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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

Description

• 

A collection of subspaces  f0, f1, ..., fN  of a Lie algebra 𝔤 defines a decreasing filtration of 𝔤  if  [i] fi fj  for ij , [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(...).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First we initialize a Lie algebra.

L1_DGLieAlgebra,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=2e1,e2,e3=e1,e2,e4=e2,e3,e4=e2+e3

(2.1)

DGsetupL1:

 

Now define a sequence of 4 subspaces.

Alg1 > 

f0e1,e2,e3,e4:f1e1,e2,e3:f2e1,e2:f3e1:f4:

 

We check that this sequence of Lie algebras defines a decreasing filtration.

Alg1 > 

Queryf0,f1,f2,f3,f4,Filtration

true

(2.2)

 

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.

Alg1 > 

f0e1,e2,e3,e4:f1e1,e2:f2e2:f3:

Alg1 > 

infolevelQuery2:

Alg1 > 

Queryf0,f1,f2,f3,f4,Filtration

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

(2.3)
Alg1 > 

 

This shows that the the Lie bracket f0,  f2 is not contained in f2

See Also

DifferentialGeometry

LieAlgebras

infolevel

Query