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StructureConstants

calculate the structure constants of a LAVF object.

StructureCoefficients

calculate the structure coefficients for a LAVF of infinite type.

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

StructureConstants( obj)

StructureCoefficients( obj)

Parameters

obj

-

a LAVF object that is a Lie algebra i.e. IsLieAlgebra(obj) returns true, see IsLieAlgebra.

Description

• 

Let L be a LAVF object which is a Lie algebra and is of finite type (see IsFiniteType). Then StructureConstants (and StructureCoefficients) methods return the structure constants of L (namely cijk), as a 3-dim array.

• 

If L is not of finite type (i.e. IsFiniteType(L) returns false), then StructureCoefficients(L) returns a sequence of two arrays of structure coefficients cijk, akirho, (k up, irho down)

• 

The output structure constants Cijk can then be displayed via the DisplayStructure command of the LieAlgebrasOfVectorFields package. See Overview of the LieAlgebrasOfVectorFields package for more detail.

• 

These methods are associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressξx,y,ηx,y:

VVectorFieldξx,yDx+ηx,yDy,space=x,y

Vξx+ηy

(1)

E2LHPDEdiffξx,y,y,y=0,diffηx,y,x=diffξx,y,y,diffηx,y,y=0,diffξx,y,x=0,indep=x,y,dep=ξ,η

E2ξy,y=0,ηx=ξy,ηy=0,ξx=0,indep=x,y,dep=ξ,η

(2)

Construct a LAVF for Euclidean group E(2).

LLAVFV,E2

Lξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(3)

IsLieAlgebraL

true

(4)

CijkStructureConstantsL

(5)

DisplayStructureCijk,X

00X200X1X2X10

(6)

Now consider a LAVF of infinite type.

SinfLHPDEdiffξx,y,y=0,ηx,y=ydiffξx,y,x,indep=x,y,dep=ξ,η

Sinfξy=0,η=yξx,indep=x,y,dep=ξ,η

(7)

LinfLAVFV,Sinf

Linfξx+ηy&whereξx=ηy,ξy=0,ηy=ηy

(8)

SolutionDimensionLinf

(9)

IsLieAlgebraLinf

true

(10)

C,AStructureCoefficientsLinf

(11)

DisplayStructureC,A,X,Z,format=commutatorList

X1,X2=X1y,X1,Z1=X2

(12)

More usually, the structure of an infinite Lie pseudogroup is displayed using Cartan structure equations, which are expressed in terms of 1-forms:

DisplayStructureC,A,ω,pi,format=cartan

dω1=ω1ω2y,dω2=ω1π1

(13)

Compatibility

• 

The StructureConstants and StructureCoefficients commands were introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields[LHPDE]

LieAlgebrasOfVectorFields[LAVF]

IsLieAlgebra

DisplayStructure

IsFiniteType