Type: \(\displaystyle 2A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 2A^{1}_1\))
Simple basis: 2 vectors: (1, 2, 2, 2, 2, 2), (1, 0, 0, 0, 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}: B^{1}_4
simple basis centralizer: 4 vectors: (0, 0, 0, 1, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1)
Number of k-submodules of g: 47
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{2}}+9V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+36V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, -1, -2, -2, -2)(0, 0, -1, -2, -2, -2)g_{-30}-\varepsilon_{3}-\varepsilon_{4}
Module 21(0, 0, -1, -1, -2, -2)(0, 0, -1, -1, -2, -2)g_{-27}-\varepsilon_{3}-\varepsilon_{5}
Module 31(0, 0, 0, -1, -2, -2)(0, 0, 0, -1, -2, -2)g_{-24}-\varepsilon_{4}-\varepsilon_{5}
Module 41(0, 0, -1, -1, -1, -2)(0, 0, -1, -1, -1, -2)g_{-23}-\varepsilon_{3}-\varepsilon_{6}
Module 51(0, 0, 0, -1, -1, -2)(0, 0, 0, -1, -1, -2)g_{-20}-\varepsilon_{4}-\varepsilon_{6}
Module 61(0, 0, -1, -1, -1, -1)(0, 0, -1, -1, -1, -1)g_{-19}-\varepsilon_{3}
Module 71(0, 0, 0, 0, -1, -2)(0, 0, 0, 0, -1, -2)g_{-16}-\varepsilon_{5}-\varepsilon_{6}
Module 81(0, 0, 0, -1, -1, -1)(0, 0, 0, -1, -1, -1)g_{-15}-\varepsilon_{4}
Module 91(0, 0, -1, -1, -1, 0)(0, 0, -1, -1, -1, 0)g_{-14}-\varepsilon_{3}+\varepsilon_{6}
Module 101(0, 0, 0, 0, -1, -1)(0, 0, 0, 0, -1, -1)g_{-11}-\varepsilon_{5}
Module 111(0, 0, 0, -1, -1, 0)(0, 0, 0, -1, -1, 0)g_{-10}-\varepsilon_{4}+\varepsilon_{6}
Module 121(0, 0, -1, -1, 0, 0)(0, 0, -1, -1, 0, 0)g_{-9}-\varepsilon_{3}+\varepsilon_{5}
Module 131(0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, -1)g_{-6}-\varepsilon_{6}
Module 141(0, 0, 0, 0, -1, 0)(0, 0, 0, 0, -1, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 151(0, 0, 0, -1, 0, 0)(0, 0, 0, -1, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 161(0, 0, -1, 0, 0, 0)(0, 0, -1, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 173(-1, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0)g_{1}
h_{1}
g_{-1}		\varepsilon_{1}-\varepsilon_{2}
0
-\varepsilon_{1}+\varepsilon_{2}
Module 181(0, 0, 1, 0, 0, 0)(0, 0, 1, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 191(0, 0, 0, 1, 0, 0)(0, 0, 0, 1, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 201(0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 1, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 211(0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 1)g_{6}\varepsilon_{6}
Module 224(-1, -1, -2, -2, -2, -2)(1, 1, 0, 0, 0, 0)g_{7}
g_{-34}
g_{2}
g_{-35}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 231(0, 0, 1, 1, 0, 0)(0, 0, 1, 1, 0, 0)g_{9}\varepsilon_{3}-\varepsilon_{5}
Module 241(0, 0, 0, 1, 1, 0)(0, 0, 0, 1, 1, 0)g_{10}\varepsilon_{4}-\varepsilon_{6}
Module 251(0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 1, 1)g_{11}\varepsilon_{5}
Module 264(-1, -1, -1, -2, -2, -2)(1, 1, 1, 0, 0, 0)g_{12}
g_{-32}
g_{8}
g_{-33}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 271(0, 0, 1, 1, 1, 0)(0, 0, 1, 1, 1, 0)g_{14}\varepsilon_{3}-\varepsilon_{6}
Module 281(0, 0, 0, 1, 1, 1)(0, 0, 0, 1, 1, 1)g_{15}\varepsilon_{4}
Module 291(0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 1, 2)g_{16}\varepsilon_{5}+\varepsilon_{6}
Module 304(-1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 0, 0)g_{17}
g_{-29}
g_{13}
g_{-31}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 311(0, 0, 1, 1, 1, 1)(0, 0, 1, 1, 1, 1)g_{19}\varepsilon_{3}
Module 321(0, 0, 0, 1, 1, 2)(0, 0, 0, 1, 1, 2)g_{20}\varepsilon_{4}+\varepsilon_{6}
Module 334(-1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 0)g_{21}
g_{-26}
g_{18}
g_{-28}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 341(0, 0, 1, 1, 1, 2)(0, 0, 1, 1, 1, 2)g_{23}\varepsilon_{3}+\varepsilon_{6}
Module 351(0, 0, 0, 1, 2, 2)(0, 0, 0, 1, 2, 2)g_{24}\varepsilon_{4}+\varepsilon_{5}
Module 364(-1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1)g_{25}
g_{-22}
g_{22}
g_{-25}		\varepsilon_{1}
-\varepsilon_{2}
\varepsilon_{2}
-\varepsilon_{1}
Module 371(0, 0, 1, 1, 2, 2)(0, 0, 1, 1, 2, 2)g_{27}\varepsilon_{3}+\varepsilon_{5}
Module 384(-1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 2)g_{28}
g_{-18}
g_{26}
g_{-21}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 391(0, 0, 1, 2, 2, 2)(0, 0, 1, 2, 2, 2)g_{30}\varepsilon_{3}+\varepsilon_{4}
Module 404(-1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 2, 2)g_{31}
g_{-13}
g_{29}
g_{-17}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 414(-1, -1, -1, 0, 0, 0)(1, 1, 1, 2, 2, 2)g_{33}
g_{-8}
g_{32}
g_{-12}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 424(-1, -1, 0, 0, 0, 0)(1, 1, 2, 2, 2, 2)g_{35}
g_{-2}
g_{34}
g_{-7}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
Module 433(-1, -2, -2, -2, -2, -2)(1, 2, 2, 2, 2, 2)g_{36}
2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-36}		\varepsilon_{1}+\varepsilon_{2}
0
-\varepsilon_{1}-\varepsilon_{2}
Module 441(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{3}0
Module 451(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{4}0
Module 461(0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0)h_{5}0
Module 471(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: 19
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 A^{1}_1
Potential Dynkin type extensions: 3A^{1}_1,