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IsIdeal

check if the solutions of a LAVF are an ideal in the Lie algebra of another LAVF

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsIdeal(L1, L2)

Parameters

L1, L2

-

LAVF objects that are Lie algebras i.e. IsLieAlgebra(obj) returns true, see IsLieAlgebra.

Description

• 

Let L1, L2 be LAVF objects that are Lie algebras. Then IsIdeal(L1,L2) checks if solutions of L1 are an ideal in the Lie algebra of solutions of L2.

• 

Internally the method returns true if L1,L2L1 (i.e. IsInvariant(L1,L2) returns true), L2 is Lie algebra (i.e. IsLieAlgebra(L2) returns true), and L1L2 (i.e. IsSubspace(L1,L2) returns true). False otherwise.

• 

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=0,diffηx,y,y=0,indep=x,y,dep=ξ,η

Sξx=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 these two LAVFs,

LLAVFV,S

Lξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(4)

LE2LAVFV,E2

LE2ξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(5)

For the solutions of L be an ideal in the Lie algebra of LE2, the following conditions must be true..

IsInvariantL,LE2

true

(6)

IsLieAlgebraLE2

true

(7)

IsSubspaceL,LE2

true

(8)

or by using a more direct method.

IsIdealL,LE2

true

(9)

conversely will be false since LE2 is not subalgebra of L.

IsIdealLE2,L

false

(10)

Compatibility

• 

The IsIdeal 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

IsInvariant

IsSubspace