Subalgebra \(A^{10}_1\) ↪ \(A^{1}_3\)
4 out of 9
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{10}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\)

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

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 7185 arithmetic operations while solving the Serre relations polynomial system.