Type: \(\displaystyle A^{2}_2\) (Dynkin type computed to be: \(\displaystyle A^{2}_2\))
Simple basis: 2 vectors: (1, 2, 3, 2), (0, 0, 0, -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}_2
simple basis centralizer: 2 vectors: (1, 0, 0, 0), (0, 1, 0, 0)
Number of k-submodules of g: 15
Module decomposition, fundamental coords over k: \(\displaystyle 3V_{2\omega_{2}}+V_{\omega_{1}+\omega_{2}}+3V_{2\omega_{1}}+8V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(-1, -1, 0, 0)(-1, -1, 0, 0)g_{-5}-\varepsilon_{1}+\varepsilon_{3}
Module 21(0, -1, 0, 0)(0, -1, 0, 0)g_{-2}-\varepsilon_{2}+\varepsilon_{3}
Module 31(-1, 0, 0, 0)(-1, 0, 0, 0)g_{-1}-\varepsilon_{1}+\varepsilon_{2}
Module 41(1, 0, 0, 0)(1, 0, 0, 0)g_{1}\varepsilon_{1}-\varepsilon_{2}
Module 51(0, 1, 0, 0)(0, 1, 0, 0)g_{2}\varepsilon_{2}-\varepsilon_{3}
Module 61(1, 1, 0, 0)(1, 1, 0, 0)g_{5}\varepsilon_{1}-\varepsilon_{3}
Module 76(-2, -3, -4, -2)(0, 1, 2, 0)g_{9}
g_{13}
g_{-12}
g_{16}
g_{-8}
g_{-24}		\varepsilon_{2}+\varepsilon_{3}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{1}
-\varepsilon_{1}+\varepsilon_{4}
Module 86(-1, -3, -4, -2)(1, 1, 2, 0)g_{11}
g_{15}
g_{-10}
g_{18}
g_{-6}
g_{-23}		\varepsilon_{1}+\varepsilon_{3}
1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{2}
-\varepsilon_{2}+\varepsilon_{4}
Module 96(-1, -2, -4, -2)(1, 2, 2, 0)g_{14}
g_{17}
g_{-7}
g_{20}
g_{-3}
g_{-22}		\varepsilon_{1}+\varepsilon_{2}
1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{3}
-\varepsilon_{3}+\varepsilon_{4}
Module 108(-1, -2, -3, -1)(1, 2, 3, 1)g_{19}
g_{-4}
g_{21}
-h_{4}
2h_{4}+3h_{3}+2h_{2}+h_{1}
g_{-21}
g_{4}
g_{-19}		1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
-\varepsilon_{4}
0
0
\varepsilon_{4}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 116(-1, -2, -2, 0)(1, 2, 4, 2)g_{22}
g_{3}
g_{-20}
g_{7}
g_{-17}
g_{-14}		\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{3}
\varepsilon_{3}+\varepsilon_{4}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{2}
Module 126(-1, -1, -2, 0)(1, 3, 4, 2)g_{23}
g_{6}
g_{-18}
g_{10}
g_{-15}
g_{-11}		\varepsilon_{2}-\varepsilon_{4}
\varepsilon_{2}
\varepsilon_{2}+\varepsilon_{4}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{3}
Module 136(0, -1, -2, 0)(2, 3, 4, 2)g_{24}
g_{8}
g_{-16}
g_{12}
g_{-13}
g_{-9}		\varepsilon_{1}-\varepsilon_{4}
\varepsilon_{1}
\varepsilon_{1}+\varepsilon_{4}
1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{3}
Module 141(0, 0, 0, 0)(0, 0, 0, 0)h_{1}0
Module 151(0, 0, 0, 0)(0, 0, 0, 0)h_{2}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 4
Heirs rejected due to not being maximally dominant: 7
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 7
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^{1}_1, A^{2}_2+A^{2}_1,