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Example 1.
We calculate the parabolic subalgebras for We use the command SimpleLieAlgebraData to initialize the Lie algebra.
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We use the command SimpleLieAlgebraProperties to obtain the Cartan subalgebra, root space decomposition etc.
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Here are the properties we need:
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| (2.2) |
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| (2.3) |
The possible subsets of the simple roots are:
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The possible parabolic subalgebras of are therefore:
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| (2.4) |
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The Query command can be used to check that these subalgebras are parabolic subalgebra.
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| (2.5) |
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With the command ParabolicSubalgebraRoots, we can find the simple roots used to create the parabolic algebra .
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Example 2.
We calculate (real) parabolic subalgebras for . We use the command SimpleLieAlgebraData to initialize the Lie algebra.
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We use the command SimpleLieAlgebraProperties to calculate the restricted root space decomposition and the restricted simple roots.
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| (2.8) |
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The possible subsets of restricted simple roots are:
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The parabolic subalgebras defined by these sets of restricted roots are:
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| (2.9) |
Check that the subalgebra defined by is parabolic.
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| (2.10) |
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Find the restricted roots used to define .
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