Subalgebra \(A^{9}_1+A^{4}_1\) ↪ \(C^{1}_5\)
39 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{9}_1+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1\) .
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: \(\displaystyle A^{9}_1\): (4, 8, 10, 10, 5): 18, \(\displaystyle A^{4}_1\): (2, 0, 0, 2, 2): 8
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-14}+g_{-15}+g_{-19}\), \(\displaystyle g_{-1}+g_{-5}+g_{-13}\)
Positive simple generators: \(\displaystyle g_{19}+2g_{15}+2g_{14}\), \(\displaystyle g_{13}+g_{5}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & 0\\ 0 & 1/2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & 0\\ 0 & 8\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}
\oplus 3V_{2\omega_{1}}\)
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 10) ; the vectors are over the primal subalgebra.\(g_{18}+g_{10}\)\(g_{19}\)\(g_{15}+g_{14}\)\(g_{12}+2g_{6}\)\(g_{16}\)\(g_{5}+g_{1}\)\(g_{13}\)\(g_{22}\)\(g_{20}\)\(g_{25}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(\omega_{1}+\omega_{2}\)\(\omega_{1}+\omega_{2}\)\(2\omega_{2}\)\(2\omega_{2}\)\(3\omega_{1}+\omega_{2}\)\(2\omega_{1}+2\omega_{2}\)\(4\omega_{1}+2\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 8 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2, 0)\(\displaystyle V_{\omega_{1}+\omega_{2}} \) → (1, 1)\(\displaystyle V_{2\omega_{2}} \) → (0, 2)\(\displaystyle V_{3\omega_{1}+\omega_{2}} \) → (3, 1)\(\displaystyle V_{2\omega_{1}+2\omega_{2}} \) → (2, 2)\(\displaystyle V_{4\omega_{1}+2\omega_{2}} \) → (4, 2)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)\(W_{7}\)\(W_{8}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-1/2g_{19}-g_{15}-g_{14}\)
\(5/2h_{5}+5h_{4}+5h_{3}+4h_{2}+2h_{1}\)
\(g_{-14}+g_{-15}+g_{-19}\)
\(g_{19}\)
\(-h_{5}-2h_{4}-2h_{3}\)
\(-2g_{-19}\)
\(g_{18}+g_{10}\)
\(-g_{4}+g_{-4}\)
\(g_{-10}+g_{-18}\)
\(g_{12}+2g_{6}\)
\(-g_{-2}+g_{-8}\)
\(-g_{8}+2g_{2}\)
\(g_{-6}+g_{-12}\)
\(g_{16}\)
\(-g_{-3}\)
\(-g_{3}\)
\(-g_{-16}\)
Semisimple subalgebra component.
\(-g_{13}-g_{5}-g_{1}\)
\(2h_{5}+2h_{4}+2h_{1}\)
\(2g_{-1}+2g_{-5}+2g_{-13}\)
\(g_{5}+g_{1}\)
\(-h_{5}-2h_{1}\)
\(-2g_{-1}-2g_{-5}\)
\(g_{22}\)
\(g_{12}-g_{6}\)
\(g_{21}\)
\(-g_{-2}-2g_{-8}\)
\(-g_{8}-g_{2}\)
\(-3g_{-21}\)
\(g_{-6}-2g_{-12}\)
\(3g_{-22}\)
\(g_{20}\)
\(g_{9}\)
\(g_{18}-g_{10}\)
\(-g_{-7}\)
\(-g_{4}-g_{-4}\)
\(-2g_{7}\)
\(g_{-10}-g_{-18}\)
\(-2g_{-9}\)
\(2g_{-20}\)
\(g_{25}\)
\(g_{17}\)
\(g_{24}\)
\(2g_{5}-g_{1}\)
\(g_{15}-g_{14}\)
\(2g_{23}\)
\(-3g_{-11}\)
\(-2h_{5}+2h_{1}\)
\(-2g_{11}\)
\(-6g_{-23}\)
\(3g_{-14}-3g_{-15}\)
\(2g_{-1}-4g_{-5}\)
\(6g_{-24}\)
\(6g_{-17}\)
\(-12g_{-25}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(3\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)		\(2\omega_{1}+2\omega_{2}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-2\omega_{1}-2\omega_{2}\)		\(4\omega_{1}+2\omega_{2}\)
\(2\omega_{1}+2\omega_{2}\)
\(4\omega_{1}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(4\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}+2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-4\omega_{1}\)
