GroupActions[InfinitesimalPseudoGroupNormalizer] - find the normalizer of a finite dimensional Lie algebra of vector fields in an (infinite-dimensional) pseudo-Lie algebra of vector fields
Calling Sequences
InfinitesimalPseudoGroupNormalizer(Gamma, options)
Parameters
Gamma - a list, a basis for a Lie algebra of vector fields on a manifold
options - (optional) arguments that can be given in any order and are described as follows
ansatz - ansatz= Z, where Z is a vector field on
unknowns - unknowns = U, where U is a list of the unknown functions appearing in the vector field
auxiliaryequations - auxiliaryequations = E, where E is a list of additional partial differential equations to be imposed on the unknowns functions U
parameters - parameters = P, where P is a set of parameters appearing in the vector fields
liealgebra - liealgebra = name, where name is the string or name for the abstract Lie algebra defined by
output - output = "general", "pde" or "list"
other options - optional arguments to be passed to pdsolve
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Examples
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Example 1.
First define a 1-dimensional manifold with coordinate .
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On , define 1 and 2-dimensional Lie algebras of vector fields respectively.
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| (2.1) |
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| (2.2) |
Find the normalizer of these two Lie algebras in the Lie algebras of all vector fields on .
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| (2.3) |
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| (2.4) |
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| (2.5) |
Example 2.
Find the normalizer for the Lie algebra of infinitesimal rotations in 3 dimensions.
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| (2.6) |
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| (2.7) |
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| (2.8) |
Now let us find the normalizer for the Lie algebra of infinitesimal rotations in 3 dimensions within the infinite dimensional Lie algebra of divergence-free vector fields.
First define a general vector field on with arbitrary coefficients and .
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| (2.9) |
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| (2.10) |
We use the keyword argument auxiliaryequations to require that the vector field be divergence-free.
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| (2.11) |
Example 3.
In this example we shall calculate the normalizers for a Lie algebra of vector fields which depends upon a parameter . We find that Nor() mod has dimension 2 for and dimension 3 for .
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| (2.12) |
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| (2.13) |
Example 4.
We calculate the normalizer of the infinitesimal Euclidean group in the infinitesimal pseudo-group of all contact transformation on a 3 dimensional contact manifold with coordinates with contact form
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| (2.14) |
Here is the standard Euclidean metric on the plane and the standard contact form on
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| (2.15) |
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| (2.16) |
We use the command InfinitesimalSymmetriesOfGeometricObjectFields to find the Lie algebra of vector fields which preserves the metric and the Pfaffian system generated by
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| (2.17) |
We define an arbitrary vector field on and again use the command InfinitesimalSymmetriesOfGeometricObjectFields, this time to find the partial differential equations which the coefficients of must satisfy in order that this vector field be an infinitesimal contact transformation.
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| (2.18) |
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| (2.19) |
Note that the factor is an additional unknown satisfying .
The sought after normalizer of in the infinitesimal pseudo-group of contact transformations can now be computed.
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| (2.20) |
We can check this result by noting that [i] the vector fieldspreserve and [ii] the normalizer is a 5 dimensional Lie algebra which contains as an ideal.
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| (2.21) |
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| (2.22) |
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| (2.23) |
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| (2.24) |
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