|
LieAlgebras[ApplyHomomorphism] - apply a Lie algebra homomorphism to a vector, form or tensor
Calling Sequences
ApplyHomomorphism(Phi, T, keyword)
Parameters
Phi - a linear transformation from a Lie algebra g to another Lie algebra k
T - a vector, a form, or a tensor defined on either the domain Lie algebra g or the range Lie algebra k
keyword - (optional) string keyword, either "domain" or "range"
|
|
Description
|
|
|
•
|
ApplyHomomorphism(Phi, T) will apply the transformation Phi to the vector, form or tensor T and return an object of the same type. The precise evaluation rules for ApplyHomomorphism depend upon the specific properties of T and whether or not Phi is invertible. The details follow.
|
|
•
|
Applied to tensors, the command ApplyHomomorphism acts as a ring homomorphism, that is, ApplyHomomorphism(Phi, T &tensor S) = ApplyHomomorphism(Phi, T) &tensor ApplyHomomorphism(Phi, S).
|
|
•
|
CASE 1. T is a vector in the domain algebra g of Phi. In this case ApplyHomomorphism(Phi, T) simply applies the linear transformation Phi to the vector T and the result is a vector in the range algebra k of the transformation Phi.
|
|
•
|
CASE 2. T is a k-form on the range algebra k of transformation Phi. In this case ApplyHomomorphism(Phi, T) simply applies the pullback of the linear transformation Phi to the k-form T and the result is a k-form in the domain g of Phi.
|
|
•
|
CASE 3. T is a tensor on g and Phi is an invertible linear transformation. Then ApplyHomomorphism(Phi, T) is the tensor on the range algebra k obtained by the pushforward by Phi of the contravariant components of T and the pullback of the covariant components of T by the inverse of Phi.
|
|
•
|
CASE 4. T is a tensor on k and Phi is an invertible linear transformation. Then ApplyHomomorphism(Phi, T) is the tensor on the domain algebra g obtained by the pushforward of the contravariant components of T by the inverse of Phi and the pullback of the covariant components of T by Phi.
|
|
•
|
CASE 5. T is a tensor on g and Phi is not invertible. Then T must be a contravariant tensor (that is, a tensor products of vectors) in which case ApplyHomomorphism(Phi, T) is the contravariant tensor defined on the range algebra k and obtained by the pushforward of Phi acting on vectors in g.
|
|
•
|
When k = g, Case 4 takes precedence over Case 5. Alternatively ApplyHomomorphism can be forced to use Case 4 or Case 5 with the third optional argument "domain" or "range".
|
|
•
|
CASE 6. T is a tensor on k and Phi is not invertible. Then T must be a covariant tensor (that is, a tensor product of 1-forms) in which case ApplyHomomorphism(Phi, T) is the covariant tensor defined on the domain algebra g and obtained by the pullback of Phi acting on 1-forms in k.
|
|
•
|
The command ApplyHomomorphism is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form ApplyHomomorphism(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-ApplyHomomorphism(...).
|
|
|
|
Examples
|
|
|
>
|
 
|
Example 1.
First initialize two copies of a Lie algebra, called Alg1 and Alg2, and display the Lie bracket multiplication tables.
|
>
|
![LC1 := _DG([["LieAlgebra", Alg1, [4]], [[[1, 4, 1], 1], [[2, 3, 1], 1], [[2, 4, 2], 1]]])](/support/helpjp/helpview.aspx?si=6605/file05781/math214.png)
|
|
>
|
![DGsetup(LC1, [x], [alpha])](/support/helpjp/helpview.aspx?si=6605/file05781/math218.png)
|
|
Alg1 >
|
![LC2 := _DG([["LieAlgebra", Alg2, [4]], [[[1, 4, 1], 1], [[2, 3, 1], 1], [[2, 4, 2], 1]]])](/support/helpjp/helpview.aspx?si=6605/file05781/math222.png)
|
|
Alg1 >
|
![DGsetup(LC2, [y], [beta])](/support/helpjp/helpview.aspx?si=6605/file05781/math226.png)
|
|
Alg2 >
|
|
|
![[[x1, x4] = x1, [x2, x3] = x1, [x2, x4] = x2], [[y1, y4] = y1, [y2, y3] = y1, [y2, y4] = y2]](/support/helpjp/helpview.aspx?si=6605/file05781/math233.png)
| (2.1) |
We use AdjointExp to construct a Lie algebra isomorphism from Alg1 to Alg2.
|
Alg2 >
|
|
|
Alg1 >
|
|
|
![Phi := [[x1, y1], [x2, -t*y1+y2], [x3, s*y1+y3], [x4, (-s*t/2+r)*y1+s*y2+y4]]](/support/helpjp/helpview.aspx?si=6605/file05781/math254.png)
| (2.2) |
We calculate the effects of the command ApplyHomomorphism in each of the following cases.
