Subalgebra \(A^{1}_3\) ↪ \(B^{1}_3\)
15 out of 16
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{1}_3\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_2\) .
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: \(\displaystyle A^{1}_3\): (1, 2, 2): 2, (0, -1, 0): 2, (-1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 15.
Negative simple generators: \(\displaystyle g_{-9}\), \(\displaystyle g_{2}\), \(\displaystyle g_{1}\)
Positive simple generators: \(\displaystyle g_{9}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{2}}\)
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.\(g_{3}\)\(g_{7}\)
weight\(\omega_{2}\)\(\omega_{1}+\omega_{3}\)
Isotypic module decomposition over primal subalgebra (total 2 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{\omega_{2}} \) → (0, 1, 0)\(\displaystyle V_{\omega_{1}+\omega_{3}} \) → (1, 0, 1)
Module label \(W_{1}\)\(W_{2}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
\(g_{3}\)
\(g_{5}\)
\(g_{-6}\)
\(g_{6}\)
\(-g_{-5}\)
\(g_{-3}\)
Semisimple subalgebra component.
\(-g_{7}\)
\(-g_{-4}\)
\(-g_{8}\)
\(-g_{-1}\)
\(g_{-2}\)
\(-g_{9}\)
\(-h_{1}\)
\(h_{2}\)
\(2h_{3}+2h_{2}+h_{1}\)
\(-g_{2}\)
\(2g_{1}\)
\(g_{-9}\)
\(-g_{-8}\)
\(-g_{4}\)
\(g_{-7}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}\)		\(\omega_{1}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(-\omega_{1}-\omega_{3}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(\omega_{2}\)
\(\omega_{1}-\omega_{2}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}\)		\(\omega_{1}+\omega_{3}\)
\(-\omega_{1}+\omega_{2}+\omega_{3}\)
\(\omega_{1}+\omega_{2}-\omega_{3}\)
\(-\omega_{2}+2\omega_{3}\)
\(-\omega_{1}+2\omega_{2}-\omega_{3}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(0\)
\(\omega_{1}-2\omega_{2}+\omega_{3}\)
\(\omega_{2}-2\omega_{3}\)
\(-2\omega_{1}+\omega_{2}\)
\(-\omega_{1}-\omega_{2}+\omega_{3}\)
\(\omega_{1}-\omega_{2}-\omega_{3}\)
\(-\omega_{1}-\omega_{3}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}
\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}}
\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus 3M_{0}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{3}}\)
Isotypic character\(\displaystyle M_{\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}
\oplus M_{-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\omega_{3}}\)		\(\displaystyle M_{\omega_{1}+\omega_{3}}\oplus M_{-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+\omega_{2}+\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}}
\oplus M_{\omega_{1}+\omega_{2}-\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+\omega_{3}}\oplus 3M_{0}\oplus M_{-\omega_{1}+2\omega_{2}-\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{2}+\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}-\omega_{3}}\oplus M_{\omega_{2}-2\omega_{3}}
\oplus M_{-\omega_{1}-\omega_{3}}\)

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 442 arithmetic operations while solving the Serre relations polynomial system.