Type: \(\displaystyle A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{1}_1\))
Simple basis: 1 vectors: (1, 2, 2, 2, 2, 2, 2, 2)
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}_6+A^{1}_1
simple basis centralizer: 7 vectors: (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1), (1, 0, 0, 0, 0, 0, 0, 0)
Number of k-submodules of g: 108
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{1}}+26V_{\omega_{1}}+81V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, -1, -2, -2, -2, -2, -2)(0, 0, -1, -2, -2, -2, -2, -2)g_{-58}-\varepsilon_{3}-\varepsilon_{4}
Module 21(0, 0, -1, -1, -2, -2, -2, -2)(0, 0, -1, -1, -2, -2, -2, -2)g_{-55}-\varepsilon_{3}-\varepsilon_{5}
Module 31(0, 0, 0, -1, -2, -2, -2, -2)(0, 0, 0, -1, -2, -2, -2, -2)g_{-52}-\varepsilon_{4}-\varepsilon_{5}
Module 41(0, 0, -1, -1, -1, -2, -2, -2)(0, 0, -1, -1, -1, -2, -2, -2)g_{-51}-\varepsilon_{3}-\varepsilon_{6}
Module 51(0, 0, 0, -1, -1, -2, -2, -2)(0, 0, 0, -1, -1, -2, -2, -2)g_{-48}-\varepsilon_{4}-\varepsilon_{6}
Module 61(0, 0, -1, -1, -1, -1, -2, -2)(0, 0, -1, -1, -1, -1, -2, -2)g_{-47}-\varepsilon_{3}-\varepsilon_{7}
Module 71(0, 0, 0, 0, -1, -2, -2, -2)(0, 0, 0, 0, -1, -2, -2, -2)g_{-44}-\varepsilon_{5}-\varepsilon_{6}
Module 81(0, 0, 0, -1, -1, -1, -2, -2)(0, 0, 0, -1, -1, -1, -2, -2)g_{-43}-\varepsilon_{4}-\varepsilon_{7}
Module 91(0, 0, -1, -1, -1, -1, -1, -2)(0, 0, -1, -1, -1, -1, -1, -2)g_{-42}-\varepsilon_{3}-\varepsilon_{8}
Module 101(0, 0, 0, 0, -1, -1, -2, -2)(0, 0, 0, 0, -1, -1, -2, -2)g_{-39}-\varepsilon_{5}-\varepsilon_{7}
Module 111(0, 0, 0, -1, -1, -1, -1, -2)(0, 0, 0, -1, -1, -1, -1, -2)g_{-38}-\varepsilon_{4}-\varepsilon_{8}
Module 121(0, 0, -1, -1, -1, -1, -1, -1)(0, 0, -1, -1, -1, -1, -1, -1)g_{-37}-\varepsilon_{3}
Module 131(0, 0, 0, 0, 0, -1, -2, -2)(0, 0, 0, 0, 0, -1, -2, -2)g_{-34}-\varepsilon_{6}-\varepsilon_{7}
Module 141(0, 0, 0, 0, -1, -1, -1, -2)(0, 0, 0, 0, -1, -1, -1, -2)g_{-33}-\varepsilon_{5}-\varepsilon_{8}
Module 151(0, 0, 0, -1, -1, -1, -1, -1)(0, 0, 0, -1, -1, -1, -1, -1)g_{-32}-\varepsilon_{4}
Module 161(0, 0, -1, -1, -1, -1, -1, 0)(0, 0, -1, -1, -1, -1, -1, 0)g_{-31}-\varepsilon_{3}+\varepsilon_{8}
Module 171(0, 0, 0, 0, 0, -1, -1, -2)(0, 0, 0, 0, 0, -1, -1, -2)g_{-28}-\varepsilon_{6}-\varepsilon_{8}
Module 181(0, 0, 0, 0, -1, -1, -1, -1)(0, 0, 0, 0, -1, -1, -1, -1)g_{-27}-\varepsilon_{5}
Module 191(0, 0, 0, -1, -1, -1, -1, 0)(0, 0, 0, -1, -1, -1, -1, 0)g_{-26}-\varepsilon_{4}+\varepsilon_{8}
Module 201(0, 0, -1, -1, -1, -1, 0, 0)(0, 0, -1, -1, -1, -1, 0, 0)g_{-25}-\varepsilon_{3}+\varepsilon_{7}
Module 211(0, 0, 0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 0, 0, -1, -2)g_{-22}-\varepsilon_{7}-\varepsilon_{8}
Module 221(0, 0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 0, 0, -1, -1, -1)g_{-21}-\varepsilon_{6}
Module 231(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, -1, -1, -1, 0)g_{-20}-\varepsilon_{5}+\varepsilon_{8}
Module 241(0, 0, 0, -1, -1, -1, 0, 0)(0, 0, 0, -1, -1, -1, 0, 0)g_{-19}-\varepsilon_{4}+\varepsilon_{7}
