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LieAlgebraData[Subalgebra] - convert a list of vectors defining a Lie subalgebra to a Lie algebra data structure

Calling Sequence

     LieAlgebraData(Subalgebra, SubSubalgebras, AlgName, "Matrix")

Parameters

     Subalgebra        - a list of linearly independent vectors defining a subalgebra h of a Lie algebra 𝔤

     SubSubalgebras    - (optional) a list of lists of linearly independent vectors defining subalgebras k1, k2, ... of the subalgebra h.

     AlgName           - a name or string, the name to be assigned to the Lie algebra

     "Matrix"          - (optional) keyword

 

Description

Examples

Description

• 

In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra, that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.  The first argument for DGsetup is a Lie algebra data structure which contains the structure constants in a standard format used by the LieAlgebras package.

• 

Let h 𝔤 be a subalgebra. The command LieAlgebraData(Subalgebra) will return the Lie algebra data structure for the subalgebra h  defined by the vectors Subalgebra which can then be initialized. In this way the subalgebra h can be studied as a Lie algebra in its own right, independent of the original ambient algebra 𝔤.

• 

With the optional argument "Matrix", LieAlgebraData(Subalgebra, "Matrix") will return the Lie algebra data structure for the subalgebra h  defined by the vectors Subalgebra and a matrix which defines the inclusion map from the subalgebra h to the Lie algebra 𝔤.

• 

Let k1, k2, ... be subalgebras of h (and hence subalgebras of 𝔤).  The command LieAlgebraData(Subalgebra, SubSubalgebras) will return the Lie algebra data structure for the subalgebra h  defined by the vectors Subalgebra and the list of lists of components for the vectors in SubSubalgebras with respect to the given basis for the subalgebra h.  With this information and the DGzip program k1, k2, ...  can then be converted to subalgebras of the Lie algebra h.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We first initialize a 5-dimensional Lie algebra called Alg1 and display its multiplication table.

L_DGLieAlgebra,Alg1,5,1,4,1,1,2,3,1,1,2,4,2,1,2,5,2,1,3,5,3,1

L:=e1,e4=e1,e2,e3=e1,e2,e4=e2,e2,e5=e2,e3,e5=e3

(2.1)

DGsetupL,x,a:

 

Let's check that S1 = x1, x2, x3, x4 is a subalgebra.

Alg1 > 

S1x1,x2,x3,x4:

Alg1 > 

QueryS1,Subalgebra

true

(2.2)

 

Now use LieAlgebraData to obtain the Lie algebra data structure for S1.  Denote the Lie algebra obtained from S1by Alg2.

Alg1 > 

L2LieAlgebraDataS1,Alg2

L2:=e1,e4=e1,e2,e3=e1,e2,e4=e2

(2.3)
Alg1 > 

DGsetupL2,y,b:

Alg2 > 

MultiplicationTableLieBracket

y1,y4=y1,y2,y3=y1,y2,y4=y2

(2.4)

 

This is the same multiplication as before.

 

Example 2.

We continue with Example 1 but this time use LieAlgebraData with the keyword "Matrix".

Alg2 > 

L2,MLieAlgebraDataS1,Alg2,Matrix

L2,M:=e1,e4=e1,e2,e3=e1,e2,e4=e2,10000100001000010000

(2.5)

 

We can use the matrix M to define a Lie algebra homomorphism from Alg2 to Alg1 which maps y1 x1 , y2x2, etc.

Alg1 > 

φTransformationAlg2,Alg1,M

φ:=y1,x1,y2,x2,y3,x3,y4,x4

(2.6)

 

Example 3.

We continue with Example 1.  The subalgebras SS1 =x1, x2 and   SS2 =x3, x4 are subalgebras of  S1 = x1, x2, x3, x4 .  We use LieAlgebraData to obtain SS1 and SS2 as subalgebras of Alg2.

