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LieAlgebras[MinimalSubalgebra] - find the smallest Lie subalgebra containing a given set of vectors from a Lie algebra, find the smallest matrix algebra containing a given set of matrices
Calling Sequences
MinimalSubalgebra(S)
MinimalSubalgebra(M)
Parameters
S - a list of vectors in a Lie algebra
M - a list of square matrices
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Description
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MinimalSubalgebra(S) calculates the smallest Lie subalgebra containing the list of vectors S from a defined Lie algebra g. A list of basis vectors for the subalgebra from the Lie algebra g is returned.
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MinimalSubalgebra(M) calculates the smallest matrix algebra containing the matrices in the list M.
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The command MinimalSubalgebra is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form MinimalSubalgebra(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MinimalSubalgebra(...).
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Examples
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Example 1.
First we initialize a Lie algebra and display the multiplication table.
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![L1 := _DG([["LieAlgebra", Alg1, [5]], [[[1, 5, 1], 2], [[2, 3, 1], 1], [[2, 5, 2], 1], [[2, 5, 3], 1], [[3, 5, 3], 1], [[4, 5, 4], 2]]])](/support/helpjp/helpview.aspx?si=6601/file05819/math65.png)
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![_DG([["LieAlgebra", Alg1, [5]], [[[1, 5, 1], 2], [[2, 3, 1], 1], [[2, 5, 2], 1], [[2, 5, 3], 1], [[3, 5, 3], 1], [[4, 5, 4], 2]]])](/support/helpjp/helpview.aspx?si=6601/file05819/math68.png)
| (2.1) |
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Find the minimal subalgebra containing [e1, e3].
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Alg1 >
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![S1 := [e1, e3]](/support/helpjp/helpview.aspx?si=6601/file05819/math82.png)
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Alg1 >
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![[_DG([["vector", Alg1, []], [[[1], 1]]]), _DG([["vector", Alg1, []], [[[3], 1]]])]](/support/helpjp/helpview.aspx?si=6601/file05819/math89.png)
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Find the minimal subalgebra containing [e2, e3].
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Alg1 >
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![S2 := [e2, e3]](/support/helpjp/helpview.aspx?si=6601/file05819/math99.png)
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Alg1 >
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![[_DG([["vector", Alg1, []], [[[1], 1]]]), _DG([["vector", Alg1, []], [[[2], 1]]]), _DG([["vector", Alg1, []], [[[3], 1]]])]](/support/helpjp/helpview.aspx?si=6601/file05819/math106.png)
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Alg1 >
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| (2.4) |
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Alg1 >
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| (2.5) |
Find the minimal subalgebra containing [e2, e5].
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Alg1 >
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![S3 := [e2, e5]](/support/helpjp/helpview.aspx?si=6601/file05819/math130.png)
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Alg1 >
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![[_DG([["vector", Alg1, []], [[[1], 1]]]), _DG([["vector", Alg1, []], [[[2], 1]]]), _DG([["vector", Alg1, []], [[[3], 1]]]), _DG([["vector", Alg1, []], [[[5], 1]]])]](/support/helpjp/helpview.aspx?si=6601/file05819/math137.png)
| (2.6) |
Example 2.
The command MinimalSubalgebra also works with matrices.
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Alg1 >
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![M := [Matrix([[1, 0, 0], [0, 1, 0], [1, 0, 0]]), Matrix([[0, 1, 0], [0, 1, 1], [0, 1, 0]])]](/support/helpjp/helpview.aspx?si=6601/file05819/math158.png)
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![M := [Matrix(3, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})]](/support/helpjp/helpview.aspx?si=6601/file05819/math161.png)
| (2.7) |
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Alg1 >
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| (2.8) |
We can use the LieAlgebraData command to verify that the set of matrices N defines a 4-dimensional Lie algebra and to determine the commutator relationships.
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Alg1 >
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![_DG([["LieAlgebra", "L2", [4]], [[[1, 2, 3], 1], [[1, 3, 3], 1], [[1, 4, 4], 1], [[2, 3, 4], 1], [[2, 4, 3], 1], [[2, 4, 4], 1]]])](/support/helpjp/helpview.aspx?si=6601/file05819/math191.png)
| (2.9) |
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Alg1 >
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Here e1, e2, e3, e4 denote the four matrices N[1], N[2], N[3], N[4].
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