Module 251(0, 0, -1, -1, -1, 0, 0, 0)(0, 0, -1, -1, -1, 0, 0, 0)g_{-18}-\varepsilon_{3}+\varepsilon_{6}
Module 261(0, 0, 0, 0, 0, 0, -1, -1)(0, 0, 0, 0, 0, 0, -1, -1)g_{-15}-\varepsilon_{7}
Module 271(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 0, -1, -1, 0)g_{-14}-\varepsilon_{6}+\varepsilon_{8}
Module 281(0, 0, 0, 0, -1, -1, 0, 0)(0, 0, 0, 0, -1, -1, 0, 0)g_{-13}-\varepsilon_{5}+\varepsilon_{7}
Module 291(0, 0, 0, -1, -1, 0, 0, 0)(0, 0, 0, -1, -1, 0, 0, 0)g_{-12}-\varepsilon_{4}+\varepsilon_{6}
Module 301(0, 0, -1, -1, 0, 0, 0, 0)(0, 0, -1, -1, 0, 0, 0, 0)g_{-11}-\varepsilon_{3}+\varepsilon_{5}
Module 311(0, 0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, 0, -1)g_{-8}-\varepsilon_{8}
Module 321(0, 0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, 0, -1, 0)g_{-7}-\varepsilon_{7}+\varepsilon_{8}
Module 331(0, 0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, 0, -1, 0, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 341(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, -1, 0, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 351(0, 0, 0, -1, 0, 0, 0, 0)(0, 0, 0, -1, 0, 0, 0, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 361(0, 0, -1, 0, 0, 0, 0, 0)(0, 0, -1, 0, 0, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 371(-1, 0, 0, 0, 0, 0, 0, 0)(-1, 0, 0, 0, 0, 0, 0, 0)g_{-1}-\varepsilon_{1}+\varepsilon_{2}
Module 381(1, 0, 0, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0, 0, 0)g_{1}\varepsilon_{1}-\varepsilon_{2}
Module 392(-1, -1, -2, -2, -2, -2, -2, -2)(0, 1, 0, 0, 0, 0, 0, 0)g_{2}
g_{-63}		\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 401(0, 0, 1, 0, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 411(0, 0, 0, 1, 0, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 421(0, 0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 431(0, 0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 441(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}\varepsilon_{7}-\varepsilon_{8}
Module 451(0, 0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 0, 1)g_{8}\varepsilon_{8}
Module 462(0, -1, -2, -2, -2, -2, -2, -2)(1, 1, 0, 0, 0, 0, 0, 0)g_{9}
g_{-62}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 472(-1, -1, -1, -2, -2, -2, -2, -2)(0, 1, 1, 0, 0, 0, 0, 0)g_{10}
g_{-61}		\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 481(0, 0, 1, 1, 0, 0, 0, 0)(0, 0, 1, 1, 0, 0, 0, 0)g_{11}\varepsilon_{3}-\varepsilon_{5}
Module 491(0, 0, 0, 1, 1, 0, 0, 0)(0, 0, 0, 1, 1, 0, 0, 0)g_{12}\varepsilon_{4}-\varepsilon_{6}
Module 501(0, 0, 0, 0, 1, 1, 0, 0)(0, 0, 0, 0, 1, 1, 0, 0)g_{13}\varepsilon_{5}-\varepsilon_{7}
Module 511(0, 0, 0, 0, 0, 1, 1, 0)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}\varepsilon_{6}-\varepsilon_{8}
Module 521(0, 0, 0, 0, 0, 0, 1, 1)(0, 0, 0, 0, 0, 0, 1, 1)g_{15}\varepsilon_{7}
Module 532(0, -1, -1, -2, -2, -2, -2, -2)(1, 1, 1, 0, 0, 0, 0, 0)g_{16}
g_{-60}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
Module 542(-1, -1, -1, -1, -2, -2, -2, -2)(0, 1, 1, 1, 0, 0, 0, 0)g_{17}
