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Example 1.
First initialize two copies of a Lie algebra, called Alg1 and Alg2, and display the Lie bracket multiplication tables.
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Alg1 >
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Alg1 >
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Alg2 >
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| (2.1) |
We use AdjointExp to construct a linear transformation (in fact, an isomorphism) from Alg1 to Alg2.
Alg2 >
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Alg1 >
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| (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.
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Alg2 >
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CASE 2: 1-forms on the range algebra Alg2.
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Alg2 >
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Alg2 >
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CASE 3. rank 1 covariant tensors on the domain algebra Alg1.
Alg1 >
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| (2.3) |
Alg1 >
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Alg1 >
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CASE 4. rank 1 contravariant vectors on the range algebra Alg2.
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Alg2 >
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Alg2 >
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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 >
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Alg1 >
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| (2.5) |
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 >
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Alg2 >
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| (2.7) |