check if a list of structure equations defines a Lie algebra by verifying the Jacobi identities - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : Query : DifferentialGeometry/LieAlgebras/Query/Jacobi

Query[Jacobi] - check if a list of structure equations defines a Lie algebra by verifying the Jacobi identities

Calling Sequences

     Query(Alg, "Jacobi")

     Query(Alg, parm, "Jacobi")

Parameters

     Alg     - (optional) the name of an initialized Lie algebra

     parm    - (optional) a set of parameters appearing in the structure equations of the Lie algebra g

Description

• 

A bracket operation , on a vector space 𝔤 defines a Lie bracket if it is bi-linear, skew-symmetric, and satisfies the Jacobi identity x,y, z +z,x,y + y,z,x =0.                

• 

In terms of the standard exterior derivative operator d defined on the exterior algebra of the dual space 𝔤 *(defined on 1-forms  ω  𝔤* by (dω)x,y =  ωx,y, the Jacobi identities are equivalent to the fundamental identity d2 =0 . 

• 

The program DGsetup does not check that its input, a Lie algebra data structure, actually defines a Lie algebra. To verify that a Lie algebra data structure does indeed define a Lie algebra, initialize the Lie algebra data structure, and run Query("Jacobi").

• 

Query(Alg, "Jacobi") returns true if the Jacobi identities hold (in which case Alg defines a Lie algebra) and false otherwise.  If the algebra is unspecified, then Query is applied to the current algebra. The Jacobi identity is checked using the exterior derivative formulation.

• 

Query(Alg, parm, "Jacobi") returns a sequence TF, Eq, Soln, AlgList. Here TF is true if Maple finds parameter values for which the Jacobi identities are valid and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for the Jacobi identities to hold; Soln is the list of solutions to the equations Eq; and AlgList is the list of Lie algebra data structures obtained from the parameter values given by various solutions in 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

restart:withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We begin by defining a bracket operation on a 3-dimensional vector space with basis x1, x2, x3. This bracket depends upon two parameters a1 and a2. We shall determine for which parameter values this bracket satisfies the Jacobi identities.

Eq:=x1,x2=a2x2,x1,x3=a1x1:

 

Convert to a Lie algebra data structure.

L:=LieAlgebraDataEq,x1,x2,x3,Alg1

L:=e1,e2=a2e2,e1,e3=a1e1

(2.1)

 

Initialize this data structure.

DGsetupL

Lie algebra: Alg1

(2.2)
Alg1 > 

TF,EQ,SOLN,AlgList:=Querya1,a2,Jacobi

TF,EQ,SOLN,AlgList:=true,0,a1a2,a1=0,a2=a2,a1=a1,a2=0,e1,e2=a2e2,e1,e3=a1e1

(2.3)

 

The equations that must be satisfied for the bracket to satisfy Jacobi are:

Alg1 > 

EQ

0,a1a2

(2.4)

 

This leads to two cases a1 =0 or a2 =0.  We initialize the resulting Lie algebra data structures and print the multiplication tables.

Alg1 > 

DGsetupAlgList1,x,α:DGsetupAlgList2,y,β:

Alg1_2 > 

printMultiplicationTableAlg1_1,LieBracket,MultiplicationTableAlg1_2,LieBracket

x1,x2=a2x2,y1,y3=a1y1

(2.5)

 

Example 2

The Jacobi identities are equivalent to the vanishing of the square of the exterior derivative.  For example:

Alg1_2 > 

ChangeLieAlgebraToAlg1:

Alg1 > 

ExteriorDerivativeExteriorDerivativeθ1

0θ1θ2θ3

(2.6)
Alg1 > 

ExteriorDerivativeExteriorDerivativeθ2

a1a2θ1θ2θ3

(2.7)
Alg1 > 

ExteriorDerivativeExteriorDerivativeθ3

0θ1θ2θ3

(2.8)

See Also

DifferentialGeometry, LieAlgebras, ExteriorDerivative, LieBracket, Query


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam