Subalgebra \(A^{3}_2\) ↪ \(B^{1}_4\)
30 out of 48
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{3}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) .
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: \(\displaystyle A^{3}_2\): (2, 3, 4, 4): 6, (-1, 0, -2, -2): 6
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \(\displaystyle g_{-8}+g_{-13}+g_{-14}\), \(\displaystyle g_{7}-g_{3}+g_{1}\)
Positive simple generators: \(\displaystyle g_{14}+g_{13}+g_{8}\), \(\displaystyle g_{-1}+g_{-7}-g_{-10}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & -1/3\\ -1/3 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & -3\\ -3 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{3\omega_{2}}\oplus V_{3\omega_{1}}\oplus 2V_{\omega_{1}+\omega_{2}}\)
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 4) ; the vectors are over the primal subalgebra.\(-g_{12}+2g_{6}+g_{5}\)\(g_{9}+2g_{6}+2g_{5}\)\(g_{16}\)\(g_{2}\)
weight\(\omega_{1}+\omega_{2}\)\(\omega_{1}+\omega_{2}\)\(3\omega_{1}\)\(3\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 4 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{1}+\omega_{2}} \) → (1, 1)\(\displaystyle V_{3\omega_{1}} \) → (3, 0)\(\displaystyle V_{3\omega_{2}} \) → (0, 3)
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.
\(-g_{12}-g_{9}-g_{5}\)
\(g_{-1}+g_{-7}-g_{-10}\)
\(-g_{14}-g_{13}-g_{8}\)
\(2h_{4}+2h_{3}+h_{1}\)
\(4h_{4}+4h_{3}+3h_{2}+2h_{1}\)
\(g_{-8}+g_{-13}+g_{-14}\)
\(-2g_{7}+2g_{3}-2g_{1}\)
\(-g_{-5}-g_{-6}-g_{-9}\)
\(-g_{12}+2g_{6}+g_{5}\)
\(-g_{-1}+2g_{-3}-g_{-10}\)
\(g_{14}-g_{13}-g_{11}+3g_{8}\)
\(-2g_{4}-2h_{3}-h_{1}-g_{-4}\)
\(-g_{4}-4h_{3}-3h_{2}-2h_{1}+g_{-4}\)
\(-3g_{-8}-g_{-11}+g_{-13}-g_{-14}\)
\(4g_{10}+4g_{7}-6g_{3}+2g_{1}\)
\(g_{-5}+3g_{-6}+2g_{-9}-2g_{-12}\)
\(g_{16}\)
\(-g_{12}-g_{6}+g_{5}\)
\(2g_{-1}+2g_{-3}+2g_{-10}\)
\(g_{14}-g_{13}-g_{11}\)
\(6g_{-15}\)
\(-2g_{4}-2h_{3}+2h_{1}+2g_{-4}\)
\(-6g_{-11}+6g_{-13}-6g_{-14}\)
\(4g_{10}+4g_{7}-4g_{1}\)
\(-12g_{-5}+12g_{-9}-12g_{-12}\)
\(36g_{-2}\)
\(g_{2}\)
\(-g_{9}+g_{6}+g_{5}\)
\(-g_{-1}+g_{-3}+g_{-7}\)
\(-2g_{14}-2g_{11}+2g_{8}\)
\(-g_{4}+2h_{4}+h_{3}-h_{1}+g_{-4}\)
\(-6g_{15}\)
\(-2g_{-8}-2g_{-11}+2g_{-14}\)
\(2g_{10}+2g_{3}+2g_{1}\)
\(-2g_{-5}+2g_{-6}+2g_{-12}\)
\(6g_{-16}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(3\omega_{1}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+3\omega_{2}\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{2}\)		\(3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(3\omega_{1}-3\omega_{2}\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(3\omega_{1}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+3\omega_{2}\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{2}\)		\(3\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(3\omega_{1}-3\omega_{2}\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{3\omega_{1}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+3\omega_{2}}
