Type: \(\displaystyle A^{1}_3+B^{1}_2\) (Dynkin type computed to be: \(\displaystyle A^{1}_3+B^{1}_2\))
Simple basis: 5 vectors: (1, 2, 2, 2, 2, 2, 2, 2), (0, -1, 0, 0, 0, 0, 0, 0), (-1, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 1, 2, 2, 2, 2), (0, 0, 0, 0, -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}_3
simple basis centralizer: 3 vectors: (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 2)
Number of k-submodules of g: 30
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{5}}+V_{\omega_{2}+\omega_{4}}+V_{\omega_{1}+\omega_{3}}+6V_{\omega_{4}}+6V_{\omega_{2}}+15V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, 0, -1, -2, -2)(0, 0, 0, 0, 0, -1, -2, -2)g_{-34}-\varepsilon_{6}-\varepsilon_{7}
Module 21(0, 0, 0, 0, 0, -1, -1, -2)(0, 0, 0, 0, 0, -1, -1, -2)g_{-28}-\varepsilon_{6}-\varepsilon_{8}
Module 31(0, 0, 0, 0, 0, 0, -1, -2)(0, 0, 0, 0, 0, 0, -1, -2)g_{-22}-\varepsilon_{7}-\varepsilon_{8}
Module 41(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 0, -1, -1, 0)g_{-14}-\varepsilon_{6}+\varepsilon_{8}
Module 51(0, 0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, 0, -1, 0)g_{-7}-\varepsilon_{7}+\varepsilon_{8}
Module 61(0, 0, 0, 0, 0, -1, 0, 0)(0, 0, 0, 0, 0, -1, 0, 0)g_{-6}-\varepsilon_{6}+\varepsilon_{7}
Module 710(0, 0, 0, -1, 0, 0, 0, 0)(0, 0, 0, 1, 0, 0, 0, 0)g_{4}
g_{32}
g_{-27}
g_{52}
-h_{8}-h_{7}-h_{6}-h_{5}
2h_{8}+2h_{7}+2h_{6}+2h_{5}+h_{4}
g_{-52}
g_{27}
g_{-32}
g_{-4}		\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{4}
-\varepsilon_{5}
\varepsilon_{4}+\varepsilon_{5}
0
0
-\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{5}
-\varepsilon_{4}
-\varepsilon_{4}+\varepsilon_{5}
Module 81(0, 0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 0, 1, 0, 0)g_{6}\varepsilon_{6}-\varepsilon_{7}
Module 91(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}\varepsilon_{7}-\varepsilon_{8}
Module 105(0, 0, 0, -1, -1, -2, -2, -2)(0, 0, 0, 1, 1, 0, 0, 0)g_{12}
g_{-44}
g_{-21}
g_{5}
g_{-48}		\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{5}-\varepsilon_{6}
-\varepsilon_{6}
\varepsilon_{5}-\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{6}
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 126(0, 0, -1, -1, -1, -2, -2, -2)(0, 0, 1, 1, 1, 0, 0, 0)g_{18}
g_{24}
g_{-56}
g_{29}
g_{-54}
g_{-51}		\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 135(0, 0, 0, -1, -1, -1, -2, -2)(0, 0, 0, 1, 1, 1, 0, 0)g_{19}
g_{-39}
g_{-15}
g_{13}
g_{-43}		\varepsilon_{4}-\varepsilon_{7}
-\varepsilon_{5}-\varepsilon_{7}
-\varepsilon_{7}
\varepsilon_{5}-\varepsilon_{7}
-\varepsilon_{4}-\varepsilon_{7}
Module 141(0, 0, 0, 0, 0, 0, 1, 2)(0, 0, 0, 0, 0, 0, 1, 2)g_{22}\varepsilon_{7}+\varepsilon_{8}
Module 156(0, 0, -1, -1, -1, -1, -2, -2)(0, 0, 1, 1, 1, 1, 0, 0)g_{25}
g_{30}
g_{-53}
g_{35}
g_{-50}
g_{-47}		\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 165(0, 0, 0, -1, -1, -1, -1, -2)(0, 0, 0, 1, 1, 1, 1, 0)g_{26}
g_{-33}
g_{-8}
g_{20}
g_{-38}		\varepsilon_{4}-\varepsilon_{8}
-\varepsilon_{5}-\varepsilon_{8}
-\varepsilon_{8}
\varepsilon_{5}-\varepsilon_{8}
