Query[Subalgebra] - check if a list of vectors defines a Lie subalgebra
Query(S, parm, "Subalgebra")
S - a list of independent vectors 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 defines a basis for a Lie subalgebra if x, y ∈ spanS for all x,y ∈S.
Query(S, "Subalgebra") returns true if the set S defines a subalgebra.
Query(S, parm, "Subalgebra") returns a sequence TF, Eq, Soln, SubAlgList. Here TF is true if Maple finds parameter values for which S is a subalgebra and false otherwise; Eq is the set of equations (with the variables in parm as unknowns) which must be satisfied for S to be a subalgebra; Soln is the list of solutions to the equations Eq; and SubAlgList is the list of subalgebras obtained from the parameter values given by the different solutions in Soln.
The program calculates the defining equations Eq for S to be a subalgebra as follows. First the list of vectors S is evaluated with the parameters set to zero to obtain a set of vectors S0. The program ComplementaryBasis is then used to calculate a complement C to S0 The list of vectors B= S, C then gives a basis for the entire Lie algebra 𝔤. For each x,y ∈S the bracket x,y is calculated and expressed as a linear combination of the vectors in the basis B. The components of [x,y[ in C must all vanish for S to be a Lie subalgebra.
We remark that the equations Eq, which the parameters must satisfy in order for S to be a subalgebra, will in general be a system of coupled quadratic equations. Maple may not be able to solve these equations or may not solve them in full generality.
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.
L ≔ _DG⁡LieAlgebra,Alg,4,1,4,1,0,2,3,1,1,2,4,2,1,3,4,3,−1
The vectors S1 = e2, e3, e4 do not determine a subalgebra while the vectors S2 = e1, e3, e4 do.
S1 ≔ e2,e3,e4:
S2 ≔ e1,e3,e4:
We find the values of the parameters a1, a2 for which S3 = e2, e1+a1e3 +a2e4 determines a Lie subalgebra.
S3 ≔ evalDG⁡e2,e1+a1⁢e3+a2⁢e4:
There are no values of the parameters a1, a2 for which S4 = e2, e3+a1e2 +a2e4 determines a Lie subalgebra.
S4 ≔ evalDG⁡e2,e3+a1⁢e2+a2⁢e4:
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