find the action of a solvable Lie group on a manifold from its infinitesimal generators - Maple Programming Help

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GroupActions[Action] - find the action of a solvable Lie group on a manifold from its infinitesimal generators

Calling Sequences

     Action(Gamma, G, options)

Parameters

     Gamma     - a list, a basis for a Lie algebras of vector fields on a manifold M

     G         - a Maple name or string, the name of a coordinate system for the abstract Lie group defined by Gamma

     options   - output = O, where O is a list of keywords "ManifoldToManifold", "GroupToManifold", "LieGroup", "Basis"

 

Description

Examples

Description

• 

Let G be a Lie group with multiplication * and identity e. An action of G on a manifold M is a smooth map μ :G.M such that μe, x = x and μa*b,x = μa, μb, xfor all a, b G and x  M.  For a given action μ,define

 μ1,a:MM  by μ1,ax = μa, x    and  μ2,x:GM  by μ2,xa = μa, x.

The infinitesimal generators for the action μ is the Lie algebra of vector fields Γμ on M defined by the pushforward by μ2,x of the right invariant vector fields on G. The infinitesimal generators Γμ can also be computed by differentiating the components of the map μ2,xwith respect to the group parameters a and evaluating the results at the identity.

• 

The command Action(Gamma, G) calculates the group action μ such that Γμ = Γ. The program returns the transformation μ1,a. With the keyword list O = ["GroupToManifold"], the transformation μ2,x is returned.

• 

 In the course of finding the action μ, the Action procedure calculates the abstract Lie algebra 𝔤 defined by the vector fields Γ. If the adjoint representation for 𝔤 is not upper triangular, then a call to the program SolvableRepresentation is made to find a new basis for 𝔤 (and hence Γ) in which the adjoint representation is upper triangular. This new basis can be retrieved by adding the keyword "Basis" to the keyword list O. The action procedure also uses the LieGroup command to find the Lie group module for the Lie algebra 𝔤. This module can be returned by adding the keyword "LieGroup" to the keyword list O.

• 

The default value for O is O = ["ManifoldToManifold"]. The option O = ["all"] is equivalent to O = ["ManifoldToManifold", "GroupToManifold", "LieGroup", "Basis"].

• 

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

Examples

withDifferentialGeometry:withGroupActions:withLieAlgebras:withLibrary:

 

Example 1.

First define a 2-dimensional manifold M with coordinates x,y.

DGsetupx,y,M:

 

On M, define a 3-dimensional Lie algebra of vector fields Γ.

M > 

ΓevalDGD_x,D_y,yD_x

Γ:=D_x,D_y,yD_x

(2.1)

 

We need a 3-dimensional manifold to represent the abstract Lie group defined by Γ.

M > 

LieAlgebraDataΓ

e2,e3=e1

(2.2)
M > 

DGsetupz1,z2,z3,G:

G > 

μ1ActionΓ,G

μ1:=x=yz3+z2z3+x+z1,y=z2+y

(2.3)

 

Use the InfinitesimalTransformation command to find the infinitesimal generators for this action. Note that they are precisely the vectors we began with.

M > 

newGammaInfinitesimalTransformationμ1,z1,z2,z3

newGamma:=D_x,D_y,yD_x

(2.4)

 

Example 2.

We continue with Example 1 but this time present the vector fields in a different order.

M > 

Γ2evalDGyD_x,D_x,D_y

Γ2:=yD_x,D_x,D_y

(2.5)
M > 

L2LieAlgebraDataΓ2,Alg2

L2:=e1,e3=e2

(2.6)
M > 

DGsetupL2

Lie algebra: Alg2

(2.7)

 

In this case the adjoint representation is not upper triangular. The Action program will force us back to the basis of Example 1.  This change of basis can be obtained using the output option.

Alg2 > 

Adjoint

000001000,000000000,000100000

(2.8)
Alg2 > 

μ1,BActionΓ2,G,output=ManifoldToManifold,Basis

μ1,B:=x=yz2+x+z1,y=z3+y,0,1,0,1,0,0,0,0,1

(2.9)
M > 

newGammaInfinitesimalTransformationμ1,z1,z2,z3

newGamma:=D_x,yD_x,D_y

(2.10)

 

This basis for the infinitesimal generators agrees with the basis whose components are given by the list B.

M > 

mapDGzip,B,Γ2,plus

D_x,yD_x,D_y

(2.11)

 

Example 3.

We take an example from the Lie algebras of vector fields in the paper by Gonzalez-Lopez, Kamran, Olver. The Lie algebra of vector fields in this paper are part of the DifferentialGeometry Library.

M > 

DGsetupx,y,M:

M > 

Γ3RetrieveGonzalez-Lopez,1,22,17,manifold=M

Γ3:=D_x,D_y,xD_y,12x2D_y,ⅇxD_y

(2.12)
M > 

DGsetupz1,z2,z3,z4,z5,G3

frame name: G3

(2.13)
G3 > 

μActionΓ3,G3

μ:=x=z5+x,y=ⅇz5+xz1+z2+z3x+12x2z4+y

(2.14)
M > 

InfinitesimalTransformationμ,z1,z2,z3,z4,z5

ⅇxD_y,D_y,xD_y,12x2D_y,D_x

(2.15)

 

Example 4.

We take an example from the Lie algebras of vector fields in the book by Petrov.

M > 

DGsetupx,y,u,v,M4:

M4 > 

Γ4RetrievePetrov,1,32,6,manifold=M4

Γ4:=D_y,D_u,D_uu+D_yyD_x,D_uyD_yu

(2.16)
M4 > 

DGsetupz1,z2,z3,z4,G4

frame name: G4

(2.17)
G4 > 

μActionΓ4,G4

μ:=x=z3+x,y=sinz4ⅇz3u+cosz4ⅇz3y+z1,u=sinz4ⅇz3y+cosz4ⅇz3u+z2,v=v

(2.18)
M4 > 

InfinitesimalTransformationμ,z1,z2,z3,z4

D_y,D_u,D_uu+D_yyD_x,D_uyD_yu

(2.19)

See Also

DifferentialGeometry

GroupActions

Library

Adjoint

DGzip

InfinitesimalTransformation

LieAlgebraData

Retrieve

 


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