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Example 1.
We find the compact roots for First we use the command SimpleLieAlgebraData to initialize the Lie algebra
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| (2.1) |
For this example we use the command SimpleLieAlgebraProperties to generate the various properties of that we need.
su52 >
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Here is the Cartan subalgebra.
su52 >
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| (2.2) |
Here is the Cartan subalgebra decomposition
su52 >
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| (2.3) |
We check that the restriction of the Killing form to the diagonal matrices with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices with real entries is positive-definite.
su52 >
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| (2.4) |
su52 >
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| (2.5) |
su52 >
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| (2.6) |
The second list of vectors in (2.3) is therefore our subalgebra as described above.
Next we find the positive roots.
su52 >
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| (2.7) |
The
compact roots are:
su52 >
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| (2.8) |
| (2.9) |
Note that these roots all have purely imaginary components.
Example 2.
We find the compact roots for First we use the command SimpleLieAlgebraData to initialize the Lie algebra
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| (2.10) |
We use the command SimpleLieAlgebraProperties to generate the various properties of that we need.
sp44 >
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Here is the Cartan subalgebra.
sp44 >
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| (2.11) |
Here is the Cartan subalgebra decomposition
sp44 >
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| (2.12) |
The restriction of the Killing form to the diagonal matrices with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices with real entries is positive-definite.
sp44 >
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| (2.13) |
sp44 >
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| (2.14) |
sp44 >
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| (2.15) |
The second list of vectors in (2.3) is therefore our subalgebra as described above.
Next we find the positive roots.
sp44 >
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| (2.16) |
The
compact roots are:
sp44 >
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| (2.17) |