Type: \(\displaystyle A^{1}_2\) (Dynkin type computed to be: \(\displaystyle A^{1}_2\))
Simple basis: 2 vectors: (1, 2, 1, 1), (0, -1, 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}: 0
simple basis centralizer: 0 vectors:
Number of k-submodules of g: 9
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{1}+\omega_{2}}+3V_{\omega_{2}}+3V_{\omega_{1}}+2V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 13(0, -1, -1, -1)(1, 0, 0, 0)g_{1}
g_{5}
g_{-10}		\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 23(-1, -1, 0, -1)(0, 0, 1, 0)g_{3}
g_{6}
g_{-9}		\varepsilon_{3}-\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 33(-1, -1, -1, 0)(0, 0, 0, 1)g_{4}
g_{7}
g_{-8}		\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
Module 43(0, 0, 0, -1)(1, 1, 1, 0)g_{8}
g_{-7}
g_{-4}		\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{3}-\varepsilon_{4}
Module 53(0, 0, -1, 0)(1, 1, 0, 1)g_{9}
g_{-6}
g_{-3}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{3}+\varepsilon_{4}
Module 63(-1, 0, 0, 0)(0, 1, 1, 1)g_{10}
g_{-5}
g_{-1}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
Module 78(-1, -1, -1, -1)(1, 1, 1, 1)g_{11}
g_{-2}
g_{12}
-h_{2}
h_{4}+h_{3}+2h_{2}+h_{1}
g_{-12}
g_{2}
g_{-11}		\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 81(0, 0, 0, 0)(0, 0, 0, 0)h_{3}-h_{1}0
Module 91(0, 0, 0, 0)(0, 0, 0, 0)h_{4}-h_{1}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: 2
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 2
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, A^{1}_2+A^{1}_1,