-\varepsilon_{4}-\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 186(0, 0, -1, -1, -1, -1, -1, -2)(0, 0, 1, 1, 1, 1, 1, 0)g_{31}
g_{36}
g_{-49}
g_{40}
g_{-46}
g_{-42}		\varepsilon_{3}-\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{8}
\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{3}-\varepsilon_{8}
Module 191(0, 0, 0, 0, 0, 1, 2, 2)(0, 0, 0, 0, 0, 1, 2, 2)g_{34}\varepsilon_{6}+\varepsilon_{7}
Module 205(0, 0, 0, -1, -1, -1, -1, 0)(0, 0, 0, 1, 1, 1, 1, 2)g_{38}
g_{-20}
g_{8}
g_{33}
g_{-26}		\varepsilon_{4}+\varepsilon_{8}
-\varepsilon_{5}+\varepsilon_{8}
\varepsilon_{8}
\varepsilon_{5}+\varepsilon_{8}
-\varepsilon_{4}+\varepsilon_{8}
Module 216(0, 0, -1, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 1, 2)g_{42}
g_{46}
g_{-40}
g_{49}
g_{-36}
g_{-31}		\varepsilon_{3}+\varepsilon_{8}
\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{8}
\varepsilon_{1}+\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{8}
Module 225(0, 0, 0, -1, -1, -1, 0, 0)(0, 0, 0, 1, 1, 1, 2, 2)g_{43}
g_{-13}
g_{15}
g_{39}
g_{-19}		\varepsilon_{4}+\varepsilon_{7}
-\varepsilon_{5}+\varepsilon_{7}
\varepsilon_{7}
\varepsilon_{5}+\varepsilon_{7}
-\varepsilon_{4}+\varepsilon_{7}
Module 236(0, 0, -1, -1, -1, -1, 0, 0)(0, 0, 1, 1, 1, 1, 2, 2)g_{47}
g_{50}
g_{-35}
g_{53}
g_{-30}
g_{-25}		\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 245(0, 0, 0, -1, -1, 0, 0, 0)(0, 0, 0, 1, 1, 2, 2, 2)g_{48}
g_{-5}
g_{21}
g_{44}
g_{-12}		\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{5}+\varepsilon_{6}
\varepsilon_{6}
\varepsilon_{5}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
Module 256(0, 0, -1, -1, -1, 0, 0, 0)(0, 0, 1, 1, 1, 2, 2, 2)g_{51}
g_{54}
g_{-29}
g_{56}
g_{-24}
g_{-18}		\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 2630(0, 0, -1, -2, -2, -2, -2, -2)(0, 0, 1, 2, 2, 2, 2, 2)g_{58}
g_{60}
g_{11}
g_{-16}
g_{61}
g_{17}
g_{37}
g_{-10}
g_{-59}
g_{23}
g_{41}
g_{55}
g_{-3}
g_{-57}
g_{-45}
g_{45}
g_{57}
g_{3}
g_{-55}
g_{-41}
g_{-23}
g_{59}
g_{10}
g_{-37}
g_{-17}
g_{-61}
g_{16}
g_{-11}
g_{-60}
g_{-58}		\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{5}
\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{5}
\varepsilon_{1}-\varepsilon_{5}
\varepsilon_{2}
\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}
\varepsilon_{1}
\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 2715(0, -1, -2, -2, -2, -2, -2, -2)(0, 1, 2, 2, 2, 2, 2, 2)g_{62}
g_{-9}
g_{63}
g_{-1}
g_{-2}
g_{64}
-h_{1}
-h_{2}
2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-64}
g_{2}
g_{1}
g_{-63}
g_{9}
g_{-62}		\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 281(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{6}0
Module 291(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{7}0
Module 301(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: 15
Heirs rejected due to not being maximally dominant: 11
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 11
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{1}_3+A^{1}_1
Potential Dynkin type extensions: A^{1}_3+B^{1}_2+A^{1}_1, A^{1}_3+B^{1}_2+A^{2}_1,