Type: \(\displaystyle A^{1}_3\) (Dynkin type computed to be: \(\displaystyle A^{1}_3\))
Simple basis: 3 vectors: (2, 3, 4, 2), (-1, 0, 0, 0), (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}: A^{2}_1
simple basis centralizer: 1 vectors: (0, 0, 0, 1)
Number of k-submodules of g: 11
Module decomposition, fundamental coords over k: \(\displaystyle V_{\omega_{1}+\omega_{3}}+2V_{\omega_{3}}+3V_{\omega_{2}}+2V_{\omega_{1}}+3V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, -1)(0, 0, 0, -1)g_{-4}1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 24(-1, -2, -3, -2)(0, 0, 1, 0)g_{3}
g_{6}
g_{8}
g_{-21}		\varepsilon_{3}
\varepsilon_{2}
\varepsilon_{1}
\varepsilon_{4}
Module 31(0, 0, 0, 1)(0, 0, 0, 1)g_{4}-1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
Module 44(-1, -2, -3, -1)(0, 0, 1, 1)g_{7}
g_{10}
g_{12}
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}
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 56(0, -1, -2, -2)(0, 1, 2, 0)g_{9}
g_{11}
g_{-20}
g_{14}
g_{-18}
g_{-16}		\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{3}+\varepsilon_{4}
\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}+\varepsilon_{4}
Module 66(0, -1, -2, -1)(0, 1, 2, 1)g_{13}
g_{15}
g_{-17}
g_{17}
g_{-15}
g_{-13}		-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}
-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}
-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 76(0, -1, -2, 0)(0, 1, 2, 2)g_{16}
g_{18}
g_{-14}
g_{20}
g_{-11}
g_{-9}		-\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 84(0, 0, -1, -1)(1, 2, 3, 1)g_{19}
g_{-12}
g_{-10}
g_{-7}		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}
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 94(0, 0, -1, 0)(1, 2, 3, 2)g_{21}
g_{-8}
g_{-6}
g_{-3}		-\varepsilon_{4}
-\varepsilon_{1}
-\varepsilon_{2}
-\varepsilon_{3}
Module 1015(-1, -2, -4, -2)(1, 2, 4, 2)g_{22}
g_{-5}
g_{23}
g_{-2}
g_{-1}
g_{24}
-h_{2}
-h_{1}
2h_{4}+4h_{3}+3h_{2}+2h_{1}
g_{-24}
g_{1}
g_{2}
g_{-23}
g_{5}
g_{-22}		\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{1}-\varepsilon_{4}
0
0
0
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{2}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{3}+\varepsilon_{4}
Module 111(0, 0, 0, 0)(0, 0, 0, 0)h_{4}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: 3
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 3
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 0
Parabolically induced by A^{1}_2
Potential Dynkin type extensions: A^{1}_4, B^{1}_4, C^{1/2}_4, D^{1}_4, A^{1}_3+A^{1}_1, A^{1}_3+A^{2}_1,