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LieAlgebras[QuotientAlgebra] - find the structure equations for a quotient algebra of a Lie algebra by an ideal

Calling Sequences

     QuotientAlgebra(h, m, Algname, keyword)

Parameters

     h       - a list of independent vectors defining an ideal 𝔥  in a Lie algebra 𝔤

     m       - a list of independent vectors defining a vector space complement to 𝔥  in 𝔤

     Algname - a name or a string, the name assigned to the quotient algebra 𝔤/𝔥

     keyword - (optional) the keyword "Matrix"

 

Description

Examples

Description

• 

Let 𝔤  be a Lie algebra and 𝔥 an ideal in 𝔤.  Then elements of the quotient algebra  𝔤/𝔥are the cosets x +𝔥 , where x  𝔤.  The Lie bracket on 𝔤/𝔥 is defined by x +𝔥, y + 𝔥 = x, y + 𝔥. If vectors y1, y2, ... ,yr form a basis for a complement to 𝔥, then the cosets y1+ 𝔥, y2 + 𝔥, ... ,yr +𝔥 form a basis for 𝔤/𝔥.

• 

The program QuotientAlgebra(h, m) creates a Lie algebra data structure for the quotient algebra 𝔤/𝔥. using the vectors in the complement m as the representative basis elements for 𝔤/𝔥.

• 

A Lie algebra data structure contains the structure constants in a standard format used by the LieAlgebras package (see LieAlgebraData). The command DGsetup is then 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.

• 

With the optional keyword present, QuotientAlgebra(h, m, "Matrix") returns the Lie algebra data structure for 𝔤/𝔥 and the matrix representation of the canonical projection map π : 𝔤 𝔤/𝔥 defined by x x + 𝔥.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra and display the multiplication table.

Alg2 > 

L1_DGLieAlgebra,Alg1,5,1,5,1,2,2,3,1,1,2,5,2,1,2,5,3,1,3,5,3,1,4,5,4,2

L1:=e1,e5=2e1,e2,e3=e1,e2,e5=e2+e3,e3,e5=e3,e4,e5=2e4

(2.1)
Alg2 > 

DGsetupL1:

 

Check that h1= e1, e3 is an ideal and find the quotient algebra (call it Alg2) using the complementary vectors e2, e4, e5

Alg1 > 

h1e1,e3:m1e2,e4,e5:

Alg1 > 

Queryh1,Ideal

true

(2.2)
Alg1 > 

L2QuotientAlgebrah1,m1,Alg2

L2:=e1,e3=e1,e2,e3=2e2

(2.3)

 

Rerun QuotientAlgebra with the keyword argument "Matrix".

Alg1 > 

DGsetupL2,f,β:

Alg2 > 

L2,AQuotientAlgebrah1,m1,Alg2,Matrix

L2,A:=e1,e3=e1,e2,e3=2e2,010000001000001

(2.4)

 

We use the DifferentialGeometrycommand Transformation to convert the matrix A into a transformation Π  from Alg1 to the quotient algebra Alg2.

Alg1 > 

ΠTransformationAlg1,Alg2,A

Π:=e1,0f1,e2,f1,e3,0f1,e4,f2,e5,f3

(2.5)

 

We can check that Π is a Lie algebra homomorphism.

Alg2 > 

QueryAlg1,Alg2,Π,Homomorphism

true

(2.6)

 

We see that Π sends e1 to 0, e2 to f1 and so on.

Alg2 > 

ApplyHomomorphismΠ,e1+2e23e3+4e45e5

2f1+4f25f3

(2.7)

 

We can verify that [e1 , e3 is a basis for the kernel of Π and that the image of Π is spanned by f1, f2, f3 (so that Π is surjective).

Alg2 > 

HomomorphismSubalgebrasΠ,Kernel

e3,e1

(2.8)
Alg1 > 

HomomorphismSubalgebrasΠ,Image

f1,f2,f3

(2.9)

See Also

DifferentialGeometry

LieAlgebras

HomomorphismSubalgebras

LieAlgebraData

Query[Homomorphism]

Query[Ideal]