Subalgebra \(B^{2}_2\) ↪ \(A^{1}_4\)
12 out of 15
Computations done by the calculator project.

Subalgebra type: \(\displaystyle B^{2}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_4\)

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

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 36146 arithmetic operations while solving the Serre relations polynomial system.