check if a Lie algebra is Abelian - Maple Help

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Query[Abelian] - check if a Lie algebra is Abelian

Calling Sequences

     Query(Alg, "Abelian")

     Query(S, "Abelian")

Parameters

     Alg     - (optional) name or string, the name of an initialized Lie algebra 𝔤

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

Description

• 

A Lie algebra 𝔤  is Abelian if x, y = 0  for all x,y 𝔤.  

• 

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

• 

Query(S, "Abelian") returns true if the subalgebra S is an Abelian 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 a pair of 3-dimensional Lie algebras.The first algebra is not Abelian and the second is, as confirmed by a call to Query.

L1:=_DGLieAlgebra,Alg1,3,1,2,1,1

L1:=e1,e2=e1

(2.1)

L2:=_DGLieAlgebra,Alg2,3,

L2:=

(2.2)

DGsetupL1,x,a:DGsetupL2,y,b:

Alg2 > 

QueryAlg1,Abelian

false

(2.3)
Alg1 > 

ChangeLieAlgebraToAlg2:

Alg2 > 

QueryAbelian

true

(2.4)

 

The subalgebra S1 =x1, x2 in Alg1 is not Abelian while the subalgebra S2 =x2, x3 is Abelian.

Alg2 > 

S1:=x1,x2:S2:=x2,x3:

Alg2 > 

QueryS1,Abelian

false

(2.5)
Alg1 > 

QueryS2,Abelian

true

(2.6)

See Also

DifferentialGeometry, LieAlgebras, ChangeLieAlgebraTo, Query


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