Type: \(\displaystyle A^{2}_1\) (Dynkin type computed to be: \(\displaystyle A^{2}_1\))
Simple basis: 1 vectors: (1, 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}_5
simple basis centralizer: 5 vectors: (0, 0, 1, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 2), (0, 0, 0, 0, 1, 0)
Number of k-submodules of g: 56
Module decomposition, fundamental coords over k: \(\displaystyle 11V_{2\omega_{1}}+45V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, -1, -2, -2, -2, -2)(0, -1, -2, -2, -2, -2)g_{-34}-\varepsilon_{2}-\varepsilon_{3}
Module 21(0, -1, -1, -2, -2, -2)(0, -1, -1, -2, -2, -2)g_{-32}-\varepsilon_{2}-\varepsilon_{4}
Module 31(0, 0, -1, -2, -2, -2)(0, 0, -1, -2, -2, -2)g_{-30}-\varepsilon_{3}-\varepsilon_{4}
Module 41(0, -1, -1, -1, -2, -2)(0, -1, -1, -1, -2, -2)g_{-29}-\varepsilon_{2}-\varepsilon_{5}
Module 51(0, 0, -1, -1, -2, -2)(0, 0, -1, -1, -2, -2)g_{-27}-\varepsilon_{3}-\varepsilon_{5}
Module 61(0, -1, -1, -1, -1, -2)(0, -1, -1, -1, -1, -2)g_{-26}-\varepsilon_{2}-\varepsilon_{6}
Module 71(0, 0, 0, -1, -2, -2)(0, 0, 0, -1, -2, -2)g_{-24}-\varepsilon_{4}-\varepsilon_{5}
Module 81(0, 0, -1, -1, -1, -2)(0, 0, -1, -1, -1, -2)g_{-23}-\varepsilon_{3}-\varepsilon_{6}
Module 91(0, 0, 0, -1, -1, -2)(0, 0, 0, -1, -1, -2)g_{-20}-\varepsilon_{4}-\varepsilon_{6}
Module 101(0, -1, -1, -1, -1, 0)(0, -1, -1, -1, -1, 0)g_{-18}-\varepsilon_{2}+\varepsilon_{6}
Module 111(0, 0, 0, 0, -1, -2)(0, 0, 0, 0, -1, -2)g_{-16}-\varepsilon_{5}-\varepsilon_{6}
Module 121(0, 0, -1, -1, -1, 0)(0, 0, -1, -1, -1, 0)g_{-14}-\varepsilon_{3}+\varepsilon_{6}
Module 131(0, -1, -1, -1, 0, 0)(0, -1, -1, -1, 0, 0)g_{-13}-\varepsilon_{2}+\varepsilon_{5}
Module 141(0, 0, 0, -1, -1, 0)(0, 0, 0, -1, -1, 0)g_{-10}-\varepsilon_{4}+\varepsilon_{6}
Module 151(0, 0, -1, -1, 0, 0)(0, 0, -1, -1, 0, 0)g_{-9}-\varepsilon_{3}+\varepsilon_{5}
Module 161(0, -1, -1, 0, 0, 0)(0, -1, -1, 0, 0, 0)g_{-8}-\varepsilon_{2}+\varepsilon_{4}
Module 171(0, 0, 0, 0, -1, 0)(0, 0, 0, 0, -1, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 181(0, 0, 0, -1, 0, 0)(0, 0, 0, -1, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 191(0, 0, -1, 0, 0, 0)(0, 0, -1, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 201(0, -1, 0, 0, 0, 0)(0, -1, 0, 0, 0, 0)g_{-2}-\varepsilon_{2}+\varepsilon_{3}
Module 213(-1, -2, -2, -2, -2, -2)(1, 0, 0, 0, 0, 0)g_{1}
g_{-22}
g_{-36}		\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{2}
Module 221(0, 1, 0, 0, 0, 0)(0, 1, 0, 0, 0, 0)g_{2}\varepsilon_{2}-\varepsilon_{3}
Module 231(0, 0, 1, 0, 0, 0)(0, 0, 1, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 241(0, 0, 0, 1, 0, 0)(0, 0, 0, 1, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 251(0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 1, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 263(-1, -1, -2, -2, -2, -2)(1, 1, 0, 0, 0, 0)g_{7}
g_{-19}
g_{-35}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 271(0, 1, 1, 0, 0, 0)(0, 1, 1, 0, 0, 0)g_{8}\varepsilon_{2}-\varepsilon_{4}
