check if a list of subspaces defines a decreasing filtration of a Lie algebra - Maple Programming Help

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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 $\mathrm{𝔤}$

Description

 • A collection of subspaces    of a Lie algebra $\mathrm{𝔤}$ defines a decreasing filtration of  if  [i]   for , [ii]  for  and [iii] for .
 • Query([f0, f1, ... fN], "Filtration") returns true if the subspaces   define a decreasing filtration of the Lie algebra 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

First we initialize a Lie algebra.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[1,4,1\right],2\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,2\right],1\right],\left[\left[3,4,2\right],1\right],\left[\left[3,4,3\right],1\right]\right]\right]\right)$
 ${\mathrm{L1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

Now define a sequence of 4 subspaces.

 Alg1 > $\mathrm{f0}≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3},\mathrm{e4}\right]:$$\mathrm{f1}≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right]:$$\mathrm{f2}≔\left[\mathrm{e1},\mathrm{e2}\right]:$$\mathrm{f3}≔\left[\mathrm{e1}\right]:$$\mathrm{f4}≔\left[\right]:$

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

 Alg1 > $\mathrm{Query}\left(\left[\mathrm{f0},\mathrm{f1},\mathrm{f2},\mathrm{f3},\mathrm{f4}\right],"Filtration"\right)$
 ${\mathrm{true}}$ (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 > $\mathrm{f0}≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3},\mathrm{e4}\right]:$$\mathrm{f1}≔\left[\mathrm{e1},\mathrm{e2}\right]:$$\mathrm{f2}≔\left[\mathrm{e2}\right]:$$\mathrm{f3}≔\left[\right]:$
 Alg1 > ${\mathrm{infolevel}}_{\mathrm{Query}}≔2:$
 Alg1 > $\mathrm{Query}\left(\left[\mathrm{f0},\mathrm{f1},\mathrm{f2},\mathrm{f3},\mathrm{f4}\right],"Filtration"\right)$
 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]
 ${\mathrm{false}}$ (2.3)
 Alg1 > 

This shows that the the Lie bracket is not contained in ${f}_{2}$

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