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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![M := map(Matrix, [[[1, 0, 0], [0, 0, 0], [0, 0, -1]], [[0, 0, 0], [0, 1, 0], [0, 0, -1]], [[0, 1, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [1, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [1, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 1, 0]]])](/support/helpjp/helpview.aspx?si=7180/file07661/math79.png)
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| (2.1) |
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![_DG([["LieAlgebra", sl3, [8, table( [ ] )]], [[[1, 3, 3], 1], [[1, 4, 4], 2], [[1, 5, 5], -1], [[1, 6, 6], 1], [[1, 7, 7], -2], [[1, 8, 8], -1], [[2, 3, 3], -1], [[2, 4, 4], 1], [[2, 5, 5], 1], [[2, 6, 6], 2], [[2, 7, 7], -1], [[2, 8, 8], -2], [[3, 5, 1], 1], [[3, 5, 2], -1], [[3, 6, 4], 1], [[3, 7, 8], -1], [[4, 5, 6], -1], [[4, 7, 1], 1], [[4, 8, 3], 1], [[5, 8, 7], -1], [[6, 7, 5], 1], [[6, 8, 2], 1]]])](/support/helpjp/helpview.aspx?si=7180/file07661/math89.png)
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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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sl3 >
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Subalgebra 2.
sl3 >
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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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