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GroupActions[IsotropyFiltration] - find the infinitesimal isotropy filtration for a Lie algebra of vector fields

Calling Sequences

     IsotropyFiltration(Gamma, pt, option)

Parameters

  Gamma     - a list of vector fields on a manifold M

  pt        - a list of equations x1 = p1, x2 = p2, ... specifying the coordinates of point p  M

  option    - the optional argument output = O, where O is a list containing the keywords "Vector" and/or the name of an initialized abstract algebra for the Lie algebra of vector fields Gamma.

  

 

Description

Examples

Description

• 

Let Γ be a Lie algebra of vector fields on a manifold M  and letp M. The isotropy filtration of  the  Lie algebra of vector fields Γ is the decreasing nested sequence of subalgebras   ΓpkΓp1 Γp0  Γ defined by

 Γpk = { X Γ | the coefficients of X and their derivatives all vanish to order k at p }.

Note that if X  Γpk and Y  Γpℓ , then X, Y  Γpk+ℓ.  The subalgebra Γp0 is called the isotropy subalgebra of Γ at p.

• 

 The command IsotropyFiltration(Gamma, pt) returns a list of list of vector fields, the first list gives a basis for Γp0, the second list gives a basis for Γp1 and so on.

• 

The command IsotropyFiltration is part of the DifferentialGeometry:-GroupActions package.  It can be used in the form IsotropyFiltration(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-IsotropyFiltration(...).

Examples

withDifferentialGeometry:withGroupActions:withLibrary:withLieAlgebras:

 

Example 1.

First we obtain a Lie algebra of vector fields from the paper by Gonzalez-Lopez, Kamran, Olver in the DifferentialGeometry Library using the Retrieve command and then we compute the isotropy filtration.

DGsetupx,y,M

frame name: M

(2.1)
M > 

ΓRetrieveGonzalez-Lopez,1,27,4,manifold=M

Γ:=D_x,2xD_x+4yD_y,x2D_x+4xyD_y,D_y,xD_y,x2D_y,x3D_y,x4D_y

(2.2)

 

We calculate the isotropy filtration as a subalgebra of Γ.

M > 

F1IsotropyFiltrationΓ,x=0,y=0

F1:=2xD_x+4yD_y,x2D_x+4xyD_y,xD_y,x2D_y,x3D_y,x4D_y,x2D_x+4xyD_y,x2D_y,x3D_y,x4D_y,x3D_y,x4D_y,x4D_y,

(2.3)

 

Example 2.

We continue with Example 1. Here we calculate the isotropy filtration as a subalgebra of the abstract Lie algebra defined by Γ. To this end, we first calculate the structure constants for Γand initialize the result as Alg1.

M > 

LLieAlgebraDataΓ,Alg1

L:=e1,e2=2e1,e1,e3=e2,e1,e5=e4,e1,e6=2e5,e1,e7=3e6,e1,e8=4e7,e2,e3=2e3,e2,e4=4e4,e2,e5=2e5,e2,e7=2e7,e2,e8=4e8,e3,e4=4e5,e3,e5=3e6,e3,e6=2e7,e3,e7=e8

(2.4)
M > 

DGsetupL:

 

Now re-run the IsotropyFiltration command with the third argument output = [Alg1].

Alg1 > 

FIsotropyFiltrationΓ,x=0,y=0,output=Alg1

F:=e2,e3,e5,e6,e7,e8,e3,e6,e7,e8,e7,e8,e8,

(2.5)

 

We check that F does indeed define a filtration of the Lie algebra Γ (note that there is an index shift  Γpk = F[k+1]).

Alg1 > 

BracketOfSubspacesF1,F1

2e3,2e5,2e7,4e8,3e6

(2.6)
Alg1 > 

BracketOfSubspacesF1,F2

2e3,2e7,4e8,3e6

(2.7)
Alg1 > 

BracketOfSubspacesF1,F3

2e7,4e8

(2.8)
Alg1 > 

BracketOfSubspacesF1,F4

4e8

(2.9)
Alg1 > 

BracketOfSubspacesF2,F3

e8

(2.10)
Alg1 > 

BracketOfSubspacesF2,F4

(2.11)

 

All these brackets can be checked at once with Query/"filtration".

Alg1 > 

QueryF,Filtration

true

(2.12)

See Also

DifferentialGeometry

GroupActions

Library

LieAlgebras

BracketOfSubspaces

IsotropySubalgebra,

LieAlgebraData

Query

 


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