Query[NaturallyReductivePair] - check if a subalgebra, subspace pair is naturally reductive with respect to an inner product on the subspace
Query(S, M, B, "NaturallyReductivePair")
Query(S, M, B, parm, "NaturallyReductivePair")
S - a list of independent vectors which defines a subalgebra in a Lie algebra g
M - a list of independent vectors which defines a reductive complement to S in g
B - a symmetric m x m matrix, which defines an inner product on M with respect to the given basis
parm - (optional) a set of parameters appearing in the list of vectors M
Let 𝔤 be a Lie algebra, S ⊂ 𝔤 a subalgebra, and M ⊂𝔤 a subspace. Let B be a non-degenerate inner product on M. Then the subalgebra, subspace pair S, M is called naturally reductive with respect to the inner product B if [i] the subspace M defines a reductive complement to the subalgebra S, and [ii] the inner product B is 𝔤 invariant, that is, Bx,yM ,z + By, x,zM = 0 for all x ∈𝔤 and y,z ∈M. Here x,yM denotes the M−component of x,y with respect to the decomposition 𝔤= S ⊕M .
Query(S, M, B, "NaturallyReductivePair") returns true if S, M is naturally reductive with respect to the inner product B, and false otherwise.
Query(S, M, B, parm, "NaturallyReductivePair") returns a sequence TF, Eq, Soln, NatRedPair. Here TF is true if Maple finds parameter values for which the pair S, M is naturally reductive and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S, M to be naturally reductive; Soln is the list of solutions to the equations Eq; and NatRedPair is the list of naturally reductive subspaces and inner products 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.
L1 ≔ _DG⁡LieAlgebra,Alg1,3,1,2,1,1,1,3,2,−2,2,3,3,1:
Define a subspace S1, a complement M1, and an inner product B1 on M1.
S1 ≔ e2:M1 ≔ e1,e3:B1 ≔ Matrix⁡0,1,1,0
Check that S1, M1 is naturally reductive with respect to B1.
Naturally reductive means that [i] the symmetric tensor g defined by B is invariant with respect to the vectors in S1 and [ii] the Lie derivative of B with respect to the vectors in M1 vanishes on pairs of vectors from M1. Thus, for the above example we have:
g ≔ evalDG⁡θ1 &t θ3+θ3 &t θ1
In this example we consider a Lie algebra containing a parameter b. We find that a certain subspace S2 admits a naturally reductive complement M2 when b =0.
First initialize a Lie algebra and display the Lie bracket multiplication table.
L2 ≔ _DG⁡LieAlgebra,Alg2,4,1,4,1,1,2,4,2,b,3,4,2,1,2,4,3,−1,3,4,3,b
For S2 we have that M2 is a reductive complement. We let the inner product B2 be arbitrary.
S2 ≔ e1,e4:M2 ≔ e2,e3:B2 ≔ Matrix⁡r,s,s,t
TF,EQ,SOLN,natRedPair ≔ Query⁡S2,M2,B2,b,r,s,t,NaturallyReductivePair
We see that the that M2 =span e2, e3 is naturally reductive only when b = 0. To check this we substitute b =0 into the Lie algebra data structure for L2 and change the name of the algebra to Alg3.
L3 ≔ subs⁡b=0,L2
Lie algebra: Alg2
S3 ≔ natRedPair11;M3 ≔ natRedPair12;B3 ≔ natRedPair13
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