Subalgebra \(A^{45}_1\) ↪ \(C^{1}_5\)
20 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{45}_1\) (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: \(\displaystyle A^{45}_1\): (10, 16, 22, 24, 13): 90
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-3}+5/2g_{-7}+g_{-9}+g_{-11}\)
Positive simple generators: \(\displaystyle 25/4g_{13}+11/2g_{11}+13/2g_{9}-15/2g_{8}+g_{7}+g_{6}+g_{5}+3g_{3}+5g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/45\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}90\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus 3V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\)
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 9) ; the vectors are over the primal subalgebra.\(-g_{11}+15/11g_{8}+26/25g_{7}+56/55g_{6}+288/275g_{5}-78/55g_{3}+8/11g_{1}\)\(g_{13}+26/33g_{8}+4/25g_{7}+4/55g_{6}+52/275g_{5}-56/165g_{3}+8/33g_{1}\)\(g_{9}-25/33g_{8}+22/25g_{7}+52/55g_{6}+236/275g_{5}-68/165g_{3}+38/33g_{1}\)\(-g_{18}+15/11g_{16}+2/11g_{15}-10/11g_{14}+6/11g_{12}+4/11g_{10}\)\(-g_{21}+70/11g_{19}+2/11g_{17}\)\(-g_{23}+75/11g_{19}+10/11g_{17}\)\(-g_{20}+48/11g_{19}+2/11g_{17}\)\(-g_{24}+5g_{22}\)\(g_{25}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(6\omega_{1}\)\(6\omega_{1}\)\(8\omega_{1}\)\(10\omega_{1}\)
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.
\(-5/4g_{13}-11/10g_{11}-13/10g_{9}+3/2g_{8}-1/5g_{7}-1/5g_{6}-1/5g_{5}-3/5g_{3}-g_{1}\)
\(13/5h_{5}+24/5h_{4}+22/5h_{3}+16/5h_{2}+2h_{1}\)
\(2/5g_{-1}+2/5g_{-3}+g_{-7}+2/5g_{-9}+2/5g_{-11}\)
\(g_{13}+26/33g_{8}+4/25g_{7}+4/55g_{6}+52/275g_{5}-56/165g_{3}+8/33g_{1}\)
\(-7/33g_{4}-24/275g_{2}-4/33h_{3}-4/5h_{2}-16/33h_{1}+2/33g_{-2}-8/275g_{-4}\)
\(-14/165g_{-1}+2/33g_{-3}-14/33g_{-5}-2/33g_{-6}-3/11g_{-7}-16/275g_{-8}+2/33g_{-9}-14/165g_{-11}-16/275g_{-13}\)
\(-g_{11}+15/11g_{8}+26/25g_{7}+56/55g_{6}+288/275g_{5}-78/55g_{3}+8/11g_{1}\)
\(-25/22g_{4}-6/275g_{2}+2h_{4}-4/11h_{3}-16/5h_{2}-16/11h_{1}+24/11g_{-2}-2/275g_{-4}\)
\(8/55g_{-1}+2/11g_{-3}-25/11g_{-5}-24/11g_{-6}-20/11g_{-7}-4/275g_{-8}+2/11g_{-9}+8/55g_{-11}-4/275g_{-13}\)
\(-g_{18}+15/11g_{16}+2/11g_{15}-10/11g_{14}+6/11g_{12}+4/11g_{10}\)
\(-25/11g_{13}+1/11g_{11}-15/11g_{8}+2/11g_{7}-4/11g_{6}+4/11g_{5}-6/11g_{3}\)
\(25/22g_{4}-6/11g_{2}-2/11h_{4}-12/11h_{2}+30/11g_{-2}-2/11g_{-4}\)
\(6/11g_{-1}-8/11g_{-3}+25/11g_{-5}-30/11g_{-6}+5/11g_{-7}-4/11g_{-8}-2/11g_{-11}-4/11g_{-13}\)
\(-20/11g_{-10}-8/11g_{-12}+8/11g_{-14}-20/11g_{-15}+8/11g_{-18}\)
\(-g_{21}+70/11g_{19}+2/11g_{17}\)
\(-g_{18}+85/22g_{16}+2/11g_{15}-g_{12}-2/11g_{10}\)
\(30/11g_{13}+g_{11}-17/11g_{9}-85/22g_{8}-4/11g_{7}+2/11g_{6}+4/11g_{5}+g_{3}+5/11g_{1}\)
\(-45/11g_{4}+6/11g_{2}+34/11h_{5}+12/11h_{4}-24/11h_{3}-2/11h_{2}-10/11h_{1}+15/11g_{-2}-8/11g_{-4}\)
\(-9/11g_{-1}-14/11g_{-3}-90/11g_{-5}-15/11g_{-6}+25/11g_{-7}+14/11g_{-8}+46/11g_{-9}-4/11g_{-11}-16/11g_{-13}\)
\(-125/22g_{-10}-60/11g_{-12}-5/11g_{-14}+30/11g_{-16}-10/11g_{-18}\)
\(-125/22g_{-17}-60/11g_{-19}+5/11g_{-20}-5/11g_{-21}+50/11g_{-23}\)
\(-g_{23}+75/11g_{19}+10/11g_{17}\)
\(-5/2g_{18}+75/11g_{16}-1/11g_{15}-10/11g_{10}\)
\(25/22g_{13}+5/2g_{11}-30/11g_{9}-75/11g_{8}-9/11g_{7}+10/11g_{6}-2/11g_{5}+25/11g_{1}\)
\(-75/44g_{4}+19/11g_{2}+60/11h_{5}+5/11h_{4}-10/11h_{3}-10/11h_{2}-50/11h_{1}+75/11g_{-2}-7/11g_{-4}\)
\(-45/11g_{-1}+40/11g_{-3}-75/22g_{-5}-75/11g_{-6}+125/11g_{-7}+26/11g_{-8}+65/11g_{-9}-20/11g_{-11}-14/11g_{-13}\)
\(-625/22g_{-10}-25/11g_{-12}-25/11g_{-14}+40/11g_{-16}-50/11g_{-18}\)
\(-625/22g_{-17}-80/11g_{-19}+25/11g_{-20}-25/11g_{-21}+250/11g_{-23}\)
\(-g_{20}+48/11g_{19}+2/11g_{17}\)
\(-g_{18}+48/11g_{16}+2/11g_{15}+g_{14}-2/11g_{10}\)
\(41/11g_{13}+2g_{11}-6/11g_{9}-48/11g_{8}-4/11g_{7}+2/11g_{6}+4/11g_{5}-6/11g_{1}\)
\(-34/11g_{4}+6/11g_{2}+12/11h_{5}-32/11h_{4}-24/11h_{3}-24/11h_{2}+12/11h_{1}+48/11g_{-2}-8/11g_{-4}\)
\(24/11g_{-1}+19/11g_{-3}-68/11g_{-5}-48/11g_{-6}+47/11g_{-7}+14/11g_{-8}+24/11g_{-9}-26/11g_{-11}-16/11g_{-13}\)
\(-35/11g_{-10}-5/11g_{-12}+50/11g_{-14}+5g_{-15}+30/11g_{-16}-10/11g_{-18}\)
\(-90/11g_{-17}-60/11g_{-19}+60/11g_{-20}-60/11g_{-21}-60/11g_{-23}\)
\(-g_{24}+5g_{22}\)
\(-2g_{23}+5g_{21}+5/2g_{20}-g_{17}\)
\(5/2g_{18}+25/2g_{16}-3g_{15}-5/2g_{14}+5g_{12}+g_{10}\)
\(75/2g_{13}-5g_{11}-25/2g_{8}+4g_{7}-g_{6}-6g_{5}-5g_{3}\)
\(-125/2g_{4}-5g_{2}+10h_{4}-10h_{2}+25g_{-2}+10g_{-4}\)
\(5g_{-1}-25/2g_{-3}-125g_{-5}-25g_{-6}+125/2g_{-7}-15g_{-8}-25g_{-11}+20g_{-13}\)
\(-75g_{-10}-25/2g_{-12}+30g_{-14}+375/2g_{-15}-35g_{-16}-75g_{-18}\)
\(-525/2g_{-17}+70g_{-19}+105g_{-20}+175g_{-21}\)
\(-280g_{-22}\)
\(g_{25}\)
\(g_{24}\)
\(2g_{23}+5/2g_{20}+g_{17}\)
\(15/2g_{18}+3g_{15}-5/2g_{14}-g_{10}\)
\(75/2g_{13}-10g_{11}+15g_{9}-4g_{7}+g_{6}+6g_{5}+5g_{1}\)
\(-125/2g_{4}+5g_{2}-30h_{5}-10h_{4}+40h_{3}+40h_{2}-10h_{1}-10g_{-4}\)
\(-30g_{-1}+75/2g_{-3}-125g_{-5}+375/2g_{-7}+15g_{-8}-50g_{-9}+75g_{-11}-20g_{-13}\)
\(-525/2g_{-10}+175/2g_{-12}-105g_{-14}+875/2g_{-15}+35g_{-16}+175g_{-18}\)
\(-700g_{-17}-70g_{-19}-280g_{-20}-350g_{-21}-1750g_{-23}\)
\(630g_{-22}+3150g_{-24}\)
\(-6300g_{-25}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
