Query[Filtration] - check if a list of subspaces defines a decreasing filtration of a Lie algebra
Calling Sequences
Query([f0, f1, ..., fN], "Filtration")
Parameters
f0, f1, - a list of independent vectors defining subspaces of a Lie algebra 𝔤
Description
Examples
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(...).
with⁡DifferentialGeometry:with⁡LieAlgebras:
Example 1.
First we initialize a Lie algebra.
L1≔_DG⁡LieAlgebra,Alg1,4,1,4,1,2,2,3,1,1,2,4,2,1,3,4,2,1,3,4,3,1
L1:=e1,e4=2⁢e1,e2,e3=e1,e2,e4=e2,e3,e4=e2+e3
DGsetup⁡L1:
Now define a sequence of 4 subspaces.
f0≔e1,e2,e3,e4:f1≔e1,e2,e3:f2≔e1,e2:f3≔e1:f4≔:
We check that this sequence of Lie algebras defines a decreasing filtration.
Query⁡f0,f1,f2,f3,f4,Filtration
true
Example 2.
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.
f0≔e1,e2,e3,e4:f1≔e1,e2:f2≔e2:f3≔:
infolevelQuery≔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]
false
This shows that the Lie bracket f0, f2 is not contained in f2
See Also
DifferentialGeometry
LieAlgebras
infolevel
Query
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