\oplus M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{2}}\)		\(\displaystyle M_{3\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{0}\oplus M_{3\omega_{1}-3\omega_{2}}
\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}}\)
Isotypic character\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{3\omega_{1}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+3\omega_{2}}
\oplus M_{0}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{2}}\)		\(\displaystyle M_{3\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{0}\oplus M_{3\omega_{1}-3\omega_{2}}
\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}}\)

Semisimple subalgebra: W_{1}
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 1039947 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:
(
g_{-8}+g_{-13}+g_{-14}, g_{14}+g_{13}+g_{8},
x_{5} g_{10}+x_{6} g_{7}+x_{7} g_{3}+x_{8} g_{1}, x_{16} g_{-1}+x_{15} g_{-3}+x_{14} g_{-7}+x_{13} g_{-10})
h: (2, 3, 4, 4), e = combination of g_{8} g_{11} g_{13} g_{14} , f= combination of g_{-8} g_{-11} g_{-13} g_{-14} h: (-1, 0, -2, -2), e = combination of g_{-10} g_{-7} g_{-3} g_{-1} , f= combination of g_{10} g_{7} g_{3} g_{1} Positive weight subsystem: 3 vectors: (1, 0), (0, 1), (1, 1)
Symmetric Cartan default scale: \begin{pmatrix}
2 & -1\\
-1 & 2\\
\end{pmatrix}Character ambient Lie algebra: V_{3\omega_{2}}+V_{3\omega_{1}}+4V_{\omega_{1}+\omega_{2}}+4V_{-\omega_{1}+2\omega_{2}}+4V_{2\omega_{1}-\omega_{2}}+V_{-3\omega_{1}+3\omega_{2}}+6V_{0}+V_{3\omega_{1}-3\omega_{2}}+4V_{-2\omega_{1}+\omega_{2}}+4V_{\omega_{1}-2\omega_{2}}+4V_{-\omega_{1}-\omega_{2}}+V_{-3\omega_{1}}+V_{-3\omega_{2}}
A necessary system to realize the candidate subalgebra.
x_{16} +x_{15} +x_{13} = 0
x_{8} +x_{7} +x_{5} = 0
x_{7} x_{15} +2x_{6} x_{14} +x_{5} x_{13} -2= 0
x_{6} x_{14} +x_{5} x_{13} -1= 0
x_{7} x_{14} -x_{6} x_{13} = 0
x_{6} x_{15} -x_{5} x_{14} = 0
x_{8} x_{16} -1= 0
x_{15} x_{16} +x_{13} x_{16} +x_{13} x_{15} +x_{14}^{2}= 0
x_{7} x_{8} +x_{5} x_{8} +x_{5} x_{7} +x_{6}^{2}= 0
The above system after transformation.
x_{16} +x_{15} +x_{13} = 0
x_{8} +x_{7} +x_{5} = 0
x_{7} x_{15} +2x_{6} x_{14} +x_{5} x_{13} -2= 0
x_{6} x_{14} +x_{5} x_{13} -1= 0
x_{7} x_{14} -x_{6} x_{13} = 0
x_{6} x_{15} -x_{5} x_{14} = 0
x_{8} x_{16} -1= 0
x_{15} x_{16} +x_{13} x_{16} +x_{13} x_{15} +x_{14}^{2}= 0
x_{7} x_{8} +x_{5} x_{8} +x_{5} x_{7} +x_{6}^{2}= 0
For the calculator:
(DynkinType =A^{3}_2; ElementsCartan =((2, 3, 4, 4), (-1, 0, -2, -2)); generators =(g_{-8}+g_{-13}+g_{-14}, g_{14}+g_{13}+g_{8}, x_{5} g_{10}+x_{6} g_{7}+x_{7} g_{3}+x_{8} g_{1}, x_{16} g_{-1}+x_{15} g_{-3}+x_{14} g_{-7}+x_{13} g_{-10}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{16} +x_{15} +x_{13} , x_{8} +x_{7} +x_{5} , x_{7} x_{15} +2x_{6} x_{14} +x_{5} x_{13} -2, x_{6} x_{14} +x_{5} x_{13} -1, x_{7} x_{14} -x_{6} x_{13} , x_{6} x_{15} -x_{5} x_{14} , x_{8} x_{16} -1, x_{15} x_{16} +x_{13} x_{16} +x_{13} x_{15} +x_{14}^{2}, x_{7} x_{8} +x_{5} x_{8} +x_{5} x_{7} +x_{6}^{2} )