Type: \(\displaystyle A^{1}_2+A^{2}_1\) (Dynkin type computed to be: \(\displaystyle A^{1}_2+A^{2}_1\))
Simple basis: 3 vectors: (1, 2, 2, 2, 2), (0, -1, 0, 0, 0), (0, 0, 0, 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}: 0
simple basis centralizer: 0 vectors:
Number of k-submodules of g: 14
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{2}+2\omega_{3}}+V_{\omega_{1}+2\omega_{3}}+3V_{2\omega_{3}}+V_{\omega_{1}+\omega_{2}}+3V_{\omega_{2}}+3V_{\omega_{1}}+2V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 13(0, -1, -2, -2, -2)(1, 0, 0, 0, 0)g_{1}
g_{6}
g_{-23}		\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 23(0, 0, 0, -1, -2)(0, 0, 0, 1, 0)g_{4}
g_{-5}
g_{-13}		\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
Module 33(-1, -1, -1, -1, -2)(0, 0, 1, 1, 0)g_{8}
g_{11}
g_{-20}		\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 43(0, 0, 0, -1, -1)(0, 0, 0, 1, 1)g_{9}
h_{5}+h_{4}
g_{-9}		\varepsilon_{4}
0
-\varepsilon_{4}
Module 53(0, 0, 0, -1, 0)(0, 0, 0, 1, 2)g_{13}
g_{5}
g_{-4}		\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
Module 63(0, 0, -1, -1, -2)(1, 1, 1, 1, 0)g_{14}
g_{-18}
g_{-16}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 73(-1, -1, -1, -1, 0)(0, 0, 1, 1, 2)g_{16}
g_{18}
g_{-14}		\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 89(-1, -1, -1, -2, -2)(0, 0, 1, 2, 2)g_{19}
g_{21}
g_{12}
g_{-10}
g_{15}
g_{3}
g_{-17}
g_{7}
g_{-22}		\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{2}
\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{1}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 93(0, 0, -1, -1, 0)(1, 1, 1, 1, 2)g_{20}
g_{-11}
g_{-8}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
Module 109(0, 0, -1, -2, -2)(1, 1, 1, 2, 2)g_{22}
g_{-7}
g_{17}
g_{-3}
g_{-15}
g_{10}
g_{-12}
g_{-21}
g_{-19}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}
-\varepsilon_{3}+\varepsilon_{4}
-\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 113(-1, 0, 0, 0, 0)(0, 1, 2, 2, 2)g_{23}
g_{-6}
g_{-1}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
Module 128(-1, -1, -2, -2, -2)(1, 1, 2, 2, 2)g_{24}
g_{-2}
g_{25}
-h_{2}
2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-25}
g_{2}
g_{-24}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{2}
0
0
-\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 131(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{4}+h_{3}-h_{1}0
Module 141(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: 0
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 0
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{1}_2
Potential Dynkin type extensions: A^{1}_2+2A^{2}_1, A^{1}_2+A^{2}_1+A^{1}_1,