Subalgebra \(A^{11}_1\) ↪ \(C^{1}_3\)
6 out of 16
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{11}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (6, 8, 5): 22
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-3}+g_{-7}\)
Positive simple generators: \(\displaystyle 4g_{7}+g_{3}+3g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\)
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 5) ; the vectors are over the primal subalgebra.\(g_{5}+3g_{4}\)\(g_{7}+3/4g_{1}\)\(g_{3}\)\(g_{6}\)\(g_{9}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)
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.
\(-4/3g_{7}-1/3g_{3}-g_{1}\)
\(5/3h_{3}+8/3h_{2}+2h_{1}\)
\(2/3g_{-1}+2/3g_{-3}+2/3g_{-7}\)
\(g_{7}+3/4g_{1}\)
\(-h_{3}-2h_{2}-3/2h_{1}\)
\(-1/2g_{-1}-1/2g_{-7}\)
\(g_{5}+3g_{4}\)
\(2g_{2}-g_{-2}\)
\(g_{-4}+g_{-5}\)
\(g_{6}\)
\(g_{5}-g_{4}\)
\(-2g_{2}-g_{-2}\)
\(g_{-4}-3g_{-5}\)
\(4g_{-6}\)
\(g_{9}\)
\(g_{8}\)
\(2g_{7}-g_{1}\)
\(-2h_{3}-4h_{2}+2h_{1}\)
\(4g_{-1}-6g_{-7}\)
\(10g_{-8}\)
\(-20g_{-9}\)
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\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 840715 arithmetic operations while solving the Serre relations polynomial system.