Query[Ideal] - check if a subalgebra defines an ideal in a Lie algebra
Query(S, parm, "Ideal")
S - a list of independent vectors which defines a basis for subalgebra in a Lie algebra 𝔤
parm - (optional) a set of parameters appearing in the list of vectors S; it is assumed that the set of vectors S is well-defined when the parameters vanish
A list of vectors S in a Lie algebra 𝔤 is a basis for an ideal in 𝔤 if x, y ∈span(S) for all x ∈ S and y ∈𝔤 .
Query(S, "Ideal") returns true if the subalgebra S defines an ideal and false otherwise.
Query(S, parm, "Ideal") returns a sequence TF, Eq, Soln, IdealList. Here TF is true if Maple finds parameter values for which S is an ideal and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S to be an ideal; Soln is the list of solutions to the equations Eq; and IdealList is the list of ideals obtained from the parameter values given by the different solutions in Soln.
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(...).
First initialize a Lie algebra; then define some subalgebras S1, S2, S3 and check to see if they are ideals.
L ≔ _DG⁡LieAlgebra,Alg,4,1,4,1,0,2,3,1,1,2,4,2,1,3,4,3,−1
S1 ≔ e1,e2
S2 ≔ e3,e4
The subalgebra S3depends on a parameter a1. We find which parameter values make S3 an ideal.
S3 ≔ evalDG⁡e2,e1+a1⁢e4:
TF,EQ,SOLN,IdealList ≔ Query⁡S3,a1,a2,Ideal
TF,EQ,SOLN,IdealList ≔ true,0,−a1,a1,a1=0,a2=a2,e2,e1
The following equations must hold for S3 to be an ideal (each expression must vanish).
S4 ≔ IdealList1
S4 ≔ e2,e1
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