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We define a manifold with coordinates.
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Example 1.
Find all vector fields which commute with the vector field .
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Find all vector fields whose coefficients depend only on which commute with the vector field .
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Example 2.
Find the infinitesimal symmetries for the metric .
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Show the defining differential equations for these symmetries. Here we explicitly define the general form of the symmetry vector and specify the unknowns.
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We can use the auxilaryequations option to find the symmetries X of the metric g for which
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Example 3.
Find the joint infinitesimal symmetries for the 0 connection and the volume form .
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Example 4.
Here is a famous calculation due to E. Cartan. See Fulton and Harris Representation Theory page 357. We find the linear infinitesimal symmetries of the 3-form defined on the 7-manifold N with coordinates .
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It is a simple matter to use the package LieAlgebras to check that this Lie algebra is indecomposable and simple and is a realization of the exceptional Lie algebra .
Example 5.
Find the point symmetries of the Lagrangian for the (2 +1) wave equation. The result is a 8-dimensional Lie algebra.
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Example 6.
Find the infinitesimal conformal symmetries of the metric . These are the vector fields such that or span.
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Note that the first argument is now a list of a list.
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The conformal symmetries of define a 10-dimensional Lie algebra.
Example 7.
Find the infinitesimal symmetries of a distribution of vector fields . These are the vector fields such that (Y) for each .
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Example 8.
Find the symmetries of a metric which depend upon 2 parameters , where .
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Example 9.
The command InfinitesimalSymmetriesOfGeometricObjectFields can also be used to calculate the symmetries of a tensor defined on a Lie algebra.
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