Subalgebra \(5A^{1}_1\) ↪ \(C^{1}_5\)
113 out of 119
Computations done by the calculator project.

Subalgebra type: \(\displaystyle 5A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle 4A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 2, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 2, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 15.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{-23}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{23}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{5}}\oplus V_{\omega_{4}+\omega_{5}}\oplus V_{\omega_{3}+\omega_{5}}\oplus V_{\omega_{2}+\omega_{5}}\oplus V_{\omega_{1}+\omega_{5}}
\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}
\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\)
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 15) ; the vectors are over the primal subalgebra.\(g_{25}\)\(g_{24}\)\(g_{23}\)\(g_{22}\)\(g_{21}\)\(g_{19}\)\(g_{20}\)\(g_{18}\)\(g_{16}\)\(g_{13}\)\(g_{17}\)\(g_{15}\)\(g_{12}\)\(g_{9}\)\(g_{5}\)
weight\(2\omega_{1}\)\(\omega_{1}+\omega_{2}\)\(2\omega_{2}\)\(\omega_{1}+\omega_{3}\)\(\omega_{2}+\omega_{3}\)\(2\omega_{3}\)\(\omega_{1}+\omega_{4}\)\(\omega_{2}+\omega_{4}\)\(\omega_{3}+\omega_{4}\)\(2\omega_{4}\)\(\omega_{1}+\omega_{5}\)\(\omega_{2}+\omega_{5}\)\(\omega_{3}+\omega_{5}\)\(\omega_{4}+\omega_{5}\)\(2\omega_{5}\)
Isotypic module decomposition over primal subalgebra (total 15 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2, 0, 0, 0, 0)\(\displaystyle V_{\omega_{1}+\omega_{2}} \) → (1, 1, 0, 0, 0)\(\displaystyle V_{2\omega_{2}} \) → (0, 2, 0, 0, 0)\(\displaystyle V_{\omega_{1}+\omega_{3}} \) → (1, 0, 1, 0, 0)\(\displaystyle V_{\omega_{2}+\omega_{3}} \) → (0, 1, 1, 0, 0)\(\displaystyle V_{2\omega_{3}} \) → (0, 0, 2, 0, 0)\(\displaystyle V_{\omega_{1}+\omega_{4}} \) → (1, 0, 0, 1, 0)\(\displaystyle V_{\omega_{2}+\omega_{4}} \) → (0, 1, 0, 1, 0)\(\displaystyle V_{\omega_{3}+\omega_{4}} \) → (0, 0, 1, 1, 0)\(\displaystyle V_{2\omega_{4}} \) → (0, 0, 0, 2, 0)\(\displaystyle V_{\omega_{1}+\omega_{5}} \) → (1, 0, 0, 0, 1)\(\displaystyle V_{\omega_{2}+\omega_{5}} \) → (0, 1, 0, 0, 1)\(\displaystyle V_{\omega_{3}+\omega_{5}} \) → (0, 0, 1, 0, 1)\(\displaystyle V_{\omega_{4}+\omega_{5}} \) → (0, 0, 0, 1, 1)\(\displaystyle V_{2\omega_{5}} \) → (0, 0, 0, 0, 2)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)\(W_{7}\)\(W_{8}\)\(W_{9}\)\(W_{10}\)\(W_{11}\)\(W_{12}\)\(W_{13}\)\(W_{14}\)\(W_{15}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{25}\)
\(h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\)
\(2g_{-25}\)
\(g_{24}\)
\(-g_{-1}\)
\(-g_{1}\)
\(-g_{-24}\)
Semisimple subalgebra component.
\(-g_{23}\)
\(h_{5}+2h_{4}+2h_{3}+2h_{2}\)
\(2g_{-23}\)
\(g_{22}\)
\(-g_{-6}\)
\(-g_{6}\)
\(-g_{-22}\)
\(g_{21}\)
\(-g_{-2}\)
\(-g_{2}\)
\(-g_{-21}\)
Semisimple subalgebra component.
\(-g_{19}\)
\(h_{5}+2h_{4}+2h_{3}\)
\(2g_{-19}\)
\(g_{20}\)
\(-g_{-10}\)
\(-g_{10}\)
\(-g_{-20}\)
\(g_{18}\)
\(-g_{-7}\)
\(-g_{7}\)
\(-g_{-18}\)
\(g_{16}\)
\(-g_{-3}\)
\(-g_{3}\)
\(-g_{-16}\)
Semisimple subalgebra component.
\(-g_{13}\)
\(h_{5}+2h_{4}\)
\(2g_{-13}\)
\(g_{17}\)
\(-g_{-14}\)
\(-g_{14}\)
\(-g_{-17}\)
\(g_{15}\)
\(-g_{-11}\)
\(-g_{11}\)
\(-g_{-15}\)
\(g_{12}\)
\(-g_{-8}\)
\(-g_{8}\)
\(-g_{-12}\)
\(g_{9}\)
\(-g_{-4}\)
\(-g_{4}\)
\(-g_{-9}\)
Semisimple subalgebra component.
\(-g_{5}\)
\(h_{5}\)
\(2g_{-5}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(\omega_{1}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}-\omega_{3}\)		\(\omega_{2}+\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{1}+\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)		\(\omega_{2}+\omega_{4}\)
\(-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{4}\)
\(-\omega_{2}-\omega_{4}\)		\(\omega_{3}+\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(-\omega_{3}-\omega_{4}\)		\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)		\(\omega_{1}+\omega_{5}\)
\(-\omega_{1}+\omega_{5}\)
\(\omega_{1}-\omega_{5}\)
\(-\omega_{1}-\omega_{5}\)		\(\omega_{2}+\omega_{5}\)
\(-\omega_{2}+\omega_{5}\)
\(\omega_{2}-\omega_{5}\)
\(-\omega_{2}-\omega_{5}\)		\(\omega_{3}+\omega_{5}\)
\(-\omega_{3}+\omega_{5}\)
\(\omega_{3}-\omega_{5}\)
\(-\omega_{3}-\omega_{5}\)		\(\omega_{4}+\omega_{5}\)
\(-\omega_{4}+\omega_{5}\)
\(\omega_{4}-\omega_{5}\)
\(-\omega_{4}-\omega_{5}\)		\(2\omega_{5}\)
\(0\)
\(-2\omega_{5}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)		\(2\omega_{2}\)
\(0\)
\(-2\omega_{2}\)		\(\omega_{1}+\omega_{3}\)
\(-\omega_{1}+\omega_{3}\)
\(\omega_{1}-\omega_{3}\)
\(-\omega_{1}-\omega_{3}\)		\(\omega_{2}+\omega_{3}\)
\(-\omega_{2}+\omega_{3}\)
\(\omega_{2}-\omega_{3}\)
\(-\omega_{2}-\omega_{3}\)		\(2\omega_{3}\)
\(0\)
\(-2\omega_{3}\)		\(\omega_{1}+\omega_{4}\)
\(-\omega_{1}+\omega_{4}\)
\(\omega_{1}-\omega_{4}\)
\(-\omega_{1}-\omega_{4}\)		\(\omega_{2}+\omega_{4}\)
\(-\omega_{2}+\omega_{4}\)
\(\omega_{2}-\omega_{4}\)
\(-\omega_{2}-\omega_{4}\)		\(\omega_{3}+\omega_{4}\)
\(-\omega_{3}+\omega_{4}\)
\(\omega_{3}-\omega_{4}\)
\(-\omega_{3}-\omega_{4}\)		\(2\omega_{4}\)
\(0\)
\(-2\omega_{4}\)		\(\omega_{1}+\omega_{5}\)
\(-\omega_{1}+\omega_{5}\)
\(\omega_{1}-\omega_{5}\)
\(-\omega_{1}-\omega_{5}\)		\(\omega_{2}+\omega_{5}\)
\(-\omega_{2}+\omega_{5}\)
\(\omega_{2}-\omega_{5}\)
\(-\omega_{2}-\omega_{5}\)		\(\omega_{3}+\omega_{5}\)
\(-\omega_{3}+\omega_{5}\)
\(\omega_{3}-\omega_{5}\)
\(-\omega_{3}-\omega_{5}\)		\(\omega_{4}+\omega_{5}\)
\(-\omega_{4}+\omega_{5}\)
\(\omega_{4}-\omega_{5}\)
\(-\omega_{4}-\omega_{5}\)		\(2\omega_{5}\)
\(0\)
