Query[Nilpotent] - check if a Lie algebra is nilpotent
Alg - (optional) the name of an initialized Lie algebra
S - a list of vectors defining a basis for a subalgebra
A Lie algebra 𝔤 is nilpotent if the k-th ideal in the lower central series for 𝔤 is 0 for some k ≥ 0.
Query(Alg, "Nilpotent") returns true if Alg is a nilpotent Lie algebra and false otherwise. If the algebra is unspecified, then Query is applied to the current algebra.
Query(S, "Nilpotent") returns true if the subalgebra spanned by the vectors S is a nilpotent Lie algebra and false otherwise.
The command Query is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).
We initialize three different Lie algebras.
L1 ≔ _DG⁡LieAlgebra,Alg1,3,
L2 ≔ _DG⁡LieAlgebra,Alg2,3,2,3,1,1:
L3 ≔ _DG⁡LieAlgebra,Alg3,4,1,4,1,2,2,3,1,1,2,4,2,1,3,4,3,1
L3 ≔ e1,e4=2⁢e1,e2,e3=e1,e2,e4=e2,e3,e4=e3
Alg1 and Alg2 are nilpotent but Alg3 is not.
The subalgebra S1 = spanz1, z2, z3 of Alg3 is nilpotent but the subalgebra S2=spanz1, z4 is not.
S1 ≔ z1,z2,z3:S2 ≔ z1,z4:
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