Subalgebra \(B^{1}_4\) ↪ \(F^{1}_4\)
58 out of 59
Computations done by the calculator project.

Subalgebra type: \(\displaystyle B^{1}_4\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_3\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle F^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_4\): (2, 3, 4, 2): 2, (-1, 0, 0, 0): 2, (0, -1, 0, 0): 2, (0, 0, -2, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 36.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{1}\), \(\displaystyle g_{2}\), \(\displaystyle g_{3}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 2 & -1 & 0\\ 0 & -1 & 2 & -1\\ 0 & 0 & -1 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 2 & -1 & 0\\ 0 & -1 & 2 & -2\\ 0 & 0 & -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{4}}\oplus V_{\omega_{2}}\)
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.\(g_{16}\)\(g_{4}\)
weight\(\omega_{2}\)\(\omega_{4}\)
Isotypic module decomposition over primal subalgebra (total 2 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{2}} \) → (0, 1, 0, 0)\(\displaystyle V_{\omega_{4}} \) → (0, 0, 0, 1)
Module label \(W_{1}\)\(W_{2}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{16}\)
\(-g_{18}\)
\(g_{-14}\)
\(-g_{20}\)
\(-g_{-11}\)
\(-g_{21}\)
\(g_{-9}\)
\(g_{-8}\)
\(-2g_{22}\)
\(-g_{-6}\)
\(2g_{-5}\)
\(-2g_{23}\)
\(g_{-3}\)
\(-2g_{-2}\)
\(2g_{-1}\)
\(-2g_{24}\)
\(2h_{3}\)
\(-2h_{2}\)
\(2h_{1}\)
\(4h_{4}+8h_{3}+6h_{2}+4h_{1}\)
\(2g_{2}\)
\(-2g_{3}\)
\(-2g_{1}\)
\(2g_{-24}\)
\(2g_{5}\)
\(-2g_{6}\)
\(-2g_{-23}\)
\(2g_{-22}\)
\(-2g_{8}\)
\(-4g_{9}\)
\(-2g_{-21}\)
\(-4g_{11}\)
\(4g_{-20}\)
\(-4g_{14}\)
\(-4g_{-18}\)
\(4g_{-16}\)
\(g_{4}\)
\(g_{7}\)
\(g_{10}\)
\(g_{12}\)
\(g_{13}\)
\(g_{-19}\)
\(g_{15}\)
\(-g_{-17}\)
\(g_{17}\)
\(g_{-15}\)
\(g_{19}\)
\(-g_{-13}\)
\(-g_{-12}\)
\(g_{-10}\)
\(-g_{-7}\)
\(g_{-4}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}+2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}+2\omega_{4}\)
\(\omega_{1}\)
\(-\omega_{2}+2\omega_{4}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{3}-2\omega_{4}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-2\omega_{4}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{3}+2\omega_{4}\)
\(-\omega_{2}+2\omega_{3}-2\omega_{4}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(0\)
\(\omega_{2}-2\omega_{3}+2\omega_{4}\)
\(\omega_{3}-2\omega_{4}\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+2\omega_{4}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}-\omega_{3}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}\)
\(\omega_{2}-2\omega_{4}\)
\(-\omega_{1}\)
\(\omega_{1}-\omega_{2}+\omega_{3}-2\omega_{4}\)
\(-\omega_{1}+\omega_{3}-2\omega_{4}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}\)		\(\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(-\omega_{4}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}+2\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}+2\omega_{4}\)
\(\omega_{1}\)
\(-\omega_{2}+2\omega_{4}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{3}-2\omega_{4}\)
\(-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}-2\omega_{4}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{3}+2\omega_{4}\)
\(-\omega_{2}+2\omega_{3}-2\omega_{4}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(0\)
\(\omega_{2}-2\omega_{3}+2\omega_{4}\)
\(\omega_{3}-2\omega_{4}\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}-\omega_{3}+2\omega_{4}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}-\omega_{3}+2\omega_{4}\)
\(\omega_{1}-\omega_{2}\)
\(\omega_{2}-2\omega_{4}\)
\(-\omega_{1}\)
\(\omega_{1}-\omega_{2}+\omega_{3}-2\omega_{4}\)
\(-\omega_{1}+\omega_{3}-2\omega_{4}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}\)		\(\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{1}+\omega_{3}-\omega_{4}\)
\(\omega_{1}-\omega_{3}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{2}+\omega_{4}\)
\(-\omega_{1}+\omega_{2}-\omega_{4}\)
\(-\omega_{2}+\omega_{3}-\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(-\omega_{4}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}-\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}+2\omega_{4}}
\oplus M_{-\omega_{3}+2\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}+2\omega_{4}}\oplus M_{\omega_{2}-2\omega_{3}+2\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{2}}\oplus 4M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}
\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{\omega_{1}+\omega_{3}-2\omega_{4}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}-2\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}-2\omega_{4}}\oplus M_{\omega_{3}-2\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}-2\omega_{4}}
\oplus M_{\omega_{2}-2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{3}-2\omega_{4}}\)		\(\displaystyle M_{\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{3}+\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{3}+\omega_{4}}
\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}
\oplus M_{-\omega_{4}}\)
Isotypic character\(\displaystyle M_{\omega_{1}-\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{2}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}+2\omega_{4}}
\oplus M_{-\omega_{3}+2\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}+2\omega_{4}}\oplus M_{\omega_{2}-2\omega_{3}+2\omega_{4}}
\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{\omega_{1}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{3}+2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}
\oplus M_{-\omega_{1}+\omega_{2}}\oplus 4M_{0}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}
\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{\omega_{1}+\omega_{3}-2\omega_{4}}\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}}\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}-2\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}-2\omega_{4}}\oplus M_{\omega_{3}-2\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}-2\omega_{4}}
\oplus M_{\omega_{2}-2\omega_{4}}\oplus M_{-\omega_{1}+\omega_{3}-2\omega_{4}}\)		\(\displaystyle M_{\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{4}}\oplus M_{\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{3}+\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}}\oplus M_{-\omega_{3}+\omega_{4}}
\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{\omega_{2}-\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}
\oplus M_{-\omega_{1}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{2}+\omega_{3}-\omega_{4}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{4}}
\oplus M_{-\omega_{4}}\)

Semisimple subalgebra: W_{1}
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 527 arithmetic operations while solving the Serre relations polynomial system.