LieAlgebras[BracketOfSubspaces] - calculate the span of the Lie bracket of two lists of vectors in a Lie algebra, calculate the span of the matrix commutator of two lists of matrices
Calling Sequences
BracketOfSubspaces(S1, S2)
BracketOfSubspaces(M1, M2)
Parameters
S1, S2 - two lists of vectors whose spans determine subspaces of a Lie algebra g
M1, M2 - two lists of square matrices
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Description
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The first calling sequence BracketOfSubspaces(S1, S2) calculates the span of the set of vectors [x, y] for all x in S1 and all y in S2.
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The second calling sequence BracketOfSubspaces(M1, M2) calculates the span of all matrices [a, b] = a.b - b.a for all a in S1 and all b in S2.
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A list of linearly independent vectors defining a basis for [S1, S2] or [M1, M2] is returned. If [S1, S2] is trivial (that is, all the vectors in S1 commute with all the vectors in S2 then an empty list is returned.
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The command BracketOfSubspaces is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form BracketOfSubspaces(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-BracketOfSubspaces(...).
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Examples
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Example 1.
First we initialize a Lie algebra.
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| (2.1) |
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We bracket the subspaces S1 = [e1, e2] and S2 = [e3, e4].
Alg1 >
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Alg1 >
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| (2.2) |
We bracket the subspace S3 = [e1, e2, e3] with itself.
Alg1 >
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Alg1 >
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| (2.3) |
Example 2.
The command also works with lists of matrices.
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| (2.4) |
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| (2.5) |
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| (2.6) |
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