Subalgebra \(A^{11}_1+A^{1}_1\) ↪ \(B^{1}_4\)
25 out of 48
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{11}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{11}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_4\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (4, 6, 7, 8): 22, \(\displaystyle A^{1}_1\): (0, 0, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-9}+g_{-10}\), \(\displaystyle g_{-3}\)
Positive simple generators: \(\displaystyle g_{10}+3g_{9}+4g_{1}\), \(\displaystyle g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\)
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 6) ; the vectors are over the primal subalgebra.\(g_{10}\)\(g_{9}+4/3g_{1}\)\(g_{3}\)\(-g_{14}+4g_{8}\)\(g_{16}\)\(g_{15}\)
weight\(2\omega_{1}\)\(2\omega_{1}\)\(2\omega_{2}\)\(3\omega_{1}+\omega_{2}\)\(6\omega_{1}\)\(5\omega_{1}+\omega_{2}\)
Isotypic module decomposition over primal subalgebra (total 6 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2, 0)\(\displaystyle V_{2\omega_{2}} \) → (0, 2)\(\displaystyle V_{3\omega_{1}+\omega_{2}} \) → (3, 1)\(\displaystyle V_{6\omega_{1}} \) → (6, 0)\(\displaystyle V_{5\omega_{1}+\omega_{2}} \) → (5, 1)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(g_{10}+3g_{9}+4g_{1}\)
\(-8h_{4}-7h_{3}-6h_{2}-4h_{1}\)
\(-2g_{-1}-2g_{-9}-2g_{-10}\)
\(g_{10}\)
\(-2h_{4}-h_{3}\)
\(-2g_{-10}\)
Semisimple subalgebra component.
\(-g_{3}\)
\(h_{3}\)
\(2g_{-3}\)
\(-g_{14}+4g_{8}\)
\(g_{7}+3g_{6}\)
\(-g_{12}-4g_{5}\)
\(-2g_{-2}+2g_{-4}\)
\(g_{4}-3g_{2}\)
\(2g_{-5}+2g_{-12}\)
\(-2g_{-6}-2g_{-7}\)
\(2g_{-8}-2g_{-14}\)
\(g_{16}\)
\(g_{11}\)
\(g_{9}-2g_{1}\)
\(-2h_{4}-2h_{3}-2h_{2}+2h_{1}\)
\(6g_{-1}-4g_{-9}\)
\(10g_{-11}\)
\(-20g_{-16}\)
\(g_{15}\)
\(g_{14}+g_{8}\)
\(g_{13}\)
\(-g_{7}+2g_{6}\)
\(g_{12}-g_{5}\)
\(2g_{-2}+3g_{-4}\)
\(-g_{4}-2g_{2}\)
\(-2g_{-5}+8g_{-12}\)
\(2g_{-6}-3g_{-7}\)
\(-10g_{-13}\)
\(-2g_{-8}-8g_{-14}\)
\(10g_{-15}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(3\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(5\omega_{1}+\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(5\omega_{1}-\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-5\omega_{1}+\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-5\omega_{1}-\omega_{2}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(3\omega_{1}+\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(5\omega_{1}+\omega_{2}\)
\(3\omega_{1}+\omega_{2}\)
\(5\omega_{1}-\omega_{2}\)
\(\omega_{1}+\omega_{2}\)
\(3\omega_{1}-\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-3\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)
\(-5\omega_{1}+\omega_{2}\)
\(-3\omega_{1}-\omega_{2}\)
\(-5\omega_{1}-\omega_{2}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{5\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{5\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}
\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}+\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}-\omega_{2}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}
\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\)		\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{5\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{5\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}
\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}
\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}+\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}-\omega_{2}}\)

Semisimple subalgebra: W_{1}+W_{3}
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 541 arithmetic operations while solving the Serre relations polynomial system.