Module 281(0, 0, 1, 1, 0, 0)(0, 0, 1, 1, 0, 0)g_{9}\varepsilon_{3}-\varepsilon_{5}
Module 291(0, 0, 0, 1, 1, 0)(0, 0, 0, 1, 1, 0)g_{10}\varepsilon_{4}-\varepsilon_{6}
Module 303(-1, -1, -1, -2, -2, -2)(1, 1, 1, 0, 0, 0)g_{12}
g_{-15}
g_{-33}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 311(0, 1, 1, 1, 0, 0)(0, 1, 1, 1, 0, 0)g_{13}\varepsilon_{2}-\varepsilon_{5}
Module 321(0, 0, 1, 1, 1, 0)(0, 0, 1, 1, 1, 0)g_{14}\varepsilon_{3}-\varepsilon_{6}
Module 331(0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 1, 2)g_{16}\varepsilon_{5}+\varepsilon_{6}
Module 343(-1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 0, 0)g_{17}
g_{-11}
g_{-31}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 351(0, 1, 1, 1, 1, 0)(0, 1, 1, 1, 1, 0)g_{18}\varepsilon_{2}-\varepsilon_{6}
Module 361(0, 0, 0, 1, 1, 2)(0, 0, 0, 1, 1, 2)g_{20}\varepsilon_{4}+\varepsilon_{6}
Module 373(-1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 0)g_{21}
g_{-6}
g_{-28}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 381(0, 0, 1, 1, 1, 2)(0, 0, 1, 1, 1, 2)g_{23}\varepsilon_{3}+\varepsilon_{6}
Module 391(0, 0, 0, 1, 2, 2)(0, 0, 0, 1, 2, 2)g_{24}\varepsilon_{4}+\varepsilon_{5}
Module 403(-1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1)g_{25}
h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}
g_{-25}		\varepsilon_{1}
0
-\varepsilon_{1}
Module 411(0, 1, 1, 1, 1, 2)(0, 1, 1, 1, 1, 2)g_{26}\varepsilon_{2}+\varepsilon_{6}
Module 421(0, 0, 1, 1, 2, 2)(0, 0, 1, 1, 2, 2)g_{27}\varepsilon_{3}+\varepsilon_{5}
Module 433(-1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 2)g_{28}
g_{6}
g_{-21}		\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 441(0, 1, 1, 1, 2, 2)(0, 1, 1, 1, 2, 2)g_{29}\varepsilon_{2}+\varepsilon_{5}
Module 451(0, 0, 1, 2, 2, 2)(0, 0, 1, 2, 2, 2)g_{30}\varepsilon_{3}+\varepsilon_{4}
Module 463(-1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 2, 2)g_{31}
g_{11}
g_{-17}		\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 471(0, 1, 1, 2, 2, 2)(0, 1, 1, 2, 2, 2)g_{32}\varepsilon_{2}+\varepsilon_{4}
Module 483(-1, -1, -1, 0, 0, 0)(1, 1, 1, 2, 2, 2)g_{33}
g_{15}
g_{-12}		\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 491(0, 1, 2, 2, 2, 2)(0, 1, 2, 2, 2, 2)g_{34}\varepsilon_{2}+\varepsilon_{3}
Module 503(-1, -1, 0, 0, 0, 0)(1, 1, 2, 2, 2, 2)g_{35}
g_{19}
g_{-7}		\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
Module 513(-1, 0, 0, 0, 0, 0)(1, 2, 2, 2, 2, 2)g_{36}
g_{22}
g_{-1}		\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{2}
Module 521(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{2}0
Module 531(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{3}0
Module 541(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{4}0
Module 551(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}0
Module 561(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: 11
Heirs rejected due to not being maximally dominant: 39
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 39
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,