Type: \(\displaystyle A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{1}_1\))
Simple basis: 1 vectors: (1, 1, 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}: A^{1}_6
simple basis centralizer: 6 vectors: (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0)
Number of k-submodules of g: 64
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{1}}+14V_{\omega_{1}}+49V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, -1, -1, -1, -1, -1, -1, 0)(0, -1, -1, -1, -1, -1, -1, 0)g_{-32}-\varepsilon_{2}+\varepsilon_{8}
Module 21(0, 0, -1, -1, -1, -1, -1, 0)(0, 0, -1, -1, -1, -1, -1, 0)g_{-29}-\varepsilon_{3}+\varepsilon_{8}
Module 31(0, -1, -1, -1, -1, -1, 0, 0)(0, -1, -1, -1, -1, -1, 0, 0)g_{-28}-\varepsilon_{2}+\varepsilon_{7}
Module 41(0, 0, 0, -1, -1, -1, -1, 0)(0, 0, 0, -1, -1, -1, -1, 0)g_{-25}-\varepsilon_{4}+\varepsilon_{8}
Module 51(0, 0, -1, -1, -1, -1, 0, 0)(0, 0, -1, -1, -1, -1, 0, 0)g_{-24}-\varepsilon_{3}+\varepsilon_{7}
Module 61(0, -1, -1, -1, -1, 0, 0, 0)(0, -1, -1, -1, -1, 0, 0, 0)g_{-23}-\varepsilon_{2}+\varepsilon_{6}
Module 71(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, -1, -1, -1, 0)g_{-20}-\varepsilon_{5}+\varepsilon_{8}
Module 81(0, 0, 0, -1, -1, -1, 0, 0)(0, 0, 0, -1, -1, -1, 0, 0)g_{-19}-\varepsilon_{4}+\varepsilon_{7}
Module 91(0, 0, -1, -1, -1, 0, 0, 0)(0, 0, -1, -1, -1, 0, 0, 0)g_{-18}-\varepsilon_{3}+\varepsilon_{6}
Module 101(0, -1, -1, -1, 0, 0, 0, 0)(0, -1, -1, -1, 0, 0, 0, 0)g_{-17}-\varepsilon_{2}+\varepsilon_{5}
Module 111(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 0, -1, -1, 0)g_{-14}-\varepsilon_{6}+\varepsilon_{8}
Module 121(0, 0, 0, 0, -1, -1, 0, 0)(0, 0, 0, 0, -1, -1, 0, 0)g_{-13}-\varepsilon_{5}+\varepsilon_{7}
Module 131(0, 0, 0, -1, -1, 0, 0, 0)(0, 0, 0, -1, -1, 0, 0, 0)g_{-12}-\varepsilon_{4}+\varepsilon_{6}
Module 141(0, 0, -1, -1, 0, 0, 0, 0)(0, 0, -1, -1, 0, 0, 0, 0)g_{-11}-\varepsilon_{3}+\varepsilon_{5}
Module 151(0, -1, -1, 0, 0, 0, 0, 0)(0, -1, -1, 0, 0, 0, 0, 0)g_{-10}-\varepsilon_{2}+\varepsilon_{4}
Module 161(0, 0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, 0, -1, 0)g_{-7}-\varepsilon_{7}+\varepsilon_{8}
Module 171(0, 0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, 0, -1, 0, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 181(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, -1, 0, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 191(0, 0, 0, -1, 0, 0, 0, 0)(0, 0, 0, -1, 0, 0, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 201(0, 0, -1, 0, 0, 0, 0, 0)(0, 0, -1, 0, 0, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 211(0, -1, 0, 0, 0, 0, 0, 0)(0, -1, 0, 0, 0, 0, 0, 0)g_{-2}-\varepsilon_{2}+\varepsilon_{3}
Module 222(0, -1, -1, -1, -1, -1, -1, -1)(1, 0, 0, 0, 0, 0, 0, 0)g_{1}
g_{-35}		\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{2}+\varepsilon_{9}
Module 231(0, 1, 0, 0, 0, 0, 0, 0)(0, 1, 0, 0, 0, 0, 0, 0)g_{2}\varepsilon_{2}-\varepsilon_{3}
Module 241(0, 0, 1, 0, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 251(0, 0, 0, 1, 0, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 261(0, 0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 271(0, 0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 281(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}\varepsilon_{7}-\varepsilon_{8}
Module 292(-1, -1, -1, -1, -1, -1, -1, 0)(0, 0, 0, 0, 0, 0, 0, 1)g_{8}
g_{-34}		\varepsilon_{8}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{8}
Module 302(0, 0, -1, -1, -1, -1, -1, -1)(1, 1, 0, 0, 0, 0, 0, 0)g_{9}
g_{-33}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{3}+\varepsilon_{9}
Module 311(0, 1, 1, 0, 0, 0, 0, 0)(0, 1, 1, 0, 0, 0, 0, 0)g_{10}\varepsilon_{2}-\varepsilon_{4}
Module 321(0, 0, 1, 1, 0, 0, 0, 0)(0, 0, 1, 1, 0, 0, 0, 0)g_{11}\varepsilon_{3}-\varepsilon_{5}
