find the normalizer of a finite dimensional Lie algebra of vector fields in an (infinite-dimensional) pseudo-Lie algebra of vector fields - Maple Programming Help

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GroupActions[InfinitesimalPseudoGroupNormalizer] - find the normalizer of a finite dimensional Lie algebra of vector fields in an (infinite-dimensional) pseudo-Lie algebra of vector fields

Calling Sequences

     InfinitesimalPseudoGroupNormalizer(Gamma, options)

Parameters

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

     options               - (optional) arguments that can be given in any order and are described as follows

     ansatz                - ansatz= Z, where Z is a vector field on M

     unknowns              - unknowns = U, where U is a list of the unknown functions appearing in the vector field Z

   auxiliaryequations    - auxiliaryequations = E, where E is a list of additional partial differential equations to be imposed on the unknowns functions U

   parameters            - parameters = P, where P is a set of parameters appearing in the vector fields Γ

     liealgebra                         -  liealgebra  = name, where name is the string or name for the abstract Lie algebra defined by Γ

   output                - output = "general", "pde"  or "list"

     other options         - optional arguments to be passed to pdsolve

     

 

Description

Examples

Description

• 

Let  Γ= X1, X2 , ... Xp be a p -dimensional Lie algebra of vector fields, defined on a manifold M.  Let 𝒢 be an infinite dimensional Lie algebra of vector fields on M, whose general element depends upon a certain number of arbitrary functions and suppose Γ  𝒢. Then the normalizer of  Γ  in 𝒢  is

NorΓ, 𝒢 = { Z  𝒢   |  Z,X  Γ for all X Γ} = {  Z  𝒢   |  Z,Xi =  ai j Xj  for  i= 1,... , p  and where the coefficients aij are constant}.

The vector fields Xi always belong to Nor(Γ, 𝒢). The procedure InfinitesimalPseudoGroupNormalizer uses the pdsolve command to calculate Nor( Γ, 𝒢),modulo the vector fields in Γ but including vector fields in the center of Γ.

• 

 With the calling sequence InfinitesimalPseudoGroupNormalizer(Gamma), the normalizer of Γ in the full infinitesimal pseudo-group of all vector fields on M is computed. With output = "general" (the default value), a single vector field, depending upon arbitrary constants and functions, is returned. If the output depends only on constants (that is, Nor( Γ, 𝒢) is a finite-dimensional algebra) and output = "list", then a list of vectors is returned. In this case the vector fields in the center of Γ are removed. With output = "pde", the determining differential equations for the normalizer are returned.

• 

With the keyword argument ansatz = Z, the procedure InfinitesimalPseudoGroupNormalizer calculates Nor( Γ, 𝒢), where 𝒢 is the infinitesimal pseudo-group defined by the vector field Z. With this keyword option, the unknown functions in Z must be explicitly listed using the keyword argument unknowns = U. Additional algebraic and differential constraints on the unknown functions in Z may be specified with the keyword argument auxiliaryequations = E. Note that the full system of differential equations for Z is likely to be inconsistent if the vector fields in Γ do not satisfy the differential constraints defined by E.

• 

 If the abstract Lie algebra determined by Γ has been calculated and initialized (see LieAlgebraData, DGsetup), then this information can be passed to InfinitesimalPseudoGroupNormalizer with the keyword argument liealgebra = name.

• 

 The keyword argument parameters = P will invoke the case-splitting capabilities of pdsolve . In this case, the output of InfinitesimalPseudoGroupNormalizer will be a sequence of normalizers for both the generic and special values of the parameters.

• 

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

Examples

withDifferentialGeometry:withGroupActions:withLieAlgebras:withLibrary:

 

Example 1.

First define a 1-dimensional manifold M with coordinate x.

DGsetupx,M:

 

On M, define 1 and 2-dimensional Lie algebras of vector fields Γ1 1, Γ1 2 , respectively.

M > 

Γ11evalDGD_x

Γ11:=D_x

(2.1)
M > 

Γ12evalDGD_x,xD_x

Γ12:=D_x,xD_x

(2.2)

 

Find the normalizer of these two Lie algebras in the Lie algebras of all vector fields on M.

M > 

N11InfinitesimalPseudoGroupNormalizerΓ11

N11:=_C1x+_C2D_x

(2.3)
M > 

N11listInfinitesimalPseudoGroupNormalizerΓ11,output=list

N11list:=xD_x

(2.4)
M > 

N12InfinitesimalPseudoGroupNormalizerΓ12

N12:=

(2.5)

 

Example 2.

Find the normalizer for the Lie algebra of infinitesimal rotations in 3 dimensions.

M > 

DGsetupx,y,z,N

frame name: N

(2.6)
N > 

Γ2evalDGzD_yyD_z,xD_yyD_x,zD_xxD_z

Γ2:=D_yzD_zy,D_xy+D_yx,D_xzD_zx

(2.7)
N > 

N2InfinitesimalPseudoGroupNormalizerΓ2

N2:=x_F1x2+y2+z2z2D_xz+y_F1x2+y2+z2z2D_yz+_F1x2+y2+z2z2D_z

(2.8)

 

Now let us find the normalizer for the Lie algebra of infinitesimal rotations in three dimensions within the infinite-dimensional Lie algebra of divergence-free vector fields.