\(-2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}-2\omega_{2}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(3\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)		\(2\omega_{1}+2\omega_{2}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-2\omega_{1}-2\omega_{2}\)		\(4\omega_{1}+2\omega_{2}\)
\(2\omega_{1}+2\omega_{2}\)
\(4\omega_{1}\)
\(2\omega_{2}\)
\(2\omega_{1}\)
\(4\omega_{1}-2\omega_{2}\)
\(-2\omega_{1}+2\omega_{2}\)
\(0\)
\(2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}+2\omega_{2}\)
\(-2\omega_{1}\)
\(-2\omega_{2}\)
\(-4\omega_{1}\)
\(-2\omega_{1}-2\omega_{2}\)
\(-4\omega_{1}-2\omega_{2}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{2\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}}\oplus M_{-2\omega_{2}}\oplus M_{-2\omega_{1}-2\omega_{2}}\)		\(\displaystyle M_{4\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}}
\oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-2\omega_{2}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\)\(\displaystyle 2M_{\omega_{1}+\omega_{2}}\oplus 2M_{-\omega_{1}+\omega_{2}}\oplus 2M_{\omega_{1}-\omega_{2}}\oplus 2M_{-\omega_{1}-\omega_{2}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{2\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}}\oplus M_{-2\omega_{2}}\oplus M_{-2\omega_{1}-2\omega_{2}}\)		\(\displaystyle M_{4\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}}
\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}}
\oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-2\omega_{2}}\)

Semisimple subalgebra: W_{1}+W_{4}
Centralizer extension: 0

Weight diagram. The coordinates corresponding to the simple roots of the subalgerba are fundamental.
The bilinear form is therefore given relative to the fundamental coordinates.
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Made total 6256102 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.
2*2 (unknown) gens:
(
x_{1} g_{-10}+x_{2} g_{-14}+x_{3} g_{-15}+x_{4} g_{-18}+x_{5} g_{-19}, x_{14} g_{19}+x_{13} g_{18}+x_{12} g_{15}+x_{11} g_{14}+x_{10} g_{10},
x_{6} g_{-1}+x_{7} g_{-5}+x_{8} g_{-9}+x_{9} g_{-13}, x_{18} g_{13}+x_{17} g_{9}+x_{16} g_{5}+x_{15} g_{1})
h: (4, 8, 10, 10, 5), e = combination of g_{10} g_{14} g_{15} g_{18} g_{19} , f= combination of g_{-10} g_{-14} g_{-15} g_{-18} g_{-19} h: (2, 0, 0, 2, 2), e = combination of g_{1} g_{5} g_{9} g_{13} , f= combination of g_{-1} g_{-5} g_{-9} g_{-13} Positive weight subsystem: 2 vectors: (1, 0), (0, 1)
Symmetric Cartan default scale: \begin{pmatrix}
2 & 0\\
0 & 2\\
\end{pmatrix}Character ambient Lie algebra: V_{4\omega_{1}+2\omega_{2}}+2V_{2\omega_{1}+2\omega_{2}}+V_{3\omega_{1}+\omega_{2}}+V_{4\omega_{1}}+4V_{2\omega_{2}}+3V_{\omega_{1}+\omega_{2}}+5V_{2\omega_{1}}+V_{3\omega_{1}-\omega_{2}}+V_{4\omega_{1}-2\omega_{2}}+2V_{-2\omega_{1}+2\omega_{2}}+3V_{-\omega_{1}+\omega_{2}}+7V_{0}+3V_{\omega_{1}-\omega_{2}}+2V_{2\omega_{1}-2\omega_{2}}+V_{-4\omega_{1}+2\omega_{2}}+V_{-3\omega_{1}+\omega_{2}}+5V_{-2\omega_{1}}+3V_{-\omega_{1}-\omega_{2}}+4V_{-2\omega_{2}}+V_{-4\omega_{1}}+V_{-3\omega_{1}-\omega_{2}}+2V_{-2\omega_{1}-2\omega_{2}}+V_{-4\omega_{1}-2\omega_{2}}
A necessary system to realize the candidate subalgebra.
