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Example 1.
First we initialize a Lie algebra.
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For this example we take h to be the trivial subspace. In this case the procedure RelativeChains simply returns a list of bases for the 1-forms on g, the 2-forms on g, the 3-forms on g, and so on.
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We pass the output of the RelativeChains program to the Cohomology program.
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To read off the dimensions of the cohomology of g, use the nops and map command.
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Example 2.
We continue with Example 1. To find the cohomology of just in degree 3, pass the Cohomology program to just the chains of degree 2 and 3 and 4.
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Example 3.
We continue with Example 1. Show that the 2-form is closed and express as a linear combination of the cohomology classes in and the exterior derivative of a 1-form.
Alg1 >
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Alg1 >
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Alg1 >
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Alg1 >
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Example 4.
First we initialize a Lie algebra.
Alg1 >
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Define a 2 dimensional subspace to be the vectors spanned by ..
Compute the relative chains with respect to the subspace .
Alg2 >
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Example 5.
In this example we compute the cohomology of a 4-dimensional Lie algebra with coefficients in the adjoint representation. First define and initialize the Lie algebra.
Rep1 >
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Define the representation space
Alg3 >
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Define the adjoint representation.
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Alg3 >
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Note that the chains are now linear functions of the coordinates on the representation space.
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Example 6.
Finally, we compute the Lie algebra cohomology of Alg3 with coefficients in the adjoint representation, relative to the subalgebra spanned bu
Rep1 >
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