Example 1
We calculate the Cartan subalgebra for the 8-dimensional Lie algebra of 3x3 trace-free matrices. The structure equations are obtained using the SimpleLieAlgebraData command.
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| (2.1) |
Initialized the Lie algebra.
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| (2.2) |
Find a Cartan subalgebra.
sl(3) >
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| (2.3) |
We can check that this subalgebra is Abelian (and hence nilpotent) and self-normalizing.
sl3 >
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| (2.4) |
sl3 >
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| (2.5) |
These properties can also be checked with the Query command
sl3 >
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| (2.6) |
For the split real forms of the simple Lie algebras, a Cartans subalgebra can always be found consisting of diagonal matrices in the standard representation.
sl3 >
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| (2.7) |
Example 2
Other Cartan subalgebras for can be found with the second calling sequence.
sl3 >
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| (2.8) |
Example 3
The Cartan subalgebra of a nilpotent Lie algebra g is g itself. Retrieve the stucture equations for a nilpotent Lie algebra from the DifferentialGeometry library.
sl3 >
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| (2.9) |
sl3 >
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Check that the algebra is nilpotent.
alg3 >
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| (2.10) |
alg3 >
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| (2.11) |
Example 4
We find the Cartan subalgebra for a solvable Lie algebra. Retrieve the structure equations for a solvable Lie algebra from the DifferentialGeometry library.
alg3 >
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| (2.12) |
alg4 >
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| (2.13) |
Check that the algebra is solvable.
alg4 >
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| (2.14) |
alg4 >
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| (2.15) |
alg4 >
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| (2.16) |
Example 5.
We find the Cartan subalgebra for a Lie algebra with a non-trivial Levi decomposition. Retrieve the structure equations for such a Lie algebra from the DifferentialGeometry library.
alg4 >
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| (2.17) |
alg4 >
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| (2.18) |
Check that the Levi decomposition is non-trivial.
alg5 >
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| (2.19) |
Calculate the Cartan subalgebra.
alg5 >
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| (2.20) |