Type: \(\displaystyle A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{1}_1\))
Simple basis: 1 vectors: (1, 2, 2)
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^{2}_1+A^{1}_1
simple basis centralizer: 2 vectors: (0, 0, 1), (1, 0, 0)
Number of k-submodules of g: 13
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{1}}+6V_{\omega_{1}}+6V_{0}\)
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}-\varepsilon_{3}
Module 21(-1, 0, 0)(-1, 0, 0)g_{-1}-\varepsilon_{1}+\varepsilon_{2}
Module 31(1, 0, 0)(1, 0, 0)g_{1}\varepsilon_{1}-\varepsilon_{2}
Module 42(-1, -1, -2)(0, 1, 0)g_{2}
g_{-8}		\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 51(0, 0, 1)(0, 0, 1)g_{3}\varepsilon_{3}
Module 62(0, -1, -2)(1, 1, 0)g_{4}
g_{-7}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 72(-1, -1, -1)(0, 1, 1)g_{5}
g_{-6}		\varepsilon_{2}
-\varepsilon_{1}
Module 82(0, -1, -1)(1, 1, 1)g_{6}
g_{-5}		\varepsilon_{1}
-\varepsilon_{2}
Module 92(-1, -1, 0)(0, 1, 2)g_{7}
g_{-4}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
Module 102(0, -1, 0)(1, 1, 2)g_{8}
g_{-2}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
Module 113(-1, -2, -2)(1, 2, 2)g_{9}
2h_{3}+2h_{2}+h_{1}
g_{-9}		\varepsilon_{1}+\varepsilon_{2}
0
-\varepsilon_{1}-\varepsilon_{2}
Module 121(0, 0, 0)(0, 0, 0)h_{1}0
Module 131(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: 5
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 5
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^{1}_2, B^{1}_2, 2A^{1}_1,