Query[Abelian] - check if a Lie algebra is Abelian
Alg - (optional) name or string, the name of an initialized Lie algebra 𝔤
S - a list of vectors defining a basis for a subalgebra
A Lie algebra 𝔤 is Abelian if x, y = 0 for all x,y ∈𝔤.
Query(Alg, "Abelian") returns true if Alg is an Abelian Lie algebra and false otherwise. If the algebra is unspecified, then Query is applied to the current algebra.
Query(S, "Abelian") returns true if the subalgebra S is an Abelian 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 a pair of 3-dimensional Lie algebras.The first algebra is not Abelian and the second is, as confirmed by a call to Query.
L1 ≔ _DG⁡LieAlgebra,Alg1,3,1,2,1,1
L2 ≔ _DG⁡LieAlgebra,Alg2,3,
The subalgebra S1 =x1, x2 in Alg1 is not Abelian while the subalgebra S2 =x2, x3 is Abelian.
S1 ≔ x1,x2:S2 ≔ x2,x3:
DifferentialGeometry, LieAlgebras, ChangeLieAlgebraTo, Query
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