check if a matrix defines a Lie algebra homomorphism between two Lie algebras - Maple Help

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Query[Homomorphism] - check if a matrix defines a Lie algebra homomorphism between two Lie algebras

Calling Sequences

     Query(Alg1, Alg2, A, "Homomorphism")

     Query(Alg1, Alg2, phi, "Homomorphism")

     Query(Alg1, Alg2, A, parm, "Homomorphism")

Parameters

     Alg1     - the name of an initialized Lie algebra 𝔤, the domain algebra for the homomorphism defined by A

     Alg2     - the name of an initialized Lie algebra 𝔥, the range algebra for the homomorphism defined by A

     A        - an m × n  matrix,where n is the dimension of the Lie algebra and m is the dimension of 𝔥

     phi      -  a transformation from Alg1 to Alg2

     parm     - a set of parameters appearing in the matrix A or in the Lie algebras 𝔤 and k

Description

• 

A matrix A defines a Lie algebra homomorphism from a Lie algebra 𝔤 to a Lie algebra 𝔥 if the linear transformation LA satisfies LAx, y = LAx, LAy  for all x, y  𝔤.

• 

Query(Alg1, Alg2, A, "Homomorphism") returns true if the matrix A defines a Lie algebra homomorphism from 𝔤 to 𝔥 and false otherwise.

• 

Query(Alg1, Alg2, phi, "Homomorphism") returns true if the transformation φ defines a Lie algebra homomorphism φ: 𝔤 -> 𝔥 and false otherwise.

• 

Query(Alg1, Alg2, parm, "Homomorphism") returns a 4-tuple TF, Eq, Soln, B.  Here TF is true if Maple finds a set of values for the parameters for which the Matrix A is a homomorphism; Eq is the defining set of equations for the parameters parm in order that the matrix A be a homomorphism; Soln is a list of solutions to the equations Eq; and B is the list of Matrices obtained by evaluating A on the solutions in the list Soln.

• 

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.

First initialize a Lie algebra.  We illustrate the fact that the Adjoint matrix Ad(x), for any x in the Lie algebra, is always a Lie algebra homomorphism (in fact, an isomorphism).

L1:=_DGLieAlgebra,Alg1,4,1,4,1,1,2,3,1,1,2,4,2,1

L1:=e1,e4=e1,e2,e3=e1,e2,e4=e2

(2.1)

DGsetupL1:

Alg1 > 

A1:=AdjointExpre1+se2+te3

A1:=1ts12ts+r010s00100001

(2.2)
Alg1 > 

QueryAlg1,Alg1,A1,Homomorphism

true

(2.3)

 

Example 2.

The matrix exponential of any outer derivation is also a Lie algebra homomorphism (isomorphism).

Alg1 > 

Outer:=DerivationsOuter

Outer:=1000000000100000

(2.4)
Alg1 > 

A2:=LinearAlgebra:-MatrixExponentialtOuter1

A2:=ⅇt000010000ⅇt00001

(2.5)
Alg1 > 

T:=TransformationAlg1,Alg1,A2

T:=e1,ⅇte1,e2,e2,e3,ⅇte3,e4,e4

(2.6)
Alg1 > 

QueryAlg1,Alg1,T,Homomorphism

true

(2.7)

 

Example 3.

In this example we construct the quotient algebra of Alg1 by the ideal e1  Call the quotient Alg2.  We check that the canonical projection map from Alg1 to Alg2 is a Lie algebra homomorphism.

Alg1 > 

L2:=QuotientAlgebrae1,e2,e3,e4,Alg2

L2:=e1,e3=e1

(2.8)
Alg1 > 

DGsetupL2,x,α:

`|`x1x2x3---------------x1`|`00x1x2`|`000x3`|`x100

(2.9)

 

The following matrix A3 maps  e10, e2x1, e3x2, e4x3.

Alg2 > 

A3:=Matrix0,1,0,0,0,0,1,0,0,0,0,1

A3:=010000100001

(2.10)
Alg2 > 

QueryAlg1,Alg2,A3,Homomorphism

true

(2.11)

 

Example 4.

In this example we shall find all the monomorphisms from the 2-dimensional solvable Lie algebra into Alg1. This effectively computes all the 2-dimensional non-Abelian subalgebras of Alg1. First initialize the 2-dimensional solvable algebra and call it Alg3.

Alg2 > 

L2:=_DGLieAlgebra,Alg3,2,1,2,1,1

L2:=e1,e2=e1

(2.12)
Alg2 > 

DGsetupL2:

 

Define a matrix A4 representing an arbitrary linear transformation from Alg1 to Alg2.

Alg3 > 

A4:=Matrixa1,a2,a3,a4,a5,a6,a7,a8

A4:=a1a2a3a4a5a6a7a8

(2.13)

 

Determine the parameter values for which A4  is a Lie algebra homomorphism.

Alg3 > 

TF,EQ,SOLN,B:=QueryAlg3,Alg1,A4,a1,a2,a3,a4,a5,a6,a7,a8,Homomorphism

TF,EQ,SOLN,B:=true,0,a5,a7,a5,a7,a3a8+a4a7+a3,a3a8a4a7a3,a1a8+a2a7a3a6+a4a5+a1,a1a8a2a7+a3a6a4a5a1,a1=a1,a2=a2,a3=a3,a4=a4,a5=0,a6=0,a7=0,a8=1,a1=0,a2=a2,a3=0,a4=a4,a5=0,a6=a6,a7=0,a8=a8,a1=a1,a2=a2,a3=0,a4=a4,a5=0,a6=a6,a7=0,a8=1,a1a2a3a40001,0a20a40a60a8,a1a20a40a601

(2.14)

 

The equations that must hold for A4 to define a Lie algebra homomorphism are given by EQ.

Alg1 > 

EQ

0,a5,a7,a5,a7,a3a8+a4a7+a3,a3a8a4a7a3,a1a8+a2a7a3a6+a4a5+a1,a1a8a2a7+a3a6a4a5a1

(2.15)

 

The possible Lie algebra homomorphisms are given by B.  Note that B2, B3, B4 can be chosen to be full rank and therefore define Lie algebra isomorphisms.

Alg1 > 

B

a1a2a3a40001,0a20a40a60a8,a1a20a40a601

(2.16)

See Also

DifferentialGeometry, LieAlgebras, Adjoint, Derivations, Query, QuotientAlgebra, Transformation


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