Example 1.
We create a Lie group module for the matrix group of upper triangular matrices .
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| (2.1) |
Since the group parameters in the matrix are , we use these for the coordinates on the Lie group.
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| (2.2) |
G1 >
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| (2.3) |
Evaluate the exports for the LG1 module:
G1 >
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| (2.4) |
G1 >
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| (2.5) |
G1 >
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| (2.6) |
G1 >
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| (2.7) |
G1 >
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| (2.8) |
Let us check the results for , , and by calculating matrix products and inverses.
G1 >
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| (2.9) |
The entries of give the formulas for the transformation :
G1 >
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| (2.10) |
The entries of give the formulas for the transformation
G1 >
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| (2.11) |
The entries of give the formulas for the transformation :
G1 >
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| (2.12) |
We can also verify that the exports for LG1 are correct with the ApplyTransformation command.
G1 >
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| (2.13) |
The output of the LieGroups command can be passed to the InvariantVectorsAndForms program to compute the left and right invariant vector fields and forms on the Lie group G1.
G1 >
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| (2.14) |
Example 2.
The second calling sequence for LieGroup is similar in concept to the first calling sequence except that the Lie group is now determined from a group of transformations on a manifold instead of a matrix group. In this example we consider a 5 parameter group of affine transformations in the plane.
G1 >
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| (2.15) |
M >
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| (2.16) |
M >
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| (2.17) |
M >
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| (2.18) |
M >
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| (2.19) |
The group parameters are .
M >
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| (2.20) |
M >
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| (2.21) |
M >
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| (2.22) |
Example 3.
If the left or right multiplication for a Lie group is known, then the third calling sequence can be used to create the corresponding Lie group module. We take, as a simple example, the left multiplication map constructed in Example 1. Then the procedure LieGroup will compute the corresponding right multiplication.
G2 >
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| (2.23) |
G3 >
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| (2.24) |
G3 >
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| (2.25) |
G3 >
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| (2.26) |
The default assumption is that the transformation given as the 1st argument to LieGroup is the left multiplication map.The keyword argument multiplication = "right" can be used to indicate that the 1st argument is the right multiplication. Then the procedure LieGroup will compute the corresponding left multiplication.
G3 >
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| (2.27) |
G5 >
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| (2.28) |
G3 >
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| (2.29) |
Example 4.
We use the 4th calling sequence to calculate the Lie group for a given abstract solvable Lie algebra. Retrieve an abstract Lie algebra from the DifferentialGeometry library and initialize it.
G2 >
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| (2.30) |
G2 >
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| (2.31) |
G2 >
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| (2.32) |
G4 >
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| (2.33) |
G4 >
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| (2.34) |
We use coordinates for tjhe Lie group
G2 >
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| (2.35) |
G4 >
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| (2.36) |
The structure equations for the right invariant vector fields on coincide with the structure equations for the Lie algebra we started with.
G4 >
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| (2.37) |
G4 >
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| (2.38) |
Example 5.
We calculate the Lie group module for the special linear group SL2, defined here as the group of fractional linear transformations on the real line.
G4 >
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| (2.39) |
M5 >
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| (2.40) |
The group parameters are subject to the constraint We solve for and re-write the group action in terms of 3 group parameters .
M5 >
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| (2.41) |
M5 >
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| (2.42) |
Note that for this group the values and both determine the identity transformation. A choice of identity point can be imposed with the keyword argument
identity .
G5 >
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| (2.43) |
G5 >
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| (2.44) |
Example 6.
In this example we calculate the Lie group multiplication for the Euclidean group of motions in the plane. Here is the standard matrix representation of this group.
G1 >
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| (2.45) |
The group parmeters are [, a, b] so we use these as coordinates for the Lie group
G5 >
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| (2.46) |
In this example, the entries of the matrix are not simple rational functions of group parameter . Therefore some simplification of the defining equations for the left and right multiplication may be required in order for the LieGroup procedure to succeed. This can be done with the DifferentialGeometry Preferences command. The combine command will apply the addition formulas for sine and cosine which will simplify the defining equations for the left and right multiplications. The multiplication rules for the Euclidean Lie group can then be determined.
G6 >
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| (2.47) |
G6 >
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| (2.48) |
G6 >
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| (2.49) |
G6 >
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