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KillingForm

calculate the Killing form of a LAVF object.

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

KillingForm( 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. Then KillingForm method returns the Killing form of L, as a KF Maple object.

• 

The returned KF object is for representing the Killing form. A valid KF object can act as a symmetric bilinear operator (form) and has access to some methods. See Overview of the KF object for more detail.

• 

This method is 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 the Euclidean Lie algebra E(2).

LLAVFV,E2

Lξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(3)

IsLieAlgebraL

true

(4)

KKillingFormL

KX,Y→2yXxyYx

(5)

To exercise the Killing form K, we introduce the following two vector fields on the same space as L

XVectorFieldξx,yDx+ηx,yDy,space=x,y

Xξx+ηy

(6)

YVectorFieldαx,yDx+βx,yDy,space=x,y

Yαx,yx+βx,yy

(7)

...and evaluate the Killing form...

KX,Y

2ξyyαx,y

(8)

Compatibility

• 

The KillingForm command was 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

KF (Object overview)