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Nilradical

 calculate the nilradical of of a LAVF object.

LowerCentralSeries

calculate the lower central series of a LAVF object.

UpperCentralSeries

calculate the upper central series of a LAVF object.

Hypercentre

calculate the hypercentre of a LAVF object.

IsNilpotent

check if a LAVF object is nilpotent.

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Nilradical( obj)

NilRadical( obj)

LowerCentralSeries( obj)

UpperCentralSeries( obj)

Hypercentre( obj)

Hypercenter( obj)

IsNilpotent( 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 Nilradical method returns the nilradical of L (i.e. its largest nilpotent ideal), as a LAVF object.

• 

The name NilRadical is provided as an alias.

• 

Let L be a LAVF object which is a Lie algebra. Then LowerCentralSeries(L) returns the lower central series of L, as a list of LAVF objects.

• 

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

• 

Similarly, the call UpperCentralSeries(L) returns the upper central series of L, as a list of LAVF objects.

• 

Let L be a LAVF object which is a Lie algebra. Then Hypercentre(L) returns the hypercentre of L (i.e. last term of the upper central series), as a LAVF object.

• 

The name Hypercenter is provided as an alias.

• 

The call IsNilpotent(L) returns true if and only if the last term of the lower central series of L is trivial (i.e. Lk=0).

• 

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)

NilradicalL

ξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(5)

UCSUpperCentralSeriesL

UCSξx+ηy&whereξ=0,η=0

(6)

LCSLowerCentralSeriesL

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

(7)

By definition, the last term of the upper central series should be identical to the hypercentre.

HypercentreL

ξx+ηy&whereξ=0,η=0

(8)

AreSameHypercentreL,UCS1

true

(9)

The last term of the lower central series of L (LCS) is not trivial. Therefore, L is not nilpotent.

IsNilpotentL

false

(10)

AreSameHypercentreL,L

false

(11)

Compatibility

• 

The Nilradical, LowerCentralSeries, UpperCentralSeries, Hypercentre and IsNilpotent 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

IsTrivial