x_{2} x_{11} +x_{1} x_{10} -2= 0
x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -4= 0
x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -5= 0
x_{4} x_{12} -x_{2} x_{10} = 0
x_{3} x_{13} -x_{1} x_{11} = 0
x_{5} x_{14} +2x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} -5= 0
x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} -5= 0
x_{4} x_{18} +x_{3} x_{17} -x_{1} x_{15} = 0
x_{4} x_{17} +x_{3} x_{16} -x_{2} x_{15} = 0
x_{10} x_{17} +x_{11} x_{16} -x_{12} x_{15} = 0
x_{10} x_{18} +x_{11} x_{17} -x_{13} x_{15} = 0
x_{1} x_{8} +x_{2} x_{7} -x_{3} x_{6} = 0
x_{1} x_{9} +x_{2} x_{8} -x_{4} x_{6} = 0
x_{6} x_{15} -1= 0
x_{9} x_{18} +2x_{8} x_{17} +x_{7} x_{16} -2= 0
x_{9} x_{17} +x_{8} x_{16} = 0
x_{8} x_{18} +x_{7} x_{17} = 0
x_{9} x_{18} +x_{8} x_{17} -1= 0
x_{9} x_{13} +x_{8} x_{12} -x_{6} x_{10} = 0
x_{8} x_{13} +x_{7} x_{12} -x_{6} x_{11} = 0
The above system after transformation.
x_{2} x_{11} +x_{1} x_{10} -2= 0
x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -4= 0
x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -5= 0
x_{4} x_{12} -x_{2} x_{10} = 0
x_{3} x_{13} -x_{1} x_{11} = 0
x_{5} x_{14} +2x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} -5= 0
x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} -5= 0
x_{4} x_{18} +x_{3} x_{17} -x_{1} x_{15} = 0
x_{4} x_{17} +x_{3} x_{16} -x_{2} x_{15} = 0
x_{10} x_{17} +x_{11} x_{16} -x_{12} x_{15} = 0
x_{10} x_{18} +x_{11} x_{17} -x_{13} x_{15} = 0
x_{1} x_{8} +x_{2} x_{7} -x_{3} x_{6} = 0
x_{1} x_{9} +x_{2} x_{8} -x_{4} x_{6} = 0
x_{6} x_{15} -1= 0
x_{9} x_{18} +2x_{8} x_{17} +x_{7} x_{16} -2= 0
x_{9} x_{17} +x_{8} x_{16} = 0
x_{8} x_{18} +x_{7} x_{17} = 0
x_{9} x_{18} +x_{8} x_{17} -1= 0
x_{9} x_{13} +x_{8} x_{12} -x_{6} x_{10} = 0
x_{8} x_{13} +x_{7} x_{12} -x_{6} x_{11} = 0
For the calculator:
(DynkinType =A^{9}_1+A^{4}_1; ElementsCartan =((4, 8, 10, 10, 5), (2, 0, 0, 2, 2)); generators =(x_{1} g_{-10}+x_{2} g_{-14}+x_{3} g_{-15}+x_{4} g_{-18}+x_{5} g_{-19}, x_{14} g_{19}+x_{13} g_{18}+x_{12} g_{15}+x_{11} g_{14}+x_{10} g_{10}, x_{6} g_{-1}+x_{7} g_{-5}+x_{8} g_{-9}+x_{9} g_{-13}, x_{18} g_{13}+x_{17} g_{9}+x_{16} g_{5}+x_{15} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{2} x_{11} +x_{1} x_{10} -2, x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -4, x_{5} x_{14} +x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} +x_{1} x_{10} -5, x_{4} x_{12} -x_{2} x_{10} , x_{3} x_{13} -x_{1} x_{11} , x_{5} x_{14} +2x_{4} x_{13} +x_{3} x_{12} +x_{2} x_{11} -5, x_{5} x_{14} +2x_{4} x_{13} +2x_{3} x_{12} -5, x_{4} x_{18} +x_{3} x_{17} -x_{1} x_{15} , x_{4} x_{17} +x_{3} x_{16} -x_{2} x_{15} , x_{10} x_{17} +x_{11} x_{16} -x_{12} x_{15} , x_{10} x_{18} +x_{11} x_{17} -x_{13} x_{15} , x_{1} x_{8} +x_{2} x_{7} -x_{3} x_{6} , x_{1} x_{9} +x_{2} x_{8} -x_{4} x_{6} , x_{6} x_{15} -1, x_{9} x_{18} +2x_{8} x_{17} +x_{7} x_{16} -2, x_{9} x_{17} +x_{8} x_{16} , x_{8} x_{18} +x_{7} x_{17} , x_{9} x_{18} +x_{8} x_{17} -1, x_{9} x_{13} +x_{8} x_{12} -x_{6} x_{10} , x_{8} x_{13} +x_{7} x_{12} -x_{6} x_{11} )