Type: \(\displaystyle A^{1}_3+A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{1}_3+A^{1}_1\))
Simple basis: 4 vectors: (1, 2, 2, 2, 1, 1), (0, -1, 0, 0, 0, 0), (0, 0, -1, 0, 0, 0), (0, 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, 0, 0, 0, 0, 1)
Number of k-submodules of g: 12
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{4}}+2V_{\omega_{3}+\omega_{4}}+2V_{\omega_{1}+\omega_{4}}+V_{\omega_{1}+\omega_{3}}+2V_{\omega_{2}}+4V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, -1)g_{-6}-\varepsilon_{5}-\varepsilon_{6}
Module 26(0, 0, -1, -2, -1, -1)(1, 0, 0, 0, 0, 0)g_{1}
g_{7}
g_{-28}
g_{12}
g_{-26}
g_{-24}		\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 33(0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 1, 0)g_{5}
h_{5}
g_{-5}		\varepsilon_{5}-\varepsilon_{6}
0
-\varepsilon_{5}+\varepsilon_{6}
Module 41(0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 1)g_{6}\varepsilon_{5}+\varepsilon_{6}
Module 58(-1, -1, -1, -1, -1, -1)(0, 0, 0, 1, 1, 0)g_{10}
g_{14}
g_{4}
g_{18}
g_{9}
g_{-22}
g_{13}
g_{-25}		\varepsilon_{4}-\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{6}
\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 68(-1, -1, -1, -1, -1, 0)(0, 0, 0, 1, 1, 1)g_{16}
g_{20}
g_{11}
g_{23}
g_{15}
g_{-17}
g_{19}
g_{-21}		\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 78(0, 0, 0, -1, -1, -1)(1, 1, 1, 1, 1, 0)g_{21}
g_{-19}
g_{17}
g_{-15}
g_{-23}
g_{-11}
g_{-20}
g_{-16}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
Module 86(-1, 0, 0, 0, 0, 0)(0, 0, 1, 2, 1, 1)g_{24}
g_{26}
g_{-12}
g_{28}
g_{-7}
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 98(0, 0, 0, -1, -1, 0)(1, 1, 1, 1, 1, 1)g_{25}
g_{-13}
g_{22}
g_{-9}
g_{-18}
g_{-4}
g_{-14}
g_{-10}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
Module 1015(-1, -1, -1, -2, -1, -1)(1, 1, 1, 2, 1, 1)g_{27}
g_{-8}
g_{29}
g_{-3}
g_{-2}
g_{30}
-h_{3}
-h_{2}
h_{6}+h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-30}
g_{2}
g_{3}
g_{-29}
g_{8}
g_{-27}		\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 111(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}+2h_{4}+h_{3}-h_{1}0
Module 121(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{6}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 8
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: 0
Parabolically induced by A^{1}_3
Potential Dynkin type extensions: A^{1}_3+A^{1}_2, A^{1}_3+2A^{1}_1,