Type: \(\displaystyle A^{1}_2\) (Dynkin type computed to be: \(\displaystyle A^{1}_2\))
Simple basis: 2 vectors: (1, 2, 2), (0, -1, 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}: 0
simple basis centralizer: 0 vectors:
Number of k-submodules of g: 6
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{1}+\omega_{2}}+2V_{\omega_{2}}+2V_{\omega_{1}}+V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 13(0, -1, -2)(1, 0, 0)g_{1}
g_{4}
g_{-7}		\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 23(-1, -1, -1)(0, 0, 1)g_{3}
g_{5}
g_{-6}		\varepsilon_{3}
\varepsilon_{2}
-\varepsilon_{1}
Module 33(0, 0, -1)(1, 1, 1)g_{6}
g_{-5}
g_{-3}		\varepsilon_{1}
-\varepsilon_{2}
-\varepsilon_{3}
Module 43(-1, 0, 0)(0, 1, 2)g_{7}
g_{-4}
g_{-1}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
Module 58(-1, -1, -2)(1, 1, 2)g_{8}
g_{-2}
g_{9}
-h_{2}
2h_{3}+2h_{2}+h_{1}
g_{-9}
g_{2}
g_{-8}		\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 61(0, 0, 0)(0, 0, 0)h_{3}-h_{1}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: 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}_1
Potential Dynkin type extensions: A^{1}_3, B^{1}_3, A^{1}_2+A^{2}_1, A^{1}_2+A^{1}_1,