(35/6g_{-6}-7/2g_{-7}-7/3g_{-16}\)
\(28/3g_{-10}-7/6g_{-18}+14/3g_{-19}\)
\(21/2g_{-20}-7/2g_{-21}\)
\(-7g_{-22}+7g_{-23}\)
\(g_{25}\)
\(g_{24}\)
\(2g_{23}+g_{22}\)
\(3g_{21}+g_{20}\)
\(6g_{19}+4g_{18}-g_{10}\)
\(10g_{16}-5g_{7}+g_{6}\)
\(20g_{13}-15g_{3}+6g_{2}-g_{1}\)
\(-20h_{5}-40h_{4}+30h_{3}-12h_{2}+2h_{1}\)
\(8g_{-1}-28g_{-2}+56g_{-3}-70g_{-13}\)
\(36g_{-6}-84g_{-7}+126g_{-16}\)
\(120g_{-10}-210g_{-18}-252g_{-19}\)
\(330g_{-20}+462g_{-21}\)
\(-792g_{-22}-924g_{-23}\)
\(1716g_{-24}\)
\(-3432g_{-25}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(14\omega_{1}\)
\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
\(-14\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{14\omega_{1}}\oplus M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}
\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}
\oplus M_{-14\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 2630405 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_{-3}+x_{4} g_{-5}+x_{5} g_{-8}+x_{6} g_{-9}+x_{7} g_{-13}, x_{14} g_{13}+x_{13} g_{9}+x_{12} g_{8}+x_{11} g_{5}+x_{10} g_{3}+x_{9} g_{2}+x_{8} g_{1})
h: (14, 24, 30, 32, 17), e = combination of g_{1} g_{2} g_{3} g_{5} g_{8} g_{9} g_{13} , f= combination of g_{-1} g_{-2} g_{-3} g_{-5} g_{-8} g_{-9} g_{-13} 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}}+2V_{10\omega_{1}}+3V_{8\omega_{1}}+5V_{6\omega_{1}}+5V_{4\omega_{1}}+7V_{2\omega_{1}}+7V_{0}+7V_{-2\omega_{1}}+5V_{-4\omega_{1}}+5V_{-6\omega_{1}}+3V_{-8\omega_{1}}+2V_{-10\omega_{1}}+V_{-12\omega_{1}}+V_{-14\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{8} -7= 0
x_{2} x_{9} -12= 0
x_{5} x_{12} +x_{3} x_{10} -15= 0
x_{7} x_{13} +x_{6} x_{11} -x_{5} x_{10} = 0
x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} -17= 0
x_{6} x_{14} +x_{4} x_{13} -x_{3} x_{12} = 0
x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} -16= 0
The above system after transformation.
x_{1} x_{8} -7= 0
x_{2} x_{9} -12= 0
x_{5} x_{12} +x_{3} x_{10} -15= 0
x_{7} x_{13} +x_{6} x_{11} -x_{5} x_{10} = 0
x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} -17= 0
x_{6} x_{14} +x_{4} x_{13} -x_{3} x_{12} = 0
x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} -16= 0
For the calculator:
(DynkinType =A^{85}_1; ElementsCartan =((14, 24, 30, 32, 17)); generators =(x_{1} g_{-1}+x_{2} g_{-2}+x_{3} g_{-3}+x_{4} g_{-5}+x_{5} g_{-8}+x_{6} g_{-9}+x_{7} g_{-13}, x_{14} g_{13}+x_{13} g_{9}+x_{12} g_{8}+x_{11} g_{5}+x_{10} g_{3}+x_{9} g_{2}+x_{8} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{1} x_{8} -7, x_{2} x_{9} -12, x_{5} x_{12} +x_{3} x_{10} -15, x_{7} x_{13} +x_{6} x_{11} -x_{5} x_{10} , x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} -17, x_{6} x_{14} +x_{4} x_{13} -x_{3} x_{12} , x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} -16 )