change the basis for a representation, either in the Lie algebra or in the representation space - Maple Help

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LieAlgebras[ChangeRepresentationBasis] - change the basis for a representation, either in the Lie algebra or in the representation space

Calling Sequences

    ChangeRepresentationBasis(ρ, B, Fr)

    ChangeRepresentationBasis(ρ, P, keyword, Fr)

Parameters

     ρ          - a representation of a Lie algebra 𝔤 on a vector space V

     B          - a list of vectors defining a new basis B for either 𝔤 or V

     Fr         - a Maple name or string, the name of the Lie algebra or vector space with the new basis B

     P          - a change of basis matrix, the columns of which are the components of the new basis vectors B with respect to the original basis

     keyword    - either "Domain" or "Range", indicating that the matrix P is a change of basis matrix for the Lie algebra or the representation space

 

Description

Examples

Description

• 

Let ρ: 𝔤  glV be a representation of a Lie algebra 𝔤 on a vector space V. Let e1, e2, ... ,enbe the given basis for 𝔤 and let E1, E2, ... ,Embe the given basis for V. Let ρei = Mi , the matrix representing the linear transformation ρeiwith respect to the basis Er. If Fs is another basis for the representation space V, then in this new basis ρei = Si , where Si=P1Mi P and P is the change of basis matrix whose columns are the components of Fs with respect to the basis Er. If fj is another basis for 𝔤, then ρfj = Tj , where Tj = in Q ji Mi and Q is the change of basis matrix whose columns are the components of the fj with respect to the ei, that is, fj = in Q ji ei .

• 

If B = Fs is a list of vectors defining a basis for V, then ChangeReperesentationBasis(ρ B, Fr) computes the matrices Si for the representation ρ with respect to the basis Fs. If P is the change of basis matrix, then the calling sequence ChangeRepresentationBasis(ρ, P, "Range", Fr) produces the same result.

• 

If B = fj is a list of vectors defining a basis for 𝔤, then ChangeRepresentationBasis(ρ, B, Fr) computes the matrices Ti for the representation ρ with respect to the basis fj. If Q is the change of basis matrix, then the command ChangeRepresentationBasis(ρ, Q, "Domain", Fr) produces the same result.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We define a representation and make a change of basis for the representation space.

LLieAlgebraDatax1,x3=x3,x1,x4=x4,x2,x4=x3,x1,x2,x3,x4,Alg1

L:=e1,e3=e3,e1,e4=e4,e2,e4=e3

(2.1)

DGsetupL

Lie algebra: Alg1

(2.2)
Alg1 > 

DGsetupx,y,z,V

frame name: V

(2.3)
V > 

MMatrix1,0,0,0,1,0,0,0,0,Matrix0,1,0,0,0,0,0,0,0,Matrix0,0,1,0,0,0,0,0,0,Matrix0,0,0,0,0,1,0,0,0:

V > 

ρRepresentationAlg1,V,M

ρ:=e1,100010000,e2,010000000,e3,001000000,e4,000001000

(2.4)

 

Define the new basis for the representation space.

Alg1 > 

BevalDGD_x+D_y+D_z,D_xD_y,D_x+2D_y+D_z

B:=D_x+D_y+D_z,D_xD_y,D_x+2D_y+D_z

(2.5)

 

Compute the representation φ1in the basis B.

V > 

φ1ChangeRepresentationBasisρ,B,V

φ1:=e1,203111203,e2,112112112,e3,101101101,e4,101000101

(2.6)

 

We can use the Query command to check that φ1is a representation of Alg1.

Alg1 > 

Queryφ1,Representation

true

(2.7)

 

Check, by example, that the matrices for are correct. We apply rho(e1) to Fr[1] and express the result as a linear combination of the vectors Fr. This should give the first column of the matrix for e1 in phi1.

Alg1 > 

aApplyRepresentationρ,e1,B1

a:=D_x+D_y

(2.8)
V > 

GetComponentsa,B

2,1,2

(2.9)

 

Example 2.

We obtain the same change of basis as in Example 1 using the other calling sequence for the procedure ChangeRepresentationBasis. We take the matrix A to be the matrix whose columns are the coefficients of the new basis in terms of the old.

V > 

B

D_x+D_y+D_z,D_xD_y,D_x+2D_y+D_z

(2.10)
V > 

PLinearAlgebra:-TransposeMatrixGetComponentsB,D_x,D_y,D_z

P:=111112101

(2.11)
V > 

φ2ChangeRepresentationBasisρ,P,Range,V

φ2:=e1,203111203,e2,112112112,e3,101101101,e4,101000101

(2.12)

 

Example 3.

Now we make a change of basis in the Lie algebra. First we use the LieAlgebraData command to create the Lie algebra in the new basis.

Alg1 > 

ChangeFrameAlg1

Alg1

(2.13)
Alg1 > 

BevalDGe1+e2,e3e2,e22e3,e1e3+e4

B:=e1+e2,e2+e3,e22e3,e1e3+e4

(2.14)
Alg1 > 

L2LieAlgebraDataB,Alg2

L2:=e1,e2=e2e3,e1,e3=2e2+2e3,e1,e4=e13e22e3+e4,e2,e4=2e2+2e3,e3,e4=3e23e3

(2.15)
Alg1 > 

DGsetupL2,f,θ

Lie algebra: Alg2

(2.16)
Alg2 > 

φ3ChangeRepresentationBasisρ,B,Alg2

φ3:=f1,110010000,f2,011000000,f3,012000000,f4,101011000

(2.17)
Alg2 > 

Queryφ3,Representation

true

(2.18)

 

Example 4. We obtain the same change of basis as in Example 3 using the other calling sequence for the procedure ChangeRepresentationBasis. We take the matrix A to be the matrix whose columns are the coefficients of the new basis in terms of the old.

Alg2 > 

B

e1+e2,e2+e3,e22e3,e1e3+e4

(2.19)
Alg1 > 

PLinearAlgebra:-TransposeMatrixGetComponentsB,e1,e2,e3,e4

P:=1001111001210001

(2.20)
Alg1 > 

φ4ChangeRepresentationBasisρ,P,Domain,Alg2

φ4:=f1,110010000,f2,011000000,f3,012000000,f4,101011000

(2.21)

See Also

DifferentialGeometry

LieAlgebras

ChangeFrame

GetComponents

LieAlgebraData

Query

 


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