Subalgebra \(2A^{1}_1\) ↪ \(B^{1}_2\)
4 out of 5
Computations done by the calculator project.

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

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

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 357 arithmetic operations while solving the Serre relations polynomial system.