Type: \(\displaystyle A^{2}_1+2A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{2}_1+2A^{1}_1\))
Simple basis: 3 vectors: (1, 1, 1, 1, 1), (0, 1, 2, 2, 2), (0, 1, 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}: 2A^{1}_1
simple basis centralizer: 2 vectors: (0, 0, 0, 1, 2), (0, 0, 0, 1, 0)
Number of k-submodules of g: 18
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{1}+\omega_{2}+\omega_{3}}+V_{2\omega_{3}}+4V_{\omega_{2}+\omega_{3}}+V_{2\omega_{2}}+5V_{2\omega_{1}}+6V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -1, -2)(0, 0, 0, -1, -2)g_{-13}-\varepsilon_{4}-\varepsilon_{5}
Module 21(0, 0, 0, -1, 0)(0, 0, 0, -1, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 33(0, -1, 0, 0, 0)(0, 1, 0, 0, 0)g_{2}
h_{2}
g_{-2}		\varepsilon_{2}-\varepsilon_{3}
0
-\varepsilon_{2}+\varepsilon_{3}
Module 41(0, 0, 0, 1, 0)(0, 0, 0, 1, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 54(0, -1, -1, -2, -2)(0, 1, 1, 0, 0)g_{7}
g_{-19}
g_{3}
g_{-21}		\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
Module 63(-1, -1, -1, -2, -2)(1, 1, 1, 0, 0)g_{10}
g_{-9}
g_{-22}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 74(0, -1, -1, -1, -2)(0, 1, 1, 1, 0)g_{11}
g_{-16}
g_{8}
g_{-18}		\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
Module 81(0, 0, 0, 1, 2)(0, 0, 0, 1, 2)g_{13}\varepsilon_{4}+\varepsilon_{5}
Module 93(-1, -1, -1, -1, -2)(1, 1, 1, 1, 0)g_{14}
g_{-5}
g_{-20}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 103(-1, -1, -1, -1, -1)(1, 1, 1, 1, 1)g_{17}
h_{5}+h_{4}+h_{3}+h_{2}+h_{1}
g_{-17}		\varepsilon_{1}
0
-\varepsilon_{1}
Module 114(0, -1, -1, -1, 0)(0, 1, 1, 1, 2)g_{18}
g_{-8}
g_{16}
g_{-11}		\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
Module 123(-1, -1, -1, -1, 0)(1, 1, 1, 1, 2)g_{20}
g_{5}
g_{-14}		\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 134(0, -1, -1, 0, 0)(0, 1, 1, 2, 2)g_{21}
g_{-3}
g_{19}
g_{-7}		\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
Module 143(-1, -1, -1, 0, 0)(1, 1, 1, 2, 2)g_{22}
g_{9}
g_{-10}		\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 153(0, -1, -2, -2, -2)(0, 1, 2, 2, 2)g_{23}
2h_{5}+2h_{4}+2h_{3}+h_{2}
g_{-23}		\varepsilon_{2}+\varepsilon_{3}
0
-\varepsilon_{2}-\varepsilon_{3}
Module 1612(-1, -2, -2, -2, -2)(1, 2, 2, 2, 2)g_{25}
g_{15}
g_{6}
g_{24}
g_{-1}
g_{-12}
g_{12}
g_{1}
g_{-24}
g_{-6}
g_{-15}
g_{-25}		\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
-\varepsilon_{3}
\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}
-\varepsilon_{1}-\varepsilon_{2}
Module 171(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{4}0
Module 181(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{5}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 12
Heirs rejected due to not being maximally dominant: 3
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 3
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{2}_1+A^{1}_1
Potential Dynkin type extensions: A^{2}_1+3A^{1}_1,