Subalgebra \(A^{20}_1\) ↪ \(A^{1}_4\)
6 out of 15
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{20}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{20}_1\): (4, 6, 6, 4): 40
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-4}\)
Positive simple generators: \(\displaystyle 4g_{4}+6g_{3}+6g_{2}+4g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/10\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}40\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}\)
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 4) ; the vectors are over the primal subalgebra.\(g_{4}+3/2g_{3}+3/2g_{2}+g_{1}\)\(g_{7}+3/2g_{6}+g_{5}\)\(g_{9}+g_{8}\)\(g_{10}\)
weight\(2\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(8\omega_{1}\)
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)\(\displaystyle V_{8\omega_{1}} \) → (8)
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.
\(-g_{4}-3/2g_{3}-3/2g_{2}-g_{1}\)
\(h_{4}+3/2h_{3}+3/2h_{2}+h_{1}\)
\(1/2g_{-1}+1/2g_{-2}+1/2g_{-3}+1/2g_{-4}\)
\(g_{7}+3/2g_{6}+g_{5}\)
\(g_{4}+1/2g_{3}-1/2g_{2}-g_{1}\)
\(-h_{4}-1/2h_{3}+1/2h_{2}+h_{1}\)
\(3/2g_{-1}+1/2g_{-2}-1/2g_{-3}-3/2g_{-4}\)
\(g_{-5}+g_{-6}+g_{-7}\)
\(g_{9}+g_{8}\)
\(g_{7}-g_{5}\)
\(g_{4}-g_{3}-g_{2}+g_{1}\)
\(-h_{4}+h_{3}+h_{2}-h_{1}\)
\(-3g_{-1}+2g_{-2}+2g_{-3}-3g_{-4}\)
\(-5g_{-5}+5g_{-7}\)
\(-5g_{-8}-5g_{-9}\)
\(g_{10}\)
\(g_{9}-g_{8}\)
\(g_{7}-2g_{6}+g_{5}\)
\(g_{4}-3g_{3}+3g_{2}-g_{1}\)
\(-h_{4}+3h_{3}-3h_{2}+h_{1}\)
\(5g_{-1}-10g_{-2}+10g_{-3}-5g_{-4}\)
\(15g_{-5}-20g_{-6}+15g_{-7}\)
\(35g_{-8}-35g_{-9}\)
\(70g_{-10}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 21786 arithmetic operations while solving the Serre relations polynomial system.