g_{-59}		\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 551(0, 0, 1, 1, 1, 0, 0, 0)(0, 0, 1, 1, 1, 0, 0, 0)g_{18}\varepsilon_{3}-\varepsilon_{6}
Module 561(0, 0, 0, 1, 1, 1, 0, 0)(0, 0, 0, 1, 1, 1, 0, 0)g_{19}\varepsilon_{4}-\varepsilon_{7}
Module 571(0, 0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}\varepsilon_{5}-\varepsilon_{8}
Module 581(0, 0, 0, 0, 0, 1, 1, 1)(0, 0, 0, 0, 0, 1, 1, 1)g_{21}\varepsilon_{6}
Module 591(0, 0, 0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 0, 0, 1, 2)g_{22}\varepsilon_{7}+\varepsilon_{8}
Module 602(0, -1, -1, -1, -2, -2, -2, -2)(1, 1, 1, 1, 0, 0, 0, 0)g_{23}
g_{-57}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
Module 612(-1, -1, -1, -1, -1, -2, -2, -2)(0, 1, 1, 1, 1, 0, 0, 0)g_{24}
g_{-56}		\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 621(0, 0, 1, 1, 1, 1, 0, 0)(0, 0, 1, 1, 1, 1, 0, 0)g_{25}\varepsilon_{3}-\varepsilon_{7}
Module 631(0, 0, 0, 1, 1, 1, 1, 0)(0, 0, 0, 1, 1, 1, 1, 0)g_{26}\varepsilon_{4}-\varepsilon_{8}
Module 641(0, 0, 0, 0, 1, 1, 1, 1)(0, 0, 0, 0, 1, 1, 1, 1)g_{27}\varepsilon_{5}
Module 651(0, 0, 0, 0, 0, 1, 1, 2)(0, 0, 0, 0, 0, 1, 1, 2)g_{28}\varepsilon_{6}+\varepsilon_{8}
Module 662(0, -1, -1, -1, -1, -2, -2, -2)(1, 1, 1, 1, 1, 0, 0, 0)g_{29}
g_{-54}		\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
Module 672(-1, -1, -1, -1, -1, -1, -2, -2)(0, 1, 1, 1, 1, 1, 0, 0)g_{30}
g_{-53}		\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 681(0, 0, 1, 1, 1, 1, 1, 0)(0, 0, 1, 1, 1, 1, 1, 0)g_{31}\varepsilon_{3}-\varepsilon_{8}
Module 691(0, 0, 0, 1, 1, 1, 1, 1)(0, 0, 0, 1, 1, 1, 1, 1)g_{32}\varepsilon_{4}
Module 701(0, 0, 0, 0, 1, 1, 1, 2)(0, 0, 0, 0, 1, 1, 1, 2)g_{33}\varepsilon_{5}+\varepsilon_{8}
Module 711(0, 0, 0, 0, 0, 1, 2, 2)(0, 0, 0, 0, 0, 1, 2, 2)g_{34}\varepsilon_{6}+\varepsilon_{7}
Module 722(0, -1, -1, -1, -1, -1, -2, -2)(1, 1, 1, 1, 1, 1, 0, 0)g_{35}
g_{-50}		\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
Module 732(-1, -1, -1, -1, -1, -1, -1, -2)(0, 1, 1, 1, 1, 1, 1, 0)g_{36}
g_{-49}		\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{8}
Module 741(0, 0, 1, 1, 1, 1, 1, 1)(0, 0, 1, 1, 1, 1, 1, 1)g_{37}\varepsilon_{3}
Module 751(0, 0, 0, 1, 1, 1, 1, 2)(0, 0, 0, 1, 1, 1, 1, 2)g_{38}\varepsilon_{4}+\varepsilon_{8}
Module 761(0, 0, 0, 0, 1, 1, 2, 2)(0, 0, 0, 0, 1, 1, 2, 2)g_{39}\varepsilon_{5}+\varepsilon_{7}
Module 772(0, -1, -1, -1, -1, -1, -1, -2)(1, 1, 1, 1, 1, 1, 1, 0)g_{40}
g_{-46}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{2}-\varepsilon_{8}
Module 782(-1, -1, -1, -1, -1, -1, -1, -1)(0, 1, 1, 1, 1, 1, 1, 1)g_{41}
g_{-45}		\varepsilon_{2}
-\varepsilon_{1}
Module 791(0, 0, 1, 1, 1, 1, 1, 2)(0, 0, 1, 1, 1, 1, 1, 2)g_{42}\varepsilon_{3}+\varepsilon_{8}
Module 801(0, 0, 0, 1, 1, 1, 2, 2)(0, 0, 0, 1, 1, 1, 2, 2)g_{43}\varepsilon_{4}+\varepsilon_{7}
Module 811(0, 0, 0, 0, 1, 2, 2, 2)(0, 0, 0, 0, 1, 2, 2, 2)g_{44}\varepsilon_{5}+\varepsilon_{6}
Module 822(0, -1, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 1, 1)g_{45}
g_{-41}		\varepsilon_{1}
-\varepsilon_{2}
Module 832(-1, -1, -1, -1, -1, -1, -1, 0)(0, 1, 1, 1, 1, 1, 1, 2)g_{46}
