Lie algebra sl(4), type \(A^{1}_3\)
Semisimple complex Lie subalgebras
Up to linear equivalence, there are total 9 semisimple subalgebras (including the full subalgebra). The subalgebras are ordered by rank, Dynkin indices of simple constituents and dimensions of simple constituents.
The upper index indicates the Dynkin index, the lower index indicates the rank of the subalgebra.
Computation time in seconds: 5.091.
952458 total arithmetic operations performed = 829927 additions and 122531 multiplications.
The base field over which the subalgebras were realized is: \(\displaystyle \mathbb Q\)
Number of root subalgebras other than the Cartan and full subalgebra: 3
Number of sl(2)'s: 4
Subalgebra \(A^{1}_1\) ↪ \(A^{1}_3\)
1 out of 9
Subalgebra type: \(\displaystyle A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_1\)
Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0), (1, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{1}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (1, 1, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-6}\)
Positive simple generators: \(\displaystyle g_{6}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus 4V_{\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+2\psi_{1}+4\psi_{2}}\oplus V_{4\psi_{1}}\oplus V_{\omega_{1}-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{0}
\oplus V_{\omega_{1}+2\psi_{1}-4\psi_{2}}\oplus V_{-4\psi_{1}}\oplus V_{\omega_{1}-2\psi_{1}-4\psi_{2}}\)
Made total 276 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1\) ↪ \(A^{1}_3\)
2 out of 9
Subalgebra type: \(\displaystyle A^{2}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{2}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{2}_1\)
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{2}_1\) .
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (1, 2, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-2}+g_{-6}\)
Positive simple generators: \(\displaystyle g_{6}+g_{2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 4V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi}\oplus V_{4\psi}\oplus 2V_{2\omega_{1}}\oplus V_{0}\oplus V_{2\omega_{1}-4\psi}\oplus V_{-4\psi}\)
Made total 142864 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1\) ↪ \(A^{1}_3\)
3 out of 9
Subalgebra type: \(\displaystyle A^{4}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\)
Basis of Cartan of centralizer: 1 vectors:
(1, 2, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 2, 2): 8
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-5}\)
Positive simple generators: \(\displaystyle 2g_{5}+2g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+8\psi}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{2\omega_{1}-8\psi}\)
Made total 184669 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1\) ↪ \(A^{1}_3\)
4 out of 9
Subalgebra type: \(\displaystyle A^{10}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (3, 4, 3): 20
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}\)
Positive simple generators: \(\displaystyle 3g_{3}+4g_{2}+3g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}\)
Made total 7185 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{1}_1\) ↪ \(A^{1}_3\)
5 out of 9
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: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\)
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (1, 1, 1): 2, \(\displaystyle A^{1}_1\): (0, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-6}\), \(\displaystyle g_{-2}\)
Positive simple generators: \(\displaystyle g_{6}\), \(\displaystyle g_{2}\)
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: \(\displaystyle V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+\omega_{2}+4\psi}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{2}-4\psi}\)
Made total 359 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{2}_1\) ↪ \(A^{1}_3\)
6 out of 9
Subalgebra type: \(\displaystyle 2A^{2}_1\) (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: \(\displaystyle A^{1}_3\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (1, 2, 1): 4, \(\displaystyle A^{2}_1\): (1, 0, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-4}+g_{-5}\), \(\displaystyle g_{-1}+g_{-3}\)
Positive simple generators: \(\displaystyle g_{5}+g_{4}\), \(\displaystyle g_{3}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0\\
0 & 1\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0\\
0 & 4\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 422676 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_2\) ↪ \(A^{1}_3\)
7 out of 9
Subalgebra type: \(\displaystyle A^{1}_2\) (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: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\)
Basis of Cartan of centralizer: 1 vectors:
(1, -2, -1)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (1, 1, 1): 2, (0, 0, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \(\displaystyle g_{-6}\), \(\displaystyle g_{3}\)
Positive simple generators: \(\displaystyle g_{6}\), \(\displaystyle g_{-3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{\omega_{1}+8\psi}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{0}\oplus V_{\omega_{2}-8\psi}\)
Made total 359 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2\) ↪ \(A^{1}_3\)
8 out of 9
Subalgebra type: \(\displaystyle B^{1}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_3\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (1, 1, 1): 2, (-1, 0, -1): 4
Dimension of subalgebra generated by predefined or computed generators: 10.
Negative simple generators: \(\displaystyle g_{-6}\), \(\displaystyle g_{3}+g_{1}\)
Positive simple generators: \(\displaystyle g_{6}\), \(\displaystyle g_{-1}+g_{-3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1\\
-1 & 1\\
\end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2\\
-2 & 4\\
\end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus V_{\omega_{1}}\)
Made total 2266 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{1}_3\) ↪ \(A^{1}_3\)
9 out of 9
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: \(\displaystyle A^{1}_3\)
Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_3\): (1, 1, 1): 2, (0, 0, -1): 2, (0, -1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 15.
Negative simple generators: \(\displaystyle g_{-6}\), \(\displaystyle g_{3}\), \(\displaystyle g_{2}\)
Positive simple generators: \(\displaystyle g_{6}\), \(\displaystyle g_{-3}\), \(\displaystyle g_{-2}\)
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: \(\displaystyle V_{\omega_{1}+\omega_{3}}\)
Made total 442 arithmetic operations while solving the Serre relations polynomial system.
Nilpotent orbit computation summary.
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