Type: \(\displaystyle A^{2}_1\) (Dynkin type computed to be: \(\displaystyle A^{2}_1\))
Simple basis: 1 vectors: (1, 1, 1, 1, 1)
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}: D^{1}_4
simple basis centralizer: 4 vectors: (0, 0, 1, 0, 0), (0, 1, 0, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 1, 2)
Number of k-submodules of g: 37
Module decomposition, fundamental coords over k: \(\displaystyle 9V_{2\omega_{1}}+28V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, -1, -2, -2, -2)(0, -1, -2, -2, -2)g_{-23}-\varepsilon_{2}-\varepsilon_{3}
Module 21(0, -1, -1, -2, -2)(0, -1, -1, -2, -2)g_{-21}-\varepsilon_{2}-\varepsilon_{4}
Module 31(0, 0, -1, -2, -2)(0, 0, -1, -2, -2)g_{-19}-\varepsilon_{3}-\varepsilon_{4}
Module 41(0, -1, -1, -1, -2)(0, -1, -1, -1, -2)g_{-18}-\varepsilon_{2}-\varepsilon_{5}
Module 51(0, 0, -1, -1, -2)(0, 0, -1, -1, -2)g_{-16}-\varepsilon_{3}-\varepsilon_{5}
Module 61(0, 0, 0, -1, -2)(0, 0, 0, -1, -2)g_{-13}-\varepsilon_{4}-\varepsilon_{5}
Module 71(0, -1, -1, -1, 0)(0, -1, -1, -1, 0)g_{-11}-\varepsilon_{2}+\varepsilon_{5}
Module 81(0, 0, -1, -1, 0)(0, 0, -1, -1, 0)g_{-8}-\varepsilon_{3}+\varepsilon_{5}
Module 91(0, -1, -1, 0, 0)(0, -1, -1, 0, 0)g_{-7}-\varepsilon_{2}+\varepsilon_{4}
Module 101(0, 0, 0, -1, 0)(0, 0, 0, -1, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 111(0, 0, -1, 0, 0)(0, 0, -1, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 121(0, -1, 0, 0, 0)(0, -1, 0, 0, 0)g_{-2}-\varepsilon_{2}+\varepsilon_{3}
Module 133(-1, -2, -2, -2, -2)(1, 0, 0, 0, 0)g_{1}
g_{-15}
g_{-25}		\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{2}
Module 141(0, 1, 0, 0, 0)(0, 1, 0, 0, 0)g_{2}\varepsilon_{2}-\varepsilon_{3}
Module 151(0, 0, 1, 0, 0)(0, 0, 1, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 161(0, 0, 0, 1, 0)(0, 0, 0, 1, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 173(-1, -1, -2, -2, -2)(1, 1, 0, 0, 0)g_{6}
g_{-12}
g_{-24}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 181(0, 1, 1, 0, 0)(0, 1, 1, 0, 0)g_{7}\varepsilon_{2}-\varepsilon_{4}
Module 191(0, 0, 1, 1, 0)(0, 0, 1, 1, 0)g_{8}\varepsilon_{3}-\varepsilon_{5}
Module 203(-1, -1, -1, -2, -2)(1, 1, 1, 0, 0)g_{10}
g_{-9}
g_{-22}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 211(0, 1, 1, 1, 0)(0, 1, 1, 1, 0)g_{11}\varepsilon_{2}-\varepsilon_{5}
Module 221(0, 0, 0, 1, 2)(0, 0, 0, 1, 2)g_{13}\varepsilon_{4}+\varepsilon_{5}
Module 233(-1, -1, -1, -1, -2)(1, 1, 1, 1, 0)g_{14}
g_{-5}
g_{-20}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 241(0, 0, 1, 1, 2)(0, 0, 1, 1, 2)g_{16}\varepsilon_{3}+\varepsilon_{5}
Module 253(-1, -1, -1, -1, -1)(1, 1, 1, 1, 1)g_{17}
h_{5}+h_{4}+h_{3}+h_{2}+h_{1}
g_{-17}		\varepsilon_{1}
0
-\varepsilon_{1}
Module 261(0, 1, 1, 1, 2)(0, 1, 1, 1, 2)g_{18}\varepsilon_{2}+\varepsilon_{5}
Module 271(0, 0, 1, 2, 2)(0, 0, 1, 2, 2)g_{19}\varepsilon_{3}+\varepsilon_{4}
Module 283(-1, -1, -1, -1, 0)(1, 1, 1, 1, 2)g_{20}
g_{5}
g_{-14}		\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 291(0, 1, 1, 2, 2)(0, 1, 1, 2, 2)g_{21}\varepsilon_{2}+\varepsilon_{4}
Module 303(-1, -1, -1, 0, 0)(1, 1, 1, 2, 2)g_{22}
g_{9}
g_{-10}		\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 311(0, 1, 2, 2, 2)(0, 1, 2, 2, 2)g_{23}\varepsilon_{2}+\varepsilon_{3}
Module 323(-1, -1, 0, 0, 0)(1, 1, 2, 2, 2)g_{24}
g_{12}
g_{-6}		\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
Module 333(-1, 0, 0, 0, 0)(1, 2, 2, 2, 2)g_{25}
g_{15}
g_{-1}		\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{2}
Module 341(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{2}0
Module 351(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{3}0
Module 361(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{4}0
Module 371(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{5}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 9
Heirs rejected due to not being maximally dominant: 23
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 23
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by 0
Potential Dynkin type extensions: A^{2}_2, B^{2}_2, 2A^{2}_1, A^{2}_1+A^{1}_1,