Alg1 > 

SS1x1,x2:

Alg1 > 

SS2x3,x4:

Alg1 > 

L2,SSLieAlgebraDataS1,SS1,SS2,Alg2

L2,SS:=e1,e4=e1,e2,e3=e1,e2,e4=e2,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1

(2.7)
Alg1 > 

BasisTools:-DGinfoAlg2,FrameBaseVectors

Basis:=y1,y2,y3,y4

(2.8)
Alg2 > 

newSS1mapDGzip,SS1,Basis,plus

newSS1:=y1,y2

(2.9)
Alg2 > 

newSS2mapDGzip,SS2,Basis,plus

newSS2:=y3,y4

(2.10)

 

Example 4.

The program LieAlgebraData can often be used to simplify the structure equations of a Lie algebra. In this example we initialize a Lie algebra Alg4 and compute its derived series.  We then use elements of the derived series to define a new basis for the Lie algebra.  In this new basis the structure equations of the Lie algebra become much simpler.

Alg2 > 

L3_DGLieAlgebra,Alg3,5,1,2,1,2,1,2,2,5,1,2,3,1,1,2,4,2,1,2,5,5,1,3,2,2,1,3,4,1,1,3,5,2,1,4,2,1,1,4,4,1,1,4,5,1,1,5,1,2,1,5,2,4,1,5,3,1,1,5,4,1,1,5,5,4,2,3,1,2,2,3,2,6,2,3,3,1,2,3,4,1,2,3,5,6,2,4,4,1,2,5,2,2,2,5,4,1,2,5,5,2,3,4,2,1,3,4,4,1,3,4,5,1,3,5,1,2,3,5,2,5,3,5,3,1,3,5,5,5,4,5,4,1:

Alg2 > 

DGsetupL3:

Alg3 > 

MultiplicationTableLieBracket

e1,e2=2e1+5e2+e3+2e45e5,e1,e3=2e2e4+2e5,e1,e4=e2e4+e5,e1,e5=2e1+4e2+e3+e44e5,e2,e3=2e1+6e2+e3+e46e5,e2,e4=e4,e2,e5=2e2+e42e5,e3,e4=e2+e4e5,e3,e5=2e15e2e3+5e5,e4,e5=e4

(2.11)
Alg3 > 

DSSeriesDerived

DS:=e1,e2,e3,e4,e5,2e1+5e2+e3+2e45e5,2e2e4+2e5,e2e4+e5,e2e4+e5,

(2.12)
Alg3 > 

DS3

e2e4+e5

(2.13)
Alg3 > 

DS2

2e1+5e2+e3+2e45e5,2e2e4+2e5,e2e4+e5

(2.14)
Alg3 > 

DS1

e1,e2,e3,e4,e5

(2.15)

 

For the new basis we take the vector in DS[3], 2 vectors in DS[2] independent of DS[3], and 3 vectors in DS[1] independent of the 3 vectors in DS[2].

Alg3 > 

newBasise2e4+e5,2e2e4+2e5,2e1+5e2+e3+2e45e5,e1,e5

newBasis:=e2e4+e5,2e2e4+2e5,2e1+5e2+e3+2e45e5,e1,e5

(2.16)

 

Now we create a Lie algebra from this adapted basis.

Alg3 > 

L4,NLieAlgebraDatanewBasis,Alg4,Matrix

L4,N:=e1,e5=2e1,e2,e3=e1,e2,e4=e1,e2,e5=2e1+e2,e3,e4=3e1e2,e3,e5=6e1e2+e3,e4,e5=e1+e3,0021012500001001120012501

(2.17)
Alg3 > 

DGsetupL4:

 

The structure equations are much simpler.

Alg4 > 

MultiplicationTableLieBracket

e1,e5=2e1,e2,e3=e1,e2,e4=e1,e2,e5=2e1+e2,e3,e4=3e1e2,e3,e5=6e1e2+e3,e4,e5=e1+e3

(2.18)

 

The Matrix N defines an isomorphism from Alg4 to Alg3.

Alg4 > 

LinearAlgebra:-DeterminantN

1

(2.19)
Alg4 > 

QueryAlg4,Alg3,N,Homomorphism

true

(2.20)

See Also

DifferentialGeometry

LieAlgebras

DGinfo

DGzip

LieAlgebraData

MultiplicationTable

Query[Homomorphism]

Query[Subalgebra]

Series

 


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