Subalgebra \(B^{1}_2+A^{11}_1\) ↪ \(C^{1}_5\)
86 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle B^{1}_2+A^{11}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) .
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: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{11}_1\): (0, 0, 6, 8, 5): 22
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-3}+g_{-5}+g_{-13}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle 4g_{13}+g_{5}+3g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2/11\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 22\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{3}}\oplus V_{4\omega_{3}}\oplus V_{\omega_{2}+3\omega_{3}}\oplus 3V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\)
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.\(g_{23}\)\(g_{15}\)\(g_{9}+3g_{8}\)\(g_{13}+3/4g_{3}\)\(g_{5}\)\(g_{21}\)\(g_{12}\)\(g_{19}\)
weight\(2\omega_{2}\)\(\omega_{2}+\omega_{3}\)\(2\omega_{3}\)\(2\omega_{3}\)\(2\omega_{3}\)\(\omega_{2}+3\omega_{3}\)\(4\omega_{3}\)\(6\omega_{3}\)
Isotypic module decomposition over primal subalgebra (total 7 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{2}} \) → (0, 2, 0)\(\displaystyle V_{\omega_{2}+\omega_{3}} \) → (0, 1, 1)\(\displaystyle V_{2\omega_{3}} \) → (0, 0, 2)\(\displaystyle V_{\omega_{2}+3\omega_{3}} \) → (0, 1, 3)\(\displaystyle V_{4\omega_{3}} \) → (0, 0, 4)\(\displaystyle V_{6\omega_{3}} \) → (0, 0, 6)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)\(W_{7}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{23}\)
\(-g_{24}\)
\(g_{-1}\)
\(-2g_{25}\)
\(2h_{1}\)
\(2h_{5}+4h_{4}+4h_{3}+4h_{2}+4h_{1}\)
\(2g_{-25}\)
\(-2g_{1}\)
\(-2g_{-24}\)
\(4g_{-23}\)
\(g_{15}\)
\(g_{17}\)
\(-g_{11}\)
\(-g_{-14}\)
\(-g_{14}\)
\(g_{-11}\)
\(-g_{-17}\)
\(g_{-15}\)
Semisimple subalgebra component.
\(-4/3g_{13}-1/3g_{5}-g_{3}\)
\(5/3h_{5}+8/3h_{4}+2h_{3}\)
\(2/3g_{-3}+2/3g_{-5}+2/3g_{-13}\)
\(g_{13}+3/4g_{3}\)
\(-h_{5}-2h_{4}-3/2h_{3}\)
\(-1/2g_{-3}-1/2g_{-13}\)
\(g_{9}+3g_{8}\)
\(2g_{4}-g_{-4}\)
\(g_{-8}+g_{-9}\)
\(g_{21}\)
\(g_{22}\)
\(g_{18}\)
\(-g_{-6}\)
\(g_{20}\)
\(-g_{7}\)
\(g_{-2}\)
\(-g_{-10}\)
\(-g_{10}\)
\(g_{2}\)
\(g_{-7}\)
\(-g_{-20}\)
\(g_{6}\)
\(g_{-18}\)
\(g_{-22}\)
\(-g_{-21}\)
\(g_{12}\)
\(g_{9}-g_{8}\)
\(-2g_{4}-g_{-4}\)
\(g_{-8}-3g_{-9}\)
\(4g_{-12}\)
\(g_{19}\)
\(g_{16}\)
\(2g_{13}-g_{3}\)
\(-2h_{5}-4h_{4}+2h_{3}\)
\(4g_{-3}-6g_{-13}\)
\(10g_{-16}\)
\(-20g_{-19}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-2\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}\)
\(-2\omega_{2}\)		\(\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{2}+3\omega_{3}\)
\(\omega_{1}-\omega_{2}+3\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}+3\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{2}+3\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(\omega_{2}-3\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(\omega_{1}-\omega_{2}-3\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-3\omega_{3}\)
\(-\omega_{2}-3\omega_{3}\)		\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)		\(6\omega_{3}\)
\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)
\(-6\omega_{3}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{2}\)
\(\omega_{1}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-2\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+2\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}\)
\(-2\omega_{2}\)		\(\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{2}+3\omega_{3}\)
\(\omega_{1}-\omega_{2}+3\omega_{3}\)
\(\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}+3\omega_{3}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{2}+3\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(\omega_{2}-3\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(\omega_{1}-\omega_{2}-3\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-3\omega_{3}\)
\(-\omega_{2}-3\omega_{3}\)		\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)		\(6\omega_{3}\)
\(4\omega_{3}\)
\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)
\(-4\omega_{3}\)
\(-6\omega_{3}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus 2M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-2\omega_{2}}\)		\(\displaystyle M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}
