Query[ParabolicSubalgebra] - check if a list of vectors defines a parabolic subalgebra of a semi-simple Lie algebra
Calling Sequences
Query()
Parameters
P - a list of vectors, defining a subalgebra of a semi-simple Lie algebra
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Description
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Let g be a semi-simple Lie algebra. A Borel subalgebra
b is any maximal solvable subalgebra. A parabolic subalgebra p is any subalgebra containing a Borel subalgebra. Alternatively, a subalgebra p is parabolic if its nilradical is the orthogonal complement of p with respect to the Killing form
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This Query command returns true if the subalgebra p defined by the vectors satisfies
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Examples
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We check to see if 3 subalgebras of are parabolic. We construct the Lie algebra directly from its standard matrix representation.
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Initialize the Lie algebra. We label the basis elements for in a manner consistent with its matrix representation.
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Subalgebra 1.
sl3 >
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| (2.4) |
sl3 >
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Subalgebra 2.
sl3 >
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| (2.6) |
sl3 >
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Subalgebra 3.
sl3 >
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| (2.8) |
sl3 >
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| (2.9) |
sl3 >
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| (2.10) |
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