NilpotentRadical - Maple Programming Help

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NilpotentRadical

calculate the nilpotent radical of a LAVF object.

IsReductive

check if a LAVF is reductive.

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

NilpotentRadical( obj)

IsReductive( 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 NilpotentRadical method returns the nilpotent radical of L, as a LAVF object.

• 

By mathematical definition, the nilpotent radical of L is the intersection of the solvable radical of L and the derived algebra of L. Note that this is not the same thing as the nilradical.

• 

Let NPR be the nilpotent radical of a LAVF object L. Then IsReductive(L) returns true if and only if L is reductive i.e. iff NPR is trivial (i.e. IsTrivial(NPR) returns true).

• 

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 the Euclidean Lie algebra E(2).

LLAVFV,E2

Lξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(3)

IsLieAlgebraL

true

(4)

NPRNilpotentRadicalL

NPRξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(5)

The nilpotent radical is not trivial, therefore L is not reductive.

IsReductiveL

false

(6)

Compatibility

• 

The NilpotentRadical and IsReductive 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

IsTrivial