\(-2\omega_{5}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{\omega_{1}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}-\omega_{3}}\)\(\displaystyle M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}\)\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{4}}\)\(\displaystyle M_{\omega_{2}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{4}}\oplus M_{\omega_{2}-\omega_{4}}\oplus M_{-\omega_{2}-\omega_{4}}\)\(\displaystyle M_{\omega_{3}+\omega_{4}}\oplus M_{-\omega_{3}+\omega_{4}}\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{-\omega_{3}-\omega_{4}}\)\(\displaystyle M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\)\(\displaystyle M_{\omega_{1}+\omega_{5}}\oplus M_{-\omega_{1}+\omega_{5}}\oplus M_{\omega_{1}-\omega_{5}}\oplus M_{-\omega_{1}-\omega_{5}}\)\(\displaystyle M_{\omega_{2}+\omega_{5}}\oplus M_{-\omega_{2}+\omega_{5}}\oplus M_{\omega_{2}-\omega_{5}}\oplus M_{-\omega_{2}-\omega_{5}}\)\(\displaystyle M_{\omega_{3}+\omega_{5}}\oplus M_{-\omega_{3}+\omega_{5}}\oplus M_{\omega_{3}-\omega_{5}}\oplus M_{-\omega_{3}-\omega_{5}}\)\(\displaystyle M_{\omega_{4}+\omega_{5}}\oplus M_{-\omega_{4}+\omega_{5}}\oplus M_{\omega_{4}-\omega_{5}}\oplus M_{-\omega_{4}-\omega_{5}}\)\(\displaystyle M_{2\omega_{5}}\oplus M_{0}\oplus M_{-2\omega_{5}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\)\(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\)\(\displaystyle M_{\omega_{1}+\omega_{3}}\oplus M_{-\omega_{1}+\omega_{3}}\oplus M_{\omega_{1}-\omega_{3}}\oplus M_{-\omega_{1}-\omega_{3}}\)\(\displaystyle M_{\omega_{2}+\omega_{3}}\oplus M_{-\omega_{2}+\omega_{3}}\oplus M_{\omega_{2}-\omega_{3}}\oplus M_{-\omega_{2}-\omega_{3}}\)\(\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\)\(\displaystyle M_{\omega_{1}+\omega_{4}}\oplus M_{-\omega_{1}+\omega_{4}}\oplus M_{\omega_{1}-\omega_{4}}\oplus M_{-\omega_{1}-\omega_{4}}\)\(\displaystyle M_{\omega_{2}+\omega_{4}}\oplus M_{-\omega_{2}+\omega_{4}}\oplus M_{\omega_{2}-\omega_{4}}\oplus M_{-\omega_{2}-\omega_{4}}\)\(\displaystyle M_{\omega_{3}+\omega_{4}}\oplus M_{-\omega_{3}+\omega_{4}}\oplus M_{\omega_{3}-\omega_{4}}\oplus M_{-\omega_{3}-\omega_{4}}\)\(\displaystyle M_{2\omega_{4}}\oplus M_{0}\oplus M_{-2\omega_{4}}\)\(\displaystyle M_{\omega_{1}+\omega_{5}}\oplus M_{-\omega_{1}+\omega_{5}}\oplus M_{\omega_{1}-\omega_{5}}\oplus M_{-\omega_{1}-\omega_{5}}\)\(\displaystyle M_{\omega_{2}+\omega_{5}}\oplus M_{-\omega_{2}+\omega_{5}}\oplus M_{\omega_{2}-\omega_{5}}\oplus M_{-\omega_{2}-\omega_{5}}\)\(\displaystyle M_{\omega_{3}+\omega_{5}}\oplus M_{-\omega_{3}+\omega_{5}}\oplus M_{\omega_{3}-\omega_{5}}\oplus M_{-\omega_{3}-\omega_{5}}\)\(\displaystyle M_{\omega_{4}+\omega_{5}}\oplus M_{-\omega_{4}+\omega_{5}}\oplus M_{\omega_{4}-\omega_{5}}\oplus M_{-\omega_{4}-\omega_{5}}\)\(\displaystyle M_{2\omega_{5}}\oplus M_{0}\oplus M_{-2\omega_{5}}\)

Semisimple subalgebra: W_{1}+W_{3}+W_{6}+W_{10}+W_{15}
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 624 arithmetic operations while solving the Serre relations polynomial system.