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AreCommuting

check if one LAVF commutes with another

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

AreCommuting(L, M)

AreCommuting(L, M, N)

Parameters

L, M, N

-

LAVF objects.

Description

• 

Let L, M, N be LAVF objects on the same space with same local coordinates. Then AreCommuting(L, M) checks if L commutes with M, i.e. if L,M=0.

• 

Similarly, the three arguments call AreCommuting(L, M, N) checks if L commutes with M mod N, i.e. if L,M is in N.

• 

This method is symmetric in the first two input arguments, that is, AreCommuting(L,M, N) is same as AreCommuting(M,L,N).

• 

Some Lie algebraic methods (IsLieAlgebra, IsAbelian, and IsIdeal) are front-ends to AreCommuting.

• 

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)

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

(2)

Construct a LAVF for E(2).

LLAVFV,E2

Lξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(3)

We can check if L is closed under commutator ...

AreCommutingL,L,L

true

(4)

or by using a more direct call.

IsLieAlgebraL

true

(5)

We can also check if L is abelian...

AreCommutingL,L

false

(6)

or by using a more direct call.

IsAbelianL

false

(7)

As we know the centre of L must be abelian,

AreCommutingCentreL,CentreL

true

(8)

Compatibility

• 

The AreCommuting 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

IsAbelian

IsIdeal