Query[Filtration] - check if a list of subspaces defines a decreasing filtration of a Lie algebra
Query([f0, f1, ..., fN], "Filtration")
f0, f1, - a list of independent vectors defining subspaces of a Lie algebra 𝔤
A collection of subspaces f0, f1, ..., fN of a Lie algebra 𝔤 defines a decreasing filtration of 𝔤 if [i] fi ⊂fj for i≥j , [ii] fi, fj ⊂fi+j for i +j ≤N, and [iii] fi, fj = 0 for i +j >N.
Query([f0, f1, ... fN], "Filtration") returns true if the subspaces f0, f1, ..., fN define a decreasing filtration of the Lie algebra 𝔤 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(...).
First we initialize a Lie algebra.
Now define a sequence of 4 subspaces.
We check that this sequence of Lie algebras defines a decreasing filtration.
Here's an example which does not define a filtration. To see the specific brackets which fail to satisfy the filtration definition, we set the infolevel for Query to 2.
bracket of subspaces with weights 0 and 0 is [2*e1, e2, e2+e3]
bracket of subspaces with weights 0 and 1 is [-e1, -e2]
bracket of subspaces with weights 0 and 2 is [-e1, -e2]
This shows that the Lie bracket f0, f2 is not contained in f2
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