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AdjointMatrix

calculate the adjoint representation of a LAVF object.

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

AdjointMatrix( L, output = out)

AdjointMatrix( L, M, output = out)

AdjointMatrix( L, M, N, output = out)

Parameters

L, M, N

-

a LAVF object of finite type (see IsFiniteType for more detail)

out

-

(optional) a string: either "matrix" or "basis"

Description

• 

In the first calling sequence, AdjointMatrix(L) returns the adjoint representation matrix of L.

• 

For AdjointMatrix(L) to make sense, the LAVF object L must be a Lie algebra (i.e. IsLieAlgebra(L) returns true. See IsLieAlgebra for more detail).

• 

In the second calling sequence,  AdjointMatrix(L,M) returns a matrix representation of the Lie algebra L on the invariant subspace M. (i.e IsInvariant(M,L) returns true. See IsInvariant for more detail).

• 

The third calling sequence is the general form of the method. AdjointMatrix(L,M,N) returns a matrix representing the action of L on M in N.

• 

For AdjointMatrix(L, M, N) to make sense, L must commute with M modulo N (i.e. AreCommuting(L,M,N) returns true. See AreCommuting for more detail).

• 

By specifying output = "basis", the output will be returned in terms of basis.

• 

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 theEuclidean Lie algebra E(2).

LLAVFV,E2

Lξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(3)

IsLieAlgebraL

true

(4)

AdjointMatrixL

0ξyξξy0η000

(5)

AdjointMatrixL,output=basis

010−100000,000001000,00−1000000

(6)

Compatibility

• 

The AdjointMatrix 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

AreCommuting

IsFiniteType