Type: \(\displaystyle A^{1}_1\) (Dynkin type computed to be: \(\displaystyle A^{1}_1\))
Simple basis: 1 vectors: (2, 3, 4, 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}: C^{1}_3
simple basis centralizer: 3 vectors: (0, 0, 0, 1), (0, 1, 0, 0), (0, 0, 1, 0)
Number of k-submodules of g: 36
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{1}}+14V_{\omega_{1}}+21V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, -1, -2, -2)(0, -1, -2, -2)g_{-16}\varepsilon_{1}+\varepsilon_{4}
Module 21(0, -1, -2, -1)(0, -1, -2, -1)g_{-13}1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 31(0, -1, -1, -1)(0, -1, -1, -1)g_{-10}1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 41(0, -1, -2, 0)(0, -1, -2, 0)g_{-9}-\varepsilon_{2}-\varepsilon_{3}
Module 51(0, 0, -1, -1)(0, 0, -1, -1)g_{-7}1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}+1/2\varepsilon_{4}
Module 61(0, -1, -1, 0)(0, -1, -1, 0)g_{-6}-\varepsilon_{2}
Module 71(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 81(0, 0, -1, 0)(0, 0, -1, 0)g_{-3}-\varepsilon_{3}
Module 91(0, -1, 0, 0)(0, -1, 0, 0)g_{-2}-\varepsilon_{2}+\varepsilon_{3}
Module 102(-1, -3, -4, -2)(1, 0, 0, 0)g_{1}
g_{-23}		\varepsilon_{1}-\varepsilon_{2}
-\varepsilon_{2}+\varepsilon_{4}
Module 111(0, 1, 0, 0)(0, 1, 0, 0)g_{2}\varepsilon_{2}-\varepsilon_{3}
Module 121(0, 0, 1, 0)(0, 0, 1, 0)g_{3}\varepsilon_{3}
Module 131(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 142(-1, -2, -4, -2)(1, 1, 0, 0)g_{5}
g_{-22}		\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{3}+\varepsilon_{4}
Module 151(0, 1, 1, 0)(0, 1, 1, 0)g_{6}\varepsilon_{2}
Module 161(0, 0, 1, 1)(0, 0, 1, 1)g_{7}-1/2\varepsilon_{1}-1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
Module 172(-1, -2, -3, -2)(1, 1, 1, 0)g_{8}
g_{-21}		\varepsilon_{1}
\varepsilon_{4}
Module 181(0, 1, 2, 0)(0, 1, 2, 0)g_{9}\varepsilon_{2}+\varepsilon_{3}
Module 191(0, 1, 1, 1)(0, 1, 1, 1)g_{10}-1/2\varepsilon_{1}+1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}
Module 202(-1, -2, -2, -2)(1, 1, 2, 0)g_{11}
g_{-20}		\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{3}+\varepsilon_{4}
Module 212(-1, -2, -3, -1)(1, 1, 1, 1)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}
Module 221(0, 1, 2, 1)(0, 1, 2, 1)g_{13}-1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}
Module 232(-1, -1, -2, -2)(1, 2, 2, 0)g_{14}
g_{-18}		\varepsilon_{1}+\varepsilon_{2}
\varepsilon_{2}+\varepsilon_{4}
Module 242(-1, -2, -2, -1)(1, 1, 2, 1)g_{15}
g_{-17}		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 251(0, 1, 2, 2)(0, 1, 2, 2)g_{16}-\varepsilon_{1}-\varepsilon_{4}
Module 262(-1, -1, -2, -1)(1, 2, 2, 1)g_{17}
g_{-15}		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 272(-1, -2, -2, 0)(1, 1, 2, 2)g_{18}
g_{-14}		-\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{2}
Module 282(-1, -1, -1, -1)(1, 2, 3, 1)g_{19}
g_{-12}		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 292(-1, -1, -2, 0)(1, 2, 2, 2)g_{20}
g_{-11}		-\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{3}
Module 302(-1, -1, -1, 0)(1, 2, 3, 2)g_{21}
g_{-8}		-\varepsilon_{4}
-\varepsilon_{1}
Module 312(-1, -1, 0, 0)(1, 2, 4, 2)g_{22}
g_{-5}		\varepsilon_{3}-\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{3}
Module 322(-1, 0, 0, 0)(1, 3, 4, 2)g_{23}
g_{-1}		\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{2}
Module 333(-2, -3, -4, -2)(2, 3, 4, 2)g_{24}
2h_{4}+4h_{3}+3h_{2}+2h_{1}
g_{-24}		\varepsilon_{1}-\varepsilon_{4}
0
-\varepsilon_{1}+\varepsilon_{4}
Module 341(0, 0, 0, 0)(0, 0, 0, 0)h_{2}0
Module 351(0, 0, 0, 0)(0, 0, 0, 0)h_{3}0
Module 361(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: 13
Heirs rejected due to not being maximally dominant: 17
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 17
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,