CASE 1: vectors in the domain algebra Alg1.
CASE 2: 1-forms on the range algebra Alg2.
CASE 3: rank 1 covariant tensors on the domain algebra Alg1.
CASE 4: rank 1 contravariant vectors on the range algebra Alg2.
In each case we show the matrix which defines the transformation.
CASE 1: vectors in the domain algebra Alg1.
|
Alg2 >
|
![Vectors := [x1, x2, x3, x4]](/support/helpjp/helpview.aspx?si=6605/file05781/math288.png)
|
|
Alg2 >
|
|
|
![Matrix([[1, -t, s, -s*t/2+r], [0, 1, 0, s], [0, 0, 1, 0], [0, 0, 0, 1]]), [y1, -t*y1+y2, s*y1+y3, (-s*t/2+r)*y1+s*y2+y4]](/support/helpjp/helpview.aspx?si=6605/file05781/math295.png)
| (2.3) |
CASE 2: 1-forms on the range algebra Alg2.
|
Alg2 >
|
![Forms := [beta1, beta2, beta3, beta4]](/support/helpjp/helpview.aspx?si=6605/file05781/math305.png)
|
|
Alg2 >
|
|
|
Alg2 >
|
|
|
![Matrix([[1, 0, 0, 0], [-t, 1, 0, 0], [s, 0, 1, 0], [-s*t/2+r, s, 0, 1]]), [alpha1-t*alpha2+s*alpha3+(-s*t/2+r)*alpha4, alpha2+s*alpha4, alpha3, alpha4]](/support/helpjp/helpview.aspx?si=6605/file05781/math316.png)
| (2.4) |
CASE 3. rank 1 covariant tensors on the domain algebra Alg1.
|
Alg1 >
|
![CovariantTensors := map(convert, [alpha1, alpha2, alpha3, alpha4], DGtensor)](/support/helpjp/helpview.aspx?si=6605/file05781/math326.png)
|
|
![CovariantTensors := [alpha1, alpha2, alpha3, alpha4]](/support/helpjp/helpview.aspx?si=6605/file05781/math329.png)
| (2.5) |
|
Alg1 >
|
|
|
Alg1 >
|
|
|
![Matrix([[1, 0, 0, 0], [t, 1, 0, 0], [-s, 0, 1, 0], [-s*t/2-r, -s, 0, 1]]), [beta1+t*beta2-s*beta3+(-s*t/2-r)*beta4, beta2-s*beta4, beta3, beta4]](/support/helpjp/helpview.aspx?si=6605/file05781/math340.png)
| (2.6) |
CASE 4. rank 1 contravariant vectors on the range algebra Alg2.
|
Alg2 >
|
![ContravariantTensors := map(convert, [y1, y2, y3, y4], DGtensor)](/support/helpjp/helpview.aspx?si=6605/file05781/math350.png)
|
|
Alg2 >
|
|
|
Alg2 >
|
|
|
![Matrix([[1, t, -s, -s*t/2-r], [0, 1, 0, -s], [0, 0, 1, 0], [0, 0, 0, 1]]), [x1, t*x1+x2, -s*x1+x3, (-s*t/2-r)*x1-s*x2+x4]](/support/helpjp/helpview.aspx?si=6605/file05781/math361.png)
| (2.7) |
We show, by way of a simple example, the extensions of the mappings in CASE 1 and CASE 3 form a mixed tensor on the range Alg2.
|
Alg1 >
|
|
|
| (2.8) |
|
Alg1 >
|
|
|
| (2.9) |
We show, by way of a simple example, the extensions of the mappings in CASE 2 and CASE 4 form a mixed tensor on the domain Alg1.
|
Alg2 >
|
|
|
| (2.10) |
|
Alg2 >
|
|
|
| (2.11) |
|
|
