Subalgebra \(A^{12}_1\) ↪ \(B^{1}_4\)
10 out of 48
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: \(\displaystyle B^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{12}_1\): (4, 6, 8, 8): 24
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}-2g_{-3}+g_{-6}+g_{-12}\)
Positive simple generators: \(\displaystyle 4g_{12}+g_{10}+2g_{6}-4g_{5}-g_{3}+4g_{1}\)
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: \(\displaystyle 2V_{6\omega_{1}}\oplus 2V_{4\omega_{1}}\oplus 4V_{2\omega_{1}}\)
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 8) ; the vectors are over the primal subalgebra.\(g_{9}+g_{7}\)\(-g_{10}+8/3g_{5}+1/2g_{3}+1/3g_{1}\)\(g_{12}+2/3g_{5}-1/4g_{3}+5/6g_{1}\)\(g_{6}-2g_{5}+1/4g_{3}+1/2g_{1}\)\(g_{11}\)\(-g_{14}-2g_{13}+2g_{8}\)\(g_{15}\)\(g_{16}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(4\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(6\omega_{1}\)
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.
\(-g_{12}-1/4g_{10}-1/2g_{6}+g_{5}+1/4g_{3}-g_{1}\)
\(2h_{4}+2h_{3}+3/2h_{2}+h_{1}\)
\(1/2g_{-1}-g_{-3}+1/2g_{-6}+1/2g_{-12}\)
\(g_{12}+2/3g_{5}-1/4g_{3}+5/6g_{1}\)
\(2/3g_{2}-2h_{4}-3/2h_{3}-h_{2}-5/6h_{1}+1/4g_{-2}\)
\(-2/3g_{-1}+2/3g_{-3}-1/4g_{-5}-1/6g_{-6}+2/3g_{-10}-2/3g_{-12}\)
\(g_{9}+g_{7}\)
\(-g_{4}-g_{-4}\)
\(-2g_{-7}\)
\(-g_{10}+8/3g_{5}+1/2g_{3}+1/3g_{1}\)
\(8/3g_{2}+h_{3}-1/3h_{1}+1/2g_{-2}\)
\(-2/3g_{-1}-4/3g_{-3}-1/2g_{-5}+1/3g_{-6}+8/3g_{-10}-2/3g_{-12}\)
\(g_{11}\)
\(g_{9}\)
\(g_{4}-g_{-4}\)
\(-2g_{-7}+2g_{-9}\)
\(-2g_{-11}\)
\(-g_{14}-2g_{13}+2g_{8}\)
\(g_{10}+2g_{6}+4g_{5}+g_{3}\)
\(8g_{2}-2h_{2}-2g_{-2}\)
\(2g_{-1}+4g_{-3}+2g_{-5}+2g_{-6}+8g_{-10}-2g_{-12}\)
\(-4g_{-8}+4g_{-13}+8g_{-14}\)
\(g_{15}\)
\(g_{14}-2g_{13}\)
\(-4g_{12}-g_{10}-g_{3}+2g_{1}\)
\(8h_{4}+2h_{3}+4h_{2}-2h_{1}+2g_{-2}\)
\(-8g_{-1}+16g_{-3}-2g_{-5}-4g_{-6}+12g_{-12}\)
\(-20g_{-13}+40g_{-14}\)
\(-80g_{-15}+20g_{-16}\)
\(g_{16}\)
\(g_{13}+g_{8}\)
\(g_{12}+g_{6}+2g_{5}-2g_{1}\)
\(4g_{2}-2h_{4}-2h_{3}-2h_{2}+2h_{1}\)
\(6g_{-1}+4g_{-3}-4g_{-6}+4g_{-10}-4g_{-12}\)
\(10g_{-8}+10g_{-13}\)
\(20g_{-15}-20g_{-16}\)
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\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 3M_{2\omega_{1}}\oplus 3M_{0}\oplus 3M_{-2\omega_{1}}\)\(\displaystyle 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\oplus 2M_{-4\omega_{1}}\)\(\displaystyle 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 2974217 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_{-5}+x_{4} g_{-6}+x_{5} g_{-7}+x_{6} g_{-9}+x_{7} g_{-10}+x_{8} g_{-12}, x_{16} g_{12}+x_{15} g_{10}+x_{14} g_{9}+x_{13} g_{7}+x_{12} g_{6}+x_{11} g_{5}+x_{10} g_{3}+x_{9} g_{1})
