check if a Lie algebra is solvable - Maple Programming Help

Online Help

All Products    Maple    MapleSim


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

Query[Solvable] - check if a Lie algebra is solvable

Calling Sequences

     Query(Alg, "Solvable")

     Query(S, "Solvable")

Parameters

     Alg     - (optional) name or string, 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 solvable if the k-th ideal 𝒟kg in the derived series for 𝔤  is 0 for some k0. Every nilpotent Lie algebra is solvable.

• 

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

• 

Query(S, "Solvable") returns true if the subalgebra S is a solvable 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,Alg1,3,2,3,1,1

L1:=e2,e3=e1

(2.1)

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

L2:=e1,e3=e1,e2,e3=e1+e2

(2.2)

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

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

(2.3)

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

 

Alg1 and Alg2 are solvable but Alg3 is not. (Alg1 is actually nilpotent while Alg3 is semisimple.)

Alg3 > 

QueryAlg1,Solvable

true

(2.4)
Alg1 > 

QueryAlg2,Solvable

true

(2.5)
Alg2 > 

QueryAlg3,Solvable

false

(2.6)

 

The subalgebra S  = spanz1, z2 is a solvable subalgebra of Alg3. (The algebra Alg3 is sl2, ℝ and S is a Borel subalgebra.)

Alg3 > 

Queryz1,z2,Solvable

true

(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