Subalgebra \(A^{8}_1+A^{3}_1\) ↪ \(A^{1}_5\)
17 out of 37
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_5\)

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

Semisimple subalgebra: W_{1}+W_{2}
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 4996057 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_{-6}+x_{2} g_{-7}+x_{3} g_{-8}+x_{4} g_{-9}, x_{11} g_{9}+x_{10} g_{8}+x_{9} g_{7}+x_{8} g_{6},
x_{5} g_{-1}+x_{6} g_{-3}+x_{7} g_{-5}, x_{14} g_{5}+x_{13} g_{3}+x_{12} g_{1})
h: (2, 4, 4, 4, 2), e = combination of g_{6} g_{7} g_{8} g_{9} , f= combination of g_{-6} g_{-7} g_{-8} g_{-9} h: (1, 0, 1, 0, 1), e = combination of g_{1} g_{3} g_{5} , f= combination of g_{-1} g_{-3} g_{-5} 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}}+2V_{4\omega_{1}}+3V_{2\omega_{2}}+4V_{2\omega_{1}}+V_{4\omega_{1}-2\omega_{2}}+2V_{-2\omega_{1}+2\omega_{2}}+5V_{0}+2V_{2\omega_{1}-2\omega_{2}}+V_{-4\omega_{1}+2\omega_{2}}+4V_{-2\omega_{1}}+3V_{-2\omega_{2}}+2V_{-4\omega_{1}}+2V_{-2\omega_{1}-2\omega_{2}}+V_{-4\omega_{1}-2\omega_{2}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{8} -2= 0
x_{2} x_{9} +x_{1} x_{8} -4= 0
x_{3} x_{10} +x_{2} x_{9} -4= 0
x_{4} x_{11} +x_{3} x_{10} -4= 0
x_{4} x_{11} -2= 0
x_{2} x_{13} -x_{1} x_{12} = 0
x_{4} x_{14} -x_{3} x_{13} = 0
x_{8} x_{13} -x_{9} x_{12} = 0
x_{10} x_{14} -x_{11} x_{13} = 0
x_{1} x_{6} -x_{2} x_{5} = 0
x_{3} x_{7} -x_{4} x_{6} = 0
x_{5} x_{12} -1= 0
x_{6} x_{13} -1= 0
x_{7} x_{14} -1= 0
x_{6} x_{9} -x_{5} x_{8} = 0
x_{7} x_{11} -x_{6} x_{10} = 0
The above system after transformation.
x_{1} x_{8} -2= 0
x_{2} x_{9} +x_{1} x_{8} -4= 0
x_{3} x_{10} +x_{2} x_{9} -4= 0
x_{4} x_{11} +x_{3} x_{10} -4= 0
x_{4} x_{11} -2= 0
x_{2} x_{13} -x_{1} x_{12} = 0
x_{4} x_{14} -x_{3} x_{13} = 0
x_{8} x_{13} -x_{9} x_{12} = 0
x_{10} x_{14} -x_{11} x_{13} = 0
x_{1} x_{6} -x_{2} x_{5} = 0
x_{3} x_{7} -x_{4} x_{6} = 0
x_{5} x_{12} -1= 0
x_{6} x_{13} -1= 0
x_{7} x_{14} -1= 0
x_{6} x_{9} -x_{5} x_{8} = 0
x_{7} x_{11} -x_{6} x_{10} = 0
For the calculator:
(DynkinType =A^{8}_1+A^{3}_1; ElementsCartan =((2, 4, 4, 4, 2), (1, 0, 1, 0, 1)); generators =(x_{1} g_{-6}+x_{2} g_{-7}+x_{3} g_{-8}+x_{4} g_{-9}, x_{11} g_{9}+x_{10} g_{8}+x_{9} g_{7}+x_{8} g_{6}, x_{5} g_{-1}+x_{6} g_{-3}+x_{7} g_{-5}, x_{14} g_{5}+x_{13} g_{3}+x_{12} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{1} x_{8} -2, x_{2} x_{9} +x_{1} x_{8} -4, x_{3} x_{10} +x_{2} x_{9} -4, x_{4} x_{11} +x_{3} x_{10} -4, x_{4} x_{11} -2, x_{2} x_{13} -x_{1} x_{12} , x_{4} x_{14} -x_{3} x_{13} , x_{8} x_{13} -x_{9} x_{12} , x_{10} x_{14} -x_{11} x_{13} , x_{1} x_{6} -x_{2} x_{5} , x_{3} x_{7} -x_{4} x_{6} , x_{5} x_{12} -1, x_{6} x_{13} -1, x_{7} x_{14} -1, x_{6} x_{9} -x_{5} x_{8} , x_{7} x_{11} -x_{6} x_{10} )