Type: \(\displaystyle A^{1}_3\) (Dynkin type computed to be: \(\displaystyle A^{1}_3\))
Simple basis: 3 vectors: (1, 1, 1, 1, 1), (0, 0, 0, 0, -1), (0, 0, 0, -1, 0)
Simple basis epsilon form:
Simple basis epsilon form with respect to k:
Number of outer autos with trivial action on orthogonal complement and extending to autos of ambient algebra: 0
Number of outer autos with trivial action on orthogonal complement: 0.
C(k_{ss})_{ss}: A^{1}_1
simple basis centralizer: 1 vectors: (0, 1, 0, 0, 0)
Number of k-submodules of g: 9
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{1}+\omega_{3}}+2V_{\omega_{3}}+2V_{\omega_{1}}+4V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, -1, 0, 0, 0)(0, -1, 0, 0, 0)g_{-2}-\varepsilon_{2}+\varepsilon_{3}
Module 24(0, -1, -1, 0, 0)(1, 0, 0, 0, 0)g_{1}
g_{-14}
g_{-11}
g_{-7}		\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{4}
Module 31(0, 1, 0, 0, 0)(0, 1, 0, 0, 0)g_{2}\varepsilon_{2}-\varepsilon_{3}
Module 44(-1, -1, 0, 0, 0)(0, 0, 1, 0, 0)g_{3}
g_{8}
g_{12}
g_{-6}		\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{3}
Module 54(0, 0, -1, 0, 0)(1, 1, 0, 0, 0)g_{6}
g_{-12}
g_{-8}
g_{-3}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{4}
Module 64(-1, 0, 0, 0, 0)(0, 1, 1, 0, 0)g_{7}
g_{11}
g_{14}
g_{-1}		\varepsilon_{2}-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{2}
Module 715(-1, -1, -1, 0, 0)(1, 1, 1, 0, 0)g_{10}
g_{-9}
g_{13}
g_{-4}
g_{-5}
g_{15}
-h_{4}
-h_{5}
h_{5}+h_{4}+h_{3}+h_{2}+h_{1}
g_{-15}
g_{5}
g_{4}
g_{-13}
g_{9}
g_{-10}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{5}+\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{6}
0
0
0
-\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{5}-\varepsilon_{6}
\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{4}
Module 81(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{2}0
Module 91(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{5}+2h_{4}+3h_{3}-h_{1}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 3
Heirs rejected due to not being maximally dominant: 2
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 2
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{1}_2
Potential Dynkin type extensions: A^{1}_4, A^{1}_3+A^{1}_1,