\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{2}+3\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+3\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+3\omega_{3}}
\oplus M_{-\omega_{2}+3\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-3\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-3\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-3\omega_{3}}
\oplus M_{-\omega_{2}-3\omega_{3}}\)		\(\displaystyle M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}}\)\(\displaystyle M_{6\omega_{3}}\oplus M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}}\oplus M_{-6\omega_{3}}\)
Isotypic character\(\displaystyle M_{2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{-2\omega_{1}+2\omega_{2}}\oplus 2M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-2\omega_{2}}\)		\(\displaystyle M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}
\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle 2M_{2\omega_{3}}\oplus 2M_{0}\oplus 2M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{2}+3\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+3\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+3\omega_{3}}
\oplus M_{-\omega_{2}+3\omega_{3}}\oplus M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}
\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-3\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}-3\omega_{3}}\oplus M_{\omega_{1}-\omega_{2}-3\omega_{3}}
\oplus M_{-\omega_{2}-3\omega_{3}}\)		\(\displaystyle M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}}\)\(\displaystyle M_{6\omega_{3}}\oplus M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}}\oplus M_{-6\omega_{3}}\)

Semisimple subalgebra: W_{1}+W_{3}
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 1149555 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.
3*2 (unknown) gens:
(
g_{-25}, g_{25},
g_{1}, g_{-1},
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})
h: (2, 2, 2, 2, 1), e = combination of g_{25} , f= combination of g_{-25} h: (-1, 0, 0, 0, 0), e = combination of g_{-1} , f= combination of g_{1} h: (0, 0, 6, 8, 5), e = combination of g_{3} g_{5} g_{8} g_{9} g_{13} , f= combination of g_{-3} g_{-5} g_{-8} g_{-9} g_{-13} Positive weight subsystem: 5 vectors: (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 2, 0)
Symmetric Cartan default scale: \begin{pmatrix}
2 & -1 & 0\\
-1 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}Character ambient Lie algebra: V_{6\omega_{3}}+2V_{4\omega_{3}}+V_{\omega_{2}+3\omega_{3}}+V_{-\omega_{1}+\omega_{2}+3\omega_{3}}+V_{\omega_{1}-\omega_{2}+3\omega_{3}}+V_{-\omega_{2}+3\omega_{3}}+5V_{2\omega_{3}}+2V_{\omega_{2}+\omega_{3}}+V_{2\omega_{2}}+2V_{-\omega_{1}+\omega_{2}+\omega_{3}}+2V_{\omega_{1}-\omega_{2}+\omega_{3}}+V_{-\omega_{1}+2\omega_{2}}+V_{\omega_{1}}+2V_{-\omega_{2}+\omega_{3}}+V_{-2\omega_{1}+2\omega_{2}}+7V_{0}+V_{2\omega_{1}-2\omega_{2}}+2V_{\omega_{2}-\omega_{3}}+V_{-\omega_{1}}+V_{\omega_{1}-2\omega_{2}}+2V_{-\omega_{1}+\omega_{2}-\omega_{3}}+2V_{\omega_{1}-\omega_{2}-\omega_{3}}+V_{-2\omega_{2}}+2V_{-\omega_{2}-\omega_{3}}+5V_{-2\omega_{3}}+V_{\omega_{2}-3\omega_{3}}+V_{-\omega_{1}+\omega_{2}-3\omega_{3}}+V_{\omega_{1}-\omega_{2}-3\omega_{3}}+V_{-\omega_{2}-3\omega_{3}}+2V_{-4\omega_{3}}+V_{-6\omega_{3}}
A necessary system to realize the candidate subalgebra.
x_{5} x_{12} +x_{3} x_{10} -3= 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} -5= 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} -4= 0
The above system after transformation.
x_{5} x_{12} +x_{3} x_{10} -3= 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} -5= 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} -4= 0
For the calculator:
(DynkinType =B^{1}_2+A^{11}_1; ElementsCartan =((2, 2, 2, 2, 1), (-1, 0, 0, 0, 0), (0, 0, 6, 8, 5)); generators =(g_{-25}, g_{25}, g_{1}, g_{-1}, 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}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{5} x_{12} +x_{3} x_{10} -3, x_{7} x_{13} +x_{6} x_{11} -x_{5} x_{10} , x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} -5, x_{6} x_{14} +x_{4} x_{13} -x_{3} x_{12} , x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} -4 )