h: (4, 6, 8, 8), e = combination of g_{1} g_{3} g_{5} g_{6} g_{7} g_{9} g_{10} g_{12} , f= combination of g_{-1} g_{-3} g_{-5} g_{-6} g_{-7} g_{-9} g_{-10} g_{-12} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: 2V_{6\omega_{1}}+4V_{4\omega_{1}}+8V_{2\omega_{1}}+8V_{0}+8V_{-2\omega_{1}}+4V_{-4\omega_{1}}+2V_{-6\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{3} x_{11} +x_{1} x_{9} -4= 0
x_{8} x_{15} +2x_{6} x_{13} +x_{4} x_{10} -x_{3} x_{9} = 0
x_{8} x_{16} +x_{7} x_{15} +2x_{6} x_{14} +2x_{5} x_{13} +x_{4} x_{12} +x_{2} x_{10} -8= 0
x_{8} x_{14} +x_{7} x_{13} -x_{6} x_{12} -x_{5} x_{10} = 0
x_{7} x_{16} +2x_{5} x_{14} +x_{2} x_{12} -x_{1} x_{11} = 0
x_{8} x_{16} +2x_{6} x_{14} +x_{4} x_{12} +x_{3} x_{11} -6= 0
x_{6} x_{16} +x_{5} x_{15} -x_{4} x_{14} -x_{2} x_{13} = 0
x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} -4= 0
The above system after transformation.
x_{3} x_{11} +x_{1} x_{9} -4= 0
x_{8} x_{15} +2x_{6} x_{13} +x_{4} x_{10} -x_{3} x_{9} = 0
x_{8} x_{16} +x_{7} x_{15} +2x_{6} x_{14} +2x_{5} x_{13} +x_{4} x_{12} +x_{2} x_{10} -8= 0
x_{8} x_{14} +x_{7} x_{13} -x_{6} x_{12} -x_{5} x_{10} = 0
x_{7} x_{16} +2x_{5} x_{14} +x_{2} x_{12} -x_{1} x_{11} = 0
x_{8} x_{16} +2x_{6} x_{14} +x_{4} x_{12} +x_{3} x_{11} -6= 0
x_{6} x_{16} +x_{5} x_{15} -x_{4} x_{14} -x_{2} x_{13} = 0
x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} -4= 0
For the calculator:
(DynkinType =A^{12}_1; ElementsCartan =((4, 6, 8, 8)); generators =(x_{1} g_{-1}+x_{2} g_{-3}+x_{3} g_{-5}+x_{4} g_{-6}+x_{5} g_{-7}+x_{6} g_{-9}+x_{7} g_{-10}+x_{8} g_{-12}, x_{16} g_{12}+x_{15} g_{10}+x_{14} g_{9}+x_{13} g_{7}+x_{12} g_{6}+x_{11} g_{5}+x_{10} g_{3}+x_{9} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{3} x_{11} +x_{1} x_{9} -4, x_{8} x_{15} +2x_{6} x_{13} +x_{4} x_{10} -x_{3} x_{9} , x_{8} x_{16} +x_{7} x_{15} +2x_{6} x_{14} +2x_{5} x_{13} +x_{4} x_{12} +x_{2} x_{10} -8, x_{8} x_{14} +x_{7} x_{13} -x_{6} x_{12} -x_{5} x_{10} , x_{7} x_{16} +2x_{5} x_{14} +x_{2} x_{12} -x_{1} x_{11} , x_{8} x_{16} +2x_{6} x_{14} +x_{4} x_{12} +x_{3} x_{11} -6, x_{6} x_{16} +x_{5} x_{15} -x_{4} x_{14} -x_{2} x_{13} , x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} +x_{5} x_{13} -4 )