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Example 1.
We obtain a Lie algebra from the DifferentialGeometry library using the Retrieve command and initialize it.
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| (2.1) |
We define a manifold M of dimension 4 (the same dimension as the Lie algebra).
Alg1 >
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M1 >
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| (2.3) |
M1 >
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We calculate the structure equations for the Lie algebra of vector fields Gamma1 and check that these structure equations coincide with those for Alg1.
M1 >
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| (2.5) |
Example 2.
We re-work the previous example in a more complicated basis. In this basis the adjoint representation is not upper triangular, in which case LiesThirdTheorem first calls the program SolvableRepresentation to find a basis for the algebra in which the adjoint representation is upper triangular. (Remark: It is almost always useful, when working with solvable algebras, to transform to a basis where the adjoint representation is upper triangular.)
M1 >
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| (2.6) |
Alg2 >
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| (2.8) |
M1 >
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| (2.9) |
Example 3.
Here is an example where one of the adjoint matrices has complex eigenvalues. The Lie algebra contains parameters and .
M1 >
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| (2.10) |
Alg3 >
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M3 >
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| (2.12) |
M3 >
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Example 4.
We calculate the Maurer-Cartan matrix of 1-forms for a solvable matrix algebra, namely the matrices defining the adjoint representation for Alg1 from Example 1.
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Note that the elements of this matrix
coincide with the appropriate linear combinations of the forms in the list from Example 1.
Alg1 >
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| (2.14) |
Alg1 >
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| (2.15) |
Alg1 >
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| (2.16) |
Alg1 >
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