Type: \(\displaystyle A^{1}_3+A^{1}_2\) (Dynkin type computed to be: \(\displaystyle A^{1}_3+A^{1}_2\))
Simple basis: 5 vectors: (1, 2, 2, 2, 2, 1, 1), (0, -1, 0, 0, 0, 0, 0), (-1, 0, 0, 0, 0, 0, 0), (0, 0, 0, 1, 2, 1, 1), (0, 0, 0, 0, -1, 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}: 0
simple basis centralizer: 0 vectors:
Number of k-submodules of g: 14
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{4}+\omega_{5}}+V_{\omega_{2}+\omega_{5}}+V_{\omega_{2}+\omega_{4}}+V_{\omega_{1}+\omega_{3}}+3V_{\omega_{5}}+3V_{\omega_{4}}+2V_{\omega_{2}}+2V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 13(0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 1, 0, 0, 0)g_{4}
g_{11}
g_{-19}		\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{5}-\varepsilon_{6}
Module 23(0, 0, 0, -1, -1, 0, -1)(0, 0, 0, 0, 0, 1, 0)g_{6}
g_{12}
g_{-18}		\varepsilon_{6}-\varepsilon_{7}
\varepsilon_{5}-\varepsilon_{7}
-\varepsilon_{4}-\varepsilon_{7}
Module 33(0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, 0, 0, 1)g_{7}
g_{13}
g_{-17}		\varepsilon_{6}+\varepsilon_{7}
\varepsilon_{5}+\varepsilon_{7}
-\varepsilon_{4}+\varepsilon_{7}
Module 43(0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 1, 1, 1, 0)g_{17}
g_{-13}
g_{-7}		\varepsilon_{4}-\varepsilon_{7}
-\varepsilon_{5}-\varepsilon_{7}
-\varepsilon_{6}-\varepsilon_{7}
Module 53(0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 1, 1, 0, 1)g_{18}
g_{-12}
g_{-6}		\varepsilon_{4}+\varepsilon_{7}
-\varepsilon_{5}+\varepsilon_{7}
-\varepsilon_{6}+\varepsilon_{7}
Module 63(0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, 1, 1, 1)g_{19}
g_{-11}
g_{-4}		\varepsilon_{5}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{5}
Module 76(0, 0, -1, -1, -1, 0, -1)(0, 0, 1, 1, 1, 1, 0)g_{22}
g_{26}
g_{-31}
g_{30}
g_{-27}
g_{-23}		\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 86(0, 0, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 0, 1)g_{23}
g_{27}
g_{-30}
g_{31}
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 98(0, 0, 0, -1, -1, -1, -1)(0, 0, 0, 1, 1, 1, 1)g_{24}
g_{-5}
g_{29}
-h_{5}
h_{7}+h_{6}+2h_{5}+h_{4}
g_{-29}
g_{5}
g_{-24}		\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{5}+\varepsilon_{6}
\varepsilon_{4}+\varepsilon_{5}
0
0
-\varepsilon_{4}-\varepsilon_{5}
\varepsilon_{5}-\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{6}
Module 1018(0, 0, -1, -2, -2, -1, -1)(0, 0, 1, 1, 1, 1, 1)g_{28}
g_{32}
g_{33}
g_{-25}
g_{34}
g_{35}
g_{3}
g_{-21}
g_{-20}
g_{37}
g_{9}
g_{-16}
g_{-15}
g_{-39}
g_{14}
g_{-10}
g_{-38}
g_{-36}		\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{6}
-\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 1118(0, 0, -1, -1, -1, -1, -1)(0, 0, 1, 2, 2, 1, 1)g_{36}
g_{38}
g_{10}
g_{-14}
g_{39}
g_{15}
g_{16}
g_{-9}
g_{-37}
g_{20}
g_{21}
g_{-3}
g_{-35}
g_{-34}
g_{25}
g_{-33}
g_{-32}
g_{-28}		\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_{6}
-\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{5}
\varepsilon_{1}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
Module 1215(0, -1, -2, -2, -2, -1, -1)(0, 1, 2, 2, 2, 1, 1)g_{40}
g_{-8}
g_{41}
g_{-1}
g_{-2}
g_{42}
-h_{1}
-h_{2}
h_{7}+h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-42}
g_{2}
g_{1}
g_{-41}
g_{8}
g_{-40}		\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 131(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{6}-h_{4}0
Module 141(0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0)h_{7}-h_{4}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 9
Heirs rejected due to not being maximally dominant: 1
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 1
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 1
Parabolically induced by A^{1}_3+A^{1}_1
Potential Dynkin type extensions: 2A^{1}_3, A^{1}_3+A^{1}_2+A^{1}_1,