find the kernel or image of a Lie algebra homomorphism; find the inverse image of a subalgebra with respect to a Lie algebra homomorphism - Maple Programming Help

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LieAlgebras[HomomorphismSubalgebras] - find the kernel or image of a Lie algebra homomorphism; find the inverse image of a subalgebra with respect to a Lie algebra homomorphism

Calling Sequences

     HomomorphismSubalgebras(φ, keyword)

     HomomorphismSubalgebras(φ,  S, keyword)

Parameters

     φ         - a transformation mapping one Lie algebra 𝔤 to another k

     keyword   - a keyword string, one of "Kernel", "Image", "InverseImage"

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

 

Description

Examples

Description

• 

Let g and kbe Lie algebras and let φ:g k  be a Lie algebra homomorphism .The kernel of φ is the ideal of vectors ker(φ) = {x  g | φx = 0}. The image of φ is the subalgebra of vectors imφ = {y k | y = φx for some x  g}. If S is a subalgebra of k, then the inverse image of S with respect to φ is the subalgebra  φ1S = {x  g | φx   S}.

• 

HomomorphismSubalgebras(φ, "Kernel") calculates kerφ.  A list of independent vectors defining a basis for the kernel is returned.  If ker(φ) = 0,then an empty list is returned.

• 

HomomorphismSubalgebras(φ, "Image") calculates im(φ).  A list of independent vectors defining a basis for the image is returned. If  im(φ) = 0,then an empty list is returned.

• 

HomomorphismSubalgebras(φ, S, "InverseImage") calculates φ1S. A list of independent vectors defining a basis for the inverse image is returned. If φ1S = 0, then an empty list is returned.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First we initialize a pair of Lie algebras and display the multiplication tables.

L1_DGLieAlgebra,Alg1,3,2,3,1,1:

DGsetupL1,x,α:

Alg1 > 

L2_DGLieAlgebra,Alg2,4,1,4,2,1,3,4,3,1:

Alg1   > 

DGsetupL2,y,β:

Alg1 > 

printMultiplicationTableAlg1,LieBracket,MultiplicationTableAlg2,LieBracket

x2,x3=x1,y1,y4=y2,y3,y4=y3

(2.1)

 

We define a transformation Phi from Alg1 to Alg2 and check that it is a Lie algebra homomorphism.

Alg2 > 

ΦTransformationx1,0&multy1,x2,y2,x3,y3

Φ:=x1,0y1,x2,y2,x3,y3

(2.2)
Alg2 > 

QueryAlg1,Alg2,Φ,Homomorphism

true

(2.3)

 

We find the kernel of Phi.

Alg2 > 

HomomorphismSubalgebrasΦ,Kernel

x1

(2.4)

 

We find the image of Phi.

Alg1 > 

HomomorphismSubalgebrasΦ,Image

y2,y3

(2.5)

 

We find the inverse image of the subalgebra spanned by y3, y4  with respect to φ.

Alg2 > 

S1y3,y4:

Alg2 > 

HomomorphismSubalgebrasΦ,S1,InverseImage

x1,x3

(2.6)

See Also

DifferentialGeometry

LieAlgebras

ApplyHomomorphism

MultiplicationTable

Query[Homomorphism]

Transformation