Type: \(\displaystyle A^{2}_2\) (Dynkin type computed to be: \(\displaystyle A^{2}_2\))
Simple basis: 2 vectors: (1, 2, 2, 2, 1), (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}: B^{1}_2
simple basis centralizer: 2 vectors: (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of k-submodules of g: 22
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{2}}+V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+4V_{\omega_{2}}+4V_{\omega_{1}}+11V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -2, -1)(0, 0, 0, -2, -1)g_{-13}-2\varepsilon_{4}
Module 21(0, 0, 0, -1, -1)(0, 0, 0, -1, -1)g_{-9}-\varepsilon_{4}-\varepsilon_{5}
Module 31(0, 0, 0, 0, -1)(0, 0, 0, 0, -1)g_{-5}-2\varepsilon_{5}
Module 41(0, 0, 0, -1, 0)(0, 0, 0, -1, 0)g_{-4}-\varepsilon_{4}+\varepsilon_{5}
Module 53(-1, -1, -1, -2, -1)(0, 0, 1, 0, 0)g_{3}
g_{7}
g_{-20}		\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 61(0, 0, 0, 1, 0)(0, 0, 0, 1, 0)g_{4}\varepsilon_{4}-\varepsilon_{5}
Module 71(0, 0, 0, 0, 1)(0, 0, 0, 0, 1)g_{5}2\varepsilon_{5}
Module 83(-1, -1, -1, -1, -1)(0, 0, 1, 1, 0)g_{8}
g_{11}
g_{-17}		\varepsilon_{3}-\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 91(0, 0, 0, 1, 1)(0, 0, 0, 1, 1)g_{9}\varepsilon_{4}+\varepsilon_{5}
Module 103(0, 0, -1, -2, -1)(1, 1, 1, 0, 0)g_{10}
g_{-18}
g_{-16}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 113(-1, -1, -1, -1, 0)(0, 0, 1, 1, 1)g_{12}
g_{15}
g_{-14}		\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
Module 121(0, 0, 0, 2, 1)(0, 0, 0, 2, 1)g_{13}2\varepsilon_{4}
Module 133(0, 0, -1, -1, -1)(1, 1, 1, 1, 0)g_{14}
g_{-15}
g_{-12}		\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 143(-1, -1, -1, 0, 0)(0, 0, 1, 2, 1)g_{16}
g_{18}
g_{-10}		\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 153(0, 0, -1, -1, 0)(1, 1, 1, 1, 1)g_{17}
g_{-11}
g_{-8}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
Module 166(-2, -2, -2, -2, -1)(0, 0, 2, 2, 1)g_{19}
g_{21}
g_{-6}
g_{23}
g_{-1}
g_{-25}		2\varepsilon_{3}
\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
2\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{2}
-2\varepsilon_{1}
Module 173(0, 0, -1, 0, 0)(1, 1, 1, 2, 1)g_{20}
g_{-7}
g_{-3}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
Module 188(-1, -1, -2, -2, -1)(1, 1, 2, 2, 1)g_{22}
g_{-2}
g_{24}
-h_{2}
h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-24}
g_{2}
g_{-22}		\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 196(0, 0, -2, -2, -1)(2, 2, 2, 2, 1)g_{25}
g_{1}
g_{-23}
g_{6}
g_{-21}
g_{-19}		2\varepsilon_{1}
\varepsilon_{1}-\varepsilon_{2}
-2\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
-2\varepsilon_{3}
Module 201(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{3}-h_{1}0
Module 211(0, 0, 0, 0, 0)(0, 0, 0, 0, 0)h_{4}0
Module 221(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: 6
Heirs rejected due to not being maximally dominant: 9
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 9
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{2}_1
Potential Dynkin type extensions: A^{2}_3, B^{2}_3, C^{1}_3, A^{2}_2+A^{2}_1, A^{2}_2+A^{1}_1,