check if a Lie algebra is nilpotent - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : Query : DifferentialGeometry/LieAlgebras/Query/Nilpotent

Query[Nilpotent] - check if a Lie algebra is nilpotent

Calling Sequences

     Query(Alg, "Nilpotent")

     Query(S, "Nilpotent")

Parameters

     Alg     - (optional) the name of an initialized Lie algebra

     S       - a list of vectors defining a basis for a subalgebra

 

Description

Examples

Description

• 

A Lie algebra 𝔤  is nilpotent if the k-th ideal in the lower central series for 𝔤  is 0 for some k  0.

• 

Query(Alg, "Nilpotent") returns true if Alg is a nilpotent Lie algebra and false otherwise.  If the algebra is unspecified, then Query is applied to the current algebra.

• 

Query(S, "Nilpotent") returns true if the subalgebra spanned by the vectors  S is a nilpotent Lie algebra and false otherwise.

• 

The command Query is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We initialize three different Lie algebras.

L1_DGLieAlgebra,Alg1,3,

L1:=

(2.1)

L2_DGLieAlgebra,Alg2,3,2,3,1,1:

L3_DGLieAlgebra,Alg3,4,1,4,1,2,2,3,1,1,2,4,2,1,3,4,3,1

L3:=e1,e4=2e1,e2,e3=e1,e2,e4=e2,e3,e4=e3

(2.2)

DGsetupL1,x,a:DGsetupL2,y,b:DGsetupL3,z,c:

 

Alg1 and Alg2 are nilpotent but Alg3 is not.

Alg3 > 

QueryAlg1,Nilpotent

true

(2.3)
Alg1 > 

QueryAlg2,Nilpotent

true

(2.4)
Alg2 > 

QueryAlg3,Nilpotent

false

(2.5)

 

The subalgebra  S1 = spanz1, z2, z3 of Alg3 is nilpotent but the subalgebra S2=spanz1, z4 is not.

Alg3 > 

S1z1,z2,z3:S2z1,z4:

Alg3 > 

QueryS1,Nilpotent

true

(2.6)
Alg3 > 

QueryS2,Nilpotent

false

(2.7)

See Also

DifferentialGeometry

LieAlgebras

Query

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam