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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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![L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]])](/support/helpjp/helpview.aspx?si=6606/file05783/math85.png)
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![_DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]])](/support/helpjp/helpview.aspx?si=6606/file05783/math88.png)
| (2.1) |
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We bracket the subspaces S1 = [e1, e2] and S2 = [e3, e4].
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Alg1 >
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![S1 := [e1, e2]](/support/helpjp/helpview.aspx?si=6606/file05783/math104.png) ![S2 := [e3, e4]](/support/helpjp/helpview.aspx?si=6606/file05783/math105.png)
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Alg1 >
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![[_DG([["vector", Alg1, []], [[[1], 1]]])]](/support/helpjp/helpview.aspx?si=6606/file05783/math112.png)
| (2.2) |
We bracket the subspace S3 = [e1, e2, e3] with itself.
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Alg1 >
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![S3 := [e1, e2, e3]](/support/helpjp/helpview.aspx?si=6606/file05783/math126.png)
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Alg1 >
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![[]](/support/helpjp/helpview.aspx?si=6606/file05783/math133.png)
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Example 2.
The command also works with lists of matrices.
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![M1 := [Matrix([[0, 1, 1], [0, 0, 0], [0, 0, 0]]), Matrix([[0, 1, 0], [0, 0, 0], [1, 0, 0]])]](/support/helpjp/helpview.aspx?si=6606/file05783/math152.png)
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![M1 := [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0})]](/support/helpjp/helpview.aspx?si=6606/file05783/math155.png)
| (2.4) |
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![M2 := [Matrix([[0, 0, 1], [0, 0, 0], [0, 0, 0]]), Matrix([[0, 0, 0], [0, 0, 1], [0, 0, 0]]), Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 1]])]](/support/helpjp/helpview.aspx?si=6606/file05783/math159.png)
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![M2 := [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1})]](/support/helpjp/helpview.aspx?si=6606/file05783/math162.png)
| (2.5) |
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![[Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = -1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = -1, (3, 2) = 0, (3, 3) = 0})]](/support/helpjp/helpview.aspx?si=6606/file05783/math169.png)
| (2.6) |
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