Module 331(0, 0, 0, 1, 1, 0, 0, 0)(0, 0, 0, 1, 1, 0, 0, 0)g_{12}\varepsilon_{4}-\varepsilon_{6}
Module 341(0, 0, 0, 0, 1, 1, 0, 0)(0, 0, 0, 0, 1, 1, 0, 0)g_{13}\varepsilon_{5}-\varepsilon_{7}
Module 351(0, 0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}\varepsilon_{6}-\varepsilon_{8}
Module 362(-1, -1, -1, -1, -1, -1, 0, 0)(0, 0, 0, 0, 0, 0, 1, 1)g_{15}
g_{-31}		\varepsilon_{7}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{7}
Module 372(0, 0, 0, -1, -1, -1, -1, -1)(1, 1, 1, 0, 0, 0, 0, 0)g_{16}
g_{-30}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{4}+\varepsilon_{9}
Module 381(0, 1, 1, 1, 0, 0, 0, 0)(0, 1, 1, 1, 0, 0, 0, 0)g_{17}\varepsilon_{2}-\varepsilon_{5}
Module 391(0, 0, 1, 1, 1, 0, 0, 0)(0, 0, 1, 1, 1, 0, 0, 0)g_{18}\varepsilon_{3}-\varepsilon_{6}
Module 401(0, 0, 0, 1, 1, 1, 0, 0)(0, 0, 0, 1, 1, 1, 0, 0)g_{19}\varepsilon_{4}-\varepsilon_{7}
Module 411(0, 0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}\varepsilon_{5}-\varepsilon_{8}
Module 422(-1, -1, -1, -1, -1, 0, 0, 0)(0, 0, 0, 0, 0, 1, 1, 1)g_{21}
g_{-27}		\varepsilon_{6}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{6}
Module 432(0, 0, 0, 0, -1, -1, -1, -1)(1, 1, 1, 1, 0, 0, 0, 0)g_{22}
g_{-26}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{5}+\varepsilon_{9}
Module 441(0, 1, 1, 1, 1, 0, 0, 0)(0, 1, 1, 1, 1, 0, 0, 0)g_{23}\varepsilon_{2}-\varepsilon_{6}
Module 451(0, 0, 1, 1, 1, 1, 0, 0)(0, 0, 1, 1, 1, 1, 0, 0)g_{24}\varepsilon_{3}-\varepsilon_{7}
Module 461(0, 0, 0, 1, 1, 1, 1, 0)(0, 0, 0, 1, 1, 1, 1, 0)g_{25}\varepsilon_{4}-\varepsilon_{8}
Module 472(-1, -1, -1, -1, 0, 0, 0, 0)(0, 0, 0, 0, 1, 1, 1, 1)g_{26}
g_{-22}		\varepsilon_{5}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{5}
Module 482(0, 0, 0, 0, 0, -1, -1, -1)(1, 1, 1, 1, 1, 0, 0, 0)g_{27}
g_{-21}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{6}+\varepsilon_{9}
Module 491(0, 1, 1, 1, 1, 1, 0, 0)(0, 1, 1, 1, 1, 1, 0, 0)g_{28}\varepsilon_{2}-\varepsilon_{7}
Module 501(0, 0, 1, 1, 1, 1, 1, 0)(0, 0, 1, 1, 1, 1, 1, 0)g_{29}\varepsilon_{3}-\varepsilon_{8}
Module 512(-1, -1, -1, 0, 0, 0, 0, 0)(0, 0, 0, 1, 1, 1, 1, 1)g_{30}
g_{-16}		\varepsilon_{4}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{4}
Module 522(0, 0, 0, 0, 0, 0, -1, -1)(1, 1, 1, 1, 1, 1, 0, 0)g_{31}
g_{-15}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{7}+\varepsilon_{9}
Module 531(0, 1, 1, 1, 1, 1, 1, 0)(0, 1, 1, 1, 1, 1, 1, 0)g_{32}\varepsilon_{2}-\varepsilon_{8}
Module 542(-1, -1, 0, 0, 0, 0, 0, 0)(0, 0, 1, 1, 1, 1, 1, 1)g_{33}
g_{-9}		\varepsilon_{3}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{3}
Module 552(0, 0, 0, 0, 0, 0, 0, -1)(1, 1, 1, 1, 1, 1, 1, 0)g_{34}
g_{-8}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{8}+\varepsilon_{9}
Module 562(-1, 0, 0, 0, 0, 0, 0, 0)(0, 1, 1, 1, 1, 1, 1, 1)g_{35}
g_{-1}		\varepsilon_{2}-\varepsilon_{9}
-\varepsilon_{1}+\varepsilon_{2}
Module 573(-1, -1, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 1, 1)g_{36}
h_{8}+h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}
g_{-36}		\varepsilon_{1}-\varepsilon_{9}
0
-\varepsilon_{1}+\varepsilon_{9}
Module 581(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{2}0
Module 591(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{3}0
Module 601(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{4}0
Module 611(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 621(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 631(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{7}0
Module 641(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{8}-h_{1}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 1
Heirs rejected due to not being maximally dominant: 54
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 54
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^{1}_2, 2A^{1}_1,