Type: \(\displaystyle A^{1}_3\) (Dynkin type computed to be: \(\displaystyle A^{1}_3\))
Simple basis: 3 vectors: (1, 2, 2, 2, 2, 2, 2), (0, -1, 0, 0, 0, 0, 0), (-1, 0, 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, 0, 1, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of k-submodules of g: 46
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{1}+\omega_{3}}+9V_{\omega_{2}}+36V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -1, -2, -2, -2)(0, 0, 0, -1, -2, -2, -2)g_{-37}-\varepsilon_{4}-\varepsilon_{5}
Module 21(0, 0, 0, -1, -1, -2, -2)(0, 0, 0, -1, -1, -2, -2)g_{-33}-\varepsilon_{4}-\varepsilon_{6}
Module 31(0, 0, 0, 0, -1, -2, -2)(0, 0, 0, 0, -1, -2, -2)g_{-29}-\varepsilon_{5}-\varepsilon_{6}
Module 41(0, 0, 0, -1, -1, -1, -2)(0, 0, 0, -1, -1, -1, -2)g_{-28}-\varepsilon_{4}-\varepsilon_{7}
Module 51(0, 0, 0, 0, -1, -1, -2)(0, 0, 0, 0, -1, -1, -2)g_{-24}-\varepsilon_{5}-\varepsilon_{7}
Module 61(0, 0, 0, -1, -1, -1, -1)(0, 0, 0, -1, -1, -1, -1)g_{-23}-\varepsilon_{4}
Module 71(0, 0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 0, -1, -2)g_{-19}-\varepsilon_{6}-\varepsilon_{7}
Module 81(0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 0, -1, -1, -1)g_{-18}-\varepsilon_{5}
Module 91(0, 0, 0, -1, -1, -1, 0)(0, 0, 0, -1, -1, -1, 0)g_{-17}-\varepsilon_{4}+\varepsilon_{7}
Module 101(0, 0, 0, 0, 0, -1, -1)(0, 0, 0, 0, 0, -1, -1)g_{-13}-\varepsilon_{6}
Module 111(0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, -1, -1, 0)g_{-12}-\varepsilon_{5}+\varepsilon_{7}
Module 121(0, 0, 0, -1, -1, 0, 0)(0, 0, 0, -1, -1, 0, 0)g_{-11}-\varepsilon_{4}+\varepsilon_{6}
Module 131(0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, -1)g_{-7}-\varepsilon_{7}
Module 141(0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, -1, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 151(0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, -1, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 161(0, 0, 0, -1, 0, 0, 0)(0, 0, 0, -1, 0, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 176(0, 0, -1, -2, -2, -2, -2)(0, 0, 1, 0, 0, 0, 0)g_{3}
g_{9}
g_{-46}
g_{14}
g_{-45}
g_{-43}		\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 181(0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 191(0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 1, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 201(0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 1, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 211(0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 1)g_{7}\varepsilon_{7}
Module 226(0, 0, -1, -1, -2, -2, -2)(0, 0, 1, 1, 0, 0, 0)g_{10}
g_{15}
g_{-44}
g_{20}
g_{-42}
g_{-40}		\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 231(0, 0, 0, 1, 1, 0, 0)(0, 0, 0, 1, 1, 0, 0)g_{11}\varepsilon_{4}-\varepsilon_{6}
Module 241(0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 1, 1, 0)g_{12}\varepsilon_{5}-\varepsilon_{7}
Module 251(0, 0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 0, 1, 1)g_{13}\varepsilon_{6}
Module 266(0, 0, -1, -1, -1, -2, -2)(0, 0, 1, 1, 1, 0, 0)g_{16}
g_{21}
g_{-41}
g_{25}
g_{-39}
g_{-36}		\varepsilon_{3}-\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
Module 271(0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 1, 1, 1, 0)g_{17}\varepsilon_{4}-\varepsilon_{7}
Module 281(0, 0, 0, 0, 1, 1, 1)(0, 0, 0, 0, 1, 1, 1)g_{18}\varepsilon_{5}
Module 291(0, 0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 0, 1, 2)g_{19}\varepsilon_{6}+\varepsilon_{7}
Module 306(0, 0, -1, -1, -1, -1, -2)(0, 0, 1, 1, 1, 1, 0)g_{22}
g_{26}
g_{-38}
g_{30}
g_{-35}
g_{-32}		\varepsilon_{3}-\varepsilon_{7}
\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
Module 311(0, 0, 0, 1, 1, 1, 1)(0, 0, 0, 1, 1, 1, 1)g_{23}\varepsilon_{4}
Module 321(0, 0, 0, 0, 1, 1, 2)(0, 0, 0, 0, 1, 1, 2)g_{24}\varepsilon_{5}+\varepsilon_{7}
Module 336(0, 0, -1, -1, -1, -1, -1)(0, 0, 1, 1, 1, 1, 1)g_{27}
g_{31}
g_{-34}
g_{34}
g_{-31}
g_{-27}		\varepsilon_{3}
\varepsilon_{2}
-\varepsilon_{1}
\varepsilon_{1}
-\varepsilon_{2}
-\varepsilon_{3}
Module 341(0, 0, 0, 1, 1, 1, 2)(0, 0, 0, 1, 1, 1, 2)g_{28}\varepsilon_{4}+\varepsilon_{7}
Module 351(0, 0, 0, 0, 1, 2, 2)(0, 0, 0, 0, 1, 2, 2)g_{29}\varepsilon_{5}+\varepsilon_{6}
Module 366(0, 0, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 2)g_{32}
g_{35}
g_{-30}
g_{38}
g_{-26}
g_{-22}		\varepsilon_{3}+\varepsilon_{7}
\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
Module 371(0, 0, 0, 1, 1, 2, 2)(0, 0, 0, 1, 1, 2, 2)g_{33}\varepsilon_{4}+\varepsilon_{6}
Module 386(0, 0, -1, -1, -1, 0, 0)(0, 0, 1, 1, 1, 2, 2)g_{36}
g_{39}
g_{-25}
g_{41}
g_{-21}
g_{-16}		\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
Module 391(0, 0, 0, 1, 2, 2, 2)(0, 0, 0, 1, 2, 2, 2)g_{37}\varepsilon_{4}+\varepsilon_{5}
Module 406(0, 0, -1, -1, 0, 0, 0)(0, 0, 1, 1, 2, 2, 2)g_{40}
g_{42}
g_{-20}
g_{44}
g_{-15}
g_{-10}		\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
Module 416(0, 0, -1, 0, 0, 0, 0)(0, 0, 1, 2, 2, 2, 2)g_{43}
g_{45}
g_{-14}
g_{46}
g_{-9}
g_{-3}		\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
Module 4215(0, -1, -2, -2, -2, -2, -2)(0, 1, 2, 2, 2, 2, 2)g_{47}
g_{-8}
g_{48}
g_{-1}
g_{-2}
g_{49}
-h_{1}
-h_{2}
2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-49}
g_{2}
g_{1}
g_{-48}
g_{8}
g_{-47}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{2}
0
0
0
-\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{2}-\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 431(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{4}0
Module 441(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 451(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 461(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{7}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 2
Heirs rejected due to not being maximally dominant: 37
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 37
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, B^{1}_4, A^{1}_3+A^{2}_1, D^{1}_4, A^{1}_3+A^{1}_1,