Subalgebra \(B^{1}_3\) ↪ \(D^{1}_4\)
21 out of 23
Computations done by the calculator project.

Subalgebra type: \(\displaystyle B^{1}_3\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) .
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: \(\displaystyle B^{1}_3\): (1, 2, 1, 1): 2, (0, -1, 0, 0): 2, (0, 0, -1, -1): 4
Dimension of subalgebra generated by predefined or computed generators: 21.
Negative simple generators: \(\displaystyle g_{-12}\), \(\displaystyle g_{2}\), \(\displaystyle g_{4}+g_{3}\)
Positive simple generators: \(\displaystyle g_{12}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-3}+g_{-4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 2 & -2\\ 0 & -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{2}}\oplus V_{\omega_{1}}\)
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 2) ; the vectors are over the primal subalgebra.\(-g_{9}+g_{8}\)\(g_{1}\)
weight\(\omega_{1}\)\(\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 2 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{1}} \) → (1, 0, 0)\(\displaystyle V_{\omega_{2}} \) → (0, 1, 0)
Module label \(W_{1}\)\(W_{2}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
\(-g_{9}+g_{8}\)
\(g_{-6}-g_{-7}\)
\(-g_{-3}+g_{-4}\)
\(h_{4}-h_{3}\)
\(-2g_{4}+2g_{3}\)
\(-2g_{7}+2g_{6}\)
\(-2g_{-8}+2g_{-9}\)
Semisimple subalgebra component.
\(-g_{1}\)
\(g_{5}\)
\(-g_{-10}\)
\(-g_{9}-g_{8}\)
\(g_{-6}+g_{-7}\)
\(-2g_{11}\)
\(-g_{-3}-g_{-4}\)
\(2g_{-2}\)
\(-2g_{12}\)
\(-h_{4}-h_{3}\)
\(2h_{2}\)
\(2h_{4}+2h_{3}+4h_{2}+2h_{1}\)
\(-2g_{2}\)
\(2g_{4}+2g_{3}\)
\(2g_{-12}\)
\(-2g_{-11}\)
\(2g_{7}+2g_{6}\)
\(2g_{-8}+2g_{-9}\)
\(4g_{10}\)
\(4g_{-5}\)
\(4g_{-1}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-\omega_{2}+2\omega_{3}\)
\(0\)
\(\omega_{2}-2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)		\(\omega_{2}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-2\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+2\omega_{3}\)
\(\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)
\(\omega_{1}-2\omega_{3}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-\omega_{2}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-\omega_{2}+2\omega_{3}\)
\(0\)
\(\omega_{2}-2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)		\(\omega_{2}\)
\(\omega_{1}-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{3}\)
\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-2\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+2\omega_{3}\)
\(\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+2\omega_{3}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)
\(\omega_{1}-2\omega_{3}\)
\(-\omega_{1}+\omega_{2}-2\omega_{3}\)
\(-\omega_{2}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{-\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}}
\oplus M_{\omega_{2}-2\omega_{3}}\)		\(\displaystyle M_{\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+2\omega_{3}}
\oplus M_{\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus 3M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-2\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}}
\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-2\omega_{3}}\oplus M_{\omega_{2}-2\omega_{3}}\oplus M_{\omega_{1}-2\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}}\)
Isotypic character\(\displaystyle M_{-\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}}
\oplus M_{\omega_{2}-2\omega_{3}}\)		\(\displaystyle M_{\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+2\omega_{3}}
\oplus M_{\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus 3M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-2\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}}
\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-2\omega_{3}}\oplus M_{\omega_{2}-2\omega_{3}}\oplus M_{\omega_{1}-2\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{2}-2\omega_{3}}\)

Semisimple subalgebra: W_{2}
Centralizer extension: 0

Weight diagram. The coordinates corresponding to the simple roots of the subalgerba are fundamental.
The bilinear form is therefore given relative to the fundamental coordinates.
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Made total 2776 arithmetic operations while solving the Serre relations polynomial system.