Query[ReductivePair] - check if a subalgebra, subspace pair defines a reductive pair in a Lie algebra
Query(S, M, "ReductivePair")
Query(S, M, parm, "ReductivePair")
S - a list of independent vectors which defines a subalgebra in a Lie algebra g
M - a list of independent vectors which defines a complementary subspace to S in g
parm - (optional) a set of parameters appearing in the list of vectors S
Let 𝔤 be a Lie algebra, S ⊂ 𝔤 a subalgebra and M ⊂ 𝔤 a subspace. Then the subalgebra, subspace pair S, M is called a reductive pair if [i] 𝔤 = S ⊕ M (vector space direct sum) and [ii] x,y ∈ M for all x ∈ S and y in M. The subspace M is called a reductive complement for the subalgebra S.
Query(S, M, "ReductivePair") returns true if the subspace M defines a reductive complement to the subalgebra S.
Query(S, M, parm, "ReductivePair") returns a sequence TF, Eq, Soln, reductiveList. Here TF is true if Maple finds parameter values for which M is a reductive complement and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for M to be a reductive complement; Soln is the list of solutions to the equations Eq; and reductiveList is the list of reductive subspaces 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.
L ≔ _DG⁡LieAlgebra,Alg,4,1,4,1,0,2,3,1,1,2,4,2,1,3,4,3,−1
We see that spane3, e4 is not a reductive complement for spane1, e2 but spane1, e2 is a reductive complement for spane3, e4.
S1 ≔ e1,e2:M1 ≔ e3,e4:
S2 ≔ e3,e4:M2 ≔ e1,e2:
Now we look for the most general reductive complement M3 for the subalgebra S3= span e3, e4.
S3 ≔ e3,e4:M3 ≔ evalDG⁡e1+a1⁢e3+a2⁢e4,e2+a3⁢e3+a4⁢e4:
TF,EQ,SOL,redPair ≔ Query⁡S3,M3,a1,a2,a3,a4,ReductivePair
The only possibility is spane1, e2.
Note that the ComplementaryBasis command can be used to generate the most general complementary subspace. This helps to calculate reductive complements for subalgebras.
S4 ≔ e4:
M4 ≔ ComplementaryBasis⁡S4,e1,e2,e3,e4,a
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