First define a general vector field on N with arbitrary coefficients Ax, y,  z,  Bx,  y,  z and Cx,  y,  z.

N > 

UA,B,Cx,y,z

U:=Ax,y,z,Bx,y,z,Cx,y,z

(2.9)
N > 

ZevalDGU1D_x+U2D_y+U3D_z

Z:=Ax,y,zD_x+Bx,y,zD_y+Cx,y,zD_z

(2.10)

 

We use the keyword argument auxiliaryequations to require that the vector field Z be divergence-free.

N > 

ExAx,y,z+yBx,y,z+zCx,y,z=0:

 

N > 

N2divInfinitesimalPseudoGroupNormalizerΓ2,ansatz=Z,unknowns=U,auxiliaryequations=E

N2div:=x_C1D_xx2+y2+z23/2+y_C1D_yx2+y2+z23/2+_C1zD_zx2+y2+z23/2

(2.11)

 

Example 3.

In this example we shall calculate the normalizers for a Lie algebra of vector fields which depends upon a parameter α.  We find that Nor(Γ3) mod Γ3 has dimension 2 for α 0 and dimension 3 for α = 0.

M > 

Γ3evalDGD_x,D_y,D_z,x+zD_x+αyD_y+zD_z

Γ3:=D_x,D_y,D_z,x+zD_x+αyD_y+zD_z

(2.12)
G > 

N3InfinitesimalPseudoGroupNormalizerΓ3,parameters=α,output=list

N3:=zD_x,D_xx+D_zz,D_y,zD_x,D_xx+D_zz,α=0,α=α

(2.13)

 

Example 4.

We calculate the normalizer of the infinitesimal Euclidean group in the infinitesimal pseudo-group of all contact transformation on a 3- dimensional contact manifold M with coordinates x,y,z with contact form ω = dy  z dx. 

N > 

DGsetupx,y,z,M

frame name: M

(2.14)

 

Here is the standard Euclidean metric on the x,y plane and the standard contact form on M.

M > 

gevalDGdx &t dx+dy &t dy

g:=dxdx+dydy

(2.15)
M > 

ωevalDGdyzdx

ω:=dxz+dy

(2.16)

 

We use the command InfinitesimalSymmetriesOfGeometricObjectFields to find the Lie algebra of vector fields which preserves the metric g and the Pfaffian system generated by ω.

M > 

Γ4InfinitesimalSymmetriesOfGeometricObjectFieldsg,ω,output=list

Γ4:=yD_x+xD_y+z2+1D_z,D_y,D_x

(2.17)

 

We define an arbitrary vector field Z on M and again use the command InfinitesimalSymmetriesOfGeometricObjectFields, this time to find the partial differential equations which the coefficients of Z must satisfy in order that this vector field be an infinitesimal contact transformation.

M > 

ZevalDGA1x,y,zD_x+A2x,y,zD_y+A3x,y,zD_z

Z:=A1x,y,zD_x+A2x,y,zD_y+A3x,y,zD_z

(2.18)
M > 

EInfinitesimalSymmetriesOfGeometricObjectFieldsω,ansatz=Z,unknowns=A1,A2,A3x,y,z,output=pde

E:=z_K111x,y,z+xA2x,y,zxA1x,y,zzA3x,y,z,_K111x,y,z+yA2x,y,zyA1x,y,zz,zA2x,y,zzA1x,y,zz,0,A1x,y,z,A2x,y,z,A3x,y,z,_K111x,y,z

(2.19)

 

Note that the factor K111x,y,z  is an additional unknown satisfying ℒZω = K111 ω.   The sought after normalizer of Γ4 in the infinitesimal pseudo-group of contact transformations can now be computed.

M > 

Nor4InfinitesimalPseudoGroupNormalizerΓ4,ansatz=Z,unknowns=E2,auxiliaryequations=E1,output=list

Nor4:=D_xx+D_yy,zD_xz2+1+D_yz2+1

(2.20)

 

We can check this result by noting that [i] the vector fields Nor4 preserve ω and [ii] the normalizer is a 5-dimensional Lie algebra which contains Γ4 as an ideal.

M > 

LieDerivativeNor4,ω

dxz+dy,0dx

(2.21)
M > 

LDLieAlgebraDataopΓ4,opNor4,alg

LD:=e1,e2=e3,e1,e3=e2,e2,e4=e2,e3,e4=e3,e4,e5=e5

(2.22)
M > 

DGsetupLD

Lie algebra: alg

(2.23)
alg > 

Querye1,e2,e3,Ideal

true

(2.24)

See Also

DifferentialGeometry

GroupActions

InfinitesimalSymmetriesOfGeometricObjectFields

LieAlgebraData

Query

 


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