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Query[DirectSumDecomposition] - check if a list of subalgebras defines a direct sum decomposition of a Lie algebra
Calling Sequences
Query([S1, S2, ...], "DirectSumDecomposition")
Query(B, [d1, d2, ...], "DirectSumDecomposition")
Parameters
S1 - a list of independent vectors defining a subalgebra of a Lie algebra g
B - a list of vectors defining a basis for g
d1 - a sequence of positive integers whose sum equals the dimension of the Lie algebra g
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Description
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A collection of subalgebras S1, S2, ... of a Lie algebra g defines a direct sum decomposition of g if g = S1 + S2 + ... (vector space direct sum) and [Si, Sj] = 0 for i <> j.
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Query([S1, S2, ... ], "DirectSumDecomposition") returns true if the subspaces S1, S2, ... define a direct sum decomposition of the Lie algebra g.
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Query(B, [d1, d2, ... ], "DirectSumDecomposition") returns true if the first d1 vectors in B, the second d2 in B, ... define a direct sum decomposition of g.
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The command Query is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).
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Examples
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Example 1.
First initialize a Lie algebra and display the Lie bracket multiplication table. One can see from the multiplication table that this Lie algebra is a direct sum of the subalgebras S1 = [e1, e2, e3], S2 = [e4, e5] and S3 = [e6]. We verify this using Query.
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![L1 := _DG([["LieAlgebra", Alg1, [6]], [[[1, 3, 1], 1], [[2, 3, 2], 1], [[4, 5, 4], 1]]])](/support/helpjp/helpview.aspx?si=7150/file07648/math111.png)
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Alg1 >
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![[[e1, e3] = e1, [e2, e3] = e2, [e4, e5] = e4]](/support/helpjp/helpview.aspx?si=7150/file07648/math122.png)
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Alg1 >
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![Query([S1, S2, S3], "DirectSumDecomposition")](/support/helpjp/helpview.aspx?si=7150/file07648/math132.png)
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| (2.2) |
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Alg1 >
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![Query([S1, S2], "DirectSumDecomposition")](/support/helpjp/helpview.aspx?si=7150/file07648/math139.png)
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| (2.3) |
Define B to be a basis for the Lie algebra which is adapted to the direct sum decomposition.
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Alg1 >
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![B := [e1, e2, e3, e4, e5, e6]](/support/helpjp/helpview.aspx?si=7150/file07648/math152.png)
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Alg1 >
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![Query(B, [3, 3], "DirectSumDecomposition")](/support/helpjp/helpview.aspx?si=7150/file07648/math156.png)
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| (2.4) |
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Alg1 >
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![Query(B, [2, 2, 2], "DirectSumDecomposition")](/support/helpjp/helpview.aspx?si=7150/file07648/math163.png)
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| (2.5) |
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