Subalgebra \(4A^{1}_1\) ↪ \(D^{1}_4\)
22 out of 23
Computations done by the calculator project.

Subalgebra type: \(\displaystyle 4A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle 3A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle D^{1}_4\)

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

Semisimple subalgebra: W_{1}+W_{2}+W_{3}+W_{4}
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.