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Transporter

find the transporter of a LAVF to another LAVF in the third LAVF

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Transporter(L, M, N)

Parameters

L, M, N

-

LAVF objects live on same space.

Description

• 

Let L, M, N be LAVF objects living on the same space. Then Transporter(L,M,N) finds the transporter of M to N in L, as a new LAVF object.

• 

By definition, the transporter of M to N in Lconsists of the vector fields in L that map vector fields in M to vector fields in N, under commutator. That is, it is the subspace XLL|XL,XMN,XMM where  L,M,N are subspaces of some Lie algebra.

• 

Some Lie algebraic structural methods (Center, Centraliser, Normaliser, and UpperCentralSeries) are front-ends to Transporter.

• 

This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressξx,y,ηx,y:

VVectorFieldξx,yDx+ηx,yDy,space=x,y

Vξx+ηy

(1)

SLHPDEdiffξx,y,x=0,diffξx,y,y=0,diffηx,y,x,x=0,diffηx,y,y=0,indep=x,y,dep=ξ,η

Sξx=0,ξy=0,ηx,x=0,ηy=0,indep=x,y,dep=ξ,η

(2)

S1LHPDEdiffξx,y,x=0,diffξx,y,y=0,ηx,y=0,indep=x,y,dep=ξ,η

S1ξx=0,ξy=0,η=0,indep=x,y,dep=ξ,η

(3)

S2LHPDEξx,y=0,diffηx,y,x=ηx,yx,diffηx,y,y=0,indep=x,y,dep=ξ,η

S2ξ=0,ηx=ηx,ηy=0,indep=x,y,dep=ξ,η

(4)

Constructing some LAVFs,

LLAVFV,S

Lξx+ηy&whereηx,x=0,ξx=0,ξy=0,ηy=0

(5)

L1LAVFV,S1

L1ξx+ηy&whereξx=0,ξy=0,η=0

(6)

L2LAVFV,S2

L2ξx+ηy&whereηx=ηx,ηy=0,ξ=0

(7)

L0LAVFV,trivial

L0ξx+ηy&whereξ=0,η=0

(8)

TransporterL,L2,L1

ξx+ηy&whereηx,x=0,ηy=0,ξ=0

(9)

This is centre of L

TransporterL,L,L0

ξx+ηy&whereηx=0,ηy=0,ξ=0

(10)

and the second centre of L

TransporterL,L,CentreL

ξx+ηy&whereηx,x=0,ξx=0,ξy=0,ηy=0

(11)

The upper central series of L is a sequences of L consisting a trivial LAVF, centre of L, and the second centre of L.

UpperCentralSeriesL

ξx+ηy&whereξ=0,η=0,ξx+ηy&whereηx=0,ηy=0,ξ=0,ξx+ηy&whereηx,x=0,ξx=0,ξy=0,ηy=0

(12)

Compatibility

• 

The Transporter command was introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields[LHPDE]

LieAlgebrasOfVectorFields[LAVF]

IsLieAlgebra

Center

Centraliser

Normaliser

UpperCentralSeries