Subalgebra \(A^{28}_1\) ↪ \(D^{1}_4\)
7 out of 23
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{28}_1\) (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: \(\displaystyle A^{28}_1\): (6, 10, 6, 6): 56
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: \(\displaystyle 6g_{4}+6g_{3}+10g_{2}+6g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/14\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}56\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus 2V_{6\omega_{1}}\oplus V_{2\omega_{1}}\)
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.\(g_{4}+g_{3}+5/3g_{2}+g_{1}\)\(-g_{9}+g_{8}\)\(-g_{10}+g_{8}\)\(g_{12}\)
weight\(2\omega_{1}\)\(6\omega_{1}\)\(6\omega_{1}\)\(10\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 3 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)\(\displaystyle V_{6\omega_{1}} \) → (6)\(\displaystyle V_{10\omega_{1}} \) → (10)
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.
\(-g_{4}-g_{3}-5/3g_{2}-g_{1}\)
\(h_{4}+h_{3}+5/3h_{2}+h_{1}\)
\(1/3g_{-1}+1/3g_{-2}+1/3g_{-3}+1/3g_{-4}\)
\(-g_{9}+g_{8}\)
\(-g_{7}+g_{6}\)
\(-g_{4}+g_{3}\)
\(h_{4}-h_{3}\)
\(-2g_{-3}+2g_{-4}\)
\(2g_{-6}-2g_{-7}\)
\(-2g_{-8}+2g_{-9}\)
\(-g_{10}+g_{8}\)
\(-g_{7}-g_{5}\)
\(-g_{4}+g_{1}\)
\(h_{4}-h_{1}\)
\(-2g_{-1}+2g_{-4}\)
\(-2g_{-5}-2g_{-7}\)
\(-2g_{-8}+2g_{-10}\)
\(g_{12}\)
\(g_{11}\)
\(g_{10}+g_{9}+g_{8}\)
\(2g_{7}+2g_{6}-2g_{5}\)
\(2g_{4}+2g_{3}-6g_{2}+2g_{1}\)
\(-2h_{4}-2h_{3}+6h_{2}-2h_{1}\)
\(-10g_{-1}+18g_{-2}-10g_{-3}-10g_{-4}\)
\(-28g_{-5}+28g_{-6}+28g_{-7}\)
\(-56g_{-8}-56g_{-9}-56g_{-10}\)
\(168g_{-11}\)
\(-168g_{-12}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle 2M_{6\omega_{1}}\oplus 2M_{4\omega_{1}}\oplus 2M_{2\omega_{1}}\oplus 2M_{0}\oplus 2M_{-2\omega_{1}}\oplus 2M_{-4\omega_{1}}\oplus 2M_{-6\omega_{1}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 21714 arithmetic operations while solving the Serre relations polynomial system.