Type: \(\displaystyle A^{2}_1\) (Dynkin type computed to be: \(\displaystyle A^{2}_1\))
Simple basis: 1 vectors: (1, 2, 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}_1
simple basis centralizer: 1 vectors: (0, 0, 1)
Number of k-submodules of g: 11
Module decomposition, fundamental coords over k: \(\displaystyle 3V_{2\omega_{1}}+4V_{\omega_{1}}+4V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, -1)(0, 0, -1)g_{-3}-2\varepsilon_{3}
Module 22(-1, -1, -1)(0, 1, 0)g_{2}
g_{-6}		\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 31(0, 0, 1)(0, 0, 1)g_{3}2\varepsilon_{3}
Module 42(0, -1, -1)(1, 1, 0)g_{4}
g_{-5}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 52(-1, -1, 0)(0, 1, 1)g_{5}
g_{-4}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
Module 62(0, -1, 0)(1, 1, 1)g_{6}
g_{-2}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
Module 73(-2, -2, -1)(0, 2, 1)g_{7}
g_{-1}
g_{-9}		2\varepsilon_{2}
-\varepsilon_{1}+\varepsilon_{2}
-2\varepsilon_{1}
Module 83(-1, -2, -1)(1, 2, 1)g_{8}
h_{3}+2h_{2}+h_{1}
g_{-8}		\varepsilon_{1}+\varepsilon_{2}
0
-\varepsilon_{1}-\varepsilon_{2}
Module 93(0, -2, -1)(2, 2, 1)g_{9}
g_{1}
g_{-7}		2\varepsilon_{1}
\varepsilon_{1}-\varepsilon_{2}
-2\varepsilon_{2}
Module 101(0, 0, 0)(0, 0, 0)h_{1}0
Module 111(0, 0, 0)(0, 0, 0)h_{3}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 3
Heirs rejected due to not being maximally dominant: 4
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 4
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by 0
Potential Dynkin type extensions: A^{2}_2, B^{2}_2, A^{2}_1+A^{1}_1, 2A^{2}_1,