Subalgebra \(A^{12}_1\) ↪ \(D^{1}_4\)
6 out of 23
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{12}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle D^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{12}_1\): (4, 6, 4, 4): 24
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-3}+g_{-6}+g_{-7}\)
Positive simple generators: \(\displaystyle 4g_{7}+2g_{6}+2g_{5}-2g_{4}+2g_{3}+4g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/6\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}24\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 2V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\)
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 6) ; the vectors are over the primal subalgebra.\(g_{6}+3/2g_{4}-5/4g_{3}+3/4g_{1}\)\(g_{7}-1/2g_{4}+3/4g_{3}+3/4g_{1}\)\(-g_{5}+3/2g_{4}-3/4g_{3}+1/4g_{1}\)\(g_{10}-g_{9}+g_{8}\)\(g_{12}\)\(g_{11}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(6\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 4 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)\(\displaystyle V_{4\omega_{1}} \) → (4)\(\displaystyle V_{6\omega_{1}} \) → (6)
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.
\(-g_{7}-1/2g_{6}-1/2g_{5}+1/2g_{4}-1/2g_{3}-g_{1}\)
\(h_{4}+h_{3}+3/2h_{2}+h_{1}\)
\(1/2g_{-1}+1/2g_{-3}+1/2g_{-6}+1/2g_{-7}\)
\(g_{7}-1/2g_{4}+3/4g_{3}+3/4g_{1}\)
\(-h_{4}-3/4h_{3}-h_{2}-3/4h_{1}-1/4g_{-2}\)
\(-1/2g_{-1}-1/2g_{-3}+1/4g_{-5}-1/4g_{-6}-1/2g_{-7}\)
\(g_{6}+3/2g_{4}-5/4g_{3}+3/4g_{1}\)
\(-g_{2}+1/4h_{3}-h_{2}-3/4h_{1}-1/4g_{-2}\)
\(-1/2g_{-1}+1/2g_{-3}-g_{-4}+1/4g_{-5}-1/4g_{-6}-1/2g_{-7}\)
\(g_{10}-g_{9}+g_{8}\)
\(g_{6}-g_{5}-g_{4}-g_{3}\)
\(-2g_{2}-h_{2}+2g_{-2}\)
\(g_{-1}-g_{-3}-2g_{-4}-2g_{-5}+g_{-6}-g_{-7}\)
\(-2g_{-8}+2g_{-9}-2g_{-10}\)
\(g_{12}\)
\(g_{9}+g_{8}\)
\(g_{7}+g_{6}-g_{5}-2g_{1}\)
\(-2g_{2}-h_{4}-h_{3}-2h_{2}+2h_{1}\)
\(6g_{-1}-2g_{-3}-2g_{-4}-4g_{-6}-4g_{-7}\)
\(10g_{-8}+10g_{-9}\)
\(-10g_{-11}-20g_{-12}\)
\(g_{11}\)
\(g_{10}+g_{9}\)
\(2g_{7}-g_{4}-g_{3}-g_{1}\)
\(-2h_{4}+h_{3}-2h_{2}+h_{1}+2g_{-2}\)
\(4g_{-1}+4g_{-3}-2g_{-5}+2g_{-6}-6g_{-7}\)
\(10g_{-9}+10g_{-10}\)
\(-20g_{-11}-10g_{-12}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 1334449 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.
1*2 (unknown) gens:
(
x_{1} g_{-1}+x_{2} g_{-3}+x_{3} g_{-4}+x_{4} g_{-5}+x_{5} g_{-6}+x_{6} g_{-7}, x_{12} g_{7}+x_{11} g_{6}+x_{10} g_{5}+x_{9} g_{4}+x_{8} g_{3}+x_{7} g_{1})
h: (4, 6, 4, 4), e = combination of g_{1} g_{3} g_{4} g_{5} g_{6} g_{7} , f= combination of g_{-1} g_{-3} g_{-4} g_{-5} g_{-6} g_{-7} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: 2V_{6\omega_{1}}+3V_{4\omega_{1}}+6V_{2\omega_{1}}+6V_{0}+6V_{-2\omega_{1}}+3V_{-4\omega_{1}}+2V_{-6\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{4} x_{10} +x_{1} x_{7} -4= 0
x_{6} x_{9} +x_{5} x_{8} -x_{4} x_{7} = 0
x_{5} x_{11} +x_{2} x_{8} -4= 0
x_{6} x_{12} +x_{3} x_{9} -4= 0
x_{3} x_{12} +x_{2} x_{11} -x_{1} x_{10} = 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -6= 0
The above system after transformation.
x_{4} x_{10} +x_{1} x_{7} -4= 0
x_{6} x_{9} +x_{5} x_{8} -x_{4} x_{7} = 0
x_{5} x_{11} +x_{2} x_{8} -4= 0
x_{6} x_{12} +x_{3} x_{9} -4= 0
x_{3} x_{12} +x_{2} x_{11} -x_{1} x_{10} = 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -6= 0
For the calculator:
(DynkinType =A^{12}_1; ElementsCartan =((4, 6, 4, 4)); generators =(x_{1} g_{-1}+x_{2} g_{-3}+x_{3} g_{-4}+x_{4} g_{-5}+x_{5} g_{-6}+x_{6} g_{-7}, x_{12} g_{7}+x_{11} g_{6}+x_{10} g_{5}+x_{9} g_{4}+x_{8} g_{3}+x_{7} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{4} x_{10} +x_{1} x_{7} -4, x_{6} x_{9} +x_{5} x_{8} -x_{4} x_{7} , x_{5} x_{11} +x_{2} x_{8} -4, x_{6} x_{12} +x_{3} x_{9} -4, x_{3} x_{12} +x_{2} x_{11} -x_{1} x_{10} , x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -6 )