Type: \(\displaystyle A^{1}_3\) (Dynkin type computed to be: \(\displaystyle A^{1}_3\))
Simple basis: 3 vectors: (1, 2, 2, 1, 1), (0, -1, 0, 0, 0), (0, 0, -1, 0, 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}: 0
simple basis centralizer: 0 vectors:
Number of k-submodules of g: 9
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{1}+\omega_{3}}+2V_{\omega_{3}}+2V_{\omega_{2}}+2V_{\omega_{1}}+2V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 16(0, 0, -1, -1, -1)(1, 0, 0, 0, 0)g_{1}
g_{6}
g_{-18}
g_{10}
g_{-16}
g_{-13}		\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 24(-1, -1, -1, 0, -1)(0, 0, 0, 1, 0)g_{4}
g_{8}
g_{11}
g_{-15}		\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 34(-1, -1, -1, -1, 0)(0, 0, 0, 0, 1)g_{5}
g_{9}
g_{12}
g_{-14}		\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 46(-1, 0, 0, 0, 0)(0, 0, 1, 1, 1)g_{13}
g_{16}
g_{-10}
g_{18}
g_{-6}
g_{-1}		\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
Module 54(0, 0, 0, 0, -1)(1, 1, 1, 1, 0)g_{14}
g_{-12}
g_{-9}
g_{-5}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
Module 64(0, 0, 0, -1, 0)(1, 1, 1, 0, 1)g_{15}
g_{-11}
g_{-8}
g_{-4}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
Module 715(-1, -1, -1, -1, -1)(1, 1, 1, 1, 1)g_{17}
g_{-7}
g_{19}
g_{-3}
g_{-2}
g_{20}
-h_{3}
-h_{2}
h_{5}+h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-20}
g_{2}
g_{3}
g_{-19}
g_{7}
g_{-17}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{2}
0
0
0
-\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 81(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{4}+1/2h_{3}-1/2h_{1}0
Module 91(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{5}+1/2h_{3}-1/2h_{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: 1
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 1
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 1
Parabolically induced by A^{1}_2
Potential Dynkin type extensions: A^{1}_4, D^{1}_4, A^{1}_3+A^{1}_1,