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KillingOrthogonal

find the subspace of a LAVF object L that is orthogonal to another LAVF w.r.t. the Killing form of L

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

KillingOrthogonal(L, M)

Parameters

L, M

-

LAVF objects in which M is subspace of L (see IsSubspace).

Description

• 

Let L, M be LAVF objects and ML. Then KillingOrthogonal(L,M) finds a new LAVF object for the subspace of L that is orthogonal to M with respect to the Killing form of L.

• 

Let M be a subspace of L. The Killing orthogonal of M in L is the subspace XLL|KXM,XL=0forallXMM where K( , ) is the Killing form of L.

• 

Some LAVF's exported methods are instance of this method, for example, KillingOrthogonal(L,L) gives KillingRadical of L, and KillingOrthogonal(L, DerivedAlgebra(L)) gives SolvableRadical of L.

• 

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 E(2).

LLAVFV,E2

Lξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(3)

IsLieAlgebraL

true

(4)

KillingOrthogonalL,L

ξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(5)

The Killing orthogonal in L to its derived algebra is the solvable radical of L.

DLDerivedAlgebraL

DLξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(6)

KOKillingOrthogonalL,DL

KOξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(7)

SRSolvableRadicalL

SRξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(8)

AreSameKO,SR

true

(9)

Compatibility

• 

The KillingOrthogonal 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

IsSubspace

KillingForm

KillingRadical

SolvableRadical