g_{-40}		\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{8}
Module 841(0, 0, 1, 1, 1, 1, 2, 2)(0, 0, 1, 1, 1, 1, 2, 2)g_{47}\varepsilon_{3}+\varepsilon_{7}
Module 851(0, 0, 0, 1, 1, 2, 2, 2)(0, 0, 0, 1, 1, 2, 2, 2)g_{48}\varepsilon_{4}+\varepsilon_{6}
Module 862(0, -1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 1, 2)g_{49}
g_{-36}		\varepsilon_{1}+\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{8}
Module 872(-1, -1, -1, -1, -1, -1, 0, 0)(0, 1, 1, 1, 1, 1, 2, 2)g_{50}
g_{-35}		\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
Module 881(0, 0, 1, 1, 1, 2, 2, 2)(0, 0, 1, 1, 1, 2, 2, 2)g_{51}\varepsilon_{3}+\varepsilon_{6}
Module 891(0, 0, 0, 1, 2, 2, 2, 2)(0, 0, 0, 1, 2, 2, 2, 2)g_{52}\varepsilon_{4}+\varepsilon_{5}
Module 902(0, -1, -1, -1, -1, -1, 0, 0)(1, 1, 1, 1, 1, 1, 2, 2)g_{53}
g_{-30}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
Module 912(-1, -1, -1, -1, -1, 0, 0, 0)(0, 1, 1, 1, 1, 2, 2, 2)g_{54}
g_{-29}		\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
Module 921(0, 0, 1, 1, 2, 2, 2, 2)(0, 0, 1, 1, 2, 2, 2, 2)g_{55}\varepsilon_{3}+\varepsilon_{5}
Module 932(0, -1, -1, -1, -1, 0, 0, 0)(1, 1, 1, 1, 1, 2, 2, 2)g_{56}
g_{-24}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
Module 942(-1, -1, -1, -1, 0, 0, 0, 0)(0, 1, 1, 1, 2, 2, 2, 2)g_{57}
g_{-23}		\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 951(0, 0, 1, 2, 2, 2, 2, 2)(0, 0, 1, 2, 2, 2, 2, 2)g_{58}\varepsilon_{3}+\varepsilon_{4}
Module 962(0, -1, -1, -1, 0, 0, 0, 0)(1, 1, 1, 1, 2, 2, 2, 2)g_{59}
g_{-17}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
Module 972(-1, -1, -1, 0, 0, 0, 0, 0)(0, 1, 1, 2, 2, 2, 2, 2)g_{60}
g_{-16}		\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 982(0, -1, -1, 0, 0, 0, 0, 0)(1, 1, 1, 2, 2, 2, 2, 2)g_{61}
g_{-10}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
Module 992(-1, -1, 0, 0, 0, 0, 0, 0)(0, 1, 2, 2, 2, 2, 2, 2)g_{62}
g_{-9}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
Module 1002(0, -1, 0, 0, 0, 0, 0, 0)(1, 1, 2, 2, 2, 2, 2, 2)g_{63}
g_{-2}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
Module 1013(-1, -2, -2, -2, -2, -2, -2, -2)(1, 2, 2, 2, 2, 2, 2, 2)g_{64}
2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-64}		\varepsilon_{1}+\varepsilon_{2}
0
-\varepsilon_{1}-\varepsilon_{2}
Module 1021(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{1}0
Module 1031(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{3}0
Module 1041(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{4}0
Module 1051(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 1061(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 1071(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{7}0
Module 1081(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{8}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 13
Heirs rejected due to not being maximally dominant: 84
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 84
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, B^{1}_2, 2A^{1}_1,