Subalgebra \(A^{28}_1\) ↪ \(B^{1}_3\)
6 out of 16
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: \(\displaystyle B^{1}_3\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{28}_1\): (6, 10, 12): 56
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}\)
Positive simple generators: \(\displaystyle 6g_{3}+10g_{2}+6g_{1}\)
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: \(\displaystyle V_{10\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{1}}\)
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 3) ; the vectors are over the primal subalgebra.\(g_{3}+5/3g_{2}+g_{1}\)\(-g_{7}+1/2g_{6}\)\(g_{9}\)
weight\(2\omega_{1}\)\(6\omega_{1}\)\(10\omega_{1}\)
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.
\(-g_{3}-5/3g_{2}-g_{1}\)
\(2h_{3}+5/3h_{2}+h_{1}\)
\(1/3g_{-1}+1/3g_{-2}+1/3g_{-3}\)
\(-g_{7}+1/2g_{6}\)
\(-1/2g_{5}-g_{4}\)
\(-1/2g_{3}+g_{1}\)
\(h_{3}-h_{1}\)
\(-2g_{-1}+g_{-3}\)
\(-2g_{-4}-g_{-5}\)
\(-g_{-6}+2g_{-7}\)
\(g_{9}\)
\(g_{8}\)
\(g_{7}+g_{6}\)
\(2g_{5}-2g_{4}\)
\(2g_{3}-6g_{2}+2g_{1}\)
\(-4h_{3}+6h_{2}-2h_{1}\)
\(-10g_{-1}+18g_{-2}-10g_{-3}\)
\(-28g_{-4}+28g_{-5}\)
\(-56g_{-6}-56g_{-7}\)
\(168g_{-8}\)
\(-168g_{-9}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
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\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


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