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\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle 3M_{6\omega_{1}}\oplus 3M_{4\omega_{1}}\oplus 3M_{2\omega_{1}}\oplus 3M_{0}\oplus 3M_{-2\omega_{1}}\oplus 3M_{-4\omega_{1}}\oplus 3M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 5158931 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_{-3}+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_{9} g_{-13}, x_{18} g_{13}+x_{17} g_{11}+x_{16} g_{9}+x_{15} g_{8}+x_{14} g_{7}+x_{13} g_{6}+x_{12} g_{5}+x_{11} g_{3} \\ +x_{10} g_{1})
h: (10, 16, 22, 24, 13), e = combination of g_{1} g_{3} g_{5} g_{6} g_{7} g_{8} g_{9} g_{11} g_{13} , f= combination of g_{-1} g_{-3} g_{-5} g_{-6} g_{-7} g_{-8} g_{-9} g_{-11} g_{-13} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: V_{10\omega_{1}}+2V_{8\omega_{1}}+5V_{6\omega_{1}}+6V_{4\omega_{1}}+9V_{2\omega_{1}}+9V_{0}+9V_{-2\omega_{1}}+6V_{-4\omega_{1}}+5V_{-6\omega_{1}}+2V_{-8\omega_{1}}+V_{-10\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{4} x_{13} +x_{1} x_{10} -5= 0
x_{8} x_{15} +x_{5} x_{11} -x_{4} x_{10} = 0
x_{8} x_{17} +x_{6} x_{15} +x_{5} x_{14} +x_{2} x_{11} -11= 0
x_{9} x_{16} -x_{8} x_{14} +x_{7} x_{12} -x_{6} x_{11} = 0
x_{9} x_{18} +2x_{7} x_{16} +x_{3} x_{12} -13= 0
x_{6} x_{17} +x_{2} x_{14} -x_{1} x_{13} = 0
x_{8} x_{17} +x_{5} x_{14} +x_{4} x_{13} -8= 0
x_{7} x_{18} -x_{5} x_{17} +x_{3} x_{16} -x_{2} x_{15} = 0
x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} -12= 0
The above system after transformation.
x_{4} x_{13} +x_{1} x_{10} -5= 0
x_{8} x_{15} +x_{5} x_{11} -x_{4} x_{10} = 0
x_{8} x_{17} +x_{6} x_{15} +x_{5} x_{14} +x_{2} x_{11} -11= 0
x_{9} x_{16} -x_{8} x_{14} +x_{7} x_{12} -x_{6} x_{11} = 0
x_{9} x_{18} +2x_{7} x_{16} +x_{3} x_{12} -13= 0
x_{6} x_{17} +x_{2} x_{14} -x_{1} x_{13} = 0
x_{8} x_{17} +x_{5} x_{14} +x_{4} x_{13} -8= 0
x_{7} x_{18} -x_{5} x_{17} +x_{3} x_{16} -x_{2} x_{15} = 0
x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} -12= 0
For the calculator:
(DynkinType =A^{45}_1; ElementsCartan =((10, 16, 22, 24, 13)); generators =(x_{1} g_{-1}+x_{2} g_{-3}+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_{9} g_{-13}, x_{18} g_{13}+x_{17} g_{11}+x_{16} g_{9}+x_{15} g_{8}+x_{14} g_{7}+x_{13} g_{6}+x_{12} g_{5}+x_{11} g_{3}+x_{10} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{4} x_{13} +x_{1} x_{10} -5, x_{8} x_{15} +x_{5} x_{11} -x_{4} x_{10} , x_{8} x_{17} +x_{6} x_{15} +x_{5} x_{14} +x_{2} x_{11} -11, x_{9} x_{16} -x_{8} x_{14} +x_{7} x_{12} -x_{6} x_{11} , x_{9} x_{18} +2x_{7} x_{16} +x_{3} x_{12} -13, x_{6} x_{17} +x_{2} x_{14} -x_{1} x_{13} , x_{8} x_{17} +x_{5} x_{14} +x_{4} x_{13} -8, x_{7} x_{18} -x_{5} x_{17} +x_{3} x_{16} -x_{2} x_{15} , x_{9} x_{18} +x_{8} x_{17} +x_{7} x_{16} +x_{6} x_{15} -12 )