Query[Solvable] - check if a Lie algebra is solvable
Alg - (optional) name or string, the name of an initialized Lie algebra
S - a list of vectors defining a basis for a subalgebra
A Lie algebra 𝔤 is solvable if the k-th ideal 𝒟kg in the derived series for 𝔤 is 0 for some k≥0. Every nilpotent Lie algebra is solvable.
Query(Alg, "Solvable") returns true if Alg is a solvable Lie algebra and false otherwise. If the algebra is unspecified, then Query is applied to the current algebra.
Query(S, "Solvable") returns true if the subalgebra S is a solvable 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,Alg1,3,2,3,1,1
L1 ≔ e2,e3=e1
L2 ≔ _DG⁡LieAlgebra,Alg2,3,1,3,1,1,2,3,1,1,2,3,2,1
L2 ≔ e1,e3=e1,e2,e3=e1+e2
L3 ≔ _DG⁡LieAlgebra,Alg3,3,1,2,1,1,1,3,2,−2,2,3,3,1
L3 ≔ e1,e2=e1,e1,e3=−2⁢e2,e2,e3=e3
Alg1 and Alg2 are solvable but Alg3 is not. (Alg1 is actually nilpotent while Alg3 is semisimple.)
The subalgebra S = spanz1, z2 is a solvable subalgebra of Alg3. (The algebra Alg3 is sl2, ℝ and S is a Borel subalgebra.)
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