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SolvableRadical

calculate the solvable radical of a LAVF object.

IsSolvable

check if a LAVF object is solvable.

DerivedSeries

calculate the derived series of a LAVF object.

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

SolvableRadical( obj)

SolubleRadical( obj)

Radical( obj)

IsSolvable( obj)

IsSoluble( obj)

DerivedSeries( obj)

Parameters

obj

-

a LAVF object that is a Lie algebra i.e.IsLieAlgebra(obj) returns true, see IsLieAlgebra.

Description

• 

Let L be a LAVF object which is a Lie algebra. Then the SolvableRadical method returns the solvable radical of L (i.e. its largest solvable ideal), as a LAVF object.

• 

Let L be a LAVF object which is a Lie algebra. Then IsSolvable(L) returns true if and only if L is solvable (i.e. SolvableRadical(L) = L).

• 

The names SolubleRadical, Radical (and IsSoluble) are provided as aliases for SolvableRadical (and IsSolvable respectively).

• 

Let L be a LAVF object which is a Lie algebra. Then the DerivedSeries method returns the derived series of L, as a list of LAVF objects.

• 

By definition, the derived series of L is a sequence of ideals L=L1L2LiLk where Li+1=Li,Li

• 

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)

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)

IsLieAlgebraL

true

(4)

SRSolvableRadicalL

SRξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(5)

We can use the method AreSame to confirm both SR and L are the same. And therefore, L is solvable.

AreSameSR,L

true

(6)

IsSolvableL

true

(7)

DerivedSeriesL

ξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0,ξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0,ξx+ηy&whereξ=0,η=0

(8)

Compatibility

• 

The SolvableRadical, IsSolvable and DerivedSeries 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

AreSame