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Centraliser

calculate the centraliser of one LAVF object in another.

Normaliser

 calculate the normaliser of one LAVF object in another.

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Centraliser( M, L)

Normaliser( L, U)

Parameters

M, L

-

LAVF objects where ML (see IsSubspace for more detail)

U

-

(optional) a LAVF object where LU

Description

• 

Let M, L be LAVF objects which are Lie algebras (see IsLieAlgebra) and  ML  . Then Centraliser(M, L) computes the centraliser of M in L, namely xL|x,M=0, as a new LAVF object.

• 

Centraliser(M, L) is equivalent to Transporter(L, M, T) where T is a trivial LAVF (i.e. its determining system has trivial solutions) associated with vector fields of L. See the method Transporter for more detail.

• 

The name Centralizer is provided as alias.

• 

Similarly, let L, U be LAVF objects that are Lie algebras and LU. Then Normaliser(L,U) computes the normaliser of L in U, namely xU|x,LL , as a new LAVF object.

• 

The second input argument U defaults to a universal LAVF associated with vector fields of L. That is, Normaliser(L) is equivalent to Normaliser(L, U) where U := LAVF(GetVectorField(L), "universal").

• 

The call Normaliser(L,U) is equivalent to Transporter(U, L, L).

• 

These methods are 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)

Example 1:

T2LHPDEdiffξx,y,x=0,diffξx,y,y=0,diffηx,y,x=0,diffηx,y,y=0,indep=x,y,dep=ξ,η

T2ξx=0,ξy=0,ηx=0,ηy=0,indep=x,y,dep=ξ,η

(2)

E2LHPDEdiffξx,y,y,y=0,diffηx,y,x=diffξx,y,y,diffηx,y,y=0,diffξx,y,x=0,indep=x,y,dep=ξ,η

E2ξy,y=0,ηx=ξy,ηy=0,ξx=0,indep=x,y,dep=ξ,η

(3)

We first construct two LAVF objects

LT2LAVFV,T2

LT2ξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(4)

LE2LAVFV,E2

LE2ξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(5)

Both LAVFs are Lie algebras and L is indeed a subalgebra of LE2

IsLieAlgebraLT2

true

(6)

IsLieAlgebraLE2

true

(7)

2-dim translation group is subspace of 2-dim Euclidean group

IsSubspaceLT2,LE2

true

(8)

The centraliser of translations in E(2) is the translations themselves...

CentraliserLT2,LE2

ξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(9)

Normaliser of translations in E(2) is E(2)

NormaliserLT2,LE2

ξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(10)

Normaliser of E(2) in 'Lie algebra' of all vector fields is 4-dim, and includes E(2) as well as the uniform scalings...

N2NormaliserLE2

N2ξx+ηy&whereξy,y=0,ηy,y=0,ξx=ηy,ηx=ξy

(11)

SolutionDimensionN2

4

(12)

Example 2:

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

Sξx,x=0,ξy=0,η=0,indep=x,y,dep=ξ,η

(13)

L is an affine group on the line (i.e. acts on x only)...

LLAVFV,S

Lξx+ηy&whereξx,x=0,ξy=0,η=0

(14)

IsLieAlgebraL

true

(15)

SolutionDimensionL

2

(16)

Normaliser of L in Lie algebra of all vector fields is an infinite Lie pseudogroup

NNormaliserL

Nξx+ηy&whereξx,x=0,ηx=0,ξy=0

(17)

SolutionDimensionN

(18)

Compatibility

• 

The Centraliser and Normaliser commands were 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

IsSubspace

Transporter