Subalgebra \(A^{2}_2+A^{3}_1\) ↪ \(A^{1}_5\)
30 out of 37
Computations done by the calculator project.

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

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 5038281 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:
(
x_{1} g_{-13}+x_{2} g_{-14}, x_{9} g_{14}+x_{8} g_{13},
x_{3} g_{9}+x_{4} g_{8}, x_{11} g_{-8}+x_{10} g_{-9},
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: (1, 2, 2, 2, 1), e = combination of g_{13} g_{14} , f= combination of g_{-13} g_{-14} h: (0, 0, -1, -2, -1), e = combination of g_{-9} g_{-8} , f= combination of g_{9} g_{8} 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: 4 vectors: (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0)
Symmetric Cartan default scale: \begin{pmatrix}
2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}Character ambient Lie algebra: V_{\omega_{1}+\omega_{2}+2\omega_{3}}+V_{-\omega_{1}+2\omega_{2}+2\omega_{3}}+V_{2\omega_{1}-\omega_{2}+2\omega_{3}}+3V_{2\omega_{3}}+2V_{\omega_{1}+\omega_{2}}+V_{-2\omega_{1}+\omega_{2}+2\omega_{3}}+V_{\omega_{1}-2\omega_{2}+2\omega_{3}}+2V_{-\omega_{1}+2\omega_{2}}+2V_{2\omega_{1}-\omega_{2}}+V_{-\omega_{1}-\omega_{2}+2\omega_{3}}+5V_{0}+V_{\omega_{1}+\omega_{2}-2\omega_{3}}+2V_{-2\omega_{1}+\omega_{2}}+2V_{\omega_{1}-2\omega_{2}}+V_{-\omega_{1}+2\omega_{2}-2\omega_{3}}+V_{2\omega_{1}-\omega_{2}-2\omega_{3}}+2V_{-\omega_{1}-\omega_{2}}+3V_{-2\omega_{3}}+V_{-2\omega_{1}+\omega_{2}-2\omega_{3}}+V_{\omega_{1}-2\omega_{2}-2\omega_{3}}+V_{-\omega_{1}-\omega_{2}-2\omega_{3}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{8} -1= 0
x_{2} x_{9} +x_{1} x_{8} -2= 0
x_{2} x_{9} -1= 0
x_{2} x_{14} -x_{1} x_{12} = 0
x_{8} x_{14} -x_{9} x_{12} = 0
x_{1} x_{7} -x_{2} x_{5} = 0
x_{4} x_{11} +x_{3} x_{10} -2= 0
x_{3} x_{10} -1= 0
x_{4} x_{11} -1= 0
x_{4} x_{14} -x_{3} x_{13} = 0
x_{10} x_{14} -x_{11} x_{13} = 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_{7} 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} -1= 0
x_{2} x_{9} +x_{1} x_{8} -2= 0
x_{2} x_{9} -1= 0
x_{2} x_{14} -x_{1} x_{12} = 0
x_{8} x_{14} -x_{9} x_{12} = 0
x_{1} x_{7} -x_{2} x_{5} = 0
x_{4} x_{11} +x_{3} x_{10} -2= 0
x_{3} x_{10} -1= 0
x_{4} x_{11} -1= 0
x_{4} x_{14} -x_{3} x_{13} = 0
x_{10} x_{14} -x_{11} x_{13} = 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_{7} x_{9} -x_{5} x_{8} = 0
x_{7} x_{11} -x_{6} x_{10} = 0
For the calculator:
(DynkinType =A^{2}_2+A^{3}_1; ElementsCartan =((1, 2, 2, 2, 1), (0, 0, -1, -2, -1), (1, 0, 1, 0, 1)); generators =(x_{1} g_{-13}+x_{2} g_{-14}, x_{9} g_{14}+x_{8} g_{13}, x_{3} g_{9}+x_{4} g_{8}, x_{11} g_{-8}+x_{10} g_{-9}, 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} -1, x_{2} x_{9} +x_{1} x_{8} -2, x_{2} x_{9} -1, x_{2} x_{14} -x_{1} x_{12} , x_{8} x_{14} -x_{9} x_{12} , x_{1} x_{7} -x_{2} x_{5} , x_{4} x_{11} +x_{3} x_{10} -2, x_{3} x_{10} -1, x_{4} x_{11} -1, x_{4} x_{14} -x_{3} x_{13} , x_{10} x_{14} -x_{11} x_{13} , x_{3} x_{7} -x_{4} x_{6} , x_{5} x_{12} -1, x_{6} x_{13} -1, x_{7} x_{14} -1, x_{7} x_{9} -x_{5} x_{8} , x_{7} x_{11} -x_{6} x_{10} )