IsNilpotent - Maple Help
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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  where

• 

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. ).

• 

These methods are associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

(1)

(2)

Construct a LAVF for E(2).

(3)

(4)

(5)

(6)

(7)

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

(8)

(9)

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

(10)

(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

 


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