Subalgebra \(2A^{6}_1\) ↪ \(B^{1}_4\)
21 out of 48
Computations done by the calculator project.

Subalgebra type: \(\displaystyle 2A^{6}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{6}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_4\)

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

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 3425923 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_{-8}+x_{2} g_{-9}+x_{3} g_{-10}, x_{9} g_{10}+x_{8} g_{9}+x_{7} g_{8},
x_{4} g_{-1}+x_{5} g_{-2}+x_{6} g_{-4}, x_{12} g_{4}+x_{11} g_{2}+x_{10} g_{1})
h: (2, 4, 6, 6), e = combination of g_{8} g_{9} g_{10} , f= combination of g_{-8} g_{-9} g_{-10} h: (2, 2, 0, 2), e = combination of g_{1} g_{2} g_{4} , f= combination of g_{-1} g_{-2} g_{-4} 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_{2\omega_{1}+4\omega_{2}}+V_{4\omega_{1}+2\omega_{2}}+V_{4\omega_{2}}+2V_{2\omega_{1}+2\omega_{2}}+V_{4\omega_{1}}+V_{-2\omega_{1}+4\omega_{2}}+3V_{2\omega_{2}}+3V_{2\omega_{1}}+V_{4\omega_{1}-2\omega_{2}}+2V_{-2\omega_{1}+2\omega_{2}}+4V_{0}+2V_{2\omega_{1}-2\omega_{2}}+V_{-4\omega_{1}+2\omega_{2}}+3V_{-2\omega_{1}}+3V_{-2\omega_{2}}+V_{2\omega_{1}-4\omega_{2}}+V_{-4\omega_{1}}+2V_{-2\omega_{1}-2\omega_{2}}+V_{-4\omega_{2}}+V_{-4\omega_{1}-2\omega_{2}}+V_{-2\omega_{1}-4\omega_{2}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{7} -2= 0
2x_{2} x_{8} +x_{1} x_{7} -4= 0
x_{3} x_{9} +2x_{2} x_{8} +x_{1} x_{7} -6= 0
x_{3} x_{9} +x_{2} x_{8} -3= 0
2x_{2} x_{12} -x_{1} x_{10} = 0
x_{3} x_{12} +x_{2} x_{11} = 0
x_{7} x_{12} -x_{8} x_{10} = 0
2x_{8} x_{12} +x_{9} x_{11} = 0
x_{1} x_{6} -x_{2} x_{4} = 0
2x_{2} x_{6} +x_{3} x_{5} = 0
x_{4} x_{10} -2= 0
x_{5} x_{11} -2= 0
x_{6} x_{12} -1= 0
2x_{6} x_{8} -x_{4} x_{7} = 0
x_{6} x_{9} +x_{5} x_{8} = 0
The above system after transformation.
x_{1} x_{7} -2= 0
2x_{2} x_{8} +x_{1} x_{7} -4= 0
x_{3} x_{9} +2x_{2} x_{8} +x_{1} x_{7} -6= 0
x_{3} x_{9} +x_{2} x_{8} -3= 0
2x_{2} x_{12} -x_{1} x_{10} = 0
x_{3} x_{12} +x_{2} x_{11} = 0
x_{7} x_{12} -x_{8} x_{10} = 0
2x_{8} x_{12} +x_{9} x_{11} = 0
x_{1} x_{6} -x_{2} x_{4} = 0
2x_{2} x_{6} +x_{3} x_{5} = 0
x_{4} x_{10} -2= 0
x_{5} x_{11} -2= 0
x_{6} x_{12} -1= 0
2x_{6} x_{8} -x_{4} x_{7} = 0
x_{6} x_{9} +x_{5} x_{8} = 0
For the calculator:
(DynkinType =2A^{6}_1; ElementsCartan =((2, 4, 6, 6), (2, 2, 0, 2)); generators =(x_{1} g_{-8}+x_{2} g_{-9}+x_{3} g_{-10}, x_{9} g_{10}+x_{8} g_{9}+x_{7} g_{8}, x_{4} g_{-1}+x_{5} g_{-2}+x_{6} g_{-4}, x_{12} g_{4}+x_{11} g_{2}+x_{10} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{1} x_{7} -2, 2x_{2} x_{8} +x_{1} x_{7} -4, x_{3} x_{9} +2x_{2} x_{8} +x_{1} x_{7} -6, x_{3} x_{9} +x_{2} x_{8} -3, 2x_{2} x_{12} -x_{1} x_{10} , x_{3} x_{12} +x_{2} x_{11} , x_{7} x_{12} -x_{8} x_{10} , 2x_{8} x_{12} +x_{9} x_{11} , x_{1} x_{6} -x_{2} x_{4} , 2x_{2} x_{6} +x_{3} x_{5} , x_{4} x_{10} -2, x_{5} x_{11} -2, x_{6} x_{12} -1, 2x_{6} x_{8} -x_{4} x_{7} , x_{6} x_